Radiation Acoustics

RADIATION ACOUSTICS

RADIATION ACOUSTICS Leonid M. Lyamshev

CRC PR E S S Boca Raton London New York Washington, D.C.

Library of Congress Cataloging-in-Publication Data Lyamshev, Leonid M. Radiation acoustics / Leonid M. Lyamshev. p. ; cm. Includes bibliographical references and index. ISBN 0-415-30999-9 (alk. paper) 1. Sound-waves. 2. Sound—Transmission. 3. Radiation sources. I. Title. [DNLM: 1. Radiation. 2. Acoustics. 3. Radiation Effects. 4. Thermodynamics. WN 100 L981r 2004] QC243.L93 2004 534—dc22 2003070031

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FOREWORD

Radiation acoustics is a new field of research developing at the interface of acoustics, nuclear physics, high energy physics, and the physics of elementary particles. It is founded upon studies and applications of radiation-acoustic effects arising in the process of interaction of penetrating radiation with matter. The thermoradiation mechanism has been the best studied among the mechanisms of sound excitation by penetrating radiation in condensed media. According to this mechanism, sound generation is caused by thermal expansion of a medium, and the acoustic field can be described as a rule within the framework of linear theory. The book considers mainly the processes of thermoradiation sound excitation in the case of continuous (modulated) and pulsed action of penetrating radiation on a substance. Basic laws of formation of acoustic signals are established and the bonds between the characteristics of these signals, radiation parameters, and thermodynamic, radiation, and acoustic properties of substances are revealed. The efficiency and optimal conditions of thermoradiation sound generation are considered. The particular features of sound generation by a particle beam moving along the surface of a liquid or solid at subsonic and supersonic velocities and an arbitrary form of modulation of radiation intensity in the beam are described. The possibility is discussed of the creation of virtual radiation-acoustic sources of sound in a wide frequency range (from sound to hypersound frequencies) with controlled parameters in liquids or solids. We consider the particular features of thermoradiation generation of sound by single particles. Experimental results on sound excitation by beams of photons (laser radiation), electrons, protons, heavy ions, X-rays, and gamma-quanta are given. Some other mechanisms of sound generation

by single particles in the process of their absorption in a medium are considered apart from the thermoradiation mechanism, i.e., the mechanisms of microshock waves and the bubble, dynamic, Cherenkov, and striction mechanisms. Applications of radiation acoustics are discussed. We have not tried to go into the details of many of them. Our purpose is to demonstrate the prospects of application of radiation acoustics to various fields from microelectronics (radiation-acoustic microscopy) to geophysics (neutrinoacoustic sounding of the Earth), and astrophysics (detection of cosmic neutrino and muons of super-high energy by huge acoustic detectors in the ocean). We have not tried to review all papers on radiation acoustics. On the contrary, we have quite deliberately not included in the book the results of studies of nonlinear radiation-acoustic phenomena arising in the process of interaction of powerful radiation beams with matter. Although the role of nonlinear effects in future radiation-acoustic technologies will undoubtedly be essential (targeted action on physical, mechanical and chemical structure of substances, radiation-acoustic destruction of materials, etc.), investigation of these effects still continues. The book may be useful not only to acousticians but also researchers and technicians specializing in adjacent and other fields as well as postgraduates and university students. I am very grateful to G. A. Askar’yan, F. V. Bunkin, and V. I. Il’ichev for many useful remarks on the manuscript. L. M. Bolotova, M. G. Lisovskaya, and B. I. Chelnokov helped me greatly in the preparation of the manuscript for publication. I am deeply grateful to them. L. M. Lyamshev

CONTENTS

INTRODUCTION

1

Chapter 1. PENETRATING RADIATION: GENERAL INFORMATION

7

1. Elementary particles: Fundamental laws of the microscopic world 7 2. Absorption of penetrating radiation in a condensed medium 13

Chapter 2. BASIC MECHANISMS OF SOUND GENERATION BY PENETRATING RADIATION IN CONDENSED MEDIA 1. Mechanisms connected with heat release 2. Thermoradiation generation of sound 3. Initiation of microshock waves 4. Bubble mechanism 5. The Cherenkov mechanism 6. Striction mechanism of sound generation 7. Sound generation in the process of pulsed radiolysis 8. Dynamic mechanism 9. Other mechanisms of sound generation

Chapter 3. THERMORADIATION EXCITATION OF SOUND IN A HOMOGENEOUS LIQUID 1. Equation of thermoradiation generation of sound 2. Reciprocity theorem in acoustics – Solution technique for boundary problems 3. Excitation of monochromatic sound in a liquid half-space with a free surface – The case of undisturbed surface 4. A liquid half-space with large-scale roughness of boundary

23 23 24 28 30 31 33 35 35 38

39 39 42

47 52

5. The case of small unevenness 6. Efficiency of thermoradiation excitation of sound in a liquid – Some estimates

Chapter 4. THERMORADIATION EXCITATION OF SOUND IN AN INHOMOGENEOUS MEDIUM 1. Sound excitation in a liquid half-space in the presence of a layer of another liquid at its boundary 2. Generation of sound in a liquid adjoining a solid layer 3. Liquid half-space with an inhomogeneous surface layer

Chapter 5. EXCITATION OF SOUND IN A LIQUID BY RADIATION PULSES 1. Sound generation in a liquid by rectangular pulses of radiation 2. A liquid with rough surface 3. Radiation pulses of arbitrary shape 4. Near wave field of thermoradiation pulsed source of sound 5. Sound generation in a liquid with gas bubbles

Chapter 6. GENERATION OF SOUND IN SOLIDS BY INTENSITY-MODULATED PENETRATING RADIATION 1. Basic equations 2. Boundary conditions 3. Method for solution of boundary problems 4. Thermoradiation generation of sound in a solid half-space with a free boundary 5. Particular features of excitation of Rayleigh waves 6. Solid half-space with a liquid layer at its surface 7. Efficiency of thermoradiation generation of sound 8. Influence of particular features of absorption of penetrating radiation on sound generation

61 72

75

75 88 96

105

106 113 116 124 130

137 137 139 141 147 151 156 160 164

Chapter 7. PULSED THERMORADIATION SOURCES OF SOUND IN SOLIDS 1. Sound generation by radiation pulses in a solid half-space 2. Excitation of Rayleigh waves by radiation pulses 3. Sound generation in a solid half-space in the presence of a liquid layer at its surface 4. Efficiency of sound generation 5. Influence of particular features of absorption of penetrating radiation 6. Thermoradiation generation of sound by pulses of non-relativistic protons

Chapter 8. MOVING THERMORADIATION SOURCES OF SOUND 1. Sound generation by a moving thermoradiation pulsed source in a liquid 2. Sound excitation by a moving thermoradiation pulsed source in solids

Chapter 9. SOUND GENERATION BY SINGLE HIGH-ENERGY PARTICLES 1. Sound generation by a particle in infinite space 2. Sound excitation by single particles in a solid half-space 3. Particular features of excitation of Rayleigh waves 4. Efficiency of sound generation

Chapter 10. EXPERIMENTAL STUDY OF THERMORADIATION EXCITATION OF SOUND 1. Laser thermooptical (thermoradiation) sources of harmonic sound oscillations in water 2. Sound excitation in water by laser pulses 3. Sound field excited by a sequence of laser pulses 4. Acoustic field of a moving laser thermoradiation source of sound in water

169

169 184 186 191 193 195

201

202 213

227 227 234 237 238

239

240 248 252 257

5. Laser thermooptical excitation of sound in solids – Excitation of surface waves 6. Sound excitation by X-rays (synchrotron radiation) in metals 7. Sound excitation by a proton beam 8. Excitation of acoustic waves in metals by electrons, positrons, and γ-quanta 9. Sound generation by an electron beam in water 10. Sound excitation by a beam of ions in metals

Chapter 11. SOME APPLICATIONS OF RADIATION ACOUSTICS 1. Scanning radiation-acoustic microscopy and visualization 2. Scanning laser-acoustic microscopy 3. Scanning electron-acoustic microscopy 4. X-ray – acoustic scanning visualization 5. Ion-acoustic microscopy and visualization 6. Acoustic detection of super-high-energy particles in cosmic rays – The DUMAND project 7. Neutrino for geoacoustics – The GENIUS project

270 281 286 298 302 306

309

310 311 314 318 321 325 330

Conclusion

339

References

343

Addendum. ACOUSTOOPTICS OF PENETRATING RADIATION

359

1. Diffraction of X-rays and neutrons by ultrasound in crystals 2. Scanning acoustic tunneling microscopy 3. Interferometers using matter waves – Atom interferometers

SUBJECT INDEX

360 361 362

367

INTRODUCTION

Investigation of radiation-acoustic effects was stimulated mainly by progress in the field of high-energy physics and the physics of elementary particles. The latter has advanced greatly during recent decades. Particles of tremendous energy of the order of magnitude of tens, hundreds, and thousands of gigaelectronvolts (GeV) were obtained with the help of accelerators and many new elementary particles subjected to amazing interactions and inter-transformations were discovered. Quantum chromodynamics and unified theory of electromagnetic and weak interactions appeared. The state-of-the-art is now that physics is on the verge of creating the unified theory of all the fundamental interactions — electromagnetic, strong (nuclear), weak, and gravitational interactions. Experimentation at even greater energy is needed for solution of this problem. This needs powerful accelerators of elementary particles, which would provide an opportunity to make the next step into the depth of the microscopic world [2, 76, 89, 176]. Such accelerators are under design and construction now. As the construction of new, more powerful accelerators opens new opportunities for advancement of investigation of elementary particles into the field of larger and larger energies, the accelerators designed initially for purely basic studies, are applied to a greater and greater extent in research into the physics of solids, biology, chemistry, and medical science. They are utilized successfully in radiation technology, defectoscopy, analysis of rare minerals, and also (as it will be demonstrated below) in radiation-acoustic research and technology. As a rule, these are low-energy (about several megaelectronvolts (MeV)) accelerators like betatrons, linear accelerators, and microtrons [13]. Now more powerful accelerators (up to several 1

2

INTRODUCTION

gigaelectronvolts) of proton, meson, and ion beams and X-rays are being tried for these purposes. The beginning of radiation acoustics is connected in a broad sense with the discovery by A. Bell, W. Roentgen, and J. Tyndall [180, 195, 245] of the optoacoustic (photoacoustic) effect, that is sound generation in a gas volume due to intermittent (modulated) light passage or, in other words, due to interaction of modulated optical radiation (modulated photon beam) with a substance (gas). At the same time Bell discussed the problem of construction of a radiophone, “a device for producing sound by radiation of any kind” [195]. Further studies of the optoacoustic effect and its applications served, as is known, as the basis for the development of optoacoustics (photoacoustics) including optoacoustic spectroscopy of gases and condensed media [74, 126]. A powerful stimulus for the development of this field in recent decades was the construction of lasers (see [96, 127] for example). The first studies of radiation-acoustic effects were conducted in the 1950s and 1960s. So, for example, in 1956 Kaganov, Lifshits, and Tanatarov considered sound radiation in a solid by a uniformly moving electron and showed that at an electron velocity greater than the sound velocity in a medium (lattice), the Cherenkov radiation of sound (phonons) occurs [101]. The analogous problem was considered earlier (1953) by Buckingham [203]. In 1955 Glaser and Rahm reported observation of tracks of particles in the process of their passage through a metastable boiling-up liquid in a bubble chamber according to sound (vibration) signals arising as a result of the birth and development of bubbles [209]. In 1957—1959 Askar’yan considered radiation of ultrasonic and hypersonic waves by charged particles in dense media due to local heating and formation of microscopic cavities along particle tracks. Excitation of surface and bulk waves under the impact of a non-relativistic electron flux upon the surface of a dense medium was considered and the problem of utilization of acoustic signals generated by particles for detection of particles was discussed [6, 7]1. In 1963 White investigated sound generation by an electron beam in a solid [260]2. Somewhat later (1967) Graham and Hutchison [216] measured 1

In contrast to Glaser and Rahm [209], who discussed sound radiation due to the rise of bubbles at particle tracks in metastable media, Askar’yan [6, 7] considered local heating arising at particle tracks in dense stable media and producing sound pulses, as well as the bubbles generating hypersonic waves. He proposed also acoustic detection of particles and noted the possibility of manifestation of hypersonic pulses in the process of biological action of radiation on cells and chromosomes (as a part of destructive effect). 2 We must note that at the same time, White conducted the first experiments on laser generation of sound in solids. Somewhat earlier the first widely known experiments

RADIATION ACOUSTICS

3

mechanical oscillations in quartz crystals, sapphire, etc., on their being irradiated by electron beam pulses, and in 1969 Beron and Hofstadter, as Askar’yan before them [6, 196], suggested that not only electrons but also other particles can generate mechanical vibrations3. Numerous studies of sound excitation in condensed media by electron and proton beams and by single particles were conducted in the 1970s by Borshkovskii, Volovik, Zalyubovskii, Kalinichenko, Lazurik, and others, and in the 1980s, by Lyamshev and Chelnokov (see [155, 156]). Various mechanisms of sound excitation in condensed media by penetrating radiation were considered. The major results of these studies and the bibliography can be found in the book [97]. The publications of many researchers on possible applications of the radiation-acoustic effects date to the same period (see, for example, F. Perry et al. on the application of these effects to the dosimetry of pulsed beams of accelerated particles and to obtaining data on the depth distribution of irradiation dose in a target [242]). A powerful stimulus for development of research in radiation acoustics were ideas to use radiation-acoustic effects for detection of super-highenergy muons and neutrinos at a large depth in the ocean [8, 199], to develop a radiation-acoustic microscope [207], and finally, the suggestion to “sound” the Earth (using a radiation-acoustic technique) by a super-highenergy neutrino beam from super-powerful (for super-high energy of particles) proton accelerators of future generation named tevatrons [248]. Further publications (see, for example, [15, 151 – 154, 184]) were connected in this or that way with these aspects. Investigations performed in the 1950s and 1980s have been described to some extent in the book [97] mentioned above and in a collection of articles [173].

on interaction of laser radiation with a liquid were conducted in the Lebedev Physical Institute of the USSR Academy of Sciences (see G. A. Askar’yan, A. M. Prokhorov, G. F. Chanturiya, and G. P. Shipulo, Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki, 1963, v. 44, No. 6, pp. 2180 – 2182) which led to the discovery of the light-hydraulic effect. This paper played a fundamental role in the development of laser and radiation acoustics. 3 Recently detection of an acoustic signal from a muon beam at the neutrino channel of the U-70 accelerator of the Institute of High-Energy Physics (Moscow) was reported (see A. B. Borisov et al., Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki, 1991, v. 100, p. 112). Acoustic signals from a muon flux in the muon filter of the neutrino channel were studied. Results of measurements of signals and theoretical estimates based on the thermoradiation mechanism of sound generation were given. A satisfactory agreement between experimental and theoretical results was noted that was the evidence of the dominant role of the thermoradiation mechanism in the process of sound generation. The possibility of application of the radiation-acoustic technique to remote determination of the characteristics of particle beams from the accelerator was considered.

4

INTRODUCTION

A definite concept has been formed by now about the mechanisms of sound generation by penetrating radiation. They are connected usually with the physical phenomena (processes) resulting in the conversion of penetrating radiation energy into acoustic energy. These processes depend on the radiation type, on the target substance in which this radiation is absorbed, and on the energy release mode in the absorption region. The mechanisms of sound generation are numerous and not equal in their efficiency [155]. Heat release is one of the most universal physical phenomena taking place due to absorption of penetrating radiation. Thermal energy can transform partially into sound wave energy in different ways. At moderate released energy densities, when no phase changes occur in a substance, the main contribution to the sound generation process arises from the thermal expansion of a medium. This is the thermoradiation (thermoelastic) mechanism of sound generation. In this case the sound fields can be described within the framework of linear theory which has been developed extensively in recent years [156]. The pattern of sound generation looks much more complex in the case of large densities of penetrating radiation energy released in a medium. In this case the phenomena arising are nonlinear. The effects caused by the growth of the expansion rate of the heated region of a medium (hydrodynamic nonlinearity) and the change of thermodynamic parameters of a substance in the process of action of penetrating radiation (thermal nonlinearity) turn out to be substantial. If the density of released thermal energy increases further, more complex processes of sound generation develop, which are connected with phase transitions, for example, in the conditions of bubble mechanism of sound generation by penetrating radiation [6, 64] and the mechanism of formation of shock waves [4, 72]4. There are also “non-thermal” mechanisms of sound generation: the Cherenkov, dynamic, striction, and other mechanisms. However, the thermoradiation mechanism is the one best-studied up to now. Research on radiation-acoustic effects could hardly have drawn the attention of physicists in recent decades if it was not connected with the prospects of its practical applications. Examples are scanning radiationacoustic microscopy of condensed media [157, 201, 206, 257], acoustic detection of super-high-energy particles (the DUMAND project — Deep Underwater Muon and Neutrino Detection) [9, 56, 67, 200, 239], research 4

We should note that discussion on the density of the energy released in a medium in the process of absorption of penetrating radiation concerns first of all laser (optical) radiation (photon beams). However, in a certain sense they are valid also for particle beams and a single particle or a group of particles. In the latter cases we can discuss the peaks of local heating and overheating leading to formation of acoustic compression waves, microshock waves, microcavitation, and microbubbles.

RADIATION ACOUSTICS

5

on the role of the radiation-acoustic mechanism in underwater noise generation in calm ocean [149], and also the opportunities opening up for the application of new generations of proton and linear super-powerful accelerators of future generations to the production of super-high-energy neutrino beams and the application of these beams in geoacoustics (neutrino geoacoustics, the GENIUS project — Geological Exploration of NeutrinoInduced Underground Sound) [185, 248] and in neutrino-acoustic ocean tomography [236]. We should note also the important role of radiationacoustic effects in investigation and prediction of radiation blistering [80]. The book consists of eleven chapters. The first two chapters give data on elementary particles, absorption of penetrating radiation in a substance, and mechanisms of radiation excitation of sound. The next seven chapters of the book contain the results of the theoretical treatment of thermoradiation sound generation in condensed media, i.e., in homogeneous and inhomogeneous liquids and solids, under the action of modulated penetrating radiation and radiation pulses on a substance. Particular features of the acoustic fields of moving thermoradiation sound sources are considered. Sound excitation by single high-energy particles is analyzed. The efficiency and optimal conditions of thermoradiation sound generation are discussed. The theoretical consideration is based on the solutions of boundary-value problems for the inhomogeneous wave equation with the right-hand side in the form of the function of power density of sound sources produced by radiation absorption in a substance. It is assumed that this absorption obeys the exponential law, which is valid for laser (optical), X-ray, and electromagnetic (in general) beams and with certain limitations for beams of relativistic electrons. This has provided an opportunity to obtain results in the final form and compare them to experimental data. At the same time, the role of the law of radiation absorption in formation of acoustic field of a thermoradiation sound source is analyzed, and the conditions when the absorption law does not play a considerable role are determined. Corresponding analytical expressions are given. Chapter 10 presents the results of numerous experiments conducted and published by many researchers and concerning thermoradiation excitation of sound by modulated laser radiation and laser pulses in the cases of stationary and moving laser beams, beams of protons and electrons in water, and by electron, positron, proton, ion, and X-ray beams and gamma-quanta in metals. We have to note here that comparison of these results to theoretical conclusions proves the validity of the thermoradiation theory. Some applications of radiation acoustics are discussed in Chapter 11. The purpose of this chapter is to demonstrate not only the variety of already existing applications but their “large scale”. We mean both radiationacoustic microscopy and immense projects of the future like the DUMAND and GENIUS projects.

CHAPTER 1

Penetrating Radiation: General Information This chapter provides information on particles constituting penetrating radiation, and absorption of penetrating radiation in the process of its interaction with a substance. Only the most general concepts are presented here. Detailed information may be found in specialized books on particle and nuclear physics.

1. ELEMENTARY PARTICLES: FUNDAMENTAL LAWS OF THE MICROSCOPIC WORLD The material world is “constructed” from elementary particles. This means that their properties, laws of motion, and forces between them determine the diversity of physical phenomena. Commonly the particles which cannot be separated into components are called elementary particles. This definition applies to electrons, protons, and neutrons, but not atoms and atomic nuclei. Protons and neutrons together are called nucleons. Another common and well-known particle is a light particle, i.e., photon. An electrically neutral particle, i.e., neutrino, is much less known. It is very difficult to detect, as it interacts with electrons and nucleons very weakly and therefore, goes through a tremendous thickness of substance almost freely. Knowledge on the structure of the microscopic world, i.e., physics of elementary particles, is the basis for the whole of modern science. Studies of atomic structure

7

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PENETRATING RADIATION: GENERAL INFORMATION

provided an opportunity to discover extraordinary properties of elementary particles and develop a theory of motion, i.e., quantum mechanics. Quantum mechanics and the theory of relativity are the two pillars supporting the whole of modern physics. Such a general concept as symmetry, which to a significant degree determines the structure of particles and their interactions, is also fundamental for modern physics. Modern models and theories of physics of elementary particles are formulated in the mathematical language of the apparatus of symmetry, i.e., theory of groups. One of the most important parameters in quantum mechanics is spin. All particles are separated into classes depending on the value of their spin: particles with half-integer spins are fermions or Fermi particles and particles with integer spins are bosons or Bose particles. The description of spin using the mathematical apparatus of the theory of groups became the starting point of many theories, i.e., so-called internal symmetries. Development of symmetry schemes unifying fermions and bosons is the goal of the supersymmetry trend and finally, the Grand Unification Theory. Let us turn to history. It was discovered in the first decade of the last century that an atom consists of a nucleus and electrons. As it turned out, a nucleus has the dimensions of 10−13 cm and the whole atomic mass is contained in it. The density of matter is extremely high in a nucleus and is equal to 1014 g/cm3. The charge of the nucleus is positive. Electrons move around a nucleus at a distance of 10−8 cm. It was determined in the 1930s that a nucleus consists of protons and neutrons. The last do not have any charge. Electrons are held within an atom by electric forces. Physicists call the forces binding positively charged protons with neutral neutrons nuclear forces, due to their nature. Pauli had predicted the existence of the neutrino already in the 1920s. It only became possible to observe this particle experimentally twenty years after it had been discovered because of its “ability” almost not to interact with matter. Physicists associated this feature with forces of weak coupling in contrast to nuclear forces, i.e., forces of strong coupling. Further research, and first of all the studies of the nature of nuclear forces, led to the discovery of a huge number of particles, their interaction, and interconversion. Quantum electrodynamics, quantum chromodynamics, and the unified theory of electric weak interaction were developed. All particles are divided into hadrons and leptons depending on processes they take part in. Fundamental interactions of only four types stand behind all processes observed up to now, i.e., electromagnetic, weak, strong, and gravitational couplings. Gravitation is universal. All elementary particles take part in it. Sources of electromagnetic field are charges. Neutral particles, which do not have any charge, interact with an electromagnetic field only due to their complex structure or quantum

RADIATION ACOUSTICS

9

effects. In this sense electromagnetic coupling is not as universal as the gravitational one. The same is also true about weak coupling. As for strong coupling, only hadrons constituting the vast majority of particles (about 200) take part in it. Multiple mesons and hyperons (both long-lived and resonance ones, i.e., with lifetime shorter than 10−20 s) belong to the family of hadrons as well as nuclons. Leptons take part in electric weak coupling and do not participate in strong coupling. There are six of them: electrons e, muons µ, tau-leptons τ, and the corresponding three neutrinos νe, νµ, and ντ1. In contrast to leptons, hadrons may be called elementary particles only in the sense that they are really indivisible. However, it was determined that they have an internal structure and behave as “loose” systems in hadron interactions. Hadrons consist of quarks. According to modern views, quarks are structureless true elementary particles like leptons. As distinguished from hadrons, leptons and quarks are called fundamental particles. However, quarks do not exist separately. They exist within hadrons in a bound state. This property of quarks is called “confinement”. Another, so to say opposite, side of this feature of quarks is the fact that they do not interact when close to each other. This property got the name of “asymptotic freedom”. A hypothesis of existence of quarks was suggested in 1964. It followed from the assumption of the existence of symmetry among leptons and quarks that there should be six quarks. This was confirmed experimentally. In the process of experimenting with high-energy particles, it became possible to “observe” quarks and determine their masses and charges. However, it was impossible (and apparently will not be possible in the future) to knock out a quark from a hadron. Five quarks have been discovered up to now and the search for the sixth one is in progress2. Quark types or “flavors” (as they are called commonly) are denoted by letters u, d, s, c, b, and t. According to theory, each quark must have a certain “color” charge (color is “chromos”; the term “chromodynamics” follows from here). There are three color charges in all. “Color” charges are introduced analogously to common yellow, red, and blue colours. Thus according to theory, there must be 18 quarks and the same amount of anti-quarks. 1

There was very little experimental data on the neutrino until recently. The situation has changed now. Multiple reports on conversion of one type of neutrino into another and non-zero mass of neutrino were published on the basis of studies of solar neutrino and experiments with accelerators. For example, see J. N. Bahcall et al., Nature, 1995, v. 375, p. 29 and A. B. Balantekin, Phys. News, 1995, AIP, p. 49. 2 The discovery of the sixth quark has been reported! See C. Quigg, Discovery of the Top Quark, Phys. News, 1995, AIP, p. 56; F. Abe et al., (CDF Collaboration), Phys. Rev. Lett., 1995, v. 74, p. 2626; and S. Abachi et al., (DO Collaboration), Phys. Rev. Lett., 1995, v. 74, p. 2632.

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PENETRATING RADIATION: GENERAL INFORMATION

Hadrons are built from them. Hadrons are divided into two big groups: barions with barionic charge and mesons without this charge. Protons, neutrons, and other particles belong to the first group, while the second group includes π-mesons, k-mesons, and so on. Barions are constructed from three quarks with different mutually complementary “color” charges. A meson consists of a quark and an anti-quark. Both barions and mesons are “colorless”. We should stress that in the case of quarks, “color” is just a convenient term to denote quantum numbers characterizing quarks. Color charges of anti-quarks differ from that of quarks. There is a strong symmetry among quarks of different flavors and leptons. Leptons and quarks include three generations of fundamental fermions. Fermions of the first generation together with photons are the construction material for modern matter. As for fermions of the second and third generations, apparently physicists begin to understand now that on the one hand they played an important role at the earliest stages of formation of the Universe and on the other hand our existence depends on the relationship between uand d-quarks and electrons. Studying the nature of the forces binding quarks in hadrons has demonstrated that here lies a deep analogy with electromagnetic forces. Interaction of an electron with another electron or charged particle is performed by photons. They have zero rest mass but do not have an electric charge. Therefore, an electron does not change its charge when it emits a photon. Photons are vector particles. They are described by a vector field (corpuscular-wave dualism). It was demonstrated totally analogously to quantum electrodynamics and its principles of symmetry that interaction between quarks is performed also by vector particles, i.e., gluons, which have zero mass and unit spin like photons. However, unlike photons, there are eight gluons and they have “color” charges corresponding to the laws of quantum chromodynamics. Moreover, they interact strongly with each other forming elementary particles called glubols. Nuclear forces between protons and neutrons (hadrons) in a nucleus are the secondary manifestation of quark-gluon interactions. A theory of quarkgluon interactions has been confirmed experimentally. It has been developed within the framework of quantum chromodynamics and describes formation of “gluon strings” and “quark, anti-quark, and gluon streams” developing when gluon strings get broken in the process of collision of high-energy hadrons. Thus, quarks and gluons (they are often called partons) are fundamental particles constituting hadrons. It was established experimentally that their dimensions are smaller than 10−16 cm, while the characteristic size of hadrons is 10−13 cm. Therefore, partons may be considered point particles like leptons.

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The nature of weak forces or weak coupling was described within the framework of the so-called theory of electric weak coupling developed in the 1960s. It was established within this theory that despite the fact that electromagnetic and weak forces are different, manifestations of one and the same coupling, i.e., electric weak coupling, constitutes their nature. According to the electric weak theory, weak couplings are caused by the interchange of intermediate vectors by W±- and Z0-bosons, just as electromagnetic coupling is caused by interchange of photons. In this case the weakness and small radius of weak coupling are explained by the fact that W±- and Z0- bosons are very heavy particles. W±-bosons are charged while Z0-bosons are neutral vector particles. Bosons were discovered in 1983 by European physicists in experiments with an accelerator with counter-propagating proton-anti-proton beams at CERN (Switzerland) and this was a solid confirmation of electric weak theory. It is possible that there exist more of such particles, i.e., intermediate vector bosons. This problem is still open. One of the fundamental properties of weak forces is the fact that they are connected with slow (weak) decays and always associated with the neutrino. Weak forces play an important role in the evolution of the Universe. Nuclear reactions in the Sun are connected with them. The fact that intermediate vector particles (photons, bosons, and gluons) take part in all interactions, successful development of a unified electric weak theory, symmetry among leptons and quarks, etc., indicate the possibility of the existence of a general bond between fundamental forces. The possibility to develop a single theory unifying these forces, i.e., electromagnetic, strong (nuclear), weak, and gravitational forces, follows from here also. The key element of different approaches to formation of the Grand Unification Theory is the development of different kinds of supersymmetry schemes. However, despite the fact that papers devoted to this problem number several thousands already, “traces” of supersymmetry have not been discovered in spectra of known elementary particles. Nevertheless, it is possible that such traces will be discovered in experiments with superpowerful accelerators of new generations. The logic of development of physics of elementary particles is for this3. Let us point out in conclusion the characteristic scales of fundamental forces. Gravitational and electromagnetic forces are long-range. Nuclear 3

It was reported recently that results of very precise measurements obtained lately support predictions on minimal supersymmetric SU (5) model, which unifies electromagnetic, weak, and strong couplings (see U. Amaldi, W. de Boer, and H. Furstenau, Phys. Lett. B, 1991, v. 260, p. 447; S. Dimoklos, S. A. Raby, and F. Wilczek, Phys. Today, 1991, v. 44, p. 25). It is expected that such “unification” (if it exists) should be observed at the energy E > 1013 TeV.

12

PENETRATING RADIATION: GENERAL INFORMATION

forces act at distances of the order of magnitude of nuclear size. A short range of coupling characterizes weak forces. It is at least two orders of magnitude less than the range of strong coupling. New theories including the Grand Unification Theory need new experiments with more powerful accelerators of new generations4. However, there are fundamental limits in construction of very powerful (“traditional”) accelerators. Estimates show that a hypothetical annular accelerator for energy 107 TeV, which could be constructed using a superconductor magnet, would be a ring with radius about that of the Earth. Even if such an accelerator was located in space, attempts to further increase of its power and size would be impossible because of synchrotron radiation. In the meantime, “natural” accelerators can “accelerate” particles, e.g., neutrino, up to the energy 108 TeV. Methods of radiation acoustics may be useful (or even extremely necessary) for detection of just such particles of super-high energy. Indeed, as will be described in detail in a special section, 7 8 the appearance of particles with energy E ≈ 10 – 10 TeV in the spectrum of cosmic rays is an extremely rare occasion. Probability and frequency of their detection increase with the increase of detector size. Huge detectors (with the size of tens of square kilometers) are needed for this. Just under such conditions, application of acoustic detection methods (i.e., utilization of arrays of hydrophones or seisms) instead of traditional techniques (scintillation counters and Cherenkov detectors in water) can be favorable and justified. Thus, according to modern concepts, six leptons, six types (“flavors”) of quarks, and five bosons (photons, gluons, and intermediate vector bosons γ, g, W±, and Z0) are the fundamental particles forming the world around us. All processes observed up to now are caused by fundamental couplings of four types: gravitational, electromagnetic, strong (nuclear), and weak. We have tried to give the most general idea of the modern state of affairs in the physics of elementary particles5. We do not need a detailed presentation here. Those who are interested in more details can find them in

4

It is useful to note that some theories of supersymmetry connect hopes for 16 development of the Grand Unification Theory with the energy 10 TeV and scales −33 at distances about 10 cm. As it follows from modern concepts, a strict symmetry exists at small distances. This symmetry is violated spontaneously in the process of transition to large scales which leads to the variety of particles and types of their interaction which we observe around us in the process of experimenting with modern large accelerators and in the spectrum of cosmic rays [73, 78,169] (see also the previous footnote). 5 According to a colorful expression by Ya. B. Zel’dovich, a presentation for “pedestrians” [98].

RADIATION ACOUSTICS

13

books by Akhiezer and Rekalo [14], Grishin [78], and Okun’ [168, 169], a paper by Zel’dovich [98], and other papers cited there6.

2. ABSORPTION OF PENETRATING RADIATION IN A CONDENSED MEDIUM In many cases the parameters of the acoustic field generated by penetrating radiation in a condensed medium are determined essentially by the characteristics of radiation absorption in a substance. Absorption depends on the type of particles (quanta), their energy, and the material of a target. Let us consider, in the most general form, some laws of radiation absorption. While moving through a substance, particles interact with its atoms, i.e., electron shells and atomic nuclei (or nuclear nucleons). As we have indicated already, there are three types of interactions involving particles (we ignore very weak gravitation coupling as it is very small and insignificant at this scale), i.e., strong (nuclear), electromagnetic, and weak. In contrast to electromagnetic coupling and in the case of not very large energy of particles, strong and weak couplings are essential only in the case of very small dimensions of the interaction region, about the dimensions of elementary particles and atomic nuclei. Therefore, considering passage of charged particles through a substance, the major role belongs to electromagnetic coupling.

TRANSITION OF HEAVY CHARGED PARTICLES THROUGH A SUBSTANCE Protons, nuclei of various atoms, e.g., nuclei of atoms of helium 4He (α-particles) consisting of two protons and two neutrons, nuclei of hydrogen isotope, isotope of helium 3He (two protons and one neutron), and some other particles taking part in strong coupling belong to heavy charged particles. Heavy charged particles of moderate energy lose it in the process of passage through a substance. This happens mostly because of inelastic collisions with bound electrons of atoms of a retarding substance. When the velocity of a particle becomes so small that it captures electrons, energy losses decrease. However, deceleration of a particle continues until its energy gets reduced down to the thermal energy of substance atoms. The 6

See also papers on accelerators [47, 176, 223, 229, 237, 256].

14

PENETRATING RADIATION: GENERAL INFORMATION

first stage of the process of deceleration, i.e., deceleration during the time when a particle is still charged, may be described theoretically in a satisfactory way [188, 190]. The second stage of the deceleration process, i.e., the stage when a particle captures electrons (e.g., at energy less than 1 MeV for an alpha-particle or 0.1 MeV for a proton), practically cannot be treated theoretically. However, there are reliable experimental data [244]. The loss of energy by multi-charge nuclei has a special feature: they can capture not a single electron but several, which makes the spectrum of energy losses more complex [232]. Theoretical calculation for the first stage of the process of deceleration of a heavy particle leads to the formula [188] dE 4πe 4 z 2 NB , = dx mV 2

(1.1)

where, for example, in the case of relativistic velocity of a particle  2mV 2  − ln(1 − β 2 ) − β 2  ; B = Z ln I   E is the particle energy; ze and V are its charge and velocity; M and m are the masses of the incident particle and electron, respectively; N is the number of atoms in the unit volume of a substance; Z is the charge of nuclei of atoms of a medium substance; I is the average excitation energy of an −19 J); β = V/u; and u is the velocity of light. The atom (I = 18.5Z⋅1.6⋅10 quantity dE/dx is called the “stopping ability” of a substance and B is called the deceleration coefficient. Sometimes the quantity B is identified as the “stopping ability” of a substance. In the general case, B is determined either theoretically or experimentally. Passing through a substance, a heavy charged particle performs ionization, which results in loss of energy. The most probable initial collisions with atomic electrons are collisions, such that a relatively slow secondary electron with kinetic energy not exceeding the ionization energy is knocked out. However, secondary electrons with relatively large energy form in the result of a small number of collisions, their maximum energy 4(v/M)E corresponding to the maximum velocity equal to the double velocity of an incident heavy particle. For example, in the case of a proton with energy E = 10 MeV, secondary electrons of different energy may be produced, the maximum energy being equal to 20 keV. These so-called delta-electrons (δ-electrons) ionize atoms of a retarding medium further. Therefore, the initial ionization produced by the charged

RADIATION ACOUSTICS

15

particle itself should be distinguished from secondary ionization produced by delta-electrons. According to experimental data, total ionization exceeds initial ionization approximately three times. Distribution of ionization losses along the path of a particle coincides to a first approximation with the stopping ability of a substance or, in other words, with the distribution of energy losses at the unit length of the path dE/dx. The main result obtained from formula (1.1) is the fact that specific energy losses of a charged particle for ionization are proportional to the particle charge, concentration of electrons in a medium, and a certain function of velocity, but they do not depend on the particle mass. Formula (1.1) shows that as the particle energy grows, specific losses for ionization decrease at first very rapidly (inversely proportional to energy), but do this more and more slowly as the particle velocity comes closer and closer to the light velocity. However, starting from a certain large enough energy of a particle, energy losses increase on account of relativistic effects. Formula (1.1) is not quite exact: in the case of a large velocity of a particle it is necessary to take into account the so-called density effect [188], which leads to decrease of specific energy losses. Due to this fact, relativistic growth of specific losses stops and the curve becomes flat. The density effect manifests itself in condensed media earlier than in rarefied ones.

ABSORPTION OF NEUTRAL HEAVY PARTICLES IN A SUBSTANCE Deceleration of neutral heavy particles (neutrons are the most important ones) in a condensed medium occurs mostly because of direct collisions with atomic nuclei of target substance. Nuclear forces play the main role in this process. It is very difficult to obtain analytical functions in this case and one has to use either experimental data or semi-empirical functions [20]7. We should note here that energy losses by heavy neutral particles are much smaller than the corresponding losses by heavy charged particles and consequently, their depth of penetration into a medium is much larger than that of heavy charged particles.

7

Another way is to use a complex computer simulation of processes of neutron transfer in a substance using the Monte Carlo method for example or other ones. Only in some cases is it possible to obtain analytical functions of a rather complex form. Some information on these functions characterizing absorption of fast neutrons in a substance can be found in a paper by Han S. Uhm, J. Appl. Phys., 1992, v. 72 (7), p. 2549 – 2555.

PENETRATING RADIATION: GENERAL INFORMATION

16

ABSORPTION OF ELECTRONS IN A SUBSTANCE Energy losses in a substance for electrons with relatively low energy (smaller than the so-called critical energy; see below) are caused by ionization and excitation of bound electrons of a retarding medium as in the case of heavy charged particles. If electron energy is larger than the average energy of atom excitation, the formula for calculation of ionization losses by such electrons does not differ too much from formula (1.1) and can be written down in the next form [188]:

−

 dE 2πe 4 mV 2 E = − (2 1 − β 2 − 1 + β 2 ) ln 2 + NZ ln dx mV 2  2 I 2 (1 − β 2 ) (1.2) 1  (1 − β 2 ) + (1 − 1 − β 2 ) 2  , 8 

where E is the energy of an incident electron (total energy minus rest energy). The difference of formula (1.1) to formula (1.2) is caused by the difference between the equivalent masses of an electron and a heavy particle or two electrons and also quantum-mechanical indiscriminability of two electrons participating in the process of collision. If electron energy is lower than the average excitation energy of atoms, it is difficult to describe the process theoretically as in the case of small energy of heavy charged particles, when they capture electrons of a medium, and one has to use experimental data [244]. In the case of high-energy electrons, we have to take into account the density effect as in the case of heavy charged particles. This effect reduces ionization losses as compared to those given by formula (1.2). If the energy is very large, electrons start to lose energy effectively since the role of bremsstrahlung increases. According to the classical electromagnetic theory, a charge under 2 2 3 acceleration a emits energy (2/3)(e a /c ) per time unit. An electron can be accelerated in the electric field of a nucleus since its mass is small and acceleration is proportional to the charge of a nucleus Z divided by the electron mass m. The bremsstrahlung has the major effect upon energy losses by fast electrons. Radiation does not play an essential role in the process of deceleration of fast heavy particles. This is caused by the fact that acceleration is proportional to 1/M, and as a result, radiation gets 2 reduced (m/M) times as against radiation by electrons. We can write down the next expression for energy losses for radiation by a relativistic electron [164, 188]:

RADIATION ACOUSTICS

1 dE 1 =− , E dx x0

17

(1.3)

where x0 is the so-called radiation length different for different substances. Comparison of expressions (1.2) and (1.3) shows that energy losses for radiation increase with energy growth almost linearly, while energy losses for ionization increase only logarithmically. Therefore, in the case of large energies, losses for radiation are dominant and as the energy of electrons decreases, and ionization becomes more and more important until at some critical energy Ecr losses for ionization and losses for radiation will become 2 comparable. We can give an approximate formula Ecr ≈ 1600 mu / Z and obtain an equation for the ratio of energy losses for radiation to losses for ionization [188], (dE / dx) rad EZ . = (dE / dx) ion 1600mu 2 It follows from here that in the case of water for example, Ecr ≈ 100 MeV and in the case of lead Ecr ≈ 10 MeV. In the case of electrons with energy higher than critical, as the distance increases, the energy losses are described (on the average) by an exponential law corresponding to expression (1.3). In the case of water and air x0 is equal approximately to 36 g/cm2, for aluminum it is equal to 24 g/cm2, and for lead it is about 6 g/cm2.

ABSORPTION OF ELECTROMAGNETIC RADIATION IN A SUBSTANCE In the process of transmission through a substance electromagnetic radiation is subjected to characteristic exponential absorption in contrast to the laws characterizing absorption of heavy charged particles or electrons. The reason for this is the fact that in the process of absorption each gammaquantum is taken away from an incident beam as a result of a single act. In this case the beam intensity at the absorber thickness x has the next form [188]: I ( x) = I 0 exp(− µx) ,

(1.4)

where I0 is the intensity of incident beam and µ is the coefficient of radiation absorption. The main processes determining absorption of gamma-

18

PENETRATING RADIATION: GENERAL INFORMATION

quanta in a substance are the photoeffect, Compton scattering, and formation of electron-positron couples. In the case of gamma-quanta (γquanta) of low energy the most essential role is played by the photoeffect; the Compton absorption is dominant in the range of intermediate values of energy; and formation of couples is most important at large values of energy.

FORMATION OF NUCLEAR-ELECTROMAGNETIC CASCADES IN A SUBSTANCE Nuclear-electromagnetic cascades (cascade showers as they are called frequently or just showers) are formed in a substance in the process of absorption of penetrating radiation with quanta of very high energy sufficient for formation of multiple secondary particles. Their rise occurs as a result of a large number of single interactions, and therefore, it is not an elementary process [119, 163, 174]. In the case of large values of energy of an initial particle, the energy distribution in a cascade is almost independent of the type of the initial particle since development of a cascade is determined basically by an electron−photon avalanche. Quantitative characteristics of electron−photon cascades are studied by the electromagnetic cascade theory. The major task of this theory is determination of the function describing the distribution of particles at a certain depth in a substance with atomic number Z with respect to energy, angles, and distances from the shower axis, which is determined as the line of motion of the initial particle. A spectrum of electrons in a shower or socalled cascade curves of a very complex shape can be obtained from the solutions of the kinetic equations of the cascade theory. About 80% of electrons in the maximum of a shower have energy smaller than the critical one, and 50% of them have energy smaller than 1/3 of the critical energy. The distance of recession of an electron from the shower axis is determined basically by Coulomb scattering at the path of the order of magnitude of the radiation length. It is inversely proportional to the electron energy. In other words, in the tail of a cascade, where the energy of electrons is small, they can go away farther from the cascade axis.

EXTENSIVE AIR SHOWERS Studies of cosmic rays lead to the discovery of nuclear-electromagnetic showers in the atmosphere, i.e., showers of particles of cosmic rays covering large areas. These showers got the name of extensive air showers. They rise as a result of interaction of high-energy particles belonging to the

RADIATION ACOUSTICS

19

so-called hard component of cosmic rays with a substance in the atmosphere. Let us present basic results concerning absorption of penetrating 10 radiation in a substance. If particle energy is very large (about 10 eV and higher), the rise of cascade showers occurs in a medium (in this case it is necessary that neutrinos and muons interact with a substance in the process of some nuclear reaction). If particle energy is smaller and showers do not arise, absorption of penetrating radiation in a substance depends on the type of particles and energy. Thus, absorption of gamma-quanta (and optical radiation) occurs according to an exponential law in the whole range of these energies. Absorption of charged heavy particles (and muons) is governed by law (1.1) up to the energies when a particle captures an electron or its energy becomes equal to the average energy of atom excitation. Further absorption is determined by more complex functions, which can be obtained in a simpler way from experimental data. As a proton continues to move in a substance, its losses grow until it captures an electron of a substance atom and then losses begin to decrease. A curve characterizing dependence of losses of a proton on the so-called residual path displays a peak. This peak got the name of the Bragg peak [188]. Absorption of relativistic electrons occurs at first according to an exponential law according to formula (1.3) until the critical energy is attained. Then its absorption is determined by formula (1.2) up to the energies when the velocity of incident electrons equals velocities of electrons of atomic shells. Further absorption is determined by complex functions obtained from experimental data. We should note that the energy range, when expressions (1.2) and (1.3) become invalid and absorption of heavy charged particles and electrons is determined by functions obtained from experiments, is usually small and we can consider a particle to be totally absorbed when it attains such small energy. The grounds for theoretical treatment of the thermoradiation mechanism of sound generation are the solutions of an inhomogeneous wave equation. The right-hand side of this equation contains a function describing the power density of thermal sources of sound, which are created by absorption of penetrating radiation in a substance. In the case of fluid media, this equation has the form

∆p −

1 ∂2 p c

or

2

∂t

2

=−

α ∂Q , C p ∂t

(1.5)

20

PENETRATING RADIATION: GENERAL INFORMATION

∂2 p ∂t 2

− c 2 ∆p = Γ

∂Q , ∂t

where p is the sound pressure, c is the velocity of sound, α is the coefficient of cubical thermal expansion, Cp is the specific heat capacity of a liquid, Q is the function characterizing energy evolution (absorption) of penetrating 2 radiation or the power density of thermal sources of sound, and Γ ≡ c α / Cp is the Grueneisen coefficient. In the case of an isotropic solid, the equation has the form   1 ∂2  (3 − 4 / n 2 ) β divF  ∆ − u = Qdt − div , ∫   2 2 2 Cε ρ c ∂ t c ρ l l  

(1.6)

2    ∆ − 1 ∂ rot u = − rotF .  ct2 ∂t 2  ct2 ρ 

(1.7)

Here cl and ct are the velocities of longitudinal and transverse waves, respectively, n = cl / ct , u is the vector of displacement of a particle of a solid, ρ is the density, β is the coefficient of linear thermal expansion, Cε is the specific heat capacity of a solid, and F is the external non-thermoelastic force applied to the unit of solid volume (the nature of this force will be explained later); F ≡ 0 if we consider only the thermoradiation mechanism of sound generation, and therefore, only thermal sources of sound are taken into account. One can see from the equations given below that the amplitude of sound pressure is proportional to the Grueneisen coefficient Γ and depends on the function of energy release Q (the power density of thermal sources of sound). The form of the function Q for different types of penetrating radiation differs. In the case of electromagnetic radiation Q( x1 , x 2 , x3 , t ) = µI ( x1 , x 2 , t ) exp(− µx3 ) ,

(1.8)

where x3 is the coordinate in the direction of propagation of radiation beam, µ is the coefficient of radiation absorption in a medium or the inverse path of radiation quanta in a medium. For example, in the case of light with the wavelength λl = 1.06 µm (radiation of a neodymium glass laser) and pure −1 water µ = 0.17 cm and in the case of CO2-laser radiation (λl = 10.6 µm) −1 and pure water µ = 800 cm . Formula (1.8) is true for electromagnetic radiation of various types, i.e., light, X-rays, synchrotron radiation, beams of γ-quanta, etc., as well as

RADIATION ACOUSTICS

21

relativistic electrons, where energy losses in the process of interaction with a substance are connected with emission of photons (radiation losses). We can write down an expression for the intensity of penetrating radiation in a beam: I ( x1 , x 2 , t ) = n p ( x1 , x 2 , t ) E ,

(1.9)

where np(x1,x2,t) is the density of particles. We have for the function of energy release Q( x1 , x 2 , t ) = n p ( x1 , x 2 , t )

dE ( x3 ) . dx3

(1.10)

One can see readily from expressions (1.3), (1.4), (1.9), and (1.10) that −1 formula (1.8) is true for relativistic electrons if we take µ = x0 . If the energy of electrons (non-relativistic) is small but larger than the average energy of excitation of substance atoms interacting with the electron beam, losses are determined by ionization. As we have mentioned above already, radiation losses increase with the growth of energy almost linearly and the losses of energy Ee increase only logarithmically. Therefore in the case of large energy Ee (relativistic velocities), radiation losses prevail and as the energy of electrons decreases, ionization plays a more and more important role until losses for ionization and losses for radiation become equal at a certain critical energy Ee,cr. For example in the case of water, Ee,cr = 100 MeV and for lead Ee,cr = 10 MeV. In the case of energies higher than the critical one, energy losses are described (on the average) by an exponential law according to formulae (1.3) and (1.8). Thus, a beam of relativistic electrons with energy of 1 GeV loses up to 90% of its energy in water and up to 99% of it in lead in compliance with exponential law (1.8), and only about 10% and 1%, respectively are lost due to ionization. In the case of heavy charged particles (protons, nuclei of various atoms, and particles in general, which take part in strong coupling), formula (1.8) is inapplicable and we can use expressions (1.1) and (1.10) to determine the form of the function of energy evolution. In the process of absorption of penetrating radiation with particles of very high energy sufficient for production of multiple secondary particles, when nuclear-electron-photon or nuclear-electromagnetic cascades rise in a substance energy absorption in a cascade can be considered as approximately exponential.

22

PENETRATING RADIATION: GENERAL INFORMATION

As for deceleration of neutral charged particles (the most important of them are neutrons) in a condensed medium, this deceleration occurs due to direct collisions with nuclei of atoms of the substance as we have mentioned above. Derivation of universal analytical functions in these cases is difficult as a rule, and one has to be content with specific calculations or empirical functions based upon experimental data. Thus, determination of the analytical form of the function of energy release, Q can be difficult. Therefore, the exponential law of absorption of penetrating radiation in a substance, and the Gaussian intensity distribution in a beam with respect to its radius, are adopted frequently when specific problems of thermoradiation generation of sound are treated. The basic conclusions of the theory also stay true in the majority of cases for other kinds of radiation, when absorption in a substance is not governed by an exponential law [156].

CHAPTER 2

Basic Mechanisms of Sound Generation by Penetrating Radiation in Condensed Media Mechanisms of sound generation are physical phenomena resulting in conversion of energy of penetrating radiation into energy of sound waves in the process of radiation absorption in a medium. Mechanisms of sound generation may depend on the type and intensity of penetrating radiation, target substance where this radiation is absorbed, and the mode of energy release. Physical phenomena of conversion of energy of penetrating radiation into sound energy (mechanisms of sound generation) are multiple and unequal in their efficiency. Here we treat only those of them which are apparently most important in the sense that they alone determine sound fields in the majority of real situations.

1. MECHANISMS CONNECTED WITH HEAT RELEASE One of the most universal physical phenomena taking place in the process of absorption of penetrating radiation in a substance is heat release. Released thermal energy can be converted partially into the energy of sound waves in different ways. 23

24

BASIC MECHANISMS OF SOUND GENERATION

In the case of moderate density of energy released in a medium when phase transitions do not occur, the major contribution into the process of sound generation belongs usually to thermal cubical expansion. This is a socalled thermal or thermoradiation mechanism of sound generation. Sometimes it is called a thermoelastic mechanism, when sound excitation by penetrating radiation in a solid is discussed. A distinctive feature of the thermoradiation mechanism is the fact that the basic fundamental features of sound fields can be described within the framework of a linear model corresponding to the acoustic approximation in the case of calculation of hydrodynamic or elastic disturbances. It has provided an opportunity to develop a very effective theory of sound generation in a liquid (e.g., by laser radiation), which agrees well with experimental data (see reviews by Bunkin and Komissarov [41], Lyamshev [128, 129], and Lyamshev and Sedov [142]). Recently a theory of thermoradiation sound generation by penetrating radiation has been developed [156]. The pattern of sound generation in the case of large intensities of radiation (and more precisely, in the case of large densities of energy delivered into a medium) looks more complex. The effects developing in this case are of nonlinear character, and the design of an exhaustive theory of these phenomena is really far from being finished. Some results concerning sound generation in a liquid by high-intensity laser radiation can be found in reviews by Lyamshev [128] and Lyamshev and Naugol’nykh [141]. Nonlinear effects similar to the ones characteristic to interaction of powerful laser radiation with a substance also arise in a medium in the process of absorption of penetrating radiation in it, when the density of energy released in a medium is sufficient for phase transitions [86]. An exhaustive theory of such phenomena has been developed to an even less extent than in the case of absorption of laser radiation. Further we consider only some examples of such phenomena. Let us treat in more detail the basic mechanisms of sound generation by penetrating radiation.

2. THERMORADIATION GENERATION OF SOUND Many studies have been devoted to the thermoradiation mechanism of sound generation by various types of penetrating radiation [8, 9, 11, 16, 23, 28, 36, 54 – 56, 58, 68, 70, 81 – 83, 85, 103, 104, 151, 153, 154, 158, 159, 193, 194, 199, 200, 209, 212, 224, 225, 231, 235, 242, 244, 255, 261]. Sets of hydrodynamic equations or equations of dynamic theory of thermoelasticity and equations of the field of radiation of a given type are initially required for derivation of the equation of thermodynamic

RADIATION ACOUSTICS

25

generation of sound in a liquid or isotropic solid by penetrating radiation. Linearizing a set of equations and ignoring the effects of thermal conductivity and viscosity in the case of a liquid, we write down an inhomogeneous wave equation of sound generation in the form [259],

∆p −

1 ∂2 p c

2

∂t

2

=−

α ∂Q . C p ∂t

(2.1)

In equation (2.1), p is the sound pressure, α is the coefficient of cubical thermal expansion, Cp is the specific heat capacity of a liquid, Q is the function characterizing the power density of thermal sound sources arising due to absorption of penetrating radiation in a liquid, and c is the sound velocity in a liquid. It follows from this equation that in the case of a sound field of high frequency or sound pulse of small duration, the amplitude of a signal is proportional to the quantity αc2/Cp or the Grueneisen parameter (coefficient) Γ. The same may be said also about sound stress in a solid. The power density of thermal sources of sound depends on the type of penetrating radiation and the target material. Many papers studied the dependence of acoustic signal amplitude on the energy of penetrating radiation and the substance and geometry of the target. Thus, Borshkovskii and Volovik [36] studied the dependence of the amplitude of acoustic signal excited in thin metal (aluminum) plates by pulsed beams of electrons and protons. Some results are given in Fig. 2.1, where dependences of acoustic signal amplitude on energy of electrons and protons are presented. As the authors note, these results are the evidence of the fact that the main contribution to acoustic vibrations from electrons and protons flying through thin metal plates derives from energy losses for ionization.

Figure 2.1 Dependences of acoustic signal amplitude on the energy of (1) protons and (2) electrons.

26

BASIC MECHANISMS OF SOUND GENERATION

Volovik and Ivanov [53] demonstrated that the ratio of amplitudes of acoustic signals excited by pulsed beams of protons and electrons in aluminum and piezoelectric ceramics is equal to the ratio of the corresponding Grueneisen coefficients (if the density of absorbed energy is the same). This was considered by the authors as the proof of the thermoelastic nature of the signal in these substances. Analogous estimations of acoustic signal amplitude (or experimental data) were made in the majority of other relatively early papers [28, 242, 261]. The authors of later papers studied the shape of a signal together with its amplitude. Blazhevich et al. [28] used the shape of an acoustic pulse excited in a solid sample by a pulse of electronic beam for determination of density of energy absorbed in the sample. In this case the shape of a one-dimensional acoustic pulse is connected with the density of energy absorbed in a sample E(x) by the relationship, v ( x, t ) ≡ −

Γ E ( x − ct ) , 2

(2.2)

where c is the sound velocity in a sample. Lyamshev and Chelnokov [152, 153, 158, 159] studied theoretically sound excitation by penetrating radiation in a liquid and solid in an approximation of the thermal mechanism. Many papers were devoted to investigation of the problem of the mechanism of generation of acoustic vibrations by penetrating radiation in a liquid [16, 68, 235, 255]. For example, Golubnichii, Kalyuzhnyi, and Korchikov [68] established that the rise of the acoustic signal produced by an absorbed electron beam in a liquid is determined by the thermal mechanism of sound generation. Performing accurate measurements, they demonstrated that the measured amplitude of acoustic signal in a liquid coincided with the one predicted theoretically and in the case of changing of water temperature around 4°C, the acoustic signal changed its polarity which was explained by the change of the sign of the absorption coefficient of water. Analogous results were obtained by Balitskii et al. [16] who studied, however, an acoustic field generated by a totally absorbed electron beam in water, and not by a beam, which was partially absorbed in a cell with water and partially transmitted through it [68]. Similar results demonstrating that, in the case of protons, the thermal mechanism of sound generation in water is dominant, were obtained by Danil’chenko et al., Levi et al., and Sulac et al. [85, 235, 254, 255]. Many papers devoted to acoustic detection of single high-energy particles (the DUMAND Project) analyze the parameters of the sound field

RADIATION ACOUSTICS

27

in the near-field zone of a nuclear-electron cascade arising in water [11, 56, 194, 200, 214]. Thus, using equation (2.1) and calculating in different ways the function Q(x, y, z, t), we can obtain the shape and amplitude of a produced acoustic pulse. The time dependence of the function Q is always taken in the form of the delta-function δ(t) since the time of the cascade rise is much smaller than other characteristic times of the process of sound generation, and the spatial dependence is determined on the basis of various direct calculations of density of energy release in the cascade taking into account possible approximations. For example, Volovik et al. [58] give the next expression for the amplitude of acoustic signal in sea water:

Pmax =

0.44ϕ (r )  E     18 Pa , r  10 

where E is the energy of a particle (in eV) and r is the distance from the cascade axis (in meters). The factor ϕ(r) allows for deviation from the law p(r) ~ r−1/2, because of absorption or sound refraction for example and it is taken approximately equal to one. For example in the case of a particle with the energy E = 1017 eV, the amplitude of acoustic signal at a distance 100 m from the cascade axis is about 4.10−3 Pa. Figure 2.2 presents shapes of acoustic pulses for various dependences of the function of power density of sound sources Q [58].

Figure 2.2 Shapes of acoustic pulses for various dependences of the function of density of energy release Q. (1) The case of distribution of energy released in an electromagnetic cascade according to the Nishimura-Kamata-Greisen model; (2) the case of uniform distribution in a cylindrical region of the length of 5 m and radius of 2 cm; (3) the area with the Gaussian distribution of density of energy release.

28

BASIC MECHANISMS OF SOUND GENERATION

3. INITIATION OF MICROSHOCK WAVES Passage of penetrating radiation through a substance may cause microshock waves. Thus, for example in the case of passage of fission fragments through a liquid, shock waves may arise in it along the track of the fragments [72, 233]. On the other hand, delta-electrons arising in the process of passage of ionizing particles through a liquid produce overheated micro-regions (thermal sources) in it. Explosive expansion of these regions gives birth to a shock wave [4]. The rise of shock waves is an essentially nonlinear effect. The theory of such processes is also far from being completed. Following Anoshin [4], let us perform estimation of parameters of shock waves produced by delta-electrons. In order to produce micro-regions capable of explosive expansion, it is necessary for the energy lost by an ionizing particle to be localized initially in a small enough volume. This condition may be satisfied in the case of delta-electrons knocked out by an ionizing projectile particle. The path l(Eδ) (in cm) of a delta-electron depends on its energy Eδ (in eV) in the following way: l ( Eδ ) = 0.58 ⋅ 10 −12 ⋅ AEδ2

1 , ρ0Z

(2.3)

where ρ0 is the medium density and Z/A is the ratio of the number of electrons to the molecular mass. As the energy of a delta-electron decreases, its trajectory differs more and more from a straight line and becomes similar to a coil. For example, already at Eδ = 10 keV the average path of an electron is a half of the path l(Eδ), i.e., 0.5l(Eδ). The number ν of water molecules per the path l(Eδ) is determined as

ν=

1/ 3

l ( Eδ ) l ( Eδ )  4πρ 0 N A  =   d 2  3A 

,

(2.4)

where d is the diameter of the spherical volume V1 occupied by a single water molecule and NA is the Avogadro constant. If the energy of a deltaelectron is small enough, its track is located within the sphere of the volume V0, which is determined according to the formula V0 = νV1 =

l ( Eδ ) A . ρ0 N Ad

RADIATION ACOUSTICS

29

The energy transmitted on average to a single molecule is equal to ∆E = Eδ /ν = Eδ d / l(Eδ). If we choose (proceeding from reasonable assumptions) for water ∆E = 30 eV when micro-explosive production of bubble nuclei occurs, the parameters making possible explosive expansion of overheated regions in water have the values Eδ = 1.23 keV, ν ≈ 41, and 3 V0 = 1.23⋅10−21 cm . The radius of the sphere of the volume V0 is equal to −8 a0 = 6.64⋅10 cm. The density of energy release Q in the thermal peak, which is determined according to a formula Q = Eδ/V0 = EδAd/[l(Eδ)ρ0N0], in this case is equal to 16.09⋅104 J/cm3. Comparing it to the caloricity of a common explosive, one can see that the latter is equal to 4.19⋅103 J/cm3 approximately. According to the theory of underwater explosions, pressure in a shock wave psh(r,t) can be written down as (2.5)

psh (r , t ) = psh max (r ) exp(−t / Θ) ,

where psh max(r) is the maximum pressure (in Pa), r is the distance from the centre of the spherical volume of thermal peak (in cm), t is time, and Θ is the time constant (in seconds). In this case it is possible to demonstrate that psh max (r ) = 1.37 ⋅ 10 2 r −1 Pa , Θ(r ) = 1.07 ⋅ 10 −10 (7.18 + log r ) at r > 102 a0. The spectrum of the pulse psh(r,t) has the form

[

S (ω ) = psh max (r ) Θ − 2 (r ) + ω 2

]

−1 / 2

.

(2.6)

If frequencies ω < 1 / Θ(r), the spectrum is uniform and can be written down as S (ω ) ≈ psh max (r ) ⋅ Θ(r ) .

Root-mean-square pressure at the distance r from a single thermal peak in water is p = 7.76 ⋅ 10 −8 (7.18 + log r ) ⋅ (1 / r ) Pa/Hz1/2 .

30

BASIC MECHANISMS OF SOUND GENERATION

The number of such thermal peaks per unit length of a track of a relativistic electron is equal to the number of delta-electrons with Eδ = 1.23 keV (according to Anoshin [4], it is equal to 17.32 cm−1). It is interesting to note that it is possible to obtain the following estimate for the intensity I (in W/cm2) of acoustic radiation in water from a single high-energy particle, which produces in the process of its entry to the atmosphere an extensive air shower consisting mostly of electrons [4]. I = 10 −29 E0 / R ,

(2.7)

where E0 is the energy of a particle (in eV) and R is the distance from the axis of an extensive air shower (in cm). Apparently, now it is possible to 17 detect extensive air showers at E0 ≥ 10 eV according to underwater acoustic radiation. When a cosmic particle of super-high energy gives birth to a shower of secondary particles directly in water, the density of radiating centers is much larger than in the case of an extensive air shower in the atmosphere. This provides an opportunity to lower the threshold of detection with respect to particle energy. Apparently, this takes place in mountain lakes, for example. The considered opportunity for acoustic detection of a high-energy cosmic particle does not concern apparently muons and neutrinos because of their large penetrating power. These particles produce cascades deep in the ocean and Earth and their sound fields are generated due to the thermal mechanism of sound excitation. It is interesting to note that the estimated value of intensity of sound field of an extensive air shower does not contradict the hypothesis by Lyamshev et al. [150] on generation of noise in the calm ocean by aggregate cosmic radiation.

4. BUBBLE MECHANISM Many authors discuss the possibility of sound generation in a liquid by penetrating radiation on account of the rise, oscillation, and collapse of microscopic bubbles at tracks of particles constituting a given type of ionizing radiation1. The theory of the bubble mechanism of sound generation is far from being completed. This is connected with the fact that a satisfactory description of arising phenomena needs to take into account the complex processes of nonlinear dynamics of a single bubble and a set of bubbles, and nonlinear effects in a liquid. For example, rough estimations 1

See [17, 57, 62, 64, 65, 150, 217, 238].

RADIATION ACOUSTICS

31

are conducted within the framework of a linear theory. A bubble arising as the result of interaction of a particle with a liquid is treated as a spherically symmetric source of sound with efficiency connected with the value of Q(x, y, z, t) in the place of “absorption” of the particle. The total effect from microscopic bubbles of a certain kind depends on both the relative amount of such microscopic bubbles and the efficiency of ultrasonic generation by a single bubble. Estimations demonstrate [57] that an essential contribution to sound radiation is made by quasistable microscopic bubbles and not only collapsing bubbles of a size of about 10−7 – 10−6 cm. In this case the acoustic signal imitates a signal from the thermal mechanism of sound generation (but it is much stronger). Recent experimental papers are evidence of the fact that the bubble mechanism of sound generation may be realized apparently (under normal experimental conditions, i.e., at atmospheric pressure and room temperature and in stable liquids) if penetrating radiation consists of heavy particles, e.g., fragments of fission nuclei [17, 71]. Further studies must clear up this question.

5. THE CHERENKOV MECHANISM When a particle moves in a medium with a velocity exceeding the phase velocity of wave propagation in this medium, it emits waves which are called the Cherenkov radiation. The intensity of the radiation and its characteristics (dispersion and polarization) depend on the nature of the waves and particle properties. The Cherenkov radiation of sound by a particle moving with a supersonic velocity in a solid can be described using a phenomenological model if a force (per unit volume) acting on a solid (lattice) and caused by a particle is introduced into the equation of the dynamic theory of elasticity [101]. The equation for the longitudinal component of displacement vector u takes the form,   1 ∂2  D  div u = ∆ − ∆δ (r − vt ) ,   2 2 2 c t c ∂ ρ l l  

(2.8)

where D is the dimensional constant coinciding in order of magnitude with the bond energy of a particle with the lattice, v is the velocity of a particle, and cl is the velocity of longitudinal waves. The transverse component of the displacement vector is identically equal to zero since the acting force is

32

BASIC MECHANISMS OF SOUND GENERATION

“longitudinal”. In this case the particle emits longitudinal sound waves and the spectral density of radiation energy Iω is proportional to the third power of sound frequency ω (see [101]:

Iω =

D2 4πρcl4

ω3 .

(2.9)

In the case of motion in a metal, a charged particle produces an electromagnetic field around itself. This field disturbs the equilibrium of conductivity electrons. The latter move ions due to the bond with the lattice. If the particle velocity is larger than the sound velocity, this mechanism leads to generation of not only longitudinal waves but also transverse sound waves [102]. Spectral densities of radiation of longitudinal Il(ω) and transverse It(ω) sound waves depend essentially on the properties of a particle and metal (the velocity of a particle, free path of electrons in a metal, Debye frequency, Fermi energy, etc.). In the case of low frequency, It(ω) ≥ Il(ω). At a certain frequency, ω = ϖ the values of It(ω) and Il(ω) become equal and at ω >> ϖ we have Il(ω) >> It(ω). The frequency ϖ depends on the particle velocity v and the free path of an electron l. The larger l and v are, the larger the value of ϖ is. The value of ϖ attains its maximum in the case of ultrarelativistic particles (v ≈ u) at l > δ0(vF cl)1/2, where vF is the Fermi velocity of conductivity electrons of a metal (~ 108 cm/s), δ0 ≈ 10−5 cm, and ϖ ≈ 104 s−1. The total intensity of radiation at all frequencies of transverse sound waves is much smaller than the analogous value for longitudinal waves It ≈ ≈ It(Vcl / u2)2 << Il and consequently, the total intensity of sound radiation (at all frequencies) is determined by radiation of longitudinal sound waves. The indirect character of excitation of sound waves (a particle – electromagnetic field – sound) does not change the Cherenkov character of their propagation along the cones cos θ l(t) = cV, the cone width for fast particles (v >> cl) being close to π/2. Deceleration of particles (finiteness of trajectory length) smears the cone to some extent. This smearing depends on frequency. It is larger, the smaller the frequency. In the case of the lowest possible frequencies (cl / ω >> L, L is the length of particle track), radiation is quite dissimilar to the Cherenkov radiation. It is determined by the average acceleration of a particle and looks like bremsstrahlung of light. The Cherenkov mechanism of sound generation contributes noticeably to the sound field emitted by a particle only at very high hypersonic frequencies (the spectral density of emitted energy is proportional to the third power of frequency). Thus, according to Borshkovskii and Volovik

RADIATION ACOUSTICS

33

[36], in the case of frequency range of hundreds of kilohertz, which may be important in practice, the ratios of acoustic energy emitted by particles due to the Cherenkov mechanism of sound generation to the acoustic energy emitted due to the dynamic (see below) and thermal mechanisms are 10−13 and 10−21. This is the evidence of relative weakness of this mechanism of sound generation in the low frequency range (in comparison with hypersound). Similar to electrodynamics, with the transition radiation connected with motion of a charged particle through media with different electromagnetic properties, the transition radiation can exist in acoustics also. This radiation is connected with the change of acoustic properties of a medium, where penetrating radiation is absorbed [171]. As in the case of the Cherenkov radiation, this radiation is quite weak. We will not discuss it in detail. There are mechanisms of sound generation specific to this or that type of radiation or target material. Let us discuss some of them briefly.

6. STRICTION MECHANISM OF SOUND GENERATION Microstiriction occurs in the field of ions in the process of medium ionization. It manifests itself noticeably in macroscopic effects [10]. A transiting charged particle (or any other particle capable of ionization) produces N1(0) pairs of ions per unit length of track in a medium. The number of these ions Nt(t) decreases sharply in time because of their recombination. Each ion attracts molecules of a medium by its field and creates local clouds. Microstrictional compression can play a significant role in the process of sound emission by charged particles in the case of a small coefficient of thermal expansion of a medium. In particular, the experimental results by Levi et al. [235] apparently can be explained with its help. It was revealed in these experiments that a sound pulse from a beam of charged particles in water vanishes and changes its sign not at T = 4°C when the coefficient of thermal expansion of water α = 0, but at T = 5.7°C when α ≈ 10−5 K−1. Amplitudes of pulses become equal at α ≈ 10−5 K−1, i.e., compensation of thermal expansion by strictional compression is possible in this case. Fast alternating striction in the field of a moving particle and in the collective field of beams also exists in the field of ions apart from microelectrostriction. Let us consider the strictional mechanism of sound generation in terms of laser excitation of sound. The equation of sound generation in the process of action of laser radiation on a liquid medium,

34

BASIC MECHANISMS OF SOUND GENERATION

which takes into account thermal and strictional mechanisms, takes on the following form [43]: ∆p −

1 ∂2 p c

2

∂t

2

=−

α ∂Q 1 + C p ∂t 8π

 ∂ε   ρ  ∆ E 2 ;  ∂ρ T

(2.10)

here ε is the dielectric constant of a liquid, E is the strength of electric field of laser (optical) radiation in a liquid, the angular brackets mean averaging with respect to the period of optical oscillations, (∂ε /∂ρ)T is the derivative of dielectric constant with respect to density in the case of constant temperature. The second term in the first part of equation (2.10) corresponds to the strictional mechanism of sound generation. Let us estimate the order of magnitude of the expression ∆〈E2〉. Monochromatic sound oscillations with the frequency ω can be excited due to the strictional mechanism in the case of two laser beams crossed under the angle θ and with frequencies ω1 and ω2, ω1 - ω2 = ω, and 2(ω1,2/u)n sin(θ/2) ~ ω / c = k. We have   1 8π 1 + (ak ) 2 + (aµ ) 2 q0 , ∆ E 2 ~  + k 2 + µ 2  E02 ~ nu a2   a2

(2.11)

where n = u1/2 is the refraction coefficient of a medium, k is the wave number, a is the radius of radiation beam, µ is the coefficient of radiation absorption in a liquid, E0 is the amplitude of the strength of electric field, u is the light velocity, c is the sound velocity in a liquid, and q0 is the intensity of light pulse in the middle of a beam. Since |∂Q / ∂ t| ~ µq0ω, the ratio of the first term in the right-hand side of equation (2.10) (corresponding to the thermal mechanism of sound generation) to the second term is equal to

αanuω µα . C p ( ρ∂ε / ∂ρ )T 1 + (ak ) 2 + (aµ ) 2 This relationship shows that the strictional mechanism of sound generation prevails over the thermal one only in the range of very high or very low frequency. In the case of water for example, the thermal mechanism is dominant with respect to the strictional mechanism at µ ≥ 0.2 cm−1 in the frequency range from 102 to 109 Hz.

RADIATION ACOUSTICS

35

7. SOUND GENERATION IN THE PROCESS OF PULSED RADIOLYSIS Pulsed radiolysis is the mechanism of release of latent energy on account of substance decomposition under the effect of pulses of penetrating radiation [52, 59 – 61]. In particular, slow electrons arising in the process of substance ionization can be absorbed effectively by some halogencontaining liquids, i.e., reactions of the following type may occur: ABC n + e − = ABC n −1 + C − , where ABCn is the formula of a molecule of a liquid containing a certain halogen C in it. Further, this released energy may be transformed into sound energy with the help of the mechanisms of sound generation already considered above. These may be, for example, the thermal or bubble mechanisms of sound generation. In this case, if the density of released latent energy of a substance exceeds the density of energy evolved on account of absorption of penetrating radiation, the determining contribution to acoustic field in such radiation-unstable substance belongs to the effect of pulsed radiolysis [60].

8. DYNAMIC MECHANISM If penetrating radiation affects a substance, transfer of momentum from radiation quanta to atoms of a medium occurs. This phenomenon accompanied by excitation of sound waves is called the dynamic mechanism of sound generation [7, 63, 165]. In crystals the effect depends on the reciprocal directions of crystal axes and the velocities of particles constituting penetrating radiation [165]. Like the thermal mechanism of sound generation, the dynamic mechanism takes place in the case of any kind of penetrating radiation and manifests itself in both liquids and solids. In solids, however, the dynamic mechanism may be of fundamental importance because of the existence of transverse sound waves there [160]. The point is that transverse waves in solids under conditions of thermal mechanism result only from transformation of longitudinal waves at interfaces and medium inhomogeneities. As a result of action of the dynamic mechanism, longitudinal and transverse waves are excited in all cases and even in homogeneous isotropic solids.

36

BASIC MECHANISMS OF SOUND GENERATION

Dynamic stress in solids in the process of transmission of beams of charged particles through them was determined by Nasonov [165]. In the case of small energy in a beam (the energy of several hundreds of electronvolts for plates of thickness of the order of magnitude of 0.01 cm), the pressure p produced by a beam incident on a plate does not depend on the target substance, and increases linearly with energy: p = n0p0ν0, where p0, ν0, and n0 are the momentum, velocity, and density of a beam of incident particles. In the case of large energy of a beam, the dependence of pressure on energy is of more complex character. Results of numerical calculation of dynamic pressure conducted by Nasonov [165] for aluminum, copper, and lead plates of thickness of 0.01 cm at different values of energy, are given in Figure 2.3.

Figure 2.3 Dynamic pressure upon a plate vs. energy of a beam in a plate. Broken curves correspond to proton beams and solid curves correspond to electron beams. (1) Copper, (2) lead, and (3) aluminum.

Sound generation under the dynamic mechanism can be treated within the framework of linear approximation. In this case, equations of sound generation in a solid have the form [160],   1 ∂2  div F  ∆ − div u = − ,  2 2  cl ∂t  cl2 ρ  (2.12) 2    ∆ − 1 ∂ rot u = − rot F ,  ct2 ∂t 2  ct2 ρ 

RADIATION ACOUSTICS

37

where u is the displacement, cl and ct are the velocities of longitudinal and transverse sound waves, ρ is the medium density, and F is the dynamic force applied to unit volume. Comparing equations (2.1) and (2.12), one can see that the dynamic mechanism of sound generation produces a source of both longitudinal and transverse waves in a solid as distinct from the thermal mechanism, which excites sources of longitudinal waves only. In the case of penetrating radiation with quanta consisting of ultrarelativistic particles (e.g., photons), the dynamic force is equal to F = Q/u, where u is the light velocity. In order to compare the efficiency of sound excitation by the thermal and dynamic mechanisms, it is necessary to rewrite expressions (2.1) in terms of the displacement vector u and add dynamic sources to the right-hand side:

  α 1 ∂2  div F  ∆ − ∆Qdt − div u = . ∫  2 2  2 Cpρ c ∂ t c ρ l l  

(2.13)

One can see readily from here that under the condition F = Q/u and for the majority of substances, the ratio of displacements in sound waves excited due to the dynamic mechanism (the second term) to analogous displacements caused due to the thermal mechanism (the first term) is of the order of magnitude of the ratio of the velocity of longitudinal sound wave to the light velocity: cl/c ≅ 10−5. Thus, accounting for the dynamic mechanism of sound generation for ideal liquids (where only longitudinal waves can propagate) gives us only an insignificant correction to the sound field produced by the thermal mechanism of sound generation. Only hypersonic frequencies may be the exception. These frequencies are excited by the thermal mechanism inefficiently because of thermal conductivity of a substance, which is not taken into account in expression (2.1). However, in the case of solids, when the field of transverse waves, which cannot arise because of reflection or scattering of longitudinal waves, is studied, the dynamic mechanism of sound generation together with the thermal mechanism may provide a significant contribution.

38

BASIC MECHANISMS OF SOUND GENERATION

9. OTHER MECHANISMS OF SOUND GENERATION As for other mechanisms of sound generation by penetrating radiation, we should note the mechanism of excitation of elastic waves with the help of the inverse piezoelectric effect [53, 122] and the transition mechanism already mentioned above. The first mechanism is connected with mechanical deformation of piezoelectric ceramics under the action of an electric field of penetrating radiation. We have to mention also that if weakly ionized plasma is produced in the area of absorption of penetrating radiation, interaction of electric and magnetic fields with plasma may violate the stable state of a medium and therefore, sound waves may be generated [123]. It is also possible that new data on as yet unknown mechanisms of sound generation may be obtained in the course of further theoretical and experimental research. Such research will provide an opportunity to use the methods of radiation acoustics for the solution of applied and basic problems even more widely. All considered mechanisms of sound generation have different restrictions in intensity, nature, and frequency range of emitted sound waves. Exact quantitative values can be obtained by the solution of specific boundary problems. It is essential in this case that in the majority of real situations all possible mechanisms of sound generation in solids contribute only small corrections to the sound field produced by the thermal mechanism of sound generation [55]. This is also true in the case of liquids if there are no phase transformations. Thus, the above discussion of the determining role of the thermal mechanism of sound generation in liquids and solids in the case of moderate density of energy released in a medium explains the importance of detailed studies of generation of sound fields in the approximation of this mechanism.

CHAPTER 3

Thermoradiation Excitation of Sound in a Homogeneous Liquid This chapter considers thermoradiation excitation of sound in a homogeneous liquid by intensity modulated penetrating radiation. Basic equations and the technique of solution of boundary problems are discussed in details. Specific features of excitation of monochromatic sound in a liquid half-space with the undisturbed surface (boundary) as well as the case when the liquid surface is characterized by large or small (in comparison with the sound wavelength) unevenness are considered. Efficiency of thermoradiation generation of sound in a liquid by penetrating radiation is discussed.

1. EQUATION OF THERMORADIATION GENERATION OF SOUND We understand the thermoradiation mechanism as a mechanism of sound excitation by penetrating radiation in a liquid, when a medium expands in the area of absorption because of heating due to radiation absorption but the aggregate state of a substance and its thermodynamic parameters do not change and the expansion velocity of the heated volume is essentially smaller than the velocity of sound propagation in the medium. This provides an opportunity to write down the set of equations of conservation 39

40

THERMORADIATION EXCITATION: HOMOGENEOUS LIQUID

in the linearized form and derive a linear equation of thermoradiation generation of sound on its basis. Thus, an initial set of linearized equations of hydrodynamics complemented with equations of state and heat conductivity of a liquid is [43, 120]:

ρ

∂v η  = −∇p ′ + η∆v +  ξ + ∇(∇ ⋅ v ) , ∂t 3 

∂ρ ′ ∂s ′ = − ρ (∇ ⋅ v) , ρT = κ∆T ′ + Q , ∂t ∂t (3.1)  ∂p  αc 2 ρT  ∂p   ρ ′ +   s ′ ≡ c 2 ρ ′ + p ′ =  s′ , Cp  ∂s  v  ∂ρ  s  ∂s s′ =   ∂T

Cp  ∂s  α  T ′ − p′ .  T ′ +   p ′ ≡ T ρ p  ∂p T

Here ρ, v, p, T, and s are the density, velocity, pressure, temperature, and specific entropy of a liquid, respectively; a stroke means a small increase of the corresponding parameter of a medium; η and ξ are the coefficients of shear and bulk viscosity; Cp and Cv are the specific heat capacities; v is the specific volume; α = v−1(∂v/∂T)p is the coefficient of cubical thermal expansion; κ and χ are the coefficients of thermal conductivity and diffusivity (χ = κv/Cp); Q is the power density of thermal sources produced due to absorption of penetrating radiation in a liquid, Q = − (∇⋅Π), and Π is the density of energy flux of penetrating radiation. We can obtain the desired equation of thermoradiation generation of sound on the basis of set (3.1): ∆p ′ −

1 ∂ 2 p′ c

2

∂t

2

+ 2Γ∆

∂p ′ αρT = Cp ∂t

 4 / 3η + ξ  ∂s ′ α ∂Q  , − χ ∆ − ρ   ∂t C p ∂t (3.2)

Γ≡

Cp  1 2  4 / 3η + ξ + χ − 1 . C  C  2 ρ  v  

RADIATION ACOUSTICS

41

Let us turn to the physical meaning of some terms in equation (3.2). First of all, one can see that the third term in the left-hand side of the equation characterizes sound attenuation caused by viscosity and heat conductivity of a liquid. In the majority of cases, this term may be omitted in the process of solution of boundary or initial problems without any restrictions to generality. It can be taken into account separately while considering the final result as is done usually in acoustics. Terms in the right-hand side of equation (3.2) describe sound sources. It is possible to ignore the role of the first of them if we ignore the influence of viscosity and heat conductivity of a liquid. This is possible if the next conditions are satisfied: l 2 l 2   τ <<  min , min  . χ   ν 

(3.3)

Here τ is the characteristic time of action of penetrating radiation on a substance, e.g., the duration of a single pulse or τ ~ ω −1, where ω is the modulation frequency of radiation intensity; and lmin is the minimum dimension of the area of absorption of penetrating radiation in a liquid. The dimension lmin ~ a is the radius of a radiation spot at the liquid surface and lmin = µ−1 is the so-called path of a particle of penetrating radiation equal to the inverse value of the coefficient of light absorption µ in the case of laser radiation or X-rays (synchrotron radiation); ν is the kinematic coefficient of viscosity. The physical sense of conditions (3.3) is evident. In fact, the quantity l2min/χ characterizes the time of equalization of temperature in the area of radiation absorption, i.e., the time of thermal relaxation. In the case of τ ≤ l2min/χ, there is no equalization of temperature in a medium. As for the condition τ >> l2min/ν, we have to note that the quantity (1/τν)1/2 ~ (ω/ν)1/2 characterizes the penetration depth of a so-called viscous wave and therefore, the condition τ << l2min/ν is equivalent to the condition lmin >> (ω/ν)1/2. In other words, the minimum dimensions of the area of heat evolution (thermal source) must be essentially larger than the depth of wave penetration. If the condition τ << l2min/χ is satisfied, the term κ∆T′ in the thermal conductivity equation in system (3.1) does not play any role and therefore, ρT ds′/dt ≈ Q(t), and under the condition τ << l2min/ν, the first term in the right-hand side of equation (3.2) is small in comparison with the second one and may be ignored within the whole region of generation and propagation of sound. Condition (3.3) is very frequently well satisfied for not very viscous liquids. The most strict limitations for the value of τ are imposed in

42

THERMORADIATION EXCITATION: HOMOGENEOUS LIQUID

the case of liquids strongly absorbing radiation; usually lmin ~ µ−1 for these liquids. For example in the case of CO2-laser radiation (λl = 10.6 µm), the absorption coefficient of water is µ = 800 cm−1 (lmin ≈ 103 cm; the penetration depth of an electron beam of energy ~ 30 keV is approximately the same) and condition (3.3) leads to the requirement τ << 10−4 s (taking into account the fact that χ << ν ≈ 0.01 cm2/s for water). For optical radiation of the near IR range lmin ~ 1 mm (in the cases interesting from the practical point of view) that also corresponds approximately to the penetration depth of an electron beam with energy ~ 1 MeV. In this case condition (3.3) may be invalid only for infrasound. Thus, condition (3.3), which is the condition for the effect of viscosity and heat conductivity upon the efficiency of a thermal source, limits sound frequency from below. The same processes limit sound frequency from above by the condition of minimum absorption of sound (i.e., Γω << 1): ω << {c2 / χ, c2 / ν}. This condition is satisfied up to hypersound frequency for the majority of liquids. The estimates given above show that in the majority of cases, the equation of thermoradiation generation of sound can be represented in the form [259], ∆p −

1 ∂2 p c 2 ∂t 2

=−

α ∂Q . C p ∂t

(3.4)

Here and further p is the sound pressure (we omit the stroke everywhere). This equation will be used further while considering various specific problems of thermoradiation excitation of sound in liquids.

2. RECIPROCITY THEOREM IN ACOUSTICS – SOLUTION TECHNIQUE FOR BOUNDARY PROBLEMS The author developed an effective technique for solution of boundary problems of sound generation by penetrating radiation under conditions of the linear thermal mechanism [130]. The solution of a boundary problem may be reduced directly to quadratures with the help of the reciprocity relationship if the solution of an auxiliary self-adjoint diffraction problem on the field of a point source of sound, when the source is located at the point of a medium, where it is necessary to determine the field of thermal sound sources due to the effect of penetrating radiation, is known.

RADIATION ACOUSTICS

43

In the majority of cases, the Fraunhofer field is of the main interest. Then, it is necessary to know the solution of an auxiliary problem of diffraction of a plane sound wave. In the majority of cases, it is quite simple to determine such a solution. Now let us consider the essence of the solution technique in more detail. We should consider some basic reciprocity relationships in acoustics first of all. As is known, Helmholtz was the first who noted the existence of the reciprocity relationship in acoustics [218], and a little later Rayleigh extended this relationship and established a certain general reciprocity principle connecting various types of external action on a linear dynamic system with the effect of these actions [177, 252]. The reciprocity principle was formulated by Rayleigh mathematically for linear systems with a finite number of degrees of freedom [177, 181, 252]. It was demonstrated also that in the case of a medium with motionless boundaries, this principle follows from the Green formula for the Laplace operator. As for a medium containing elastic plates, shells, membranes, etc., Rayleigh indicated only generally the validity of the reciprocity principle in this case also. There was no mathematical formulation of the reciprocity principle describing the connection between volumetric sources in a medium, certain external forces affecting shells, membranes, etc., and radiation fields produced by these forces and bodies. Such a relationship was obtained by the author [131]1. Further, we restrict our consideration within the framework of harmonic oscillations. As is known, this does not limit its generality, but makes mathematical manipulation simpler. We omit the time factor of the form exp (−iωt) everywhere. We take an arbitrary volume Ω filled with any kind of combination of acoustic media and elastic shells (rods, membranes, etc.), which are closed or limited and fixed in motionless screens. We denote a shell surface by Si and a fixation contour by Γi. Let us treat the field p1(r) produced by a certain system of volumetric harmonic sources Q(1)(r), which is distributed continuously over Ω. Then, p(1)(r) is the solution of an equation2,

1

See L. M. Lyamshev, Proceedings of 14th International Congress on Acoustics, Beijing (China), 1992, vol. 4, p. 5, for more details. 2 As is known, equation (3.5) is valid also for an inhomogeneous medium, when the sound velocity in a medium is a function of coordinates and the medium density is constant. However, there are no basic difficulties in consideration of a more general case of an inhomogeneous medium with density and velocity depending on coordinates if we proceed from an equation for such medium given in the form provided by Landau and Lifshits [121].

44

THERMORADIATION EXCITATION: HOMOGENEOUS LIQUID

∆p (1) (r ) + k 2 p (1) (r ) = −Q (1) (r )

(3.5)

and satisfies boundary conditions, lim p (1) (r ) → 0 , Im k > 0 ,

(3.6)

∂p (1) (r ) (1) = wi (r ) , 2 n | ∂ s1 ω ρ

(3.7)

−

1

(1)

(1)

Li wi (r ) = Fi

(r ) − p (1) (r ) | S i ,

(1) (1) (1) Tij ( wi ) = g ij |Γi′ , Rij ( wi ) = − f ij | Γi′′ , Γi′ + Γi′′ = Γi .

(3.8) (3.9)

Let another system of continuously distributed sources Q(2)(r) be set. The field p(2)(r) produced by these sources obeys an equation, ∆p ( 2) (r ) + k 2 p ( 2) (r ) = −Q ( 2) (r ) ,

(3.5′)

and boundary conditions (3.5) and (3.6), where index (1) is changed for (2) as in expression (3.5). In expressions (3.7) − (3.9), n is the external normal to Si, wi is the normal displacement of the shell surface, Fi are the external mechanical forces affecting the shell in the direction of the normal, gij and fij are the external forces, moments of displacements, etc., affecting the shell along its contour Γ (at the shell border), Li is the self-adjoint differential operator3, which is consistent with the Green formula,

∫ wi

(1)

Si

N

∫ ∑  Rij ( wi

Γi j =1 3

( 2)

( 2)

Li wi

( 2)

− wi

( 2) 

Li wi



dS i =

)Tij ( wi(1) ) − Tij ( wi( 2) ) Rij ( wi(1) ) dΓi , 

(3.10)

The conditions, when the operator Liwi is self-adjoint in the case of thin elastic shells, coincide essentially with the conditions of satisfiability for the Betti theorem in the case of shells.

RADIATION ACOUSTICS

45

where Rij and Tij are also certain differential operators. We multiply equation (3.5) by p(2)(r) and equation (3.5’) by p(1)(r), add them, and integrate both sides of the obtained equality over the volume Ω using the Green formula, boundary conditions (3.6) – (3.9), and formula (3.10). Finally, we obtain k

∫

N

m

i =1 S i

Ω m

∑∫

∂p (1) (r ) ( 2) Fi (r )dS i + ∂n i

( 2) (1) ∫ Q (r) p (r)dΩ + ∑

∑ gij(2) Rij

i =1 Γ′i j =1

∂p (1) (r ) dΓi + ∑ ∫ ∂ni

N

∑ fij(2)Tij

i =1 Γi′′ j =1

∂p (1) (r ) dΓi = ∂ni (3.11)

∫Q

(1)

∑∫

(r ) p ( 2) (r )dΩ + ∑ ∫

∂p ( 2) (r ) ∂ni

i =1 Si

Ω m

k

N

m

∂p ( 2) (r ) (1) gij dΓi + ∑ ∫ ∂ni

∑ Rij

i =1 Γ′i j =1

N

Fi(1) (r )dSi +

∑ Tij

i =1 Γi′′ j =1

∂p ( 2) (r ) (1) f ij dΓi . ∂ni

Expression (3.11) can be treated as the mathematical formulation of the reciprocity principle in acoustics. Let us analyze some particular cases. Let (1)

Fi

( 2)

= Fi

( 2)

= g ij

(1)

= f ij

= 0 , i = 1, 2, ..., k , j = 1, 2, ..., N .

Then, the next relationship follows from expression (3.11):

∫Q

Ω

(1)

(r ) p ( 2) (r )dΩ = ∫ Q ( 2) (r ) p (1) (r )dΩ .

(3.12)

Ω

This reciprocity relationship is obtained usually under the assumption that the sources and their fields exist in a free space or a space with motionless (free) boundaries. Expression (3.12) can be considered as a certain integral relationship connecting the solutions of two self-adjoint boundary problems of sound diffraction at elastic shells (membranes, rods, etc.) in a liquid.

46

THERMORADIATION EXCITATION: HOMOGENEOUS LIQUID

Let us assume that (1)

Q ( 2) (r ) = Fi

(1)

( 2)

(r ) = g ij = g ij

( 2)

= f ij

= 0,

i = 1, 2, ..., k , j = 1, 2, ..., k . We have

∫Q

(1)

(r ) p ( 2) (r )dΩ =

Ω

∫ Si

∂p (1) (r ) ( 2) Fi (r )dSi . ∂ni

(3.13)

Relationship (3.13) connects the solutions of boundary problems of diffraction and sound radiation. We should note some opportunities for application of the reciprocity principle in the form of integral relationships obtained above. Let it be necessary to solve a problem on sound scattering by an elastic shell when the incident field is produced by a system of sources distributed in space. Then, if the problem on the field of a point source in the presence of a shell is already solved, the desired solution is reduced directly to quadratures as it follows from expression (3.12). Indeed, if it is necessary to determine the field of scattering at a certain point r1 of the space Ω, then positioning an auxiliary point source Q0(1)δ(r − r1) with the known field p(1)(r) at this point we obtain p ( 2) (r1 ) =

1

∫p

(1) Q0 Ω

(1)

(r1 , r2 )Q ( 2) (r )dΩ(r ) .

(3.14)

We can determine the solution of a problem on transmission of sound field through a shell as has been done above, if this field is produced by a system of distributed sources. Let us assume that it is necessary to determine the field of a shell oscillating under the action of forces distributed over its surface F(2)(r). If we know the solution of the corresponding diffraction problem as earlier, we have from expression (3.13), p ( 2) (r1 ) =

∂p (1) (r1 , r ) ( 2) F (r )dS (r ) . (1) ∫ ∂n Q0 S 1

(3.15)

RADIATION ACOUSTICS

47

Formulae (3.14) and (3.15) will be used further in the process of consideration of problems of thermodynamic generation of sound.

3. EXCITATION OF MONOCHROMATIC SOUND IN A LIQUID HALF-SPACE WITH A FREE SURFACE – THE CASE OF UNDISTURBED SURFACE Let us consider the simplest case of thermoradiation generation of sound. We assume that a beam of penetrating radiation (e.g., a laser beam, a beam of relativistic electrons or γ-quanta or X-rays) propagating from the upper half-space in the positive direction of the axis z of the Cartesian coordinate system x, y, z (Fig. 3.1) is incident upon a free surface of a liquid (the equation of the surface has a general form z = ξ (x, y) and the beam intensity changes according to a harmonic law with the frequency ω). In this case thermal sources of sound arise in a liquid. We can write down an expression for the power density of these sources, Q( x, y , z , t ) = Aµ ( x, y ) exp{−[ z − ξ ( x, y )]}[1 + m cos ωt ] ,

(3.16)

where I(x, y) is the intensity distribution of particles in the beam, which we assume Gaussian, I(x, y) = I0 exp [−(x2 + y2)/a2]; m is the modulation index (0 ≤ m ≤ 1); µ is the absorption coefficient of penetrating radiation in a liquid; A is the coefficient of radiation transmission at the liquid boundary; and a is the radius of radiation spot at the liquid surface.

Figure 3.1 Scheme of sound generation by penetrating radiation in a liquid. (1) Direction of incidence of a beam of modulated penetrating radiation; (2) boundary “air – liquid”; (3) region of absorption of penetrating radiation in a liquid; (4) front of sound waves in a liquid.

48

THERMORADIATION EXCITATION: HOMOGENEOUS LIQUID

Here and below we use the exponential law of penetrating radiation absorption in a substance in order to be more specific, and assume the intensity distribution in the beam to be Gaussian. The question on the role of the absorption law of penetrating radiation in formation of the acoustic field is considered separately. As we have noted already (Chapter 1, Section 3), the exponential law describes absorption of electromagnetic radiation (light, laser radiation, Xrays, a beam of γ-quanta). Let us consider the process of transmission of a laser beam through an infinitely thin plane-parallel layer of a substance. We denote a radiation flux incident perpendicularly on the layer by I. Being transmitted through the layer of the thickness dz, the flux decreases by dI. If the layer is infinitely thin, i.e., absorption is small, we can assume that the dependence between radiation attenuation and the layer thickness is linear and radiation attenuation is proportional to the incident flux: dI = −Iµ dz, where µ is the constant characterizing the layer absorption, calculated per unit thickness, and called by the name of absorption coefficient. Integration of this equation with respect to z gives the dependence of the transmitted flux of optical radiation on the absorption coefficient of a substance, layer thickness z, and incident flux I = I0, i.e., I = I0exp(−µz). This is the Buger-Lambert law (see [187] for example). The exponential law describes absorption of relativistic electrons until their velocity becomes small enough and electrons stop to be “relativistic”. However, it may turn out in many interesting cases that less than 10% of the total energy of electron beam is released at this last “non-relativistic” stage of absorption of electrons in a substance and absorption may be considered as exponential for the whole path of electron absorption. Now the equation of thermoradiation generation of sound takes on the form, ∆p −

1 ∂2 p c 2 ∂t 2

=

αωm AµI ( x, y ) exp{− µ[ z − ξ ( x, y )]} sin ωt . Cp

(3.17)

Taking the time factor in the form exp(−iωt), we have finally (∆ + k 2 ) p = i

αωm AµI ( x, y ) exp{− µ[ z − ξ ( x, y )]} , Cp

(3.18)

where k = ω/c is the wave number of sound in a liquid. This equation is basic for consideration of thermoradiation generation of harmonic sound oscillations in liquids. The factor exp(−iω t) is omitted as usual.

RADIATION ACOUSTICS

49

The solution of equation (3.18) can be written down on the basis of the reciprocity theorem, p(r ) = i

αωm Aµ ∫ I ( x ′, y ′) exp{− µ[ z ′ − ξ ( x ′, y ′)]} × Cp Ω

(3.19) p * ( x ′, y ′, z ′; x, y , z )dx ′dy ′dz ′ , where p*(r′, r) is the solution of a self-adjoint problem on the diffraction of the sound field of a point source positioned at the point r, where it is necessary to determine the acoustic field of thermal sources produced by the action of penetrating radiation upon a medium (liquid). In fact, p*(r′, r) is the source or Green function of the reduced wave equation, which satisfies the boundary conditions for a corresponding self-adjoint boundary problem. Let us consider the sound field of a monochromatic radiation-acoustic source in the far wave field in a liquid half-space with the even boundary (z = 0). Let us analyze the particular features of the sound field in the far wave zone. In the case under consideration, the auxiliary solution has the form, dp * (r ′, r ) =

exp(ikr ) {exp[−i (αx ′ + βy ′ + γz ′)] − 4πr (3.20) exp[−i (αx ′ + βy ′ − γz ′)]} .

Expression (3.20) is the asymptotic representation of the Green function at kr → ∞. Solution (3.19) of equation (3.18) takes on the form,

p(r ) = i

exp(ikr ) αωm AµI 0 Cp 4πr

∞

∞

 x′ 2 + y ′ 2 − ′ exp( z ) exp µ − ∫ ∫∫  a2  −∞ 0

 ×   (3.21)

{exp[−i (αx ′ + βy ′ + γz ′)] − exp[−i (αx ′ + βy ′ − γz ′)]}dx ′dy ′dz ′ . 2

2

2

2

The α-component of the wave vector k, i.e., α + β + γ = k , is under the exponent in expressions (3.19) and (3.21). Integrating, we obtain p( r ) =

 k 2a 2  exp(ikr ) αωm µk cosθ exp − sin θ  . AI 0 a 2   2C p r 4 µ 2 + k 2 cos2 θ  

(3.22)

50

THERMORADIATION EXCITATION: HOMOGENEOUS LIQUID

Here θ is the angle between the direction of incidence of a penetrating radiation beam and the direction from the observation point to the coordinate origin, and r is the distance from the observation point to the coordinate origin. One can see form expression (3.22) that the amplitude of sound pressure increases with the growth of the power of penetrating radiation (~ I0a2), frequency, and modulation index. Directivity of radiation depends on the parameters ka and kµ−1, i.e., on the relation between the dimensions of a radiation-acoustic source and the wavelength of sound. If kµ−1 << 1 and ka << 1, we observe the dipole radiation of sound since satisfaction of these conditions means that a monopole source is operating at the free surface of a liquid. The sound field of this source represents the field of a dipole because of the influence of the free surface (Fig. 3.2a). If kµ −1 >> 1 and ka << 1, sound is emitted mainly along the surface. A set of volume thermal sources forms a vertical array, which is thin in the transverse direction and long in comparison with the sound wavelength in the direction of the beam of penetrating radiation, i.e., a so-called “rod-like” array (Fig. 3.2b). If kµ −1 << 1 and ka >> 1, the array has the shape of a disk with diameter much larger than the sound wavelength. Acoustic radiation is emitted mainly in the propagation direction of penetrating radiation (Fig. 3.2c).

Figure 3.2 Directivity patterns of thermoradiation sound sources. (a) Source dimensions are small, (b) and (c), directivity of radiation of sources of “rod-like” and “disk” types.

Analysis of expression (3.22) shows also that the optimal mode of acoustic radiation in the case of thermoradiation excitation of sound is observed when k ≈ µ. This requires a certain type of penetrating radiation, i.e., a certain type of quanta-particles, their energy, and laser light wavelength. A problem of optimal conditions of thermoradiation (laser) generation of sound in the case of the linear thermal mechanism was considered in detail from the point of view of acquisition of the maximum amplitude of sound pressure [29]. Now let us analyze the sound field of a radiation-acoustic source in the near wave zone (the Fresnel zone), when the observation point r = {ρ, L}

RADIATION ACOUSTICS

51

(where ρ = (x′2 + y′2)1/2 and L is the coordinate of the observation point along the axis z) is located under the irradiation spot (ρ ≤ a) at depth L ≤ ka2. We write down the auxiliary solution in a general form (and not in an asymptotic form as has been done above while considering the field in the Fraunhofer zone), p(r ′, r ) =

1  exp(ik | r ′ − r |) exp(ik | r ′ * − r |)  −  , 4π  | r′ − r | | r ′ * −r | 

(3.23)

where r′ = {x′, y′, z′} and r′* = {x′, y′, −z′}. The solution of equation (3.18) takes on the form

p(r ) =

1 4π

∞

∞

 x′2 + y′2  αωm −  exp(− µz ) × µ A I exp 0 ∫ 2   Cp a   −∞ 0

∫ ∫i

(3.24)  exp(ik | r ′ − r |) exp(ik | r ′ * − r |)  − dxdydz .  | r′ − r | | r′ * −r |   As L >> µ −1 and L >> a in all cases, we assume that in the near wave field L >> ka2/(µL) or L >> ka2/(a/L) that is the condition of the Fresnel diffraction. Taking into account these conditions and the fact that radiation intensity rapidly decreases already at ρ = a, we may change |r′ − r| and |r′* − r| in the denominator of the integrand in expression (3.13) for L and change them in the exponential factor for L = z′ + (ρ − ρ′)2 / 2L. After this substitution and integration, the solution takes on the form [43],

p=i

 ρ2  mAαc µk  exp[(ik − Γ) L] . I 0 exp −  a2  Cp µ2 + k2  

(3.25)

Naturally, a sound beam in the near wave zone is not subjected to divergence and its width (with respect to pressure amplitude) coincides with the width of the penetrating radiation beam. Thus, we arrive at an important conclusion that while in the far wave field, where sound waves are spherical, the amplitude of sound pressure p is determined by the total power P of penetrating radiation, in the near wave field the amplitude of sound is determined by the beam intensity according to expression (3.25).

52

THERMORADIATION EXCITATION: HOMOGENEOUS LIQUID

4. A LIQUID HALF-SPACE WITH LARGE-SCALE ROUGHNESS OF BOUNDARY Above we have considered the particular features of a sound field of a radiation-acoustic source in a liquid bounded by an even surface. In real conditions a liquid surface is uneven because of many reasons. Therefore, it is useful to estimate the effect of boundary roughness upon the sound field generated by a radiation-acoustic source. At first we consider the influence of unevennesses which are large in comparison with the sound wavelength in a liquid [108, 133]. We take random unevennesses in order to consider a general case. Let a penetrating radiation beam intensity-modulated with the sound frequency ω be incident along the axis z upon the uneven surface of the liquid half-space determined by the equation z = ξ(x, y). Thermal sound sources and a sound field arise in a liquid due to absorption of penetrating radiation in it. As we have mentioned above, they are described by the inhomogeneous reduced wave equation (∆ + k 2 ) p = i

αA mωµI ( x, y ) exp[− µ ( z − ξ ( x, y ))] . Cp

We write down the solution of this equation on the basis of the reciprocity principle (see equation (3.19)),

p ( x, y , z ) =

1 4π

 iαA p x y z x y z mωµI ( x1 , y1 ) × ( , , ; , , )  0 1 1 1 ∫  C p Ω (3.26)

 exp{− µ [ z1 − ξ ( x1, y1 )]} dx1dy1dz1 ,  where p0(x, y, z; x1, y1, z1) is the solution of the regular problem of sound scattering at the boundary when the source of the regular field is located at the point (x, y, z), i.e., at the point, where it is necessary to determine the desired field; and Q is the volume occupied by thermal sound sources produced by the effect of penetrating radiation. Assuming that the point (x, y, z) is located in the far wave field, we can represent the incident regular wave by an expression, pi =

e ikR exp[i (αx1 + βy1 + γz1 )] , R

(3.27)

RADIATION ACOUSTICS

53

where R = (x2 + y2 + z2)1/2. Here α, β, and γ are the components of the wave vector directed from the point (x, y, z) to the coordinate origin. In the considered case of the boundary with large-scale unevenness, the solution of the diffraction problem can be written down in the Kirchhoff approximation, i.e., under the assumption that reflection at each point of the surface occurs in the same way as from the infinite plane tangential to the boundary surface at this point [18, 125]. In this case the total field at the boundary consists of the incident pi and reflected pr = Wpi waves, where W is the reflection coefficient. We may take W = −1 at the liquid−air boundary. The scattered field in a liquid half-space in the general case can be represented by the Kirchhoff integral. However, in the case under consideration, when the distance from the point (x1, y1, z1) to the boundary is not larger than the sound wavelength (just this length of the region of the thermal sound sources along the axis z provides the optimal mode of sound generation as has been mentioned before), the scattered field in the layer of the depth k−1 can be represented at the boundary in an approximate form, pr = −

e ikR exp{i (α ′x1 + β ′y1 + γ ′z1 ) − i[γz1 + γξ ( x1 , y1 )]} , R

(3.28)

α ′ = k sin θ ′ cos ϕ ′ , β ′ = k sin θ ′ sin ϕ ′ , γ ′ = k cos θ ′ , and the angles θ′ and ϕ′ are expressed with the help of the coordinate angles of the wave vector of the incident wave θ and ϕ in the following way:

ϕ ′ = arctan

(tan θ sin ϕ + 2∂ξ / ∂y )(1 − tan θ cos ϕ ⋅ 2∂ξ / ∂x) , (1 − tan θ sin ϕ ⋅ 2∂ξ / ∂y )(tan θ cos ϕ + 2∂ξ / ∂x)

θ ′ = arctan

tan θ cos ϕ + 2∂ξ / ∂x 1 + tan 2 ϕ ′ . 1 − tan θ cos ϕ ⋅ 2∂ξ / ∂x

The validity of expression (3.28) is limited by the conditions of applicability of the Kirchhoff approximation and, in particular, by the condition of the absence of multiple reflection of the incident wave at the boundary. As can be demonstrated readily, this corresponds to imposing a certain restriction upon the slip angle of the incident wave, i.e., the following condition must be satisfied: (π/2 − θ) > 3δ , where δ is the meansquare angle of inclination of boundary unevennesses.

54

THERMORADIATION EXCITATION: HOMOGENEOUS LIQUID

As we consider only mildly sloping unevennesses, that is equivalent to the satisfaction of the condition ∂ξ/∂x << 1 and ∂ξ/∂n << 1, then we can ignore the influence of the tangential plane at the point of reflection of the sound beam from the boundary in the layer of depth 1/k with precision up to the order of magnitude of δ 2 << 1 that can be demonstrated easily in the process of averaging. Thus, we can assume that reflection from the boundary occurs in the same way as from the plane parallel to the coordinate plane (x, y) and passing through the given point of the boundary surface. Therefore, the solution of the regular problem of scattering of a sound wave generated by a distant source can be represented approximately in the region of thermal sources by an expression, p0 = −2i

e ikR exp[i (αx1 + βy1 ) − γξ ( x1 , y1 )] sin γ [ z1 − ξ ( x1 , y1 )] . R

(3.29)

This expression and expression (3.26) are the starting points for further considerations. The coefficient of transmission of penetrating radiation A in formula (3.26) depends, generally speaking, on the angle of liquid surface inclination. However, estimates show that the angular dependence of A in the considered case of mildly sloping unevennesses is very weak and we can take the value of A to be equal to one. Substitution of solution (3.29) to formula (3.26) gives the next expression for sound pressure:

p=

∞ e ikR αµmω I0 2πC p R ∫

∞

  exp ∫ ∫  − −∞ 0 

ρ12   exp(−iγξ1 ) exp[− µ ( z1 − ξ1 )] × a 2  (3.30) sin[γ ( z1 − ξ1 )] exp[i (αx1 + βy1 )]dx1dy1dz1 ,

where a is the effective radius of the radiation beam with intensity distributed according to the law I ( x, y ) = I 0 exp(− ρ12 / a 2 ) , ξ1 = ξ ( x1 , y1 ) , ρ1 =

x12 + y12 .

It is interesting to determine the average sound pressure. After the change of the variable z = z1 − ξ1, the average field in the far wave zone can

RADIATION ACOUSTICS

55

be determined by averaging of expression (3.30) with respect to all possible realizations of the inhomogeneous surface: ∞ I αµmω e ikR 〈 p〉 = 0 2πC p R ∫

∞

  exp ∫ ∫  − −∞ 0 

ρ12  − iγξ1 〉×  exp(− µz ) sin γz 〈 e a 2 

exp[i (αx1 + βy1 )]dx1dy1dz1 .

(3.31)

−iγξ

Here 〈e 1〉 = f (−γ) is the characteristic function of the random quantity ξ which is the displacement of the boundary surface. After integration we obtain an expression for 〈p〉,  a 2 k 2 sin 2 θ  I αµmω e −ikR a 2 k cos θ 〈 p〉 = 0 exp −  f (−γ ) 2C p R µ 2 + k 2 cos 2 θ 4  

(3.32)

(here θ is the angle between the axis z and the direction to the observation point (x, y, z)). Using formula (3.32) we can calculate the pressure field if we know the function f(−γ) in the explicit form. Let us assume that surface displacements obey the normal distribution law. We substitute the function f(−γ) in the explicit form into formula (3.32). We obtain the next expression for the density of probability distribution: w(ξ ) =

 ξ2  exp − , 2πσ  2σ 2  1

where σ is the mean-square height of unevennesses. Now the characteristic function is expressed by the formula  γ 2σ 2   k 2σ 2 cos 2 θ   = exp − f (−γ ) = exp −  = e − p / 8 , (3.33) 2  2      where p = 2kσ cosθ is the Rayleigh parameter. One can see from expression (3.33) that if the Rayleigh parameter tends to zero, the characteristic function becomes equal to one and formula (3.32) transforms into the expression describing the pressure field in the case of thermoradiation excitation of sound in a liquid half-space with even (plane) boundary. Formulae (3.32) and (3.33) show that the average radiation field is

56

THERMORADIATION EXCITATION: HOMOGENEOUS LIQUID

characterized by certain directivity. In the case of the observation angles θ < arctan(21/2σ/a) the directivity characteristic is affected mainly by the Rayleigh parameter and in the case of the angles θ > arctan(21/2σ / a), the directivity is determined by the ratio of the radius of the penetrating radiation beam to the sound wavelength, i.e., the influence of the Rayleigh parameter on the average sound field decreases as the angle θ grows. The average field in the direction of the axis z decreases rapidly with increase of the Rayleigh parameter. Polar diagrams obtained according to formulae (3.32) and (3.33) and under the condition (ka)2 = 10 are given in Fig. 3.3 (ka is the wave dimension of the radiation spot).

Figure 3.3 Polar diagrams of average pressure distribution for (ka)2 = 10. Curves 1 – 4 correspond to the next values of the parameter σ/a: o.01, 0.1, 0.5, and 1. Dashed lines indicate the angles determined by the conditions θ = arctan(21/2σ / a).

It is necessary to note that, as in the theory of sound scattering by rough surfaces, the average field in this case (under the conditions of the selected approximate solution of the problem) depends only on the distribution of unevenness heights and does not depend on the shape of the uneven surface. Another important characteristic of random sound field is the average intensity. We can write an expression for it in the next form on the basis of expression (3.30): 2  ρ2 + ρ2   I 0αµmω  ∞ ∞ ∞ ∞ 2 2 ×   exp − 1 〈 p〉 = 2  2πC p R    a   −∞ 0 −∞ 0  

∫ ∫∫ ∫ ∫∫

e − µz sin γz1e − µz 2 sin γz2 〈 exp[−iγ (ξ1 − ξ 2 )]〉 ×

(3.34)

RADIATION ACOUSTICS

57

exp[i (α ( x1 − x2 ) + β ( y1 − y2 ))]dx1dy1dz1dx2 dy2 dz2 . Here 〈exp[−iγ(ξ1 − ξ2)]〉 = f (−γ, γ) is the two-dimensional characteristic function. If we consider statistically homogeneous unevenness, when the two-dimensional characteristic function depends on the difference of the coordinates ξ = x1 − x2 and η = y1 − y2 , and assume that unevennesses are isotropic, i.e., the function f (−γ, γ) depends only on the distance between the points ρ = (ξ 2 + η2)1/2, then we obtain an expression for the average field intensity, ∞ 2π  I 0αµmω  γ 2a 2   f (−γ , γ ; ρ ) × 〈| p | 〉 =  2C p R  ( µ 2 + γ 2 ) ∫ ∫ ∫ ∫   0 0 2

2

  exp − 

ρ 22 a

2

−

ρ2 a

2

−

  (cos ϕ cosψ + sin ϕ sinψ )  × 2 a 

2 ρρ 2

(3.35)

exp[iρ (α cos ϕ + β sin ϕ )]ρρ 2 dρ dρ 2 dψ dϕ , where x1 − x2 = ρ cos ϕ, y1 − y2 = ρ sin ϕ, x2 = ρ2 cos ψ, and y2 = ρ2 sin ψ. After integration expression (3.35) is reduced to the form 2 ∞  I 0αµmω  γ 2a 2   f (−γ , γ ; ρ ) × 〈| p | 〉 =  2C p R  ( µ 2 + γ 2 ) 2 ∫   0 2

(3.36)

ρ2 

  I  ρ α 2 + β 2  ρ dρ , exp −  2a 2  0     where I0(x) is the Bessel function of the zero order. As in the previous case, we assume that the two-dimensional distribution law is valid for boundary displacement,  ξ 2 − 2ξ ξ N ( ρ ) + ξ 2  1 2  2 , exp− 1 w(ξ1 , ξ 2 ; ρ ) =  2 2 2πσ (1 − N ( ρ ))   2πσ 2 1 − N 2 ( ρ ) 1

(3.37)

58

THERMORADIATION EXCITATION: HOMOGENEOUS LIQUID

where N(ρ) is the coefficient of boundary unevenness. In this case the twodimensional characteristic function takes on the form f (−γ , γ ; ρ ) = exp[−γ 2σ 2(1 − N ( ρ ))] .

(3.38)

In the case of large unevennesses meeting the condition γσ >> 1, the integrand in expression (3.36) is essential only for the values of ρ close to zero. Then the correlation coefficient can be expanded into a series and we can take only two terms of the expansion: | N ′′(0) |  ρ  N (ρ ) ≈ 1 − ρ 2  0

2

  ,  

where ρ0 is the correlation length of the boundary unevenness. Let us determine the mean-square angle of unevenness inclination by a relationship,  ∂ξ   tan 2 σ =   ∂ρ 

2

=

σ2 ρ 02

| N ′′(0) | ,

(3.39)

and obtain an expression for the characteristic function,  γ 2 tan 2 δ 2  f (−γ , γ ; ρ ) ≈ exp − ρ .   2  

(3.40)

Substituting expression (3.40) to formula (3.36) and assuming that tan2σ = 2(σ/ρ0)2 for the normal distribution law, we represent finally the average intensity of the sound field of thermoradiation sources in the case of largescale homogeneous isotropic unevenness of the boundary in a liquid in the form, 2  I 0αmc  1 µ 2 a 4 cos 2 θ 2  〈| p | 〉 =  ×  2C p R  [( µ / k ) 2 + cos 2 θ ] 2 1 + ∆2 cos 2 θ  

(3.41)   ρ 2  ∆2 sin 2 θ , exp  −  0    σ  4( ∆2 cos2 θ + 1) 

RADIATION ACOUSTICS

59

where ∆ = 21/2kaσ / ρ0 is the dimensionless parameter. We have to note that the average field intensity can be also obtained readily for the case of anisotropic unevenness. It is necessary to note that the quantity tan2δ in the index of the characteristic function determines the bond between displacements of two points of the uneven boundary. As one can see from expression (3.41), in the process of calculation of the average intensity, tan2δ is essential in the exponential index, though we may ignore fluctuations of surface inclination while determining the scattered field (see expression (3.29)). Now let us analyze expression (3.41) and consider two limiting values of the parameter ∆ (∆ << 1 and ∆ >> 1). Let ∆ << 1. In this case the adopted condition kσ >> 1 corresponds to the smallness of the light spot as against the correlation length of the surface unevenness, i.e., the relationship a/ρ0 << 1. If ∆ tends to zero, the average intensity tends in the limit to the intensity of the sound field arising in the case of penetrating radiation absorption in a liquid with the plane surface, 2  k 2 a 2 sin 2 θ   I 0αmc  µ 2 a 4 cos 2 θ 2   lim 〈| p | 〉 = exp − .  2C p R  [( µ / k ) 2 + cos 2 θ ] 2 2 ∆ →0    

It follows from here that in the case ∆ << 1, we may ignore the effect of boundary unevenness on the sound intensity and calculate the intensity in the approximation of plane boundary. Now let ∆ >> 1 that at kσ >> 1 corresponds to the radius of the light spot comparable to or larger than the correlation length of the boundary unevenness, i.e., the condition a/ρ0 ≥ 1. The average intensity is described by an expression, 2  I 0αmc  a2  µρ 0 2   〈| p |〉 ≈   2C p R  [( µ / k ) 2 + cos 2 θ ] 2  kσ  

2

  ×  (3.42)

 1  ρ 2  exp −  0  tan 2 θ  .  4  σ   As one can see from formula (3.42), in this case the angular characteristic is determined by the scale of surface unevenness. The intensity decreases e times for the observation angle corresponding to the relationship tanθ = 2σ/ρ0, i.e., the angular beam of the directivity pattern is approximately

60

THERMORADIATION EXCITATION: HOMOGENEOUS LIQUID

equal to the mean-square angle of inclination of the uneven surface θk = 21/2σ. Directivity patterns for the average intensity are given in Fig. 3.4 for µ = k, σ/ρ0 = 0.17, and ∆2 = 0.1, 1, and 10. The limited angular width of the directivity pattern θ 0 = arctan(21/2σ/ρ0) = 14° for the selected ratio σ/ρ0 is indicated at the plot at ∆2 = 10.

Figure 3.4 Polar diagrams of average intensity distribution. Curves 1 – 3 correspond to the next values of the parameter ∆2: 0.1, 1, and 10.

It is easy to determine the dispersion of acoustic field fluctuations using formulae (3.32) and (3.41): 2  I 0αmc  1 µ 2 a 4 cos 2 θ  2 2   × D = 〈| p | 〉− | 〈 p〉 | =   2C p R  [( µ / k ) 2 + cos 2 θ ]  ∆2 cos 2 θ + 1  

  ρ 2   a 2 k 2 sin 2 θ   ∆2 sin 2 θ  exp − − k 2σ 2 cos2 θ   . exp  −  0  2   σ  4( ∆2 cos2 θ + 1)     (3.43) In the case of the dimension of the radiation spot small being compared with the correlation length of boundary unevenness, the dispersion is proportional to the factor [1 − exp(−p/4)], i.e., the dispersion is maximal in the direction of the axis z and increases with the growth of the Rayleigh parameter. If the dimension of the radiation spot is large in comparison with the correlation length, the angular dependence of the dispersion is close to

RADIATION ACOUSTICS

61

the angular dependence of the square of the average pressure. In this case the dispersion is minimal in the direction of the axis z and tends to zero in this direction with the growth of the Rayleigh parameter. According to this consideration, we note the next characteristic features of the sound field arising due to absorption of modulated penetrating radiation in a liquid half-space with large-scale unevenness of the boundary. The average pressure represents the product of the sound pressure in the half-space with even boundary and the characteristic distribution of the height of boundary unevennesses. The average pressure depends essentially on the Rayleigh parameter as in the problems of sound scattering. The influence of boundary unevenness on the average intensity of sound field is given by the exponential dependence on the parameter ∆, i.e., in the case of large unevennesses (kσ >> 1), the intensity depends exponentially on the ratio of the radius of the radiation spot at the boundary to the correlation length of boundary unevenness. If the radius of the radiation spot is small as against the correlation length of unevennesses, we may ignore the effect of boundary unevenness on the average intensity of the field. In this case, the larger the Rayleigh parameter, the larger the peak of the directivity pattern along the axis z of the dispersion. If the radius of the radiation spot is larger than the correlation length of unevenness, the angular width of the directivity pattern is approximately equal to the average mean-square angle of inclination of the uneven surface. In this case the dispersion has a directivity pattern with the minimum along the axis z.

5. THE CASE OF SMALL UNEVENNESS Now let us consider the effect of an uneven liquid boundary on thermoradiation generation of sound when unevennesses are assumed to be mildly sloping, statistically homogeneous, and isotropic and their height is small in comparison with the sound wavelength. We consider the developed generation mode as above. We will obtain the expressions providing an opportunity to calculate the average field and the intensity of sound field fluctuations in the far wave zone. Simple relationships connecting the average sound field with the Rayleigh parameter, mean-square height, and spatial correlation length of unevennesses will be given for some limiting cases. As one will be able to see, the boundary unevenness affects sound field fluctuations in two ways: first, the field of volumetric thermal sources is scattered at random boundary unevennesses, and second, the intensity of these sources fluctuates as the track length of particles (quanta) of penetrating radiation in a liquid changes randomly.

62

THERMORADIATION EXCITATION: HOMOGENEOUS LIQUID

Let a penetrating radiation beam propagating in the positive direction of the axis z be incident upon the free uneven boundary of a liquid half-space z ≥ ξ (x, y). The boundary unevenness is random and ξ ( x, y ) = 0. Here and further, the line above means averaging with respect to the statistical ensemble. Using the reciprocity principle, we write down the solution of the equation of thermoradiation sound generation in the form p(r0 ) =

iωmαAµ Cp

∫ I ( x, y) exp{−µ[ z − ξ ( x, y)]}G(r, r0 )dV ,

(3.44)

V

where G(r, r0) is the function, which is the solution of the inhomogeneous reduced wave equation (the equation of thermoradiation sound generation in a liquid) with the right-hand side in the form of the δ-function, satisfying the condition G (r, r0 ) | z =ξ ( x, y ) = 0 ,

(3.45)

at the uneven surface, where r(x, y, z) is the radius-vector of the current point and r0(x0, y0, z0) is the radius-vector of the observation point. Integration in expression (3.44) is performed over the part of the half-space z ≥ ξ (x, y), where thermal sources exist. The function G(r, r0) is the solution of the problem, i.e., the field of a point source located at the point r0 of the half-space with the uneven boundary, where it is necessary to determine the radiation field. It is known that the exact analytical representation for G(r, r0) has not been obtained yet. Therefore, making certain assumptions on the problem parameters, we use an approximate expression for G(r, r0). We need to determine the sound field in the far wave zone4. In this case the function G(r, r0) can be represented as the solution of the problem on the diffraction of a plane monochromatic wave at an uneven boundary, and the spherical divergence of the field can be taken into account with the help of the common factor exp(ikr0) / (4π r0). We assume that the height of boundary unevennesses is small as against the sound wavelength. We restrict ourselves also for simplicity to 4

As in the previous section, we mean here the far wave zone with respect to both the dimensions of the region of effective heat release (and therefore, effective sound generation) and the uneven boundary. The conditions determining the far wave zone with respect to the uneven boundary have been considered in detail by Lysanov [125] for example.

RADIATION ACOUSTICS

63

consideration of mildly sloping, statistically homogeneous, and isotropic unevennesses of the boundary. The problem of scattering of a plane sound wave at the boundary with small mildly sloping unevennesses has been considered in details by many researchers (see [18, 125] for example). We will basically follow these considerations to determine the solution. Let us expand boundary condition (3.45) into a series with respect to the power of the small parameter kσ, where σ is the mean-square height of unevennesses, and keep in the expansion only the terms of order of smallness not higher than the second. Then the exact boundary condition is changed for the approximate one at z = 0. We represent the field described by the function G(r, r0) in the form of the sum of the average field Gav(r, r0) and the random addition Gran(r, r0), G (r, r0 ) = Gav (r , r0 ) + G ran (r , r0 ) ,

where G ran (r , r0 ) = 0. Let us write down an approximate expression for Gav(r, r0) keeping the terms of the order of smallness not higher than the second order with respect to kσ: Gav (r, r0 ) =

exp(ikr0 ) exp[i (k x x + k y y )] × 4πr0 (3.46)

[exp( −ik z z ) + w exp(ik z z )] , where (kx, ky, kz) are the components of the wave vector k coinciding in its direction with the radius-vector r0 and w is the average coefficient of reflection of a plane sound wave from the uneven boundary, which is represented (as it is possible to demonstrate) with the help of the normalized function of correlation of statistically homogeneous and isotropic unevennesses N(ρ), where ρ is the horizontal distance between two points at the boundary: w = −1 + η cosθ ;  2 ∞ 2 2  ∂ exp(ik ρ + z )  η = 2ikσ 2 ∫ N ( ρ ) J 0 (kρ sin θ )  ρ dρ .  ∂z 2  2 2 ρ +x 0   z = 0

(3.47)

Here J0(kρ sinθ) is the Bessel function of the zero order and θ is the angle between the axis z and the radius-vector r0. Thus, we have

64

THERMORADIATION EXCITATION: HOMOGENEOUS LIQUID

Gav (r , r0 ) = G0 (r , r0 ) + O(k 2σ 2 ) , where

G0 (r, r0 ) =

exp(ikr0 ) exp[i (k x x + k y y )][exp(−ik z z ) − exp(ik z z )] . 4πr0

(3.48)

Within the framework of the above assumptions, the term Gran(r, r0) has the form Gran (r , r0 ) =

exp(ikr0 ) Ψ ( x, y , z ) , 4πr0

(3.49)

where Ψ(x, y, z) is the random component of the field of scattering of a plane monochromatic sound wave incident at the angle θ on the statistically uneven boundary of the half-space. As is known (see [125]), the function Ψ(x, y, z) can be represented in the form of the double Fourier integral, Ψ ( x, y , z ) = ∫

∞

∫ A(α , β ) exp[i(αx + βy + γz )]dαdβ ,

(3.50)

−∞

where γ = (k2 − α2 −β 2)1/2, and an expression for the random amplitude A(α, β) has the form A(α , β ) =

i 2π 2

k cos θ ∫

∞

∫ ξ ( x, y ) ×

−∞

(3.51) exp{−i[(α − k x ) x + ( β − k y ) y ]}dxdy . Let us make one more assumption. Let µσ << 1. We represent exp[µξ(x, y)] in the form of a series keeping the terms of the order of smallness with respect to µσ not higher than the second order:

exp[µξ ( x, y )] = 1 + µξ ( x, y ) +

µ2 2 ξ ( x, y ) . 2

(3.52)

RADIATION ACOUSTICS

65

Such an assumption is quite natural since, as we have mentioned above, the most efficient conversion of optical energy into sound energy occurs at k ~ µ, and therefore this case is most interesting from the practical point of view. Keeping the terms of the order of smallness with respect to kσ and µσ not higher than the second order, we obtain on the basis of relationships (3.44) – (3.52) the approximate expression describing the solution p(r0): p(r0 ) =

iωmαAµ Cp

∫ I ( x, y) exp(−µz ){Gav (r , r0 ) + µξ ( x, y)G0 (r, r0 ) + V

(3.53)

µ2 2

ξ 2 ( x, y )G0 ( r, r0 ) + Gran ( r, r0 ) + µξ ( x, y )Gran ( r, r0 )}dV .

Let us proceed in expression (3.53) from integration with respect to the volume V(z ≥ ξ(x, y)) to integration with respect to the volume V0(z ≥ 0) and estimate the error arising in this case. One can see that the error is

∆=

iωmαAµ Cp

ξ ( x, y )

∫ ∫ F ( x, y, z)dzdS ,

(3.54)

0 S0

where F(x, y, z) denotes the integrand in expression (3.53), and S0 is the area in the plane x, y, which limits the volume occupied by thermal sources. Taking into account the restrictions imposed above, we represent F(x, y, z) in expression (3.54) in the approximate form F ( x, y, z ) = F ( x, y,0) + z

∂F ( x, y, z ) . ∂z z =0

Integration with respect to z in expression (3.54) can be performed easily now. We have the result ∆=

iωmαAµ Cp



 1 ∂F ( x, y, z ) ξ 2 ( x, y )dS . ∂z  z =0

∫ F ( x, y,0)ξ ( x, y) + 2

S0

Substituting here F(x, y, z) from expression (3.53), using expressions (3.46) – (3.50), and keeping the terms of the order of smallness with respect

66

THERMORADIATION EXCITATION: HOMOGENEOUS LIQUID

to kσ and µσ not higher than the second order, we obtain finally that, substituting integration with respect to the volume V0 for integration with respect to the volume V in expression (3.53), it is necessary to add the following term: ∆=

exp(ikr0 ) ωmαA kµ cos θ Cp 4πr0

∫ξ

2

( x, y ) I ( x, y ) ×

S0

(3.55) exp[i ( k x x + k y y )]dS .

Expressions (3.53) and (3.54) determine with precision up to the terms of the second order of smallness with respect to kσ and µσ, the sound field generated by a thermoradiation source in a liquid occupying a half-space with an uneven boundary under the condition that the height of unevennesses is small compared with the sound wavelength. Let us represent this field in the form of the sum of the average field pav(r0) and the random addition pran(r0): p (r0 ) = p av (r0 ) + p ran (r0 ) ,

so that p ran (r0 ) = 0 . Now let us consider for definiteness the specified form of the intensity distribution I(x,y) in the penetrating radiation beam. It is presumed commonly that the real distribution is close to the Gaussian one. Therefore, we take I ( x, y ) = I 0 exp[−( x 2 + y 2 ) / a 2 ] .

(3.56)

Substituting expression (3.56) into expressions (3.53) and (3.55) and averaging with respect to the total realizations of the function ξ(x, y), we obtain for the average field

pav (r0 ) =

 k 2a 2  ωmαAa 2 I 0 exp(ikr0 ) exp − sin θ  ×   2r0 4 Cp   (3.57) kµ cos θ 2

µ + k 2 cos 2 θ

f (θ ) ,

RADIATION ACOUSTICS

f (θ ) = 1 −

67

η µ η (kσ cos θ ) 2 cos θ + i − − µ ( µ 2 + k 2 cos 2 θ ) × k 2 2 2 ∞

∫ ℑ(κ )κ 0

2π

dκ dΨ

∫

2

2 2 0 µi k − (κ cos Ψ + k sin θ ) − κ Ψ

(3.58)

,

where the function ℑ(κ) is the spatial spectrum of statistically homogeneous and isotropic unevennesses of the boundary. The spatial spectrum is connected with the function of correlation of unevennesses by a wellknown relationship ∞

σN ( ρ ) = 2π ∫ ℑ(κ ) J 0 (κρ )κ dκ . 0

Here J0(κ, ρ) is the Bessel function of the zero order. The function f(θ ) characterizes the effect of boundary unevenness on the sound field in a liquid half-space. In the absence of unevenness, f(θ ) ≡ 1 and expression (3.57) describes the field generated by an optical source in a half-space with an even boundary. This coincides with a corresponding expression (see expression (3.22)). Expressions (3.47), (3.57), and (3.58), provide an opportunity to calculate the parameters of the average component of the sound field in the half-space according to known characteristics of boundary unevennesses. We have to note some particular cases when it is possible to make expression (3.58) simpler to some extent. Let kρ0 >> 1, where ρ0 is the correlation coefficient of unevennesses. Let us consider the double integral in expression (3.58). At the essential interval of integration, κ << k. Therefore, let us take approximately κ = 0 in the denominator of the integrand. As ∞

∫ ℑ(κ )κ dκ = σ

2

/ 2π ,

0

we obtain f (θ ) = 1 −

η µ η (kσ cos θ ) 2 cos θ + i − − ( µσ ) 2 − iµkσ 2 cos θ . 2 k 2 2

68

THERMORADIATION EXCITATION: HOMOGENEOUS LIQUID

2

In the case of non-slip observation angles (kρ0 cos θ >> 1) 3 η = 2(kσ ) 2 cos θ , f (θ ) = 1 − P 2 − ( µσ ) 2 , 8 where P = 2kσ cosθ is the Rayleigh parameter. Thus in this case, the decrease of the average field in comparison with the field in a half-space with an even boundary is determined not only by the Rayleigh parameter as in the case of large-scale unevenness, but also by the quantity µσ. It is necessary to note that the latter does not depend on the observation angle. In the case of slip observation angles such that kρ0 cos2θ << 1, ∞

( 2 kσ ) 2

1 dN ( x*)  π exp i  ∫ dx * , η=− 2πkρ 0 x * dx *  4 0

where N(x*) is the normalized correlation function of boundary unevenness and x* is the distance normalized to the correlation length. In particular, we have at N(x*) = exp(−x*2) f (θ ) = 1 − 1.2kσ 2

µ + k cos θ (kσ cos θ ) 2 − − ( µσ ) 2 + 2 πkρ 0

  k cosθ − µkσ 2 cos θ  . i 1.2kσ 2   πkρ 0   In the case of small-scale boundary unevenness (kρ0 << 1)

η = −2i

kσ 2

ρ0

∞

1 dN ( x*) dx * , dx *

∫ x* 0

and if N(x*) = exp(−x*2) as above, then we obtain from expression (5.58), f (θ ) = 1 − π

kσ 2 µσ 2 (kσ cos θ ) 2 − − i π cos θ − µ ( µ 2 + k 2 cos 2 θ ) × ρ0 2 ρ0

σ 2 ρ 02 ∞  ρ 02κ 2  2π dκ dΨ . κ ∫ exp − ∫   4π 4 2 2 2 2 i k k µ − − κ Ψ + θ − κ Ψ ( cos sin ) sin   0 0

RADIATION ACOUSTICS

69

The integral here cannot be reduced and one has to calculate it numerically. It is interesting that in the considered approximation, the modulus of the function f(θ) is smaller than one, though the modulus of the average coefficient of reflection of a plane sound wave from the boundary with small-scale unevennesses (kρ0 << 1) is equal to one. Thus, we can see that the average component of the sound field depends on the characteristics of boundary unevenness and, in particular, on the Rayleigh parameter and the spatial correlation of unevenness. In the case of large-scale unevenness and observation angles close to the normal angle, the average component of the field is described by a simple analytical expression. In other cases detailed calculation is needed to determine the parameters of the average sound field. Let us consider the random component of the field pran(r0). We obtain from expression (3.53) for pran(r0):  iωmαAµ  pav (r0 ) =  I ( x, y ) exp(− µz )Gran (r , r0 )dV + Cp  ∫ V0 (3.59)   µ ∫ ξ ( x, y ) I ( x, y ) exp( − µz )G0 ( r, r0 )dV  .  V0  One can see from this expression that within the framework of the perturbation method, the random component of the field is represented by the sum of two integrals. Each of these integrals describes separately different mechanisms of unevenness influence on sound generation due to absorption of laser radiation. The first integral in expression (3.59) is the random component of the field of stationary thermal volumetric source of sound in a half-space with a rough boundary, and the second integral is the random component of the field of thermal volumetric sound sources with the intensity changing randomly in a half-space with an even boundary. Fluctuations of intensity of thermal volumetric sources are caused here by the fact that the path length changes randomly in the case of propagation of a particle beam in a liquid to a certain point of the half-space. Multiplying expression (3.59) by the complex conjugate and using relationships (3.48) and (3.49), we can obtain readily a formula for calculation of the dispersion of fluctuations of the sound field:  ωmαAµ  × | p ran (r0 ) | 2 =   4πC p r0   

70

THERMORADIATION EXCITATION: HOMOGENEOUS LIQUID

  ∗  ∫ ∫ I ( x, y ) I ( x1, y1 ) exp( − µz − µz1 ) Ψ( x, y, z ) Ψ ( x, y , z )dVdV1 − V  0   ∫ ∫ I ( x, y ) I ( x1, y1 ) exp(ik x x + ik y y − µz1 ) × µ 2 + k 2 cos2 θ V S  0 0 2iµk cos θ

ξ ( x, y ) Ψ ∗ ( x1, y1, z1 ) dSdV1 − (3.60)

∫ ∫ I ( x, y ) I ( x1, y1 ) exp( −ik x x1 − ik y y1 − µz ) × V0 S 0

]

ξ ( x1, y1 ) Ψ( x, yz ) dVdS1 +  2 µk cos θ   µ 2 + k 2 cos2 θ 

   

2

∫ ∫ I ( x, y ) I ( x1, y1 ) exp[ik x ( x − x1 ) + ik y ( y − y1 )] × S0

 ξ ( x, y )ξ ( x1, y1 ) dSdS1  .  Sound field fluctuations are given above in the form of the sum of two random quantities. Therefore, the fluctuation dispersion must be represented by the sum of the dispersions of each of these fluctuations plus cross terms. The first integral in expression (3.60) describes the dispersion of fluctuations of the field of stationary volumetric thermal sources in a half-space uneven boundary. The co-factor Ψ( x, y, z)Ψ * ( x1, y1, z1) in the integrand is nothing other than the correlation function of the random component of the scattered field in the case of incidence of a plane sound wave on a rough boundary. This function has been thoroughly studied. The last integral in expression (3.60) is the dispersion of fluctuations of the field caused by fluctuation of intensity of thermal volumetric sources. By the way, we have to note that just the same integral (see [132]) describes the

RADIATION ACOUSTICS

71

dispersion of the random component of the sound field arising due to absorption of laser radiation in a half-space. This component is produced due to random fluctuations of the transverse distribution of light intensity in the laser beam if we assume that the relative fluctuation of this distribution is equal to µξ(x, y). The other two integrals in expression (3.60) are the cross terms, which are nonzero, since the random functions describing sound field fluctuations are statistically independent. In the case of statistically homogeneous and isotropic unevenness, we obtain from expression (3.60) using relationships (3.50), (3.51), and (3.56) | p ran (r0 ) | 2 2 | pav (r0 ) |

= ( µ 2 + k 2 cos 2 θ ) 2 ×

 κ 2 a 2  2π − κ exp[ −kκa 2 sin θ cos Ψ ] × ℑ κ ( ) exp ∫  2  ∫   0 0

∞

(3.61) µ   ∗ [2 µ − i ( q − q )] 1 − 2  µ2 µ + k 2 cos2 θ   +  dκ dΨ . 2 2 2 2 ∗ ( µ − iq)( µ − iq ) ( µ + k cos θ )     2 2 1/2 Here q = (k cosθ − κ − 2kκ sinθ cos Ψ) and q* is the complex conjugate to q. We obtain for the field under the source (θ = 0) 2    1 − 2µ  k   | p ran (r0 ) | 2 µ2 + k2 µ2 = 2π ( µ 2 + k 2 ) 2 ∫ ℑ(κ ) + 2  µ 2 + k 2 − κ 2 (µ 2 + k 2 ) 2 | p av (r0 | 0 θ =0     

 κ 2a 2  κ dκ + exp −  2   

   ×    

(3.62)

 2µ   (µ + κ 2 − k 2 ) 1− 2   2 2 2 2   µ κ a  µ +k     + exp − κ dκ  . ∫ ℑ(κ )  2 2 2   2 (µ + k )   µ + κ 2 − k2    k    

∞

72

THERMORADIATION EXCITATION: HOMOGENEOUS LIQUID

Expressions (3.61) and (3.62) allow us to calculate the dispersion of intensity fluctuations of the sound field for different forms of the spatial spectrum of unevenness. We should note that, even for the simple forms of ℑ(κ), it is impossible to obtain analytical expressions, but expressions (3.61) and (3.62) can be calculated rather easily.

6. EFFICIENCY OF THERMORADIATION EXCITATION OF SOUND IN A LIQUID – SOME ESTIMATES It is interesting to clarify what is the efficiency of conversion of the penetrating radiation energy into acoustic energy in conditions of thermoradiation sound excitation. This question with respect to sound generation by laser radiation was addressed by Bozhkov and Bunkin [29]. The efficiency can be characterized by the quantity η = Psound / Prad, where Psound = ∫ IsoundR2dΩ is the power of the sound generated in a liquid in the case of incidence of a harmonically modulated beam of penetrating radiation with power Prad at its surface and Isound = | p2 | / 2pc is the sound intensity. Calculating this integral (over all spatial angles of the half-space z > 0), it is possible apparently to use expression (3.22) for the amplitude p, which is valid for the far wave field. For example, in the case of a wide laser beam (ka >> 1) with Gaussian profile, it is easy to obtain a general expression for η:

η = (c / ρ )(mAα / 2C p ) 2 [ µk ( µ 2 + k 2 )]2 I 0 .

(3.63)

Thus, the efficiency of radiation-acoustic conversion η depends linearly on the penetrating radiation efficiency I0 = Prad/(πa2). The maximum efficiency corresponds to the condition k = µ and is determined by a formula,

η max = (c / ρ )[mAα / 4C p ] 2 I 0 .

(3.64)

Let us give some estimates of generation of monochromatic sound in water (in many cases the necessary radiation-acoustic and thermophysical constants are well known for water) and in particular, the estimates for the cases of sound generation by CO2- and YAG-lasers. In these cases the thermal expansion coefficient α = 3⋅10−4 K−1 and according to expression (3.64), the maximum conversion efficiency ηmax ≈ 5⋅10−12 W/cm2.

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73

CO2 laser (λ = 10.06 µm). Let us calculate the maximum value of the ultrasonic amplitude that can be obtained at the axis of a laser beam at a distance R = 1 m from the water surface while using a laser pulse with length 10 µs and modulated with an ultrasonic frequency (intra-pulse modulation). The absorption coefficient µ ≈ 800 cm−1 [99, 119, 179]. At the optimal sound frequency, the far wave field is realized in the case of focusing of laser radiation into a spot a ≤ 2 mm. Assuming a = 1 mm and Popt = 103 W, we obtain according to expression (3.22), p ≈ 500 dyne/cm2 at the sound frequency 6 MHz and in the sound spot ∆l ≈ 15 cm. In this case the intensity I0 = Popt/(πa2) = 3⋅104 W/cm2 and according to expression (3.64), the conversion efficiency η ≈ 5⋅10−8. YAG laser (λ = 1.06 µm). Let us calculate the amplitude of ultrasound emitted in the quasi-CW mode along the water surface at the frequency ω/2π = 100 kHz at distance R = 1 m from the radiation spot of radius a = 2.5 mm. The absorption coefficient µ = 0.18 cm−1. As k ≈ 4 cm−1, then ka ≈ 1, ka2 ≈ 2.5 mm, and k/µ >> 1, i.e., the conditions for sound radiation along the liquid surface are satisfied. In this case the direction of the maximum of the directivity pattern constitutes the angle θ ≈ µ/k ≈ 4.5⋅10−2 rad = 2.6° to the liquid surface. Assuming Popt = 50 W, we obtain p ≈ 1.3 dyne/cm2 or 0.13 Pa according to expression (3.22).

CHAPTER 4

Thermoradiation Excitation of Sound in an Inhomogeneous Medium Inhomogeneity of liquid may influence sound excitation by penetrating radiation. Firstly, parameters of thermoradiation sources of sound may change because of the change of the path of radiation particles in a medium. Secondly, sound waves may be refracted and scattered by inhomogeneities and reflected by boundaries. In this chapter we consider some examples of thermoradiation excitation of sound in an inhomogeneous liquid.

1. SOUND EXCITATION IN A LIQUID HALF-SPACE IN THE PRESENCE OF A LAYER OF ANOTHER LIQUID AT ITS BOUNDARY Let us consider specific features of thermoradiation generation of sound in a liquid half-space in the presence of a layer of another liquid at its boundary, and in the case of absorption of intensity-modulated penetrating radiation in a two-layer medium. The presence of a liquid layer with acoustic (radiation) parameters, which differ from the parameters of the liquid in the half-space, can substantially influence characteristics of sound emission and intensity of the sound field. Both increase and decrease of intensity are possible in this case [145]. Let a beam of penetrating radiation be incident normally upon a free boundary of a liquid layer with density ρ and sound velocity c. Let the layer 75

76 THERMORADIATION EXCITATION: INHOMOGENEOUS MEDIUM

of the thickness H be located on a liquid half-space with density ρ1 and sound velocity c1. It is necessary to determine the sound field in the halfspace produced by absorption of intensity-modulated penetrating radiation in such a two-layer medium. As usual we select a coordinate system in such a way that the plane (x,y) coincides with the free surface of the layer. The axis z is directed within the liquid, and the coordinate origin is positioned in the center of the place of incidence of penetrating radiation (radiation spot) at the free surface of the liquid. We assume that the intensity of penetrating radiation is lower than the value necessary for changes of aggregate state in the region of absorption. Let us consider the stable regime of sound generation. The field of sound pressure p in the layer is described by the solution of the equation, ∆p + k 2 p =

iωmα µE ( x, y , z ) , 0 < z < H , Cp

(4.1)

where ω is the frequency of generated sound, k = ω /c is the wave number, m is the modulation index of intensity of penetrating radiation, α, Cp, and µ are the coefficient of cubical thermal expansion, specific heat capacity, and absorption coefficient of penetrating radiation in the layer. The function E(x,y,z) describes distribution of intensity of penetrating radiation in the liquid layer. The sound pressure p1 in the half-space is described by the solution of the equation ∆p1 + k12 p1 =

iωmα1 µ1E1 ( x, y, z ) , z > H , Cp

(4.2)

1

where k1 = ω /c1 and α1, Cp1, and µ1 are the coefficient of cubical thermal expansion, specific heat capacity, and absorption coefficient of penetrating radiation in a liquid occupying the half-space z > H, respectively. The function E1(x, y, z) describes distribution of intensity of penetrating radiation in the half-space. Solutions of equations (4.1) and (4.2) must satisfy the boundary conditions p = 0, p = p1, z = 0,

1 ∂ρ 1 ∂ρ1 = , z = H, ρ ∂z ρ1 ∂z

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77

and the condition of termination at infinity. The desired solution p1 of equation (4.2) can be written down on the basis of the reciprocity principle. The mathematical formulation of the reciprocity principle in acoustics was obtained by the author [131] (see Chapter 3, Section 2) under very general assumptions for the case of an inhomogeneous medium with constant density. An opportunity to generalize relationships obtained for the case of media with density depending on coordinates was indicated there also. In the case of a twolayer medium, such generalization is attained in a rather simple way described by the author [131]. Therefore, we will not dwell on this technique. Let us write down an expression for the desired sound pressure p1(r0) in the form

ρ iωmα p1 (r0 ) = 1 µ ∫ E ( x, y, z ) p * (r, r0 )dV − ρ Cp Vl

(4.3) iωmα1 µ Cp

∗

∫ E1( x, y, z ) p1 (r, r0 )dV ,

Vhs

where r0(x0, y0, z0) is the radius-vector of the observation point and r(x, y, z) is the instant radius-vector. Integration in expression (4.3) is performed over the layer region Vl and the region of half-space Vhs, where thermal sources of sound exist. The functions p* and p1* describe the field of a point source of unit amplitude located in the observation point r0. Thus, the problem is reduced to determination of these auxiliary functions, and the functions E(x, y, z) and E1(x, y, z) describing the distribution of the intensity of penetrating radiation in the layer and in the half-space. Firstly let us obtain an expression determining the intensity distribution of penetrating radiation in the layer E(x, y, z). In order to do this, it is necessary to sum all reflections of the initial beam of penetrating radiation from the layer boundaries. Thus, a direct beam produces at some level z in the layer the distribution of radiation intensity, which is determined by the expression AI(x, y) exp(−µz), where A is the coefficient of penetrating radiation transmission through the free boundary of the layer, and the function I(x, y) describes the transversal distribution of radiation intensity in a beam. The radiation intensity in the beam at the level z after partial reflection at the boundary “layer – half-space” is described by the expression A(1 − A1 ) I ( x, y ) exp[− µ (2 H − z )] ,

78 THERMORADIATION EXCITATION: INHOMOGENEOUS MEDIUM

where A1 is the transmission coefficient of radiation through the boundary “layer – half-space”. Then, after a partial reflection from the free boundary of the layer, the radiation intensity at the level z decreases down to the value A(1 − A)(1 − A1 ) I ( x, y ) exp[− µ (2 H + z )] , and so on. It is easy to sum the series obtained using the formula for the sum of an infinitely decreasing geometric progression. In the result we can write down an expression for E(x, y, z): E ( x, y , z ) = A

exp(− µz ) + (1 − A1 ) exp[− µ (2 H − z )] I ( x, y ) , 1 − (1 − A)(1 − A1 ) exp(2 µH )

(4.4)

at 0 ≤ z ≤ H. The intensity of penetrating radiation in the half-space is calculated in an analogous way. A corresponding expression for E1(x, y, z) can be presented in the form

E1 ( x, y, z ) =

AA1 exp[− µH − µ1 ( z − H )] I ( x, y ) , z ≥ H . 1 − (1 − A)(1 − A1 ) exp(−2µH )

(4.5)

It is necessary to determine the excited sound field p1(r0) at the point r0 in the liquid half-space at distances greater than the dimensions of the region occupied by thermal sources of sound. In this case we assume that the observation point r0 is located not too close to the boundary “layer – half-space”1. In this case the auxiliary solutions within this area can be represented approximately in the form

p * (r , r0 ) =

1

exp(ik1r0 ) exp(−ik1 x sin θ )W [exp(−iγz ) − exp(iγz )] , 4πr0

We have to note that under certain conditions, a normal mode relatively weakly emitting energy into the half-space can exist in the layer, and therefore it relatively weakly attenuates with distance from the source. The solution given below is valid also in the case when the distance from the source to the observation point is large in comparison with the dimensions of the layer region, where the existence of a weakly attenuating normal mode is noticeable.

RADIATION ACOUSTICS

p * (r , r0 ) =

exp(ik1r0 ) exp(ik1 x sin θ ){exp(−ik1 z cos θ ) − 4πr0

79

(4.6)

V exp[ik1 ( z − 2 H ) cos θ ]} . Here W =

2 β exp[i (γ − k1 cosθ ) H ] 1 − β + (1 + β ) exp(2iγH ) , V = , 1 + β + (1 − β ) exp(2iγH ) 1 + β + (1 − β ) exp(2iγH ) (4.7)

γ = k 2 − k12 sin 2 θ , β =

ρk1 cosθ , ρ1γ

θ is the angle between the radius-vector r0 and the axis z, and the plane x, y is assumed to include the observation point. Substituting auxiliary solutions in the form of expressions (4.6) and E(x, y, z) and E1(x, y, z) in the form of expressions (4.4) and (4.5) into expression (4.3), we obtain a relationship determining the sound pressure p1(r0):

p1 (r0 ) =

 ρ α WµγQ exp(ik1r0 ) Aωm + F (θ ) 1 1 − (1 − A)(1 − A1 ) exp(−2µH ) 2πr0  ρ C p µ 2 + γ 2 (4.8)

µ1k1 cos θ (1 + V ) + iµ12 (1 − V )  α 2 A1 exp[− µH − ik1H cos θ ] , C p1 2 µ12 + k12 cos2 θ  where

F (θ ) = ∫

∞

∫ I ( x, y) exp(ik1 x sin θ )dxdy ,

−∞

Q = 1 − 2 exp(− µH ) cos(γH ) −

80 THERMORADIATION EXCITATION: INHOMOGENEOUS MEDIUM

µ  A1 exp( − µH )  sin(γH ) − cos(γH ) − (1 − A1 ) exp( −2 µH ) . γ   Expression (4.8) determines the sound field in a liquid half-space in the far wave zone. The field is produced due to absorption of intensity-modulated penetrating radiation in the case when a layer of another liquid is present at the half-space boundary. Using this expression, we can calculate directivity patterns of thermoradiation sources of sound in various situations. As we have noted before, a transverse distribution of intensity in a beam of penetrating radiation is close to the Gaussian one. Therefore, we assume I ( x, y ) = I 0 exp[−( x 2 + y 2 ) / a 2 ] . Integrating, we obtain the next expression for F(θ ): F (θ ) = I 0πa 2 exp[−(k12 a 2 / 4) sin 2 θ ] .

(4.9)

Figure 4.1 shows examples of directivity patterns calculated according to formulae (4.8) and (4.9) for a laser source (photon beam) and the Gaussian distribution of intensity in a laser beam. Directivity patterns in Fig. 4.1a correspond to the situation in which a layer of benzene is present at the surface of water. The wavelength of optical radiation is 0.3 µm. In this case µ = 2.3 cm−1 and µ1 = 0.18 cm−1 (for example, see [179]). The coefficients of light transmission are A = 0.96 and A1 = 0.99. The radius of light spot is taken to be equal to a = 0.4 cm. The ratios are ρ/ρ1 = 0.88, α1/Cp = 63.3⋅10−12 g/erg, and α1/Cp1 = 4.7⋅10−12 g/erg. The directivity patterns are calculated for the three values of thickness of the benzene layer: H = 0.33 cm (kH = π/2), H = 0.66 cm (kH = π), and H = 1.99 cm (kH = 3π). The observation angle is counted off from the vertical axis, where the modulus of the amplitude of sound pressure p1(r0) is normalized to (2ωmI0a2/2r0)⋅10−12 g/(cm⋅s2). We should note that under such conditions the directivity pattern of a laser thermoradiation source in water without a benzene layer is strongly extended along the surface, and the amplitude of sound pressure in the vertical direction (θ = 0) is approximately two orders of magnitude smaller than that in the considered case with a layer. Thus, the presence of a layer of another liquid on the surface of a liquid half-space can lead to a significant change of intensity of sound field. If a liquid in the layer is characterized by the value of α /Cp large in comparison with a liquid in the half-space, the intensity of sound field increases

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81

substantially. The difference in the ratios µ/k for liquids in the layer and in the half-space leads also to significant differences in directivity patterns in the cases with a layer and without it. As we have noted already, in the absence of a layer, the directivity pattern is strongly extended along the free surface of water. And when a benzene layer is present, it displays a peak in the vertical direction. Finally, if the values of kH are large enough, the resonance properties of a layer begin to manifest themselves which also leads to changes (irregularity) in directivity patterns.

Figure 4.1 Directivity pattern of a thermooptical source of sound in a two-layer medium. (a) Benzene – water, thickness of benzene layer H = 0.33, 0.66, and 1.99 cm (curves 1 – 3, respectively); (b) water – benzene; and (c) warm water – cold water correspond to the thickness H = 0, 1.86, and 2.23 cm (1 – 3).

Directivity patterns given in Fig. 4.1b describe a reversed situation: a water layer is located on a benzene half-space. Calculations are conducted

82 THERMORADIATION EXCITATION: INHOMOGENEOUS MEDIUM

for three values of layer thickness, i.e., H = 0 (a half-space without a layer), H = 1.86 cm (kH = 2.5π), and H = 2.23 cm (kH = 3π). Such arrangement of liquids may be of theoretical interest only. However, it is important to note the following fact. Only about 30% of the energy of laser radiation is absorbed in a water layer in this case. Moreover, the ratio α/Cp for benzene is more than one order of magnitude larger than this ratio for water. Therefore, acoustic radiation is produced almost exceptionally by thermal (thermooptical) sources of sound located in the half-space, i.e., benzene. Nevertheless, the shapes of directivity patterns in the cases with a layer and without it are essentially different. This fact is connected with the change of conditions at the boundary of the half-space: a layer is present with wave resonance properties instead of a free boundary. Directivity patterns given in Fig. 4.1c correspond to a situation in which a layer of warm water (20°C) is located on top of cold water (0°C). The wavelength of laser radiation is selected to be equal to 1 µm. In this case µ = µ1 = 0.18 cm−1. The frequency of generated sound is ω /2π = 105 Hz. The next values of wave numbers are k = 4.23 cm−1 and k1 = 4.38 cm−1. Furthermore, A = 0.98, A1 = 1, ρ/ρ1 ≈ 1, α/Cp = 4.76⋅10−12 g/erg, and α1/Cp = 1.19⋅10−12 g/erg. The radius of the light spot is taken to be equal to 0.4 cm. The patterns are calculated for three values of layer thickness: H = 0 (a homogeneous half-space), H = 1.86 cm (kH = 2.5π), and H = 2.23 cm (kH = 3π). In the case considered, the liquids in the layer and the half-space differ noticeably only in the ratio α /Cp. Nevertheless, the presence of the layer leads not only to an increase of the intensity of generated sound, but also changes significantly the shapes of directivity patterns. It is difficult to conduct a more detailed analysis of the dependence of the parameters of the sound field generated in a two-layer medium by a source of penetrating radiation on the problem parameters in a general form, as expression (4.8) is rather complex. Therefore, we restrict ourselves to consideration of a particular case which may be of practical interest. Let exp(−µH) << 1, i.e., the energy of penetrating radiation is absorbed in the main layer. Further, let ρ ≤ ρ1 and c ≤ c1. Then, ignoring the terms of the order of exp(−µH) in expression (4.8), we obtain an approximate expression for p1(r0):

ρ ωmαA exp(ik1r0 ) µγ p1 (r0 ) ≈ 1 WF (θ ) . 2πr0 ρ Cp µ2 +γ 2

(4.10)

Let us compare the sound field generated by a thermoradiation source in a two-layer medium with the sound field in a homogeneous liquid halfspace without a layer. An expression describing a sound field in the case of

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83

thermoradiation generation of sound in a homogeneous half-space is obtained above and can be rewritten in the form

p0 (r0 ) =

ωmα1 A2 exp(ik1r0 ) µ1k1 cos θ F (θ ) , C p1 2πr0 µ12 + k12 cos 2 θ

(4.11)

where A2 is the coefficient of light transmission through the free boundary of a half-space. The influence of a layer on the amplitude of sound field and the directivity pattern of a thermoradiation sound source can be evaluated comparing expressions (4.10) and (4.11). We have to note that the shape of the directivity pattern depends essentially on the ratio of the wave number of sound to the absorption coefficient of penetrating radiation in a liquid. In the case of a homogeneous half-space, this dependence was investigated in detail. Reasoning and conclusions are practically the same in the considered case with a layer. To make it simpler, let us take µ = µ1 and αA/Cp = α1A2/Cp2. Let us consider the modulus of the ratio p1(r0)/p0(r0), which we denote by the letter q. Taking into account expressions (4.7), (4.10), and (4.11), we obtain

q=

µ 2 + k12 cos 2 θ p1 (r0 ) 1 . =2 2 2 + β + − β | 1 ( 1 ) exp(2iγH ) | p0 (r0 ) µ +γ

(4.12)

At the beginning let c = c1 and ρ = ρ1, i.e., liquids in the layer and halfspace differ only in density. In this case an expression for q takes on an especially simple form q=

2 | 1 + ρ / ρ1 + (1 − ρ / ρ1 ) exp(2ikH cosθ ) |

.

(4.13)

Analyzing expression (4.13), one can see that q ≥ 1 at any value of θ. The maximum of q is equal to the ratio ρ/ρ1 and attained when kH cos θ = (n + 1/2)π, where n is a certain natural number. The minimum of q is equal to one and attained at kH cosθ = nπ. Thus, selecting a corresponding layer thickness (or the frequency of generated sound) it is possible to obtain an increase of sound pressure in a

84 THERMORADIATION EXCITATION: INHOMOGENEOUS MEDIUM

set direction by the factor ρ/ρ1 as against the pressure generated in a homogeneous liquid half-space in the absence of a layer of another liquid. In the case ρ < ρ1 and c < c1, the dependence of q on the observation angle θ is much more complex. Let us consider alteration of q depending on parameters of liquids only for θ = 0. In this case expression (4.12) takes on the form,

q=2

µ 2 + k12 µ

2

1

+ k 2 | 1 + ρc / ρ1c1 + (1 − ρc / ρ1c1 ) exp(2ikH ) |

.

As ρc/ρc1 > 1, the maximum value of q is attained at kH = (n + 1/2)π and is equal to 2 2 ρ c µ + k1 . q= 1 1 ρc µ 2 + k 2

It is necessary to note that the value of q may be larger than one in this case. Everything discussed is illustrated by the directivity patterns calculated according to approximate formula (4.10) and given in Fig. 4.2. Calculations were conducted for the Gaussian distribution of the intensity of penetrating radiation in a beam (in a laser beam) as above. The observation angle was counted off the vertical axis where the level of sound pressure p1(r0) normalized to the value of

ωmαA 2 I 0 µk a 2 Cp 2r0 µ + k 2 is displayed. The next relationships between the problem parameters are taken: ρ/ρ1 = 0.79, µ/k1 = 1, and ka = 2. Directivity patterns in Fig. 4.2a correspond to the value k/k1 = 1 and in Fig. 4.2b k/k1 = 1.27. These directivity patterns were calculated for the values kH = 2.5π and kH = 3π. The directivity pattern of sound radiation generated by a radiation-acoustic source in a homogeneous half-space without a layer is given for comparison. We have to note that the values k/k1 = 1.27 and ρ/ρ1 = 0.79 correspond roughly to the case when a layer of alcohol is present on the water surface. One more particular case is interesting. Let ρ = ρ1 and k2 = k12 + ib2, while b/k1 << cosθ. In other words, a liquid in the layer differs from a liquid in the half-space by the presence of a small attenuation of sound. This sound

RADIATION ACOUSTICS

85

attenuation may be caused by a small amount of air bubbles in the surface layer for example. Restricting ourselves to the terms of the order of smallness not higher than the second order of smallness with respect to b/k1, let us write down expression (4.10) in an approximate form,   i (b / k1 ) 2 b  p1 (r0 ) = p 0 (r0 )1 − + i 2 2  ( µ / k1 ) + cos θ  2k1 cos θ  1 − exp( 2ik1H cos θ ) − k1H cos θ  

b    k1 

2 

   exp  − k1H  2   

2

  × 

 b   k1 cos θ

  

2

.  

One can see from the analysis of this expression that if the values of k1H are large enough, the amplitude of the field generated by a radiation-acoustic source in this case can be essentially smaller than in the case of a homogeneous half-space. Moreover, the presence of an absorbing layer can make the directivity pattern of generated sound field essentially narrower.

Figure 4.2 Directivity patterns of a thermooptical source of sound in a two-layer medium. The patterns are calculated according to approximate formula (4.10). (1) kH = 2.5π; (2) kH = 3π; and (3) a homogeneous half-space without a layer.

86 THERMORADIATION EXCITATION: INHOMOGENEOUS MEDIUM

EXTENSION TO THE CASE OF UNEVEN FREE BOUNDARY. The above results may be extended to the case when the free boundary of a layer is uneven. Let us consider for example the case when the height of unevenness is small in comparison with the wavelength of generated sound. A problem of sound generation by penetrating radiation in a liquid halfspace with uneven boundary in the case of small (in comparison with the sound wavelength) height of unevenness is considered in detail by Lyamshev and Sedov [144] (see Chapter 3, Section 5). Using the results of this study and calculation technique described above, we can obtain pav(r0) for the average field in the case of uneven boundary. Let us assume for simplicity that almost the total energy of penetrating radiation is absorbed in a layer, where thermal (thermoradiation) sources are located. In this case the auxiliary solution is determined by the expression, p * (r , r0 ) =

exp(ik1r0 ) w exp(−ik1 x sin θ )[exp(−iγz ) + v exp(iγz )] , 4πr0

where w=

2 β exp[i (γ − k1 cosθ ) H ] , 1 − β − v(1 − β ) exp(2iγH )

and v is the average coefficient of reflection of a plane sound wave from an uneven surface. After calculation analogous to calculation described in detail in Chapter 3 and by Lyamshev and Sedov [144], we obtain an expression

ρ ωmαA exp(ik1r0 ) µγ pav (r0 ) = 1 wF (θ ) f (θ ) . 2πr0 ρ Cp µ2 +γ 2

(4.14)

The function f(θ) in expression (4.14) characterizes the effect of the free boundary of the layer on the sound field generated by a thermoradiation source. In Chapter 3, Section 5 we obtained a rather general relationship connecting the function f(θ) with various parameters of boundary unevenness and analyzed some limiting cases. An expression determining the average coefficient of reflection of a plane sound wave from an uneven boundary was also given there. Here we consider only one particular case. Let the correlation coefficient of boundary unevenness ρ0 be greater than the wavelength of generated sound. Furthermore, let us restrict ourselves to

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87

the case of not too flat observation angles satisfying the condition kρ0 cos2θ >> 1. Then, as is demonstrated by Lyamshev and Sedov [144], f(θ) = 1 – 3P2/8 – (µσ)2, where P = 2kσ cosθ is the Rayleigh parameter and σ is the mean-square-root height of unevenness. In this case the relationship v = − 1 + P2/2 is valid for the average reflection coefficient. Substituting these values of f(θ) and v into expression (4.14) and keeping the terms of the order of smallness not higher than the second order with respect to kσ and µσ, we obtain an expression,  3  p2 (1 − β ) exp(2iγH ) pav (r0 ) = p1 (r0 )1 − p 2 − ( µσ ) 2 +  , (4.15) 2 1 + β + (1 − β ) exp(2iγH )   8  (expression (4.10) is used here for p1(r0)). Expression (4.15) is obtained under the assumption p2 (1 − β ) exp(2iγH ) << 1 . 2 1 + β + (1 − β ) exp(2iγH ) Analyzing expression (4.15), one can see that naturally, the average field in the case of uneven boundary is rather smaller than the field under an even boundary. The degree of field decrease is different in different directions, and this difference depends on the layer thickness. The maximum attenuation under the source (θ = 0) takes place at kH = (n + 1/2)π, where n is a certain number of natural series. It is determined by the factor p av (r0 ) p2  1 1   +  − ( µσ ) 2 . = 1− 4  2 β  p1 (r0 ) The minimum attenuation occurs at kH = nπ and it is determined by the factor p av (r0 ) p2  1  2 = 1−  + β  − ( µσ ) . p1 (r0 ) 4 2  Thus, in some cases the presence of a layer of another liquid at the surface of a liquid half-space may influence essentially the characteristics of the sound field generated by intensity-modulated penetrating radiation in the half-space.

88 THERMORADIATION EXCITATION: INHOMOGENEOUS MEDIUM

2. GENERATION OF SOUND IN A LIQUID ADJOINING A SOLID LAYER We have considered already sound generation by penetrating radiation in a liquid half-space in the presence of a layer of another liquid on its surface due to absorption of intensity-modulated penetrating radiation in such a two-layer medium. Situations may arise in practice where a liquid adjoins a solid layer. In this connection it is expedient to clarify the specific features of thermoradiation generation of sound in a liquid half-space adjoining a thin elastic plate for example [134]. We will consider two limiting cases: the case when a plate is transparent to radiation and penetrating radiation is absorbed in a liquid and the case when absorption of penetrating radiation occurs in the surface layer of a plate and sound field in a liquid is caused by plate vibrations under the effect of penetrating radiation. As it turns out in both cases, plate vibrations affect the characteristics of sound field in a liquid mainly in the directions close to the so-called critical angles determined by the ratio of the sound velocity in a liquid to the velocity of flexural or longitudinal oscillations in a plate. Let a thin elastic plate be present at the boundary of a liquid half-space, and let an intensity-modulated beam of penetrating radiation be incident upon the plate. The radiation propagates in the positive direction of the axis z of a rectangular system of coordinates (x, y, z). Let us investigate the specific features of the sound field generated in the liquid half-space by a thermoradiation source. First, we assume that the plate is radiation transparent, i.e., penetrating radiation is almost not absorbed in the plate, 2h << µ−1 << µ1−1, where 2h is the plate thickness and µ and µ1 are the coefficients of radiation absorption in the liquid and the plate. We can write down an expression for sound pressure,  iαAmω  r p(r1 ) = ∫∫  I ( x, y ) exp(− µz ) p *   dΩ(r ) .  Cp   r1  Ω

(4.16)

Here α is the coefficient of thermal expansion; Cp is the specific heat capacity of liquid; c is the sound velocity in a liquid; A is the transmission coefficient of penetrating radiation for the liquid limited by a plate; and p*(r/r1) is the solution of the diffraction problem on the field of a point source of sound in a liquid half-space limited with a thin elastic plate, when the source is located at the point of the half-space, where it is necessary to determine the acoustic field generated by a thermoradiation source.

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89

We need to determine an acoustic field produced due to absorption of penetrating radiation in a liquid in the far wave zone. In this case the auxiliary solution can be written down in the approximate form, p * (r / r1 ) =

1 exp(ikR1 ) {exp[i (k x x + k y y − k z z )] + R1 4π (4.17)

+V (k x , k y , k z ) exp[i (k x x + k y y + k z z )]} ,

where R1 = (x12 + y12 + z12)1/2, V(kx, ky, kz) is the reflection coefficient of a plane sound wave in the case of a thin elastic plate positioned at the boundary of a liquid half-space with vacuum, and kx, ky, and kz are the components of the wave vector k of a sound wave in a liquid. Two types of vibrations are possible in a thin plate: vibrations symmetrical with respect to its median plane, or so-called transverse pressure vibrations connected with longitudinal vibrations, and antisymmetrical vibrations, i.e., flexural vibrations. Using the equations of plate motion and solving a problem on reflection of a plane sound wave from a plate positioned at the boundary of a liquid half-space with vacuum, we obtain an expression for the reflection coefficient V(kx, ky, kz) (for example, see [135]), V (k x , k y , k z ) =

2 ZZ1 cos θ − 2 ρcZ1 − ρcZ . 2 ZZ1 cos θ + 2 ρcZ1 + ρcZ

(4.18)

Here Z and Z1 are the plate impedances for flexural vibrations and transverse pressure vibrations: 4  c   f    Z = −iωm1 1 −  cosθ  , cosθ =   c   

Z1 =

k 2 − (k x2 + k y2 ) k2

2  c 1 −  1 sin θ   , 2 c    c    ωh 1 − ν 12 −  1 sin θ      c   

iE1

,

(4.19)

(4.20)

90 THERMORADIATION EXCITATION: INHOMOGENEOUS MEDIUM

where m1 = 2hρ1 is the plate mass per unit area, ρ1 is the material density, ν1 is the Poisson’s ratio of the plate, c1 = (E1/ρ1)1/2 is the velocity of longitudinal waves, E1 is the elasticity modulus, cf = [2ω2E1h3/3m1]1/4 is the velocity of flexural waves in the plate, θ is the angle between the direction to the observation point r1(x1, y1, z1) and the coordinate axis z, ρ is the liquid density, and c is the sound velocity in the liquid. Substituting expression (4.17) into expression (4.16) and assuming that the distribution of intensity of penetrating radiation in the beam obeys the Gaussian law I(x, y) = I0 exp[−(x2 + y2)/a2] , we obtain after integration

p(r1 ) = i

 k 2 a 2 sin 2 θ αAmωµ exp(ikR1 ) I 0 a 2 exp −  4C p 4 R1 

 ×   (4.21)

  1 1  ik cos θ − µ − V (θ ) ik cos θ + µ  .   Expression (4.21) is basic and its analysis allows us to determine the effect of the plate on the field of a radiation-acoustic source in a liquid. Let us consider some particular cases. Let | z | → ∞. This means that transverse pressure vibrations of the plate are “forbidden”. In this case we obtain an expression for the reflection coefficient, Z cos θ − ρc . V z →∞ = 1 Z cosθ + ρc

(4.22)

Further, let Z cos θ << ρc .

(4.23)

One can see from expression (4.19) that this condition is valid when the plate thickness is very small. Now V Z → ∞, Z cos θ << ρc ≈ −1 . 1

(4.24)

Taking into account the assumptions made for sound pressure in a liquid, we obtain

RADIATION ACOUSTICS

p (r1 ) =

91

exp(ikR1 ) ωmαA µk cos θ × I0a 2 2 2C p R1 k cos 2 θ + µ 2

(4.25)  k 2 a 2 sin 2 θ exp −  4 

 .  

Naturally, expression (4.25) characterizes the field of a radiation-acoustic source in a liquid with a free boundary. Now let the plate thickness satisfy the condition Z cos θ >> ρc .

(4.26)

We have to note that the last condition may be valid also if the velocity of flexural waves cf is small in comparison with the sound velocity in a liquid. We obtain an expression for the reflection coefficient, V Z → ∞, Z cos θ >> ρc ≈ 1 , 1

(4.27)

and the formula for sound pressure in a liquid takes on the form p(r1 ) ≈ −

exp(ikR1 ) iωmαA µ2 I0a 2 × 2C p R1 k 2 cos 2 θ + µ 2 (4.28)  k a sin θ  . exp −   4   2 2

2

Expression (4.28) characterizes the far field of a radiation-acoustic source acting in a liquid half-space with a rigid, motionless but radiationtransparent boundary. One can see from expressions (4.25) and (4.28) that in the first case if k < µ and ka < 1, the dipole radiation of sound takes place and in the second case there is the monopole radiation. The condition k ≈ µ is connected usually with the optimal regime of thermooptical radiation of sound in a liquid. If this condition is satisfied, the amplitude of sound pressure in the case of a liquid with a free boundary, as a matter of fact, differs from the amplitude in the case of a rigidly fixed

92 THERMORADIATION EXCITATION: INHOMOGENEOUS MEDIUM

boundary. However, it is interesting to note that in the case k = µ, the amplitude of sound pressure in the direction of incidence of a beam of penetrating radiation (θ = 0°) does not depend on the character of the boundary. In the considered case of a heavy thin plate, the reflection coefficient V is always equal to one in its absolute value and changes only in its phase within the range from 0 to π depending on the parameters of the plate and liquid. Analyzing expression (4.18), one can see that the reflection coefficient takes on the value V = −1 if the conditions Z = 0 or Z1 = 0 are satisfied. These conditions correspond to such critical angles of observation when θf = arcsin(c/cf) or θ1 = arcsin(c/c1). Considerable changes including “gaps” are observed in the directivity pattern of a radiation-acoustic source in the directions close to critical angles. It is easy to explain them. The point is that the conditions Z = 0 and Z1 = 0 or (cf/c) sinθ = 1 and (c1/c) sinθ = 1 that is equivalent (see expressions (4.19) and (4.20)) are the conditions of the socalled spatial resonance, when a plate becomes almost sound-transparent [135]. If we imagine that thermal sources of sound in the area of absorption of penetrating radiation produce a field of plane sound waves propagating under various angles, then the plate behaves (with respect to plane waves propagating under critical angles) as almost totally sound-transparent and does not influence the amplitudes and phases of these waves. In other words, interaction of these waves with the boundary of a liquid half-space occurs as if this boundary were free. What is said above is illustrated by Fig. 4.3a–d where angular dependences of the level of sound pressure in a liquid half-space with a free boundary (curve 1) and limited by a thin plate (curve 2) are shown. Calculations were performed for water and a glass plate. The plate thickness is 1 mm, the modulation frequency of radiation intensity of a photon beam (laser 6 radiation) is 10 Hz, the beam radius is 1 mm, and the laser power is 1 W. The values of critical angles are θf = arcsin(c/cf) ≈ 40° and θ1 = arcsin(c/c1) ≈ 31°. The results given in Fig. 4.3a correspond to the case k = µ = 42 cm−1 and the case 420 cm−1 = µ > k = 42 cm−1 is given in Fig. 4.3b. One can see that the “gaps” in angular patterns not only attain the value of sound pressure in the case of a free boundary, but are also much deeper. Figures 4.3c and d illustrate the cases µ << k, namely 4.2 cm−1 = µ < k = 42 cm−1 (Fig. 4.3c) and 0.42 cm−1 = µ < k = 42 cm−1 (Fig. 4.3d). One can see that at µ << k, the maxima and minima of angular patterns are observed in the vicinity of critical angles. The maxima are observed when the field amplitude is equal to its value in the case of the free boundary. Such a pattern is explained by the interference structure of the field of vibrations in a plate and sound waves in a liquid. Now let us consider the case when absorption of penetrating radiation occurs almost entirely in a thin surface layer of a plate, i.e., when µ -1 << h.

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93

Figure 4.3 Dependence of sound pressure in a liquid on observation angle. (1) Liquid boundary is free; (2) a glass plate is present on the liquid surface.

Now sound field in a liquid is produced only by plate vibrations caused by the effect of external forces. The role of these forces is played by a thermoelastic stress arising in a surface layer of a plate when intensitymodulated penetrating radiation is absorbed there. The acoustic field of a vibrating plate can be calculated using the reciprocity theorem in the formulation by the author [131] (see Chapter 3, Section 2), p(r1 ) =

∫ S

∂p * (r / r1 ) F (r )dS (r ) . ∂n

(4.29)

Here, as in expression (4.16), p*(r/r1) is the solution of a diffraction problem on the field of a point source of sound in a liquid half-space limited by a plate, when the source is located at the point r1, where it is necessary to determine sound pressure in the acoustic field emitted by a plate vibrating under the action of penetrating radiation; and n is the normal to the plate, which is directed along the positive direction of the coordinate axis z.

94 THERMORADIATION EXCITATION: INHOMOGENEOUS MEDIUM

We assume that the density of absorbed energy of penetrating radiation is essentially smaller than the specific heat capacity of evaporation of plate material. Let us consider the thermal mechanism of sound generation. We can write down an expression for thermoelastic stress playing the role of external forces2, F (r, t ) = − Eα1T1 (r, t ) ,

(4.30)

where E is the elasticity modulus of plate material, α1 is the coefficient of thermal expansion of plate material, and T(r,t) is the function characterizing distribution of temperature and its change with time. If the conditions of interaction of penetrating radiation with the plate provide an opportunity to ignore heat conductivity and losses for heat emission from the plate surface, and the penetration depth of radiation is very small while the distribution of radiation intensity in a beam has the Gaussian character, we can write down an expression for T(r, t) (for example, see [40]),  x2 + y2  µm  exp( −iωt ) , T (r, t ) = 1 I 0 exp −  C p1 a 2  

(4.31)

where Cp1 is the specific heat capacity of plate material. We need to determine the sound field in a liquid in the far wave zone. We obtain the next expression for the normal derivative of pressure at the plate surface in the field of a point source located at the spatial point where it is necessary to determine the acoustic field of a plate vibrating under the action of penetrating radiation: exp(ikR1 ) ∂p * (r / r1 ) . = −ik z (1 − V ) exp[i (k x x + k y y )] 4πR1 ∂n

(4.32)

Substituting expressions (4.30) – (4.32) into expression (4.29) and integrating, we obtain p (r1 ) =

2

exp(ikR1 ) iωa1mµ1 cos θ × I 0a 2 2C p c R1

Expression (4.30) is a particular case of the relationship of thermal elasticity (for example, see [166]).

RADIATION ACOUSTICS

×

95

 k 2 a 2 sin 2 θ ρcZ + 2 ρcZ1 exp −  2ZZ1 cos θ + ρcZ + 2 ρcZ1 4 

 .  

(4.33)

A plate can influence considerably the parameters of the acoustic field. Let us consider several particular cases. Let |z1| → ∞ as above, i.e., a plate cannot perform transverse pressure vibrations. In this case we have p (r1 )| Z |→ ∞ =

exp(ikR1 ) iωα1 mµ1 cos θ × I 0a 2 2C p1 c R1

(4.34)  k 2 a 2 sin 2 θ ρc exp −  Z cos θ + ρc 4 

 .  

If Z = 0, pressure in the acoustic field of the plate attains its maximum value and naturally, dipole radiation takes place at ka << 1 since the field in the liquid is produced due to the effect of an external periodic source created by an intensity-modulated beam of penetrating radiation incident upon the plate surface. The condition Z = 0 is satisfied for a so-called critical angle for the plate when the velocity of flexural waves in the plate is larger than the sound velocity in the ambient medium and (cf/c) sinθ = 1. Thus, the maximum of plate radiation is observed in the direction of the critical angle θf = arcsin (c/cf). It is easy to see that the radiation maximum is observed in the direction of another critical angle also. These considerations are illustrated by results of calculation given in Fig. 4.4. Levels of sound pressure in a liquid (water) with the free surface (curve 1) and limited from the top by a thin steel plate (curve 2) are given here. Calculation is performed for the next parameters: the plate thickness 2h = 1 mm, the beam radius a = 1 mm, the modulation frequency of laser radiation ~ 106 Hz, and the laser power 1 W. Thus, the presence of a thin elastic layer (elastic plate) at the surface of a liquid half-space can affect essentially the parameters of the acoustic field excited by intensity-modulated penetrating radiation in a liquid. In the case of a radiation-transparent layer and the absorption coefficient of radiation in a liquid µ considerably larger than the wave number of sound k, the presence of a layer leads to the increase of the amplitude of the field of a radiation-acoustic source in a liquid as compared to the case when the liquid boundary is free, and only in the direction of so-called critical angles the amplitude of the acoustic field decreases essentially. On the contrary, if penetrating radiation is weakly absorbed in a liquid, i.e., µ << k, and a plate

96 THERMORADIATION EXCITATION: INHOMOGENEOUS MEDIUM

is radiation-transparent or penetrating radiation is completely absorbed in it close to its external boundary, then the maximum values of the amplitude of acoustic field are observed in the directions close to critical angles.

Figure 4.4 Dependence of sound pressure in a liquid on observation angle.

These conclusions can be extended to the case of an elastic layer of finite thickness or a system of layers. Basic formulae stay valid and it is necessary only to substitute corresponding expressions for the reflection coefficients V in their final form (see [40]). It is possible to demonstrate that in the case of a layer of finite thickness, the number of critical angles is determined by the number of normal (symmetrical and antisymmetrical) modes in the layer.

3. LIQUID HALF-SPACE WITH AN INHOMOGENEOUS SURFACE LAYER Thermoradiation generation of sound by intensity-modulated penetrating radiation in the process of its absorption in a liquid half-space with a layer of a liquid with different parameters at the half-space boundary was considered in Section 1. However, parameters of a liquid in the surface layer can change not stepwise but continuously due to nonuniform heating of a liquid by penetrating radiation, in the process of an experiment for example. In this connection it is interesting to analyze the effect of alteration of temperature of a liquid in the surface layer with depth on thermoradiation generation of sound.

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97

In this section we consider specific features of sound generation by intensity-modulated penetrating radiation in a liquid half-space with a surface layer where the coefficient of cubical thermal expansion, specific heat capacity, and sound velocity change with depth according to a linear law. It will be demonstrated that the amplitude of sound field and directivity of acoustic radiation may differ significantly from those in a homogeneous liquid half-space if the ratio of the coefficient of cubical expansion to the specific heat capacity of a liquid changes essentially at the characteristic dimensions of the region of interaction of penetrating radiation with a liquid. Alteration of sound velocity with depth in a surface layer affects only propagation of sound waves [145]. Let a laser beam or a beam of relativistic electrons, γ-quanta, or X-rays be incident upon the even free boundary of a liquid half-space. We select a system of coordinates in such way that the plane x, y coincides with the free surface of the half-space, the axis z is directed inside the liquid, and the origin of coordinates is located in the center of the radiation (light) spot at the liquid surface (see Fig. 4.5a).

Figure 4.5 Generation of sound in an inhomogeneous liquid (a); patterns of alteration of sound velocity (b); and ratio of coefficients α/Cp (c).

Let us assume that the temperature of the liquid directly at the surface is t1 and it changes continuously with depth to the value t2 at the depth h. The liquid temperature is t2 at z > h. It is necessary to determine the sound field in the inhomogeneous part of the half-space, which is generated due to absorption of intensity-modulated penetrating radiation in such medium. Alteration of liquid temperature with depth leads to a change of the ratio α/Cp (where α and Cp are the coefficient of cubical thermal expansion and specific heat capacity of a liquid, respectively) and the sound velocity in the liquid c. Let us assume for definiteness that the liquid temperature changes with depth according to a linear law. If we restrict ourselves to consideration of not very large values of temperature (for example, for water t1 = 25°C and

98 THERMORADIATION EXCITATION: INHOMOGENEOUS MEDIUM

t2 = 10°C), the law of alteration of the ratio α/Cp with depth is also approximately linear,

α ( z ) / C p ( z ) = α 0 / C p0 − ∆(α / C p ) z , 0 < z < h , where α0 and Cp are the corresponding parameters of the liquid near the surface and ∆(α/Cp) is the alteration of the ratio α/Cp per unit depth. We take the law of alteration of sound velocity with depth in the form c( z ) = c0 / 1 + bz , 0 < z < h , bh < 1 , where c0 is the sound velocity near the surface. Since |bh| << 1 in all cases of interest from the practical point of view, this law of alteration of sound velocity with depth is very close to a linear one. We also restrict ourselves to the case when almost the total energy of penetrating radiation is absorbed in the surface layer exp(−µh) << 1, where µ is the absorption coefficient of penetrating radiation in the surface layer of the liquid. The field of sound pressure p1 in the layer is described by the solution of the equation, ∆p1 + k 02 (1 + bz ) p1 = iωm[α 0 / C p0 − ∆(α / C p ) z ] × AµI ( x, y ) exp( − µz ) ,

(4.35)

where ω is the frequency of generated sound (the modulation frequency of intensity of penetrating radiation), k = ω /c0, m is the modulation index, A is the coefficient of penetrating radiation transmission through the liquid surface, and I(x, y) is the function describing the distribution of radiation intensity at the liquid surface. The sound pressure p in the inhomogeneous part of the half-space obeys the equation, ∆p + k12 p = 0 , z > h ,

(4.36)

where k1 = k0(1 + bh)1/2. Solutions of equations (4.35) and (4.36) must be consistent with the boundary conditions p1 = 0, z = 0, p = p1, and ∂p/∂z = ∂p1/∂z, z = h as well as the condition of suppression at infinity. The desired

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99

solution of equation (4.36) can be written down immediately based on the reciprocity principle: p(r0 ) = iωmAµ ∫ [α 0 / C p0 − ∆ (α / C p ) z ] exp(− µz ) × Ω

(4.37) I ( x, y ) p * (r, r0 )dΩ(r ) , where r0(x0, y0, z0) is the radius-vector of the observation point and r(x, y, z) is the current radius-vector. Integration in expression (4.37) is conducted over the region Ω of the inhomogeneous layer where thermal sources of sound exist, and the function p*(r,r0) describes the field in the layer which is produced by a point source of unit amplitude and located in the observation point r0. We need to determine the sound field p(r0) at distances exceeding essentially both the dimensions of the region occupied by thermal sources of sound and the thickness of the inhomogeneous layer. We assume that the observation point is located not too close to the boundary of the layer with the inhomogeneous part of the half-space. In this case the auxiliary solution within the limits of the region of action of thermal sources of sound can be represented as the field of a plane sound wave incident in the direction of the vector – r0/r and spherical divergence may be taken into account with the help of the factor exp(ikir0)/4πr0. Taking into account everything said before, we can obtain the next expression for the auxiliary field in the layer: p * (r, r0 ) =

2Qq[u (τ 0 )ν (τ ) − ν 0 (τ 0 )u (τ )] exp(−ik1 x sin θ ) , u (τ 0 )[ν ' (τ 1 ) + qν (τ 1 )] − ν (τ 0 )[u ' (τ 1 ) + qu (τ 1 )]

(4.38)

where Q = exp(ikr0 − ik1h cos θ ) / 4πr0 , q = ik1 H cos θ , H = (bk 02 ) −1 / 3 ,

τ 0 = H 2 ( k12 sin 2 θ − k02 ) , τ1 = τ 0 − h / H , τ = τ 0 − z / H , u, ν, u′, and ν′ are the Airy functions and their first derivatives and θ is the angle between the axis z and the radius-vector r0.

100 THERMORADIATION EXCITATION: INHOMOGENEOUS MEDIUM

Substituting expression (4.38) into expression (4.37), we obtain an expression for the sound pressure p(r0) in the observation point r0, p(r0 ) =

2iωmAQqE (θ ) × u (τ 0 )[ν ' (τ 1 ) + qν (τ 1 )] − ν (τ 0 )[u ' (τ 1 ) + qu (τ 1 )]

(4.39)  α    0 S − ∆ (α / C p ) S  − ν (τ )  α 0 S − ∆(α / C p ) S   . u(τ 0 )  1 3 0 2 4  C C µ µ   p0   p0   Here F (θ ) = ∫

∞

∞

∫ I ( x, y ) exp(ik1x sin θ )dxdy,

−∞

∞ ζ − S1 = ν (τ 0 − ζ / µH )e dζ , S 2 = u(τ 0 − ζ / µH )e −ζ dζ , 0 0

∫

∫

∞

∞

0

0

S3 = ∫ ζν (τ 0 − ζ / µH )e −ζ dζ , S4 = ∫ ζu(τ 0 − ζ / µH )e −ζ dζ .

It is very difficult to analyze expression (4.39) in a general form. At the beginning let us consider a simplified version of this expression. We have to note that, for example, in the case of water, the ratio α/Cp changes essentially with temperature. Thus, α/Cp = 6.6⋅10−12 g/erg at 25°C but α/Cp = 2.0⋅10−12 g/erg at 10°C. The change of sound velocity is only 3% in this case. Therefore, let us consider first the effect of the change of the ratio α/Cp on the sound field assuming the sound velocity constant within the whole half-space. A corresponding expression for the sound field we obtain directly from expression (4.37) by substitution of the auxiliary solution in the form of the sum of incident plane waves and plane waves reflected by the free surface into it, and taking into account the factor exp(ik1r0)/4πr0, p(r0 ) = ωmA

exp(ik1r0 ) α0 µk1 cosθ × F (θ ) 2 2 2 2πr0 µ + k cos θ C p0 1

RADIATION ACOUSTICS

  ∆(α / C p ) µ 1 − 2 . α 0 / C p µ 2 + k 2 cos2 θ   1  

101

(4.40)

One can see from expression (4.40) that the maximum sound amplitude in a certain direction is observed at µ = k1cosθ, i.e., as in the case of thermoradiation generation of sound in a homogeneous half-space. It is natural to compare the sound field described by expression (4.40) with the acoustic field generated in a homogeneous half-space p0(r0). An expression for p0(r0) is well known, and the ratio p(r0)/p0(r0) can be written down in the form,  ∆(α / C p ) p(r0 ) α C p1  µ , 1− 2 = 0 p0 (r0 ) C p 0 α1  α 0 / C p 0 µ 2 + k 2 cos 2 θ  1  

(4.41)

where α1 and Cp1 are the values of the corresponding liquid parameters in the homogeneous part of the half-space. If the value of α/Cp decreases with depth α0/Cp0 > α1/Cp1, then ratio (4.41) is larger than one and naturally, as follows from general physical concepts, the presence of an inhomogeneous layer in this case leads to an increase of the amplitude of the sound field generated by modulated penetrating radiation. Analyzing expression (4.41) we can note the following. At certain values of µ, k1, and ω the increase of amplitude of sound field is a maximum at θ = 0, i.e., in the propagation direction of the beam of penetrating radiation, and the amplitude decreases as the observation angle θ grows. Thus, the presence of an inhomogeneous layer changes the shape of the directivity pattern of the sound field to some extent. However, if the value of the ratio α/Cp changes little at the characteristic size of the region of action of sound sources 1/µ, the square bracket in expression (4.41) differs little from one and sound generation occurs in the same way as in a homogeneous liquid half-space with the parameters α0 and Cp0. The last is evident from physical considerations. Let us turn again to analysis of expression (4.39), which must be performed numerically. In this case we restrict ourselves to consideration of not too “gliding” observation angles. Let us note first that as calculation shows, it is possible to use the relation µ ≈ k1 in order to determine the optimum generation mode to good precision. The validity of this relation was demonstrated for a homogeneous liquid half-space. Let us estimate the effect of change of sound velocity on the amplitude of generated acoustic field for the values µ ≈ k1, i.e., for generation modes close to the optimal one. It turns out that if the sound velocity decreases

102 THERMORADIATION EXCITATION: INHOMOGENEOUS MEDIUM

with depth, the field amplitude in these directions increases to some extent, the maximum increase being attained in the direction of the beam of penetrating radiation, i.e., at θ = 0. However, the relative increase of amplitude is about several percent. If the sound velocity increases with depth, the amplitude of acoustic field in the indicated directions decreases, being attenuated most significantly at θ = 0, while the relative value of this decrease is also about (c1 – c0)/c0. These effects are almost independent of the layer thickness and determined only by the difference of sound velocities at the layer boundaries, which mainly follows from general physical concepts connected with the particular features of sound propagation in a homogeneous medium. Finally, if the sound velocity decreases with depth, then a certain “critical” angle of observation exists. In the case of observation angles larger than this “critical” one, the sound field is practically absent. The critical angle is determined by the condition τ0 = 0 or sinθ = c1/c0. Directivity patterns calculated according to expressions (4.39) and (4.40) are given in Fig. 4.6 in order to illustrate this. Directivity patterns have been calculated for the case of laser excitation of sound and the Gaussian distribution of light intensity in a laser beam.

Figure 4.6 Directivity pattern of a thermooptical sound source in a liquid half-space. (1) A homogeneous half-space with the parameters of liquid α1, Cp1, and c1; (2) the ratio α/Cp changes in a surface layer of liquid; (3) the ratio α/Cp and the sound velocity change in the surface layer.

The next parameters have been taken for calculation: µ = 3 cm−1, α/Cp0 = 6.6⋅10−12 g/erg, α1/Cp1 = 2⋅10−12 g/erg, k0 = 3 cm−1, k1 = 3.08 cm−1, h = 30 cm and the radius of the light spot at the liquid surface a = 0.4 cm. The

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103

observation angle is counted off from the vertical axis, where the absolute value of sound pressure p(r0) normalized to the value (ωmAα0/2πr0Cp0)P is indicated, where P is the power of laser radiation. Thus, the change of the ratio α/Cp0 with depth in the surface layer of a liquid affects essentially the amplitude and shape of the directivity pattern of the sound field generated due to absorption of modulated penetrating radiation in a liquid if this ratio changes noticeably within the characteristic dimensions of the region of sound generation. The change of sound velocity with depth affects the amplitude of generated sound field weakly for all observation angles except for the angles larger than the critical one, when the field is almost nonexistent. The latter follows also from general physical considerations. Indeed, the amplitude of sound pressure is directly proportional to the Grueneisen parameter of a liquid Γ = αc2/Cp. If c = c0(1 + ε), where ε << 1 and c0ε is a small addition to sound velocity, then Γ = αc0(1 + 2ε)/Cp, i.e., the change of the amplitude of sound field is proportional to ε << 1.

CHAPTER 5

Excitation of Sound in a Liquid by Radiation Pulses Excitation of sound by radiation pulses is especially interesting since large radiation power (and therefore, high efficiency of radiation-acoustic (optoacoustic) conversion) and acoustic pulses of large amplitude may be produced in the process of operation of accelerators and lasers in a pulsed mode. Many theoretical and experimental papers have been devoted to the study of the process of thermooptical generation of sound by laser pulses. First theoretical papers (see for example [200, 215]) considered onedimensional problems, and it is not surprising that in many cases the sound pulses observed experimentally differed in their shape from the ones predicted theoretically. For example, Hunter and Jones [220] tried to explain this discrepancy by the nonlinear effects taking place in the process of generation and propagation of sound pulses in liquids. In this connection, attempts were made to solve nonlinear equations of hydrodynamics numerically [204]. However, as detailed theoretical studies demonstrated [109, 110], in the case of small density of optical energy in a medium the linear theory describes the real process of sound generation by laser pulses correctly, and the data obtained theoretically agree well with the results of previous experiments including the data obtained by Hunter and Jones [220] and special experiments [75, 90, 91, 93]. Further we consider the laws of sound generation by radiation (laser) pulses in a liquid. The major attention will be given to the characteristics of sound pulses in the far wave field. We should note that we mean pulses of laser radiation, γ-quanta, X-rays, relativistic electrons, and synchrotron 105

106

EXCITATION OF SOUND IN A LIQUID BY RADIATION PULSES

radiation, i.e., the cases when radiation absorption in a medium is described by an exponential law. The conclusions are valid also for pulses of other kinds of penetrating radiation with some exceptions, which we consider in Chapter 7.

1. SOUND GENERATION IN A LIQUID BY RECTANGULAR PULSES OF RADIATION Let us consider the far sound field generated in a liquid half-space by rectangular pulses of radiation and analyze the dependence of the shape and duration of signals on the characteristic parameters in the case of “long” and “short” laser pulses [109]. Let a radiation pulse of the duration τ be incident from the air along the axis z upon the boundary of a liquid halfspace z > 0 (Fig. 5.1). The field of sound pressure in a liquid is described by an inhomogeneous wave equation,

∆p −

1 ∂2 p c

2

∂t

2

=

α ∂Q , C p ∂t

(5.1)

where the power density of thermal sources is determined by the expression, Q( x, y, z , t ) = AµI ( x, y )e − µz [Θ(t ) − Θ(t − τ )] ,

(5.2)

and Θ(t) is the Heaviside function. We perform the Fourier transformation with respect to time and obtain an inhomogeneous reduced wave equation, (∆ + k 2 ) F [ p] = −

α Cp

µI ( x, y ) exp(− µz )[1 − exp(iωτ )] ,

(5.3)

where F[p] is the spectrum of sound pressure, k = ω/c is the wave number, and ω is the circular frequency. As usual, we take the Gaussian distribution of penetrating radiation intensity over the cross-section of a particle or laser beam, I(x, y) = I0exp(−ρ2/a2), where ρ = (x2 + y2)1/2 and a is the effective radius of the beam. Under these assumptions and in the case of a free plane boundary, the solution of equation (5.3) is given by the formula,

RADIATION ACOUSTICS

 a 2 k 2 sin θ iI µα e ikR a 2 k cos θ exp − F[ p] = − 0  2C p R µ 2 + k 2 cos 2 θ 4 

107

 ×   (5.4)

(1 − e ), iωt

where R = (x2 + y2 + z2)1/2 and θ is the angle between the axis z and the direction to the observation point (x, y, z).

Figure 5.1 Correlation of problem parameters in the scheme of pulsed thermoradiation generation of sound in a liquid.

The sound pressure characterized by spectrum (5.4) is given by the expression,

p ( x, y , z , t ) = −

iI 0αµa 2 c 4πC p R cos θ

∞

∫

−∞

exp[(− a 2 k 2 sin 2 θ ) / 4]

µ 2 cos 2 θ + k 2

{exp[ik ( R − ct )] − exp[ik ( R − ct + cτ )]}kdk .

× (5.5)

After integration of expression (5.5) (see [21], p. 74 (26)), we obtain the next formula for the sound pressure generated by a radiation pulse of rectangular shape:

p ( x, y , z , t ) =

 s 2   R / c − t  I 0αa 2 Erfc s − R / c − t  − exp  exp − 2  4   8C p Rτ µ τ τ a  µ     

EXCITATION OF SOUND IN A LIQUID BY RADIATION PULSES

108

  R/c −t  Erfc s + R / c − t  − exp − R / c − t + τ − exp     τµ  τa  τµ 2     s R / c − t +τ Erfc − τa 2

 R / c − t +τ   + exp  τµ  

 ×  

  Erfc s + R / c − t + τ 2  τa  

(5.6)

  , 

where

τ cos θ a sin θ ,τ = , s = a = µa tan θ , τ a = µc c τµ and ∞

Erfc( x) = 2 π ∫ exp(−t 2 )dt x

is the complementary error function. It follows from formula (5.6) that the basic parameters determining the shape of a sound pulse in a liquid are the length of penetrating radiation pulse τ, the characteristic delay time of sound from elementary thermal sources located in the horizontal section of the region of sound generation τa, which is determined by the effective radius of the beam a, the characteristic delay time of sound from elementary thermal sources located in the vertical section of the region of sound generation τµ, which is determined by the efficient radiation length or the length of light absorption in a liquid 1/µ, and the ratio s = τa/τµ (Fig. 5.1). Relative levels of sound pulses calculated according to formula (5.6) are given in Fig. 5.2a and b. According to them we can judge the characteristic features of pulse shapes according to the relation between the problem parameters. If the length of laser pulse is large, i.e., τ >> τa, τµ, then the shape of a sound pulse is determined mainly by the value of τ (Fig. 5.2a) and depends weakly on the parameter s. If τ << τa (Fig. 5.2b), i.e., in the case of short pulses of penetrating radiation, the shape of sound pulses depends strongly on the characteristic times τa and τµ and their ratio s. The amplitude of sound pressure decreases as the parameter s grows. Let us analyze the dependence of the shape of a sound pulse on the parameter s = τa/τµ in more detail. Let us consider the limiting values of the parameter s.

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109

Figure 5.2 Relative levels of sound pressure in water in the cases of “large” (a) and “small” (b) length of radiation (laser) pulse τ for three observation angles. (a) τ = 5 µs; (b) τ = 0.05 µs; θ = 30°, 45°, and 60° (curves 1 – 3, respectively); γ* = (ct – −1 R)/l, l is the unit length; µ = 5 cm and a = 0.1 cm.

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EXCITATION OF SOUND IN A LIQUID BY RADIATION PULSES

Let s >> 1 that corresponds to the region of thermal sources in the form of a flat disk or to observation under large angles θ. In this case the expression under the integral in formula (5.5) decreases exponentially as frequency grows and may be ignored at k >> 2/(a sinθ ). Therefore, we can ignore the quantity k2 as against µ2/cosθ in the denominator of the integrand. Integrating (see [21], p. 73 (19)) we obtain an expression for sound pressure at s >> 1 in the form   R/c −t 2   p ( x, y , z , t ) = ( R − ct ) exp − 2  2 π C p Rs sin θ  τ a2   I 0αc

   −  (5.7)

 R / c − t + τ 2   . ( R − ct + cτ ) exp −  2    τa   Sound pulses determined by expression (5.7) are close to the ones given by curves 3 in Fig. 5.2a and b. The pulse shape is determined by the characteristic times τ and τa in this case. In the case of small τ satisfying the condition cτ τ = << 1 , τ a a sin θ expression (5.7) becomes even simpler and takes on the form

p ( x, y , z , t ) =

 R/c −t 2  exp − 2  τ a2 2 π C p Rs sin θ  I 0αc 2τ

 R/c−t 2  .  ⋅ 2 2 τ a 

(5.8)

The length of the pulse of negative excessive pressure (the distance between zero points in the curve) and the time interval between the peaks of positive excessive pressure (the distance between the symmetrical peaks in the curve) can be estimated readily for the pulses determined by expression (5.8) (Fig. 5.2b, curve 3). They are equal to 21/2τa and 61/2τa, respectively. Therefore, in the case of short radiation pulses and under the condition s >> 1 the shape and length of sound pulses are determined completely by the characteristic time τa. The pressure amplitude is inversely proportional to the parameter s.

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111

If s >> 1, which corresponds to the region of thermal sources in the shape of a narrow cylinder or to observation under small angles θ, then the decrease of the integration element in expression (5.5) with growth of frequency is determined by the denominator, and for k >> µ/cosθ the integration element becomes close to zero. Therefore, the exponent in the integration element can be changed to one. Integration gives an expression for sound pressure in the case s << 1,

p ( x, y , z , t ) =

I 0αa 2 2C p Rτ µ

  R / c − t exp − τµ  

 sign ( R − ct ) −   (5.9)

 R / c − t +τ exp −  τµ 

  sign( r − ct + cτ )  .   

Expression (5.9) can be obtained also directly if we proceed to the limit s → 0 in formula (5.6). In this limiting case, the length of negative pressure pulse is equal to the length of the laser pulse τ (Fig. 5.2b). In the case of small τ satisfying τ /τµ = µcτ /cosθ << 1, expression (5.9) takes on the form,

p ( x, y , z , t ) =

I 0αa 2 2C p Rτ µ

  R / c − t exp − τµ  

 sign ( R − ct ) −   (5.10)

 R/c−t exp −  τµ 

   sign( R − ct + cτ ) sign( R − ct + cτ ) + τ  .  τ µ   

The sound pulse determined by formula (5.10) is presented basically by a pulse of negative excessive pressure with the length τ (Fig. 5.3, curve 2). We have to note that expression (5.9) is exact for the direction along the axis z (the axis of the laser beam). Thus, on the basis of the conducted analysis of solution (5.6) and its limiting cases, we can separate the next characteristic features of the sound field generated in a liquid with an even boundary by laser pulses. If the length of a radiation pulse is large in comparison with the characteristic times τa and τµ, i.e., τ >> τa, τµ, the length of a sound pulse is determined by the length of a radiation pulse independently of the change of the observation angle θ and the ratio of the characteristic times s. In this

112

EXCITATION OF SOUND IN A LIQUID BY RADIATION PULSES

case the pressure amplitude decreases as the observation angle (i.e., the parameter s) grows.

Figure 5.3 Levels of sound pressure (relative) in water at s << 1. (1) τ = 13 µs, µ = 5 cm−1; (2) τ = 0.05 µs, µ = 1 cm−1.

In the opposite case of short pulses of penetrating radiation, if the condition τ << τa, τµ is satisfied, the length of sound pulse changes strongly with the change of observation angle depending on the value of s. In the case of small θ (s << 1), the length of sound pulse is determined by the value of τ, the sound pulse being presented basically by a pulse of negative excessive pressure, while in the case of large θ (s >> 1), it is determined by the characteristic time τa. In the latter case the amplitude of sound pressure decreases as the observation angle increases, and at large values of s, it is inversely proportional to this parameter.

Figure 5.4 Sound pulses detected experimentally by Hutcheson, Roth, and Barnes 3 −1 −1 [221] in the case of observation along the axis z. (1) µ = 10 cm , (2) µ = 5 cm , τ = 0.05 s.

It is interesting to note that sound pulses given in Fig. 5.3 coincide in their shape depending on the parameter τ/τµ with the pulses observed by Hutcheson, Roth, and Barnes [221] in the direction of the laser beam (Fig. 5.4). In particular, at τ/τµ = 7.5 a pulse given by curve 1 in Fig. 5.4 was

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113

observed experimentally. This pulse coincides in its shape with curve 1 in Fig. 5.3. At τ/τµ = 4⋅10−8 a negative pressure pulse was observed given by curve 2 in Fig. 5.4 and coinciding in its shape with curve 2 in Fig. 5.3. Hutcheson, Roth, and Barnes noticed the discrepancy in the shape of the pulses obtained experimentally and the shape of the pulses predicted theoretically by Chia-lun Hu [210] and Gournay [215] and erroneously attributed this discrepancy to nonlinear effects. In fact, as has been noted earlier, the discrepancy is connected with the fact that Chia-lun Hu [210] and Gournay [215] have considered only onedimensional problems of optical generation of sound waves. Naturally, the solution of one-dimensional problems cannot describe real spatial sound fields in the presence of a boundary. The coincidence of experimental data obtained by Hutcheson, Roth, and Barnes [221] with the results of the solution of the boundary problem of optical generation of sound by radiation (laser) pulses considered here (see [109]) is evidence of the validity of a linear approximation for description of processes of thermoradiation (optical) generation of sound. Experimental and theoretical results will be discussed in more detail in Chapter 10.

2. A LIQUID WITH A ROUGH SURFACE Let us extend the solution obtained above to the case of a rough surface of a liquid [109]. Let the boundary be described by the equation z = ζ(x, y, t), the velocity of boundary motion satisfying the condition max(τ , τ a , τ µ )

∂ζ << g , ∂t

i.e., we can assume the liquid boundary to be motionless during the process of sound generation. As has been shown in Chapter 3, a spectrum of sound pressure averaged with respect to realizations of random surface ζ is represented by the product of the spectrum of sound pressure obtained for an even boundary by the characteristic function of distribution of height of unevennesses. In the case considered, averaging with respect to surface realizations can be equalized to averaging with respect to the number of pulses repeated with the period T during which the liquid surface changes its shape. For definiteness, we consider the normal distribution of heights of unevennesses, when the average spectrum of sound pressure has the form

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EXCITATION OF SOUND IN A LIQUID BY RADIATION PULSES

iI αµa 2 e ikR k cosθ F [ p] = − 0 (1 − e iωt ) × 2 2C p R µ + k 2 cos 2 θ (5.11)  k a sin θ k σ cos θ  , − exp −   4 2   2 2

2

2 2

2

where σ is the mean-square height of waves. The inverse Fourier transform of spectrum (5.11) for determination of average sound pressure assumes integration with respect to frequency starting from zero, but expression (5.11) is true only for wave numbers k > ε, where ε is determined by the condition of the large scale of unevennesses with respect to the sound wavelength. However, it is possible to demonstrate that the error from the replacement of the lower limit of integration by zero is proportional to the squared ratio ε/µ and may be arbitrarily small in the case of corresponding restrictions for the dimensions of unevennesses. As a result we obtain the next formula for the sound pressure average with respect to pulses:

p =

 s 2   R / c − t  I 0αa 2 Erfc sσ − R / c − t  − exp σ exp −  2  4   8C p Rτ µ τ µ  τ aσ       R/c−t   Erfc sσ + R / c − t  − exp  2  τµ  τ aσ     (5.12)  R / c − t +τ exp −  τµ 

  Erfc sσ − R / c − t + τ  2  τ aσ  

  + 

 R / c − t +τ exp  τµ 

  Erfc sσ + R / c − t + τ  2  τ aσ  

    .  

If sσ << 1 (s << 1), i.e., the effective radiation length or the length of light absorption is large in comparison with the mean-square height of unevennesses, the length of sound pulse is determined by the length of

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115

radiation pulse τ, while in the case of small τ the sound pulse is presented basically by a negative peak. If sσ >> 1, i.e., the mean-square height of unevennesses is large in comparison with the effective length of absorption of penetrating radiation in a liquid, the shape and length of sound pulses are determined completely by the characteristic time τ (similar to the case of a plane boundary when for s >> 1 the shape and length are determined by the value of τa). In particular this is also true for observation along the axis of the radiation beam. We can determine the time-average intensity of pressure field for pulses repeated with period T. The pulse-average field intensity averaged with respect to period T is evidently the time-average intensity and it is represented by the next expression:

p

2

∞

 2  2 2 ≡  p  = ∫ F [ p] dω , T  T 0

(5.13)

where 〈|F[p]|2〉 is the spectral power average with respect to surface realizations. Let us consider as an example the normal law of distribution of heights of unevennesses large in comparison with the sound wavelength, i.e., the case kσ >> 1. Using the results obtained in Chapter 3, Section 4 and taking into account the remarks on the inverse Fourier transform, it is possible to write down the time-average intensity:

p

2

1  I 0 µα =  T  C p R

2  a 4c ∞ 1 − cos(kcτ )  ×  cos 2 θ ∫ ( µ 2 / cos 2 θ + k 2 ) 2  0

(5.14)    2 ∆2 sin 2θ k 2 dk    ρ0  exp − ,    2 2 2 2   2  4(1 + ∆ cos θ )  1 + ∆ cos θ   2

where ∆ = 21/2kaσ/ρ0 is the parameter of scattering and ρ0 is the correlation coefficient of boundary unevenness. As has been demonstrated in Chapter 3, Section 4, at ∆ << 1 (i.e., under the condition that the radius of radiation beam a is small in comparison with the correlation coefficient of boundary unevenness ρ0), we can ignore the

116

EXCITATION OF SOUND IN A LIQUID BY RADIATION PULSES

effect of boundary unevenness on the average intensity and use the results obtained for an undisturbed surface. Under the condition ∆ >> 1 (the radius of radiation spot at the liquid surface is comparable to the correlation coefficient of unevenness of liquid boundary or larger), the effect of unevenness on the average intensity of sound field is essential and we obtain the next formula for the time-average intensity:

p

2

=

π T

2  1  tan θ  I 0α  a 2c 1   exp−   C p R  4µ cos θ tan 2 β  2  tan δ  

  

2

 ×  (5.15)

  τ 1 − 1 +  τ   µ

   exp − τ   τµ  

  ,  

where tanβ = 21/2σ /ρ0 is the mean-square angle of surface slope. The average intensity of field (5.15) in the case of large radiuses of penetrating radiation beam (a/ρ0 ≥ 1) has an angular distribution. The angular width of the lobe of a polar directivity pattern is equal to 21/2β and determined by the mean-square angle of surface slope β. The average intensity depends essentially on the ratio τ/τµ: if the length of radiation pulse is small τ << τµ, the average intensity is proportional to the square of this parameter, and if τ ≥ τµ, it attains quickly the maximum value, which does not depend on this ratio.

3. RADIATION PULSES OF ARBITRARY SHAPE The influence of the parameters of penetrating radiation (laser beam) and a medium on the sound field generated by radiation pulses of rectangular shape in a liquid half-space with even and uneven boundaries, has been analyzed in detail in previous sections. The conclusions obtained are qualitatively true also for pulses of various triangular shapes [111]. In reality, the shapes of real radiation pulses (see [175]) can be only approximately represented by the simple functions of the models considered earlier. The examples are laser pulses in the Q-switch mode, which can be approximated by a rectangular function [109], and a triangular shape of pulses in the millisecond mode [175].

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117

Now we will consider specific features of sound generation by radiation pulses of arbitrary shape with the following natural restrictions: a pulse is positive, the pulse envelope satisfies the existence condition of integrals and the convergence conditions of series obtained in the process of problem solution. It will be demonstrated that in the case when the length of radiation pulse is large in comparison with the delay time of sound from thermal sources in the horizontal section of the generation region, a pulse of sound pressure in the far wave field does not depend on geometrical parameters of generation region and is determined completely by the shape of penetrating radiation pulse. In the opposite case when the length of radiation pulse is small, the shape of sound pulse does not depend on the shape of radiation pulse and is determined by the ratio of characteristic delay times of sound from thermal sources in the horizontal and vertical sections of generation region. In this case the sound pulse is represented by a family of universal curves [110]. Let a beam of penetrating radiation be incident along the axis z at the boundary of a liquid half-space z > 0. Let the beam intensity at the liquid boundary be described by an expression, I ( x, y , t ) = I ( x, y ) f (t ),

(5.16)

where f(t) is the function of time, which determines the shape of a radiation pulse in such a way that max f(t) = 1, and I(x, y) is the surface distribution of radiation intensity at the boundary. The solution of the reduced wave equation for the spectrum of sound pressure generated by penetrating radiation may be written down in the form of an integral,

pω ( x, y, z ) =

1 4π

 iAα  ωµI ( x , y ) × p0 ( x, y, z; x1 , y1 , z1 ) − 1 1  Cp    V1

∫

(5.17) exp( − µz1 ) F (ω )dV1 , where p0 is the solution of the diffraction problem for a point source, V1 is the volume occupied by thermal sources of sound, which are produced by the action of penetrating radiation, and ∞

F (ω ) =

∫ f (t ) exp(iωt )dt

−∞

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EXCITATION OF SOUND IN A LIQUID BY RADIATION PULSES

is the spectrum of radiation pulse. A solution of the stationary problem for a liquid half-space with a free boundary can be obtained readily in the far wave field for a specific distribution of intensity I(x1, y1). Let us consider the Gaussian distribution of intensity as usual, I ( x1 , y1 ) = I 0 exp[−( x12 + y12 ) / a 2 ] , where a is the effective beam radius. In this case the spectrum of sound pressure has the form

pω = −

2 2 2 AI 0αa 2 e ikR ω τ µ e −ω τ a / 4 F (ω ) , 2C p R 1 + ω 2τ 2 µ

(5.18)

where R is the distance from the center of the region of thermal sources to the observation point, τµ = cosθ/(µc) and τa = a sinθ/c are the characteristic delay times of sound from elementary sources in the vertical and horizontal sections of the region of thermal sources, respectively, θ is the angle between the axis z and the direction to the observation point, and c is the sound velocity. An expression for sound pressure is obtained by the inverse Fourier transform of spectrum (5.18),

p=−

∞

 κ 2s2 exp −  4 4πC p Rτ µ2 − ∞  AI 0αa 2

∫

 κ2  κ  exp(iκγ ) F  τµ 1+ κ 2  

  dκ ,  

(5.19)

where the substitution κ = ωτµ, γ = (R/c – t)/τµ, and s = τa/τµ is performed. Expression (5.19) is the initial one for further analysis. Let us determine now the characteristic features of acoustic field without setting a specific shape of radiation pulse or its spectrum. Let τ be the length of radiation pulse, which is determined in a certain way, e.g., according to the drop of pulse envelope, the portion of energy contained in a pulse, etc. Let us represent the integral in expression (5.19) in the form of the sum of two integrals,

RADIATION ACOUSTICS

p=−

∞  κ 2s2 AI 0αa 2  − exp  2 ∫  4 4πC p Rτ µ − ∞ 

   exp(iκγ ) F  κ τµ   

119

 dκ −   (5.20)

∞

 κ 2s2   κ ∫ exp − 4  exp(iκγ ) F  τ µ    −∞

 dκ   . 1+κ 2   

The spectral width of functions in the integration element in expression (5.20) can be determined in the following way: the upper frequency limit for the spectrum of radiation pulse F(ω) is obtained on the basis of a known relationship [182] ω = c1/τ, the upper limiting frequency for the exponential function exp(−κ2s2/4) is ω = c2/τ, and the frequency limit for the rational function 1/(1 + κ2) is ω = c3/τµ, where c1, c2, and c3 are the constants depending on the way of determination of the spectrum width and the width of radiation pulse. Thus, depending on the relation of the characteristic parameters of the problem (τ, τa, τµ), some function under the integral determines the character of the drop of spectral density. Let us consider the limiting relationships τ >> τa and τ << τµ, which mean that the delay time of sound from elementary sources in the crosssection of the generation region is very small or very large, respectively in comparison with the length of radiation pulse. Let τ >> τa, i.e., we consider the region of thermal sources in the form of a narrow cylinder or radiation along the directions close to the axis z. Under this condition we may assume that the exponent exp(−κ2s2/4) in expression (5.20) is equal to one within the frequency range, where the spectrum of a pulse of penetrating radiation is essential, and write down an expression

p=−

∞  AI 0αa 2  κ  R τ f t F − −   ∫ 2  µ   τ c  4πC p Rτ µ  −∞  µ

   exp(iκγ ) dκ  .  1 + κ 2  

(5.21)

The second integral can be estimated for the cases τ >> τµ and τ << τµ, i.e., for different limiting relationships of the length of radiation pulse and delay time of sound from elementary sources in the longitudinal section of generation region. If τ >> τµ, i.e., the drop rate of spectral density in integral (3.6) determines the spectrum of a pulse of penetrating radiation F(ω), the rational function can be expanded in a series and integrated:

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EXCITATION OF SOUND IN A LIQUID BY RADIATION PULSES

∞

 κ ∫ F  τ µ −∞ 

∞ ∞  R   exp(iκγ ) (−1) n κ 2n dκ = τ µ2n +1 f 2n  t −  . ∑ ∑  c   n=0 n=0

Let us assume as has been agreed earlier that the series converges with respect to the power of the parameter τµ /τ << 1 and restrict ourselves to the first two terms of the series. As a result, the sound pressure is represented by the following expression: p≈−

AI 0αa 2  R τ µ f ′′ t −  , c 4πC p R 

(5.22)

instead of expression (5.21). And if τ << τµ , i.e., the region of thermal sources has the shape of a long narrow cylinder, the spectrum of radiation pulse in the second integral can be represented under the condition ω < 1/τµ or ωτ < τ /τµ << 1 by the formula τ

τ

0

0

F (ω ) = ∫ f (t )e iωt dt ≈ ∫ f (t )dt = σ

(5.23)

(here σ is the “area” of radiation pulse). After that, an expression for sound pressure (5.21) takes on the form

p≈−

AI 0αa 2 4πC p Rτ µ

  R  πσ  R  exp − t −  .  f t −  − c  τµ c    

(5.24)

Thus, in the considered case of radiation pulses long in comparison with the characteristic time τa or in other words, for narrow beams of penetrating radiation, the shape of a sound pulse does not depend on the parameter s and is determined completely by the envelope of radiation pulse f(t). In the case of radiation pulses long in comparison with τµ (i.e., long in comparison with all characteristic times of the problem), the shape of sound pulse is determined almost completely by the second derivative of the envelope of radiation pulse (5.22). In the case of radiation pulses short in comparison with τµ, a sound pulse is a rarefaction pulse repeating, in fact, the envelope of the “overturned” radiation pulse –f(t) with a certain positive addition

RADIATION ACOUSTICS

121

proportional to the small parameter τ/τµ (expression (5.24)). It is necessary to note that formula (5.21) is exact for θ = 0 and formulae (5.22) and (5.24) describe the field along the axis z. These laws were observed experimentally while investigating sound generation by laser pulses [220]. This consideration confirms the validity of the experimental data. In particular, the fact that in the case of short laser pulses (τ << τµ) a rarefaction pulse with length τ propagates along the axis z, is fundamental. Positive “splashes” preceding the rarefaction pulse and following it (as one can see from expression (5.24)) are also the features proper to this physical phenomenon and not distortions introduced by measuring devices as it might seem. Now let us consider the opposite case τ << τµ, i.e., the case of radiation pulses short in comparison with the delay time of sound from sources in the cross-section of the generation region. In this case the spectral density is limited to frequencies ω < 1/τa and consequently, ωτ < τ/τa << 1. Therefore, expression (5.23) is valid and an expression for sound pressure (5.20) takes on the form

p=−

∞  κ 2s2 AI 0αa 2σ  exp −  ∫  4 4πC p Rτ µ2 − ∞ 

  exp(iκγ )dκ −   (5.25)

∞

  κ 2s2  −  exp(iκγ ) dκ  . exp ∫  4  1 + κ 2    −∞ Expression (5.25) can be integrated readily:

p=−

 γ2 AI 0αa 2σ  4 exp −   s2 16C p Rτ µ2  π s 

  2  − exp s   4  

 ×   (5.26)

 s γ   s γ e −γ Erfc −  + eγ Erfc +   ,  2 s   2 s   

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EXCITATION OF SOUND IN A LIQUID BY RADIATION PULSES

where Erfc( z ) =

2

∞

2

−t ∫ e dt is the complementary error function.

π z Thus, as we can see from expression (5.26), in the case of a wide beam τ << τa, the shape of a sound pulse does not depend on the shape of radiation pulse and is determined only by the ratio of characteristic times s. Shapes of sound pulses determined by formula (5.26) are given in Fig. 5.5a and b (the coefficient before the curly bracket in expression (5.26) is assumed to be equal to one).

Figure 5.5 Shapes of a pulse of relative sound pressure. (a) A pulse determined by formula (5.26) for s = (1) 1.0 and (2) 5; (b) s >> 1, τ << τa.

Let us consider limiting cases. Let s >> 1, which corresponds to the region of thermal sources in the shape of a wide disk and observation under not too small angles θ. In this case the exponent is the determining function under the integral and sound pressure is given by the formula

p≈

  2 2   2(t − R / c) − 1 exp − (t − R / c)   π C p Rτ µ2 s 3  τ µ2 τ a2   AI 0αa 2σ

 .  

(5.27)

It is interesting to note that expression (5.2) turns into formula (5.8) in the particular case of a rectangular pulse and corresponding substitution of the quantity σ. If s << 1 which corresponds to the region of thermal sources in the shape of a long cylinder and observation under small angles θ, the exponent under the second integral in expression (5.27) can be considered equal to one. As a result we have

RADIATION ACOUSTICS

p≈−

 (t − R / c) 2 AI 0αa 2σ  2  exp −  8C p Rτ µ2 s  π τ a2 

   − s exp − t − R / c   τµ  

123

  .  

(5.28)

As one can see from expressions (5.27) and (5.28), the shape of sound pulse is universal for any envelope of a radiation pulse. In particular in the case s >> 1, τ << τa, sound pressure given by expression (5.27) is presented by a curve in Fig. 5.5b (the coefficient at the square bracket in expression (5.27) is assumed to be equal to one). Zeros and the points of extremum at the curve in Fig. 5.5b are completely determined by the value of τa (see Section 1). Only the amplitude of sound pulse depends on the envelope of radiation pulse. Thus, within the framework of the thermoradiation mechanism, sound pulses generated by radiation pulses of arbitrary shape obey the next laws in the far wave field. In the case of narrow radiation beams (τ >> τa) or observation along the axis z, the shapes of sound pulses do not depend on the ratio of characteristic times s and are determined completely by the envelopes of radiation pulses (see expressions (5.22) and (5.24)). However, in the case of short radiation pulses τ << τa or for long regions of generation, a sound pulse is presented mainly by a rarefaction pulse repeating the shape of a radiation pulse (expression (5.24)). In the case of wide beams of penetrating radiation (τ << τa) and observation under not too small angles, the shapes of sound pulses are absolutely independent of the envelopes of radiation pulses and are presented by a universal curve (Fig. 5.5b) depending on the ratio of the characteristic times s. Only the amplitude of sound pressure depends on the shape of a radiation pulse in this case (τ << τa). In the case of limiting values of s, we obtain universal formulae for sound pressure (5.27) and (5.28). The first of them determines the shape of a sound pulse according only to the characteristic time τa. The conditions of pairwise equality of characteristic times τ, τa, and τµ determine correspondingly the characteristic angles of the problem θ0 = arctan µa (s = 1), θ1 = arccos µcτ (τ = τµ), and θ2 = (arcsin cτ)/a (τ = τa). The next three cases are possible (see Fig. 5.6): (1) τ < τ0, (2) τ = τ0, and (3) τ > τ0, where τ0 = a/[c(1 + µ2a2)1/2. In the first case, formula (5.24) is valid for the angles θ < θ 2 (τa < τ1, τµ > τ), expression (5.28) is true for the angles θ 2 < θ < θ 0 (τa > τ, s < 1), and formula (5.27) is valid for the angles θ > θ 0 (τa > τ, s > 1). In the second case when θ 0 = θ 1 = θ 2, expression (5.24) is true for the angles θ < θ 0 (τ > τa, τ < τµ) and expression (5.27) is valid for θ > θ 0 (τ < τa, s > 1). In the third case at θ < θ 1 (τ > τa, τ < τµ), formula (5.24) is true, expression (5.22) is valid at θ 1 < θ < θ 2 (τ > τa, τ > τµ), and at θ > θ 2 (τ < τa, s < 1) expression (5.27) is true.

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EXCITATION OF SOUND IN A LIQUID BY RADIATION PULSES

Figure 5.6 Ranges of application of expressions (5.22), (5.24), (5.27), and (5.28). (a) τ < τ0; (b) τ = τ0; and (c) τ > τ0. Digits in brackets are the numbers of expressions.

We have to note in conclusion that the conducted theoretical consideration can be extended to the case of a liquid with rough surface using the results obtained in Section 2 of this chapter.

4. NEAR WAVE FIELD OF THERMORADIATION PULSED SOURCE OF SOUND In the previous section we have discussed the characteristics of acoustic radiation far from the region of sound generation. Meanwhile, the situation when the track length of penetrating radiation in a liquid is large enough and detection of an acoustic signal occurs in the near wave field of a thermoacoustic source arising due to absorption of penetrating radiation in a liquid, is quite common. Such a case is considered in this section (see [190]). Let a beam of penetrating radiation be incident perpendicularly on the free surface of a liquid, and the radiation track length l = µ−1, where µ is the coefficient of radiation absorption in the liquid, be large compared with the radius of the beam cross-section a (l >> a). We assume that the intensity distribution of penetrating radiation is even (and not Gaussian as usual) over the cross-section. A cylindrical region emitting a sound wave due to thermal expansion is formed in the liquid under the action of radiation pulse (Fig. 5.7). Taking equality of sound pressure to be zero as the boundary condition at the free surface of a liquid, we can write down the solution of the wave equation of thermoradiation generation of sound in the form p=

1 α 4π C p

1 ∂Q(t − r1 / c, r ) dV − ∂t

∫ r1

V

RADIATION ACOUSTICS

−

1 α 4π C p

1 ∂Q(t − r2 / c, r ) dV . ∂t

∫ r2

V

125

(5.29)

We assume that the length of radiation pulse τ is so large that cτ >> a, and we can ignore the difference between the arrival times of signals from different points of the beam cross-section.

Figure 5.7 To calculation of the near wave field of a pulsed thermoradiation source. (1) A beam of penetrating radiation; (2) the surface of a liquid; and (3) a thermoradiation source.

Let us consider the characteristics of a sound field in the near wave field of the radiating region, i.e., at distances x << l2/λ = 1/(µ2cτ). In this case in the cylindrical system of coordinates r1 = ( z − z A ) 2 + x 2 and r2 = ( z + z A ) + x 2 (see Fig. 5.7). Taking also into account the fact that Q = ∂I/∂z, where I = I(t)exp(−µz) is the intensity of penetrating radiation, we obtain z 4 − µz z2  e µαs  e − µz  r1   r0   p= I  t − dz − ∫ I  t − dz . 4πC p  ∫ r1  c r c    z3 2  z1 

(5.30)

Here s = πa2. It is necessary to know the function I(t) in order to calculate the integrals in formula (5.30). This function describes the time dependence of intensity of penetrating radiation. A typical experimental dependence in a laser pulse radiated in the free-running mode is given in Fig. 5.8a. Irregularity of the

126

EXCITATION OF SOUND IN A LIQUID BY RADIATION PULSES

pulse is caused by chaotic modulation of laser radiation. The envelope of such a pulse can be approximated by the formula I (t ) = 5.4 I m (t / τ ) exp(−2t / τ ) .

(5.31)

It is plotted in Fig. 5.8b. Here Im is the peak intensity connected with the total pulse energy E by the expression I = 0.74E/τs, which follows from the normalization requirement ∞

∫ Isdt = E .

0

Figure 5.8 Oscillogram of a laser pulse (a) and the plot of the function I/Im (b).

Figure 5.9. Time dependence of pressure in an acoustic signal in the observation point with the coordinates zA = 12 cm, xA = 10 cm. (a) µcτ = 22.5, τ’ = 0.01τ; (b) µcτ = 0.15, τ’ = 7.5τ; T = t /τ − t1/τ.

Formula (5.31) is true at t ≥ 0. We assume that I = 0 at t < 0. Correspondingly, integrating in expression (5.30), we have to take into consideration the fact that only those sections of the axis z, where the conditions t – r/c ≥ 0 are satisfied, make nonzero contributions. These conditions provide an opportunity to determine the limits on the integral. In the case xA/c ≤ t ≤ (xA2 + zA2)1/2/c only the first integral in expression (5.30) is nonzero, while z1 = zA − [(ct)2 − xA2]1/2, z2 = zA + [(ct)2 − xA2]1/2. If (xA2 + zA2)1/2/c ≤ t, the limits on integral are the following:

RADIATION ACOUSTICS

127

z1 = 0 , z 2 = z A + [(ct ) 2 − x 2A ]1 / 2 , z 3 = 0 , z 4 = [(ct ) 2 − x 2A ]1 / 2 − z A . Figure 5.9a gives the shape of an acoustic signal at the point with the coordinates xA = 10 cm, zA = 12 cm. This plot was obtained by numeric integration of expression (5.30) taking into account expression (5.31) under the condition cτ >> a, µcτ = 22.5. The acoustic signal at the point A starts at the moment t1 = xA /c when a compression wave from the closest region of sources arrives at this point. Then, at the moment t2 ≈ (xA2 + zA2)1/2 a rarefaction wave arrives. This wave is produced by addition of the pulse radiated by sources located in the vicinity of the point 0 and the wave reflected from the free surface. The rarefaction wave is added to the compression wave. Therefore, the pulse length depends on the difference of arrival times of the compression and rarefaction waves. This difference is determined by the relationship

τ ' ≈ t2 − t1 .

(5.32)

Oscillograms of an acoustic signal excited by a laser pulse in water and detected at the point with the coordinates xA = 10 cm, zA = 12 cm under the conditions corresponding to the calculation are given in Fig. 5.10a and b.

Figure 5.10 Oscillograms of an acoustic signal at the observation point with the coordinates zA = 12 cm, xA = 10 cm. (a) A signal in a wide frequency band; (b) a signal in a low frequency band.

The upper picture demonstrates a signal detected by a wide-band receiver. One can see that the acoustic pulse consists of a leading and a high-frequency component, which are specified by the spiking structure of

128

EXCITATION OF SOUND IN A LIQUID BY RADIATION PULSES

the laser pulse. The lower picture gives the low-frequency part of the signal separated by a filter. As one can see, the leading part of the pulse is similar to the theoretical dependence describing the shape of an acoustic pulse in the near wave field of a thermoacoustic array. If the distance from sound sources in the direction perpendicular to the axis of the laser beam grows, the difference between the arrival times of compression and rarefaction waves decreases and they eliminate each other. As the result of this process under the condition τ1 < τ′ << τ, where τ1 is the characteristic period of chaotic modulation, the leading low-frequency part of the signal vanishes and only the high-frequency component exists. An example of such a signal in a smaller time scale is given in Fig. 5.11. If the distance from the source in the transverse direction grows even more, the high-frequency component of the signal vanishes also at distances xA >> zA, where τ = 0.

Figure 5.11 Oscillogram of an acoustic signal close to the surface at xA > zA.

A different pattern is observed in the case when the difference of the arrival times of compression and rarefaction waves τ′ exceeds the characteristic length of a radiation pulse τ. Such situation may arise in the case of short radiation pulses or in the case of moving away from the observation point along the axis z. Figure 5.9b gives as an example the shape of an acoustic pulse at the same observation point as in Fig. 5.9a in the case of a shorter laser pulse when τ′ = 7.5τ. If the distance from the generation region increases, the amplitude of a leading pulse decreases proportionally to exp(−µz) and the amplitude of a closing pulse decreases proportionally to 1/(x2 + z2)1/2.

RADIATION ACOUSTICS

129

If a medium is spatially inhomogeneous and its properties change noticeably at distances smaller than the value of cτ, the structure of an acoustic pulse at the observation point becomes even more complex. For example, let the properties of a medium at the depth z = H change in such a way that the parameter µαS/Cp increases. Then, an additional signal arises at the observation point at the time moment t = (xA2 + H2)1/2/c. The leading part of this signal is a compression wave with amplitude proportional to the degree of medium inhomogeneity. We should note in conclusion that the characteristic properties of the sound pulses excited by pulsed penetrating radiation in the near wave field of a thermoacoustic array can be revealed in the process of consideration of the next simplest one-dimensional model (see [141] and also [41, 210, 215]). Let the energy E be released in the spherical region of the radius R0 during the time τ. This leads to the increase of the volume V = (4π /3)R03 by the value ∆V =

Eα , ρC p

(5.33)

where as usual α is the cubical coefficient of thermal expansion of a medium, ρ is the medium density, and Cp is the specific heat capacity. If R0 >> cτ (c is the sound velocity), the region does not have time to expand during the time of energy release and pressure in the region increases by the value

p = ρc

2 ∆V

V

=

c 2 Eα . C pV

(5.34)

Thus, a spherical region of increased pressure arises in a medium at the initial moment. This leads to radiation of a spherical sound wave. The profile of such wave is given in Fig. 5.12a and the peak value of pressure is equal to a half of the initial excessive pressure multiplied by the ratio R0/r, which takes into account spherical divergence of the wave:

p=

PR0 3 Eαc 2 . = 2r 16π C R 2 r p 0

(5.35)

In this case the pressure amplitude is determined by the released energy and does not depend on its release rate, while the pulse length is determined

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EXCITATION OF SOUND IN A LIQUID BY RADIATION PULSES

mainly by the travel time of sound over the disturbed region (the time of volume discharge) R0/c. If R0 << cτ, the dimensions of radiating region are small as against the characteristic length of sound wave cτ. In this case the calculation according to Lyamshev and Naugol’nykh [141] and Chia-lun Hu [210] leads to the following expression for the peak value of pressure: p=

3Eαc . 16πR0τC p r

(5.36)

This expression can be obtained with precision to a numeric coefficient from the expression p = ρV″/(4πr) if we take for estimation V′ ~ ∆V/τ, V ~ V′/R0. Thus, the pressure amplitude is determined in this case by the rate of energy release and the pulse length is determined by the time of energy release τ (see Fig. 5.12b).

Figure 5.12 Profile of a wave propagating from a thermal source. (a) R0 >> cτ; (b) R0 << cτ.

Thus, one can see from this discussion that a sound signal in the near wave field of a thermoacoustic array in a liquid with a free surface has a characteristic form, i.e., the form or shape of a so-called N-wave. Specific properties of sound signals in the near wave field of a laser thermoacoustic source of sound were discussed in detail by Burmistrova et al. [44] and Karabutov, Rudenko, and Cherepetskaya [106].

5. SOUND GENERATION IN A LIQUID WITH GAS BUBBLES Let us consider one more specific example of formation of an acoustic signal under the effect of pulsed penetrating radiation. We will discuss the specific features of thermoradiation generation of sound in a liquid with gas bubbles.

RADIATION ACOUSTICS

131

Bubbles are always present in real liquids and in water in particular. Therefore the question of the effect of bubbles on thermoradiation excitation of sound in a liquid with bubbles can be of certain practical interest. We will not use the discussed earlier results of general consideration of processes of thermoradiation excitation of sound in inhomogeneous media but following Egerev and Naugol’nykh [94], we will consider this problem proceeding from general physical concepts. Let a flow of penetrating radiation be incident normally from a transparent medium on the plane boundary of a liquid. Let the beam diameter be large enough in comparison with the “radiation” length in a liquid l so that an arising sound wave may be considered as a plane one. We assume that penetrating radiation has the form of rectangular pulses of intensity I and length τ. It is known that the process of sound generation has certain differences in the cases cτ < l and cτ > l in the sense that the first of them corresponds, as it were, to the problem with the initial conditions, and the second one corresponds to the problem with the boundary conditions. In particular, in the case of short radiation pulses (cτ < l), the excessive pressure P arises as a result of energy evolution in a liquid. This pressure is proportional to the increase of the volume ∆V of the region of heat discharge, which is caused by the effect of thermal expansion P = ρc 2V −1dV .

(5.37)

Here ρ is the liquid density, c is the sound velocity in the liquid, V = lS is the volume of the medium where energy is released, S is the sectional area of a radiation beam, ∆V = αE / ρC p ,

(5.38)

where α is the coefficient of thermal expansion, E = IτS is the energy of radiation pulse, and Cp is the specific heat capacity. If the medium-liquid interface is transparent and acoustically rigid, the energy absorbed in the medium region V is radiated in the form of a compression pulse with amplitude proportional to the energy of radiation pulse [110]:

p=

P ρc 2 ∆V c 2 Eα , = = 2 2 V 2lSC p

(5.39)

and its length is determined by the travel time of sound over the region of energy release T = l/c.

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EXCITATION OF SOUND IN A LIQUID BY RADIATION PULSES

In the case of long pulses (cτ > l), continuous generation of sound from the liquid volume V occurs in the process of absorption of penetrating radiation, and the pressure in a sound wave is determined by the expansion rate of the region V, which is proportional to the intensity of penetrating radiation [110], p = ρcV = ρc

∆V αIc . = Sτ Cp

(5.40)

The coefficient of thermal expansion in liquids is small and therefore, the intensity of generated sound can change strongly in the presence of a large amount of gas bubbles affecting noticeably the degree of expansion of a heated volume. The effect of bubbles is determined mainly by their size and concentration in the volume absorbing penetrating radiation. Small bubbles, which are always present in liquids, influence under real conditions the process of thermoradiation generation of sound. This effect plays an essential role even at very small concentrations of bubbles and small (in comparison with the heat of vaporization) density of released energy, when penetrating radiation scattering (caused by bubbles) from the region of energy release is negligible, and the increase of pressure of gas and vapor in a bubble resulting from its heating is small. It is caused by a strong change of compressibility of a liquid containing gas bubbles. Large bubbles have a low resonance frequency f0 and during the time of radiation energy release τ, they behave as incompressible ones. This makes a medium apparently more rigid in the region of heat evolution and increases the amplitude of acoustic pulse. Small bubbles (f0 >> 1/τ) have enough time to change their volume under the effect of increasing pressure and by virtue of large compressibility of gases as against compressibility of liquids, the pressure in an emitted compression wave decreases. The degree of bubble compression may be determined from the energybalance equation: the work of external pressure on a bubble A is equal to the increase of internal energy of gas in a bubble W. The quantity A is expressed by the formula, A = ( P0 + P)(V0 − V1 ) ,

(5.41)

where P is the excessive pressure in a compression wave, which is determined by formulae (5.37) and (5.38), V0 and V1 are the initial and final values of the bubble volume, and P0 is the hydrostatic pressure. The value of W for an adiabatic process is equal to

RADIATION ACOUSTICS

W =

QV0  V0  γ − 1  V1 

  

γ −1

 − 1 ,  

133

(5.42)

where Q is the initial gas pressure in a bubble and γ is the adiabatic exponent. Equating expressions (5.41) and (5.42), we obtain an equation for determination of the degree of bubble compression z = V0/V1 in the form ( P0 + P)(γ − 1) z γ − z . = Q z −1

(5.43)

If the change of the radius of each bubble is small, i.e., z = 1 + ε, where ε << 1 and bubbles are not very small such that it is possible to ignore the surface tension forces, equation (5.43) leads to the simple result

ε = 2 P /(γQ) .

(5.44)

Expression (5.44) is true if the temperature wavelength in a gas Λ = (4πχτ)1/2 is small compared with the bubble radius R0 (here χ is the temperature conductivity). In the opposite case (Λ > R0), the process of bubble compression is isothermal and we have to take γ = 1 in the expression for ε. Now it is possible to determine the amplitude of a sound wave emitted in a liquid with bubbles. Let M be the volume of gas phase in the generation zone. Then, the change of this volume under the effect of excessive pressure at moderate values of intensity of penetrating radiation is Mε. Thus, the absorbing volume increases by the value ∆V – Mε. Let cτ << l. Using formula (5.39), we obtain an equation pc =

1 2 ∆V − Mε ρcc . 2 V

Taking into account expressions (5.38) and (5.44), we obtain

pc =

αEcc2 1 . 2 SC p l 1 + ρcc2 n / γQ

(5.45)

134

EXCITATION OF SOUND IN A LIQUID BY RADIATION PULSES

Here n = M/V is the volume concentration of gas and cc is the sound velocity in a two-phase medium: cc ≈

2πf 0 ( 2πf 0 ) 2 − 3nc 2 / R02

.

And if cτ >> l, then using formulae (5.38), (5.40), and (5.44), we obtain pc = ρcc

∆V − Mε . Sτ

It follows from here that pc = αcc I

1 C p (1 + ρcc ⋅ 2nl / τγQ)

.

(5.46)

The dependence of the amplitude of acoustic pressure in a wave emitted from the generation zone on the volume concentration of gas bubbles n for cτ << l and cτ >> l is plotted in Fig. 5.13 (it is assumed that f0 >> 1/τ in both cases). The values of pc are normalized to the amplitude of acoustic pressure p in a single-phase liquid to make the plot more illustrative. The next values of constants are taken: 1/l = 0.18 cm−1 and c = 1.5⋅105 cm/s.

Figure 5.13 Dependence of sound pressure amplitude on the volume concentration of gas bubbles. (1) τ = 1 µs, short pulses; (2) τ = 1 µs, long pulses.

The plot illustrates the effect of bubbles on the process of generation of a compression pulse. One can see that the effect becomes noticeable if the concentration of small enough bubbles is not too small.

RADIATION ACOUSTICS

135

Now let us consider the case when the action of penetrating radiation leads to generation of a wave, which has a rarefaction pulse together with a compression pulse. For example, let the energy E be released as the result of absorption of a radiation pulse in a spherical region of radius a during the time τ << a/c, where c is the sound velocity. This causes a fast increase of pressure by the value [44, 141]

P=

c 2αE , C pV

(5.47)

where V = (4/3)πa3. The rise of a region of increased pressure leads to radiation of an N-wave with the peak value at a distance from the center of generation zone determined by a relationship p = Pa/2r and the length determined by the time of sound travel over the disturbed region T = a/c (see Fig. 5.14a). In the process a discharge wave propagates inside the disturbed region from the zone of energy release. Its peak value at the point r < a is determined by the relationship p=−

Pa , 2 r

(5.48)

and the characteristic length is τ = 2r/c (Fig. 5.14b). As r decreases, the wave amplitude increases and the length decreases. A rarefaction region arises in the liquid. Bubbles in this region expand. The characteristic radius of a bubble attained as the result of action of the negative phase of the discharge N-wave can be estimated in the roughest approximation as follows. In the order of magnitude R = uτ ' ,

(5.49)

where u ≈ (∆p/ρ)1/2 is the rate of bubble expansion, τ′ ≈ 2r/c is the time of its expansion, ∆p is the difference of pressures inside and outside of a bubble, which causes its expansion, ∆p ~ Pa/2r. Substituting the values of τ′ and ∆p into expression (5.49), we obtain R = 2 Par /( ρc 2 ) .

(5.50)

136

EXCITATION OF SOUND IN A LIQUID BY RADIATION PULSES

As one can see, the maximum radius of a bubble arising under the action of a rarefaction wave decreases with the decrease of r since the duration of action diminishes.

Figure 5.14 Pressure profiles in (a) a radiated acoustic pulse and (b) discharge wave.

Taking for estimation the value of r at which the amplitude of rarefaction wave attains p = − P (r = a/2), we obtain R ≈ a P /( ρc 2 ) = a Eα /(VC p ρ ) .

(5.51)

For example, let the energy E = 10–2 J be released in water in the volume V = 1 cm3. Then, at α = 1.4⋅10−4 K−1 and Cp = 4.2 J/(g⋅K), we obtain the value of R = 2⋅10−3 cm, i.e., the nuclei present in a liquid can expand to a noticeable size. We have to stress that this estimation is based on the assumption of the absence of bubble interaction. We may assume that despite the sketchiness, the obtained results reflect correctly the main features of the phenomenon of sound generation by pulses of penetrating radiation in a liquid with gas bubbles. A more detailed consideration might be conducted on the basis of the reciprocity method and the theory of propagation of sound waves in a liquid with bubbles [114].

CHAPTER 6

Generation of Sound in Solids by Intensity-Modulated Penetrating Radiation Shear waves can propagate in solids together with longitudinal waves. Thermal (volumetric) sources of longitudinal waves arise under the action of penetrating radiation due to the thermoelastic mechanism of sound excitation. Shear waves result only from reflection of longitudinal waves from borders or their scattering by inhomogeneities. In practice, solids are always limited and both longitudinal and shear waves are always present in solids under the effect of penetrating radiation. Further we will consider specific features of generation of monochromatic acoustic waves in solids by intensity-modulated penetrating radiation.

1. BASIC EQUATIONS Let us write down the equation of thermoelasticity [168] for a homogeneous and isotropic solid,    A 1 ∂2  4      ∆ − 2 2 Φ =  3 − 2 αT − 2 , сl ∂t  n  cl ρ   137

138 GENERATION OF SOUND IN SOLIDS: MODULATED RADIATION

(3 − 4 / n 2 )αcl2T0 ∂Φ  1 ∂ Q  ∆ − T − ∆ =− , ∂ ∂ χ χ χ t c t c   ε ερ

(6.1)

2    ∆ − 1 ∂ Ψ = − M ,  ct2 ∂t 2  ct2 ρ 

where Φ and Ψ are the scalar and vector potentials of the displacement vector of a solid u = grad Φ + rot Ψ, respectively, cl and ct are the propagation velocities of longitudinal and transverse sound waves, respectively, n = cl/ct, α is the linear thermal expansion coefficient of a solid, T is the difference of temperatures of a heated and unheated solid, T0 is the temperature of an unheated solid, χ is the thermal diffusivity, cε is the specific heat capacity of a solid, ρ is its density, Q is the quantity of heat released per unit time in unit volume, and A and M are the potential and solenoidal components of the external non-thermoelastic force F applied to unit volume of a solid (F = grad A + rot M). The first equation of system (6.1) is a wave equation for longitudinal waves. The sources of these waves are the change of solid volume caused by the change of solid temperature and the potential component of the non-thermoelastic volumetric force. The second equation of this system is the balance of heat. Its first term is a common diffusion term and the second one is present due to the change of solid temperature in the process of propagation of a longitudinal wave in it, and is called a dilatation term. The right-hand side of the second equation is the source of thermal energy. The third equation of the system is a wave equation for transverse waves. The source of these waves is the solenoidal component of the non-thermoelastic volumetric force. Now let us apply the operator ∆ – 1/∂t to the first equation of system (6.1) and change the right-hand side of the obtained equation according to the second equation of system (6.1). In the result we get a system of equations equivalent to system (6.1),   4  2   1 + 3 − ∆ −   n2    

=−

 2 α T0  

2 3 cl2  1 ∂  ∆ − 1 ∆ ∂ + 1 ∂ Φ = cε  χ ∂t c 2 ∂t 2 χc 2 ∂t 3  l l  

(3 − 4 / n 2 )αQ  1 ∂ A −  ∆ − ,  χcε ρ χ ∂t  c 2 ρ  l

(6.2)

RADIATION ACOUSTICS

139

2    ∆ − 1 ∂ Ψ = − M .  ct2 ∂t 2  ct2 ρ 

The dilatation term (3 – 4/n2)2α2T0c02/cε lies within the limits 10−3 – 10−5 for the majority of substances, which provides grounds to ignore it further. Now basic equations of sound generation in solids by penetrating radiation take on the form   1 ∂2 1 ∂3   2 1 ∂ div u = ∆ − ∆ − ∆ +  χ ∂t c 2 ∂t 2 χc 2 ∂t 3  l l   =−

(3 − 4 / n 2 )α∆Q 1  1 ∂  ∆ −  div F , − 2 χcε ρ χ ∂t  cl ρ 

(6.3)

2    ∆ − 1 ∂  rot u = − rot F .  ct2 ∂t 2  ct2 ρ 

If we ignore the influence of heat conductivity on sound generation in a solid (this question with respect to liquids was discussed earlier; see also [43]), equations (6.3) take the form   1 ∂2  (3 − 4 / n 2 )α div F  u div , ∆ − = ∆Qdt − ∫   2 2 2 cε ρ c ∂ t c ρ l l   (6.4) 2    ∆ − 1 ∂  rot u = − rot F .  ct2 ∂t 2  ct2 ρ 

2. BOUNDARY CONDITIONS In the process of solution of boundary problems with the help of equations (6.4), the boundary conditions at the boundary surfaces are determined by the character of allowed displacements and stress at these surfaces. However, if we use equations (6.3), the situation becomes more complex

140 GENERATION OF SOUND IN SOLIDS: MODULATED RADIATION

because of the presence of a thermal field in their solutions apart from the field of sound waves. In this connection, let us determine the character of the boundary conditions necessary for solution of corresponding boundary problems. Let us perform the Fourier transformation of equations (6.3) with respect to time:    iω 3  iω  A ω ω2 (3 − 4 / n 2 )α  2 iω  Φω = − Qω −  ∆ + , ∆+ ∆+ ∆ +  2 2 c χ χ ρ χ  c 2 ρ  ε cl cl   l (6.5) 2  ∆ + ω  ct2 

 Ψ = − M ω ,  ω ct2 ρ 

where ω is the circular frequency and ∞

pω =

∫ p exp(iωt )dt

−∞

is the Fourier transform of the function p. Let us write down the Green function (for a free space) for the equation   2 iω iω 3   2 ω G = δ ( x, y , z ) . ∆ + ∆ + ∆ +  2 2  χ χ c c l l  

(6.6)

Let us perform the Fourier transformation of equation (6.6) with respect to the coordinates x, y, z in order to do this. As a result we obtain the equation for the Fourier transform of G, (ω 2 / c 2 − iω / χ ) −1 (ω 2 / c 2 − iω / χ ) −1 l l G∗ = − + . 2 2 2 2 2 ω / cl − k x − k y − k z iω / χ − k x2 − k 2y − k z2

(6.7)

Here kx , ky , and kz are the components of the wave vector in the axes x, y, and z, respectively and

RADIATION ACOUSTICS

G∗ = ∫

141

∞

∫ ∫ G exp(−ik x − ik y − ik z )dxdydz

−∞

is the Fourier transform of the function G. The first term in the right-hand side of expression (6.7) is the Fourier transform of the Green function of a reduced wave equation with an additional factor – (ω2/cl2 − iω /χ), and the second term differs from the first one only in the sign and formal substitution of the quantity iω/χ instead of ω2/cl2 in the denominator. Therefore, the Green function of equation (6.6) for a free space has the form  2  iω  ω G= − χ   c2  l 

−1

exp[i (ω / cl ) R] − exp[(i − 1) ω / 2 χ R] 4πR

,

(6.8)

where R = (x2 + y2 + z2)1/2 is the distance from the source to the observation point. The fact that the Green function of equation (6.6) represents two independent terms, one of them describing a sound wave and another one describing a thermal wave, is explained by neglecting the dilatation term. If we take into account the dilatation term, both parts of the Green function are interconnected, but such a bond must be small because of the smallness of the dilatation term. Thus, if we ignore dilatation (that is implied everywhere further), sound and thermal fields are independent of each other, and therefore, the boundary conditions may be determined independently for sound and thermal fields. Basically, we will consider characteristics of sound waves in the far wave field (the Fraunhofer field). Therefore, we will ignore the second term of the Green function (6.8) describing the thermal field decreasing with distance from the source much faster than the sound field.

3. METHOD FOR SOLUTION OF BOUNDARY PROBLEMS A technique based on the reciprocity principle, which provides an opportunity to write down the solutions of boundary problems in quadratures on the basis of the solution of an auxiliary diffraction problem, was used earlier for solution of boundary problems. This technique was

142 GENERATION OF SOUND IN SOLIDS: MODULATED RADIATION

proposed by the author [131] and has been widely used in the theory of laser generation of sound [128]. Let us obtain analogous relationships for solids. The reciprocity relationship for solids can be written down in the following form [117]:

∫u

(1)

( x1 , y1 , z1 ; x, y, z )F ( x, y, z )dΩ =

Ω

(6.9)

∫ u( x, y, z; x1, y1, z1 )F

(1)

( x1, y1, z1 )dΩ1 ,

Ω1

where u(1)(x1, y1, z1; x, y, z) is the harmonic vector of displacements of a solid at the point x, y, z, which originates from the action of forces in the region Ω1; u(x, y, z; x1, y1, z1) is the harmonic vector of displacement of a solid at the point x1, y1, z1, which originates from the action of forces in the region Ω; and F(1)(x1, y1, z1) and F(x, y, z) are the harmonic bulk forces applied to the regions Ω1 and Ω. Now let us present the displacement vector and the bulk force in the form of the sum of potential and solenoidal components u = grad Φ + rot Ψ, F = grad A + rot M and rewrite the expressions obtained as the result of pairwise multiplication of the potential and solenoidal parts of the vector of bulk force: grad Φ ⋅ grad A = div [Φ ⋅ gradA] − Φ∆A , grad Φ ⋅ rot M = div [Φ ⋅ M ] , (6.10) rot Ψ ⋅ grad A = div [Ψ ⋅ grad A] , rot Ψ ⋅ rot M = div [Ψ ⋅ rot M ] + Ψ rot ⋅ rot M . In the processes of substitution of expressions (6.10) in reciprocity relationship (6.9) and integration over the volume, the terms containing the operator “div” transform into surface integrals and vanish by virtue of the boundary conditions. Therefore, reciprocity relationship (6.9) takes on the form,

∫ [Φ

Ω

(1)

( x1 , y1 , z1 ; x, y, z )∆A( x, y, z ) −

RADIATION ACOUSTICS

143

Ψ (1) ( x1, y1, z1; x, y, z ) rot rot M( x, y, z )]dΩ = (6.11)

∫ [Φ( x, y, z; x1, y1, z1)∆A

(1)

( x1, y1, z1 ) −

Ω1

Ψ( x, y , z; x1, y1, z1 ) rot rot M (1) ( x1, y1, z1 )dΩ1 . Now let the region Ω be located in the far wave field and the source of waves in it has a delta-like character: ∆A(x, y, z) = δ(x, y, z) and rot rot M(x, y, z) = γ1δ(x, y, z) + γ2δ(x, y, z) (γ1 and γ2 are the unit vectors in the incidence plane and the boundary plane). Then the transverse wave Ψ has two polarizations Ψ1 and Ψ2 in the planes of incidence and boundary, respectively. In this case the expressions for Φ(x, y, z; x1, y1, z1), Ψ1(x, y, z; x1, y1, z1), and Ψ2(x, y, z; x1, y1, z1) for a half-space can be presented in the next form: Φ (1) ( x, y, z; x1 , y1 , z1 ) = −

exp(ik l R) (exp Ql + V1 exp Pl ) − 4πR

exp(ikl R ) V2 exp Pl , 4πR (1)

Ψ1 ( x, y, z; x1 , y1 , z1 ) = −

exp(ik t R ) (exp Qt + V3 exp Pt ) − 4πR

exp(ikt R ) V4 exp Pt , 4πR (1)

Ψ2 ( x, y, z; x1 , y1 , z1 ) = −

exp(ik t R ) (exp Qt + V5 exp Pt ) , 4πR

Ql = −ik lx x1 − ik ly y1 − ik lz z1 , Pt = −ik tx x1 − ik ty y1 + ik tz z1 , Qt = −ik tx x1 − ik ty y1 − ik tz z1 , Pl = −ik lx x1 − ik ly y1 + ik lz z1 ,

(6.12)

144 GENERATION OF SOUND IN SOLIDS: MODULATED RADIATION

where R = [(x1 – x)2 + (y1 – y)2 + (z1 – z)2]1/2; kl and kt are the wave vectors of longitudinal and transverse waves directed from the origin of coordinates to the point x, y, z; and V1, V2, V3, V4, and V5 are the reflection coefficients of plane waves from the boundary of a solid (in this case the boundary equation is z = 0) for a longitudinal wave in the case of incidence of a longitudinal wave, a transverse wave in the case of incidence of a longitudinal wave, a transverse wave polarized in the plane perpendicular to the boundary plane in the case of an analogous transverse wave, in the case of incidence of a transverse wave, and a transverse wave polarized in the boundary plane in the case of incidence of an analogous transverse wave, respectively. The coefficients of reflection of plane waves from the boundaries of a solid depend on the angle of wave incidence at the boundary and on the boundary form. Using a delta-like character of sources at the point x, y, z, expression (6.11) can be presented in the form Φ ( x, y, z ) − Ψ1 ( x, y, z ) − Ψ2 ( x, y, z ) =

∫ {Φ( x, y, z; x1, y1, z1 )A

(1)

( x1, y1, z1 ) −

Ω1

(6.13) Ψ1 ( x, y, z; x1, y1 , z1 )[ rot rot M (1) ( x1 , y1, z1 )]γ 1 − Ψ2 ( x, y , z; x1 , y1, z1 )[ rot rot M (1) ( x1, y1, z1 )]γ 2 }dΩ1 , where [rot rot M(1)(x1, y1, z1)]γ1 and [rot rot M(1)(x1, y1, z1)]γ2 are the projections of the vector rot rot M(1)(x1, y1, z1) at the planes of incidence and boundary, respectively. Dividing expression (6.13) in the only way according to the principle of different polarizations of displacement, we obtain the next expressions: Φ ( x1 , y1 , z1 ; x, y, z ) = −

exp(ik l R ) 4πR

∫ {[exp Ql + V1 exp Pl ] ×

Ω1

∆A (1) ( x1, y1, z1 ) + V4 exp Pt [ − rot rot M (1) ( x1, y1, z1 )]γ 1 }dΩ ,

RADIATION ACOUSTICS

Ψ1 ( x1 , y1 , z1 ; x, y, z ) = −

exp(ik l R ) 4πR

145

∫ {[exp Qt + V3 exp Pt ] ×

Ω1

(6.14) [ rot rot M (1) ( x1 , y1, z1 )]γ 1 + V2 exp Pl ∆A (1) ( x1 , y1, z1 )}dΩ1 , Ψ2 ( x1 , y1 , z1 ; x, y , z ) = −

exp(ik t R ) 4πR

∫ [exp Qt + V5 exp Pt ] ×

Ω1

[ − rot rot M (1) ( x1 , y1, z1 )]γ 2 dΩ1 . If we consider only the thermal mechanism of sound generation by penetrating radiation, sources of transverse waves are absent. Therefore, taking into account expression (6.8), we obtain for the Fourier transforms with respect to the times of potentials (omitting the index (1)), Φ ( x, y , z ) =

∫ g ( x, y, z; x1 , y1 , z1 )∆A( x1 , y1 , z1 )dΩ1 ,

Ω1

(6.15) Ψ1 ( x, y, z ) =

∫ h( x, y, z; x1 , y1 , z1 )∆A( x1 , y1 , z1 )dΩ1 ,

Ω1

where  2  iω  ω g ( x, y, z; x1 , y1 , z1 ) = − − χ   c2  l 

−1

exp[i (ω / cl ) R] (exp Ql + V1 exp Pl ) , 4πR (6.16)

 2  iω  ω h( x, y, z; x1 , y1 , z1 ) = − − χ   c2  l 

−1

exp[i (ω / ct ) R] V2 exp Pl . 4πR

Let us write down in conclusion, basic equations of sound generation (we omit the factor exp(– iωt) in the case of harmonic oscillations):

146 GENERATION OF SOUND IN SOLIDS: MODULATED RADIATION

 ω2   ∆ + 2 cl 

2  div F (3 − 4 / n 2 )α 1 − iωχ / cl i  , u ∆Q − div =  2 2 4 ω 2 cε ρ c c ω χ ρ 1 / + l l 

(6.17) 2  ∆ + ω  ct2 

  rot u = − rot F .  ρct2 

We have to note also that it is convenient to consider sound fields in a solid in terms of the components of the strain tensor σik, which have the next form in spherical coordinates in the far wave field:  ∂ 2Φ 2  ∂u σ RR , σ RR = ρcl2 R = ρcl2 , σ θϕ = 0 , σ θθ = σ ϕϕ = 1 − ∂R ∂R 2  n2  (6.18) 2 2 ∂uϕ ∂u ∂ Ψ1 ∂ Ψ2 = ρct2 , σ Rϕ = ρct2 , σ Rθ = ρct2 θ = ρct2 2 ∂R ∂R ∂R ∂R 2 where uR is the solid displacement in the propagation direction of longitudinal wave, uθ is the solid displacement in the observation plane perpendicularly to the propagation direction of transverse wave, and uϕ is the displacement in the boundary plane perpendicularly to the propagation direction of transverse wave. One can see from expressions (6.18) that in fact, only the components σRR, σRθ, and σRϕ, which originate from longitudinal waves and transverse waves with polarization in the boundary plane, respectively, are independent. Figure 6.1 presents the geometry of the vector of displacement in a solid, which is caused by the presence of sound wave.

Figure 6.1 Geometry of displacement vector in a solid.

RADIATION ACOUSTICS

147

4. THERMORADIATION GENERATION OF SOUND IN A SOLID HALF-SPACE WITH A FREE BOUNDARY Let a beam of penetrating radiation be incident from the air (vacuum) upon the boundary of a solid half-space z > 0 under a certain angle with respect to the axis z. Let the beam be intensity-modulated with the sound frequency ω. It is necessary to note that the specific features of sound generation in a liquid in the case of oblique incidence of laser radiation were considered by Bozhkov, Bunkin, and Gyrdev [30]. Thermal sources of sound arise in a solid in the process of absorption of energy of penetrating radiation, Q( x, y, z , t ) = µI ( x, y ) exp(− µz / cos θ )(1 + m cos ωt ) ,

(6.19)

where µ is the coefficient of radiation absorption, I(x,y) is the radiation intensity at the boundary taking into account the coefficient of radiation transmission through this boundary, m is the modulation coefficient of radiation intensity, θ is the angle of inclination of the axis of the radiation source in a solid with respect to the axis z, which does not coincide, generally speaking, with the angle of incidence on the boundary because of the possible effect of refraction. Here we turn our attention again to the fact that the law of absorption of penetrating radiation in a solid is assumed to be exponential. This is true for photons, relativistic electrons, synchrotron radiation, and X-rays but not for protons, ions, neutral particles, low-energy electrons, etc. The effect of particular features of absorption of these particles on the sound field is considered further. It is necessary to note that dynamic sources of sound arise also apart from thermal sources. These sources have the bulk force F directed along the axis of a thermoacoustic source and equal in its absolute value (under the assumption that quanta of penetrating radiation can be treated as ultrarelativistic particles) to F = Q/u, where u is the light velocity. The presence of a surface dynamic recoil force due to partial reflection of penetrating radiation from the solid surface does not play any considerable role, as it will be seen from further consideration, since this force is directed along the normal to the surface. Using equations (6.17), we can show that the amplitude of displacements in sound waves caused by the presence of thermal sources of sound is approximately five orders of magnitude larger than the amplitudes of displacements in sound waves produced by dynamic sources of sound. Indeed, comparing the first and the second terms in the right-hand side of

148 GENERATION OF SOUND IN SOLIDS: MODULATED RADIATION

the second equation of system (6.17), we can see readily that under the condition F = Q/u and for the majority of solids, the ratio of the displacements in sound waves produced due to the thermal mechanism (the first term) to the analogous displacements caused by the dynamic mechanism (the second term) corresponds to the ratio of the light velocity to the velocity of longitudinal sound waves, i.e., c/cl ≈ 105. Therefore, dynamic sources of sound play a more or less significant role only in the cases when thermal sources do not excite all possible components of sound fields. For example, in the case of a liquid where propagation only of longitudinal waves is possible, accounting for the dynamic mechanism is senseless. At the same time, in the case of a solid, transverse waves exist also apart from longitudinal waves. Transverse waves polarized in the plane of incidence may arise in the process of reflection of longitudinal waves from a boundary and therefore, one may ignore the effect of the dynamic mechanism on sound generation for this type of waves. As for transverse waves polarized in the boundary plane, they cannot arise in the process of reflection of longitudinal waves from a boundary. Transverse waves of this polarization are excited only due to the dynamic mechanism of sound generation. Assuming now that I(x, y, z) = I0 exp [–(y2 + x2cos2ϑ)/a2] (a is the characteristic radius of penetrating radiation beam) and using the technique of boundary problem solution described above, which is based on the reciprocity principle, we obtain the next expressions for the components of strain tensor in the far wave field:

σ RR = G1G2

exp[i (ω / cl ) R] × R

exp[−ω 2 a 2 sin 2 θ (sin 2 ϕ + cos2 ϕ / cos2 ϑ ) / 4cl2 ] 1 + ω 2 cos2 θ cos2 ϑ / µ 2 cl2   cos θ cosϑ  ω K (θ ) + iM (θ ) ω , µcl  

σ Rθ = G1G2

exp[i (ω / ct ) R ] × R

×

RADIATION ACOUSTICS

exp[−ω 2 a 2 sin 2 θ (sin 2 ϕ + cos2 ϕ / cos2 ϑ ) / 4ct2 ] 1 + ω 2 cos2 θ cos2 ϑ / µ 2 ct2

149

× (6.20)

 cos θ 1 / n 2 − sin 2 θ  1 − ω  µct 

σ Rϕ = −iG3

  V2 (θ )ω , 

exp[i (ω / ct ) R] × R

exp[−ω 2 a 2 sin 2 θ (sin 2 ϕ + cos2 ϕ / cos2 ϑ ) / 4ct2 ] 1 + ω 2 cos2 θ cos2 ϑ / µ 2 ct2

G1 =

1 − iωχ / cl2 1 + ω 2 χ 2 / cl2

, G2 =

ω,

m(3 − 4 / n 2 )αa 2 I 0 ma 2 I 0 sin 2 ϕ sin ϑ , G3 = , 4cε 2cl c t

K (θ ) = 1 − V1 (θ ) , M (θ ) = 1 + V1 (θ ) ,

ϕ is the angle in the boundary plane between the projections of the axis of the thermoradiational array (source) and the direction to the observation point to the boundary. The reflection coefficients V1(θ) and V2(θ) in the case of free boundary can be written down explicitly [121], V1 (θ ) =

2 sin θ sin 2θ n 2 − sin 2 θ − (n 2 − 2 sin 2 θ ) 2 2

2 sin θ sin 2θ n −

sin 2

,

θ + (n 2 − 2 sin 2 θ ) (6.21)

V2 (θ ) = −

4n sin θ cos 2θ 1 − n 2 sin 2 θ

.

2 sin θ sin 2θ 1 − n 2 sin 2 θ + n cos 2 2θ

One can see from expressions (6.20) that the transverse waves polarized in the boundary plane exist only in the case of inclined position of the axis of the thermoradiational array (i.e., within the region of radiation absorption) with respect to the surface, i.e., under the condition that the dynamic force acts under some angle with respect to the surface. It is clear from here that

150 GENERATION OF SOUND IN SOLIDS: MODULATED RADIATION

the presence of a surface dynamic recoil force does not play a significant role because of partial reflection of penetrating radiation from the surface of a solid. This happens since the force is directed along a normal to the surface. We have to note that one can readily obtain an expression for the sound pressure p in a liquid half-space from expressions (6.20) if one takes into account the fact that the analog of sound pressure in a liquid is the component of strain tensor (–σRR) and the reflection coefficient V1(θ ) must be taken equal identically to minus one in the case of a liquid. In this case we obtain the next expression for sound pressure from expressions (6.20):

p=−

1 − iωχ / c 2 mαa 2 I 0 2 cos θ cos ϑ exp[i (ω / c) R] × ω R µc 1 + ω 2 χ 2 / c 4 2C p (6.22) exp[−ω 2 a 2 sin 2 θ (sin 2 ϕ + cos2 ϕ / cos2 ϑ )] / 4c 2 . 1 + ω 2 cos2 θ cos2 ϑ / µ 2 c 2

Here c is the sound velocity in a liquid. This formula coincides with an analogous expression by Bozhkov, Bunkin, and Gyrdev [30] if we ignore the effect of heat conductivity on the process of sound generation (χ → 0) and there is no refraction of radiation beam at the liquid boundary. In the case of normal incidence of a beam of penetrating radiation on a free liquid boundary, an expression for sound pressure takes on the form (we ignore heat conductivity), p=−

mαa 2 I 0ω 2 cos θ exp[i (ω / c) R ] exp(−ω 2 a 2 sin 2 θ ) / 4c 2 . 2C p µc R 1 + ω 2 cos 2 θ / µ 2 c 2

(6.23)

It is easy to see that this formula is analogous to expression (3.22) from Chapter 3 (see also [40]). As one can see from expressions (6.20), heat conductivity affects considerably the generated sound fields only if the frequency ω ≥ c2/χ. In this case the effect is reduced to the decrease of their amplitudes. Figure 6.2 gives the directivity patterns (in polar coordinates) of the components of strain tensor σRR, σRθ, and σRϕ determined by expressions (6.20) under the condition ω2a2/cl2 = 10 and n = 2. The directivity patterns for the components σRR and σRθ are obtained for the case of the vertical position of the axis of the thermoacoustic array, while the directivity pattern for the component σRϕ is obtained for the case ϑ = 5° and ϕ = 90°.

RADIATION ACOUSTICS

151

Figure 6.2 Directivity of a thermoradiational source in the case of normal incidence of penetrating radiation on the surface of a solid (a and b) and in the case of oblique incidence of a radiation beam (c).

The directivity pattern of the component of strain tensor σRθ due to transverse waves polarized in the observation plane has a sharp peak for small observation angles since the coefficient of reflection of these waves from the boundary is equal to zero at the zero observation angle, and, in the case of large observation angles, this component is decreased by the exponential factor, which is larger than in the case of longitudinal waves because of the smaller velocity of transverse waves ct in comparison with the velocity of longitudinal waves cl. The directivity pattern of the component of strain tensor σRϕ due to transverse waves polarized in the boundary plane, is similar to the corresponding pattern for σRR, but it decreases faster with the growth of observation angle because of the fact that the velocity ct is smaller than the velocity cl as in the previous case.

5. PARTICULAR FEATURES OF EXCITATION OF RAYLEIGH WAVES Surface waves may exist also at corresponding interfaces in solids apart from bulk waves. The simplest example of such waves is the Rayleigh wave at a free surface of a solid homogeneous isotropic half-space. This wave arises in the process of incidence of longitudinal or transverse waves polarized in the plane perpendicular to the boundary plane at this boundary. If there is some other kind of interface between two solid half-spaces or solid and liquid half-spaces instead of a free surface, the existence of

152 GENERATION OF SOUND IN SOLIDS: MODULATED RADIATION

corresponding types of surface waves is possible also. Like the Rayleigh wave, these waves are produced by superposition of incident and reflected inhomogeneous waves of the types indicated above. Another type of surface waves can exist in a solid half-space if there is a solid layer with different acoustic properties at its boundary. These are the so-called Love waves, which transform into transverse waves polarized in the boundary plane if the layer width tends to infinity. It is clear that it is necessary to take into account the dynamic mechanism of sound generation in order to consider generation of Love waves, while in the process of consideration of surface waves of the Rayleigh type, it is possible to restrict oneself to taking account of the thermal mechanism of sound generation only since the role of the dynamic mechanism is inessential in this case. As the thermal mechanism of sound generation produces only sources of longitudinal waves, the incidence angle of a “plane” longitudinal wave, which initiates the Rayleigh wave resulting from superposition of incident and reflected waves, is determined from the relationship, 2 1/ 2 2 − 1)1 / 2 , cos χ 0 = (ω 2 / cl2 − ω 2 / v R ) (ω / cl ) = i (cl2 / v R

and the incidence angle χ0 itself is expressed as follows:

χ 0 = π / 2 − i arccos(cl2 / v R2 − 1)1 / 2 , where vR is the propagation velocity of the Rayleigh wave determined from equation [100], 6 vR

ct6

−

4 8v R

 2 + 241 − ct4  3n 2

2

  vR 1      2 − 161 − 2  = 0 .  n   ct

The reflection coefficients V1(θ ′) and V2(θ ′), where θ ′ is the incidence angle of a longitudinal wave (see expressions (6.21)), we write down in the form V1(θ ′) = B(θ ′)/A1(θ ′) and V2(θ ′) = D(θ ′)/A2(θ ′). (Using expressions (6.21) we have to remember that θ ′ = θ for the coefficient V1(θ ′) and θ ′ = arcsin (n sin θ ) for the coefficient V2(θ ′) in this case. The denominator A1,2(χ0) for the incidence angle χ0 vanishes naturally.) It is necessary to expand an elementary spherical wave into plane waves in order to apply the solutions of boundary problems determined in Chapter 1 to the problem of generation of Rayleigh waves [40]:

RADIATION ACOUSTICS

exp[i (ω / cl ) R] iω / cl = R 2π

153

π / 2 − i∞ 2π

∫

∫ exp[i(ω / cl ) x sin χ cosψ +

0

0

(6.24) y sin χ sinψ ± z cos χ ] sin χ dϕ dψ ,

where the sign + or – is taken in order for the principle of radiation decay in infinity to be satisfied. The part connected with the Rayleigh wave is separated in expansion (6.24) for the incidence angle χ0: 2π

exp[i (ω / cl ) R] iω = exp[i (ω / v R )( x cosψ + R 2πv R ∫ 0 (6.25) 2 2 1/ 2 y sinψ ± i (1 − v R / cl ) z )] dψ ,

and in this case expressions (6.21) for the reflection coefficients V1(θ ′) and V2(θ ′) in the corresponding range of angles take on the form: V1* ( χ 0 ) = − B( χ 0 ) / A1' ( χ 0 ) , V2* ( χ 0 ) = − D( χ 0 ) / A2' ( χ 0 ) , where

i vR V1* ( χ 0 ) = − 2 L cl

2   2 2c l − n  2 vR 

1/ 2

 c2    L =  l − n2   v2   R 

2 1/ 2  2 2     n  cl    2 2cl  *  , V2 ( χ 0 ) = L  2 − 1  n − 2  , v R   v R     

2  3c 2 (c / v 2 − 1)cl2 / v R 2 − l − l R + 2 2 2 − 2   v c / v n R R l  

1/ 2

 4c 2   l − 1  2   vR 

 2c 2   l − n2  .  2   vR 

(6.26)

154 GENERATION OF SOUND IN SOLIDS: MODULATED RADIATION

Further direct application of the method of solution of boundary problems developed in Section 3 and integration of the expressions obtained using the saddle-point technique leads us to the next expressions for vertical and radial displacements in the far wave field of the Rayleigh wave (we should remember that radial displacements in the far wave field uR = ∂Φ/∂R – ∂Ψ1/∂z and vertical displacements uz = ∂Φ/∂z + ∂Ψ1/∂R):

uR =

 ω 2a 2 exp  − 2  4v R

2 (3 − 4 / n 2 )αa 2 I 0 m 1 − i 1 − iωχ / cl 3/ 2 8 π cε ρv R

2 2

R 1+ ω χ / c4 l

exp

  2 ϕ   2 cos  sin ϕ +  ω iV * ( χ ) + ω 1 − 1 0    cos2 ϑ     

  ω exp −  v   R 

  1−  

2 vR cl2

1/ 2  

   

cosϑ   z  1 + ω 1− µv R    

2 vR cl2

iωR × vR 1/ 2

2  vR   2 cl 

 

1/ 2 

   



 V2* ( χ 0 ) ×

−1

   

. (6.27)

2 (3 − 4 / n 2 )αI 0 m 1 − i 1 − iωχ / cl iωR × exp uz = 3 / 2 2 2 4 vR R 1+ω χ / c 8 π cε ρv R l

 ω 2a 2 exp  − 2  4v R

  2 2 ϕ   2 cos  sin ϕ +  ω iV * ( χ ) − 1 − v R 2 0    cl2 cos2 ϑ     

1/ 2   ω  ω cosϑ  v R2   1+ 1− ω  µv R  c 2     l     *

*

−1

  ω exp  − v  R 

1/ 2

   

 v2 1 − R  cl2 



 V1* ( χ 0 ) ×  

1/ 2 

   

 z ,  

where V1 (χ0) and V2 (χ0) are determined by expressions (6.26), R = (x2 + y2)1/2 is the distance from the origin of coordinates to the observation point, and other notations correspond to the notations introduced earlier.

RADIATION ACOUSTICS

155

We consider all displacements at the solid surface at z = 0, therefore expressions (6.27) may be presented in the form

uR =

 ω 2a 2 exp  − 2  4v R

2 (3 − 4 / n 2 )αa 2 I 0 m 1 − i 1 − iωχ / cl 3/ 2 8 π cε ρv R

2 2

R 1+ ω χ / c4 l

exp

  2   2  sin ϕ + cos ϕ  ω iV * ( χ ) + ω 1 − 1 0    cos2 ϑ        ω cosϑ 1 + µv R  

 1 −  

2 vR cl2

1/ 2 

   

iωR × vR 1/ 2

2  vR   cl2 



 V2* ( χ 0 ) ×  

−1

   

, (6.28)

2 (3 − 4 / n 2 )αI 0 m 1 − i 1 − iωχ / cl iωR exp × uz = 3/ 2 2 2 4 vR R 1+ω χ / c 8 π cε ρv R l

 ω 2a 2 exp  − 2  4v R

  2 ϕ  2  2 cos  sin ϕ +  ω iV * ( χ ) − 1 − v R 2 0     cl2 cos2 ϑ         ω cosϑ  1 1− +  µv R    

1/ 2 

2  vR  2  cl 

   

1/ 2

   



 V1* ( χ 0 ) ×  

−1

.

We have to note that thermoradiational excitation of surface acoustic waves in solids was considered by Askar’yan et al. [11]. Figure 6.3 presents a polar directivity pattern of radial and vertical displacements in the Rayleigh wave at different inclinations of the axis of thermoacoustic array to the normal to the surface. This diagram gives displacements at the body surface at z = 0 as a function of the observation angle ϕ with respect to the direction x. Stretching of the directivity pattern along the axis x, which accompanies increasing inclination of the axis of

156 GENERATION OF SOUND IN SOLIDS: MODULATED RADIATION

thermoacoustic array with respect to the normal to the surface can be explained by the fact that in this case the projection of the region occupied by sound sources onto the body surface differs more and more from a circle that increases the factor [−(ω2a2/4vR2) (sin2ϕ + cos2ϕ/cos2ϑ)] in expression (6.28).

Figure 6.3 Polar directivity pattern of radial and vertical displacements in a surface (Rayleigh) wave at different angles of incidence of a radiation beam at the surface of a solid. (1) Normal incidence (θ = 0°); (2) θ = 30°; and (3) θ = 45°.

6. SOLID HALF-SPACE WITH A LIQUID LAYER AT ITS SURFACE A situation may arise in some cases when a layer with different properties is present at the surface of a solid. In this connection we note a study by Lyamshev and Sedov [143] who investigated sound generation by modulated radiation in a liquid half-space with a layer of another liquid at its boundary. The authors treated the problem for the case of a liquid halfspace with a solid layer at the boundary.

Figure 6.4 Geometry of the problem of sound generation in a solid half-space with a liquid layer at its boundary.

RADIATION ACOUSTICS

157

The problem of sound generation by penetrating radiation in a solid with a liquid or solid layer with different properties at its boundary is also important. For precision, let us consider the problem of sound generation in a solid half-space with a liquid layer at its boundary. We assume that penetrating radiation is incident vertically at the free boundary of the layer. The geometry of this problem is shown in Fig. 6.4. As we assume that a beam of penetrating radiation has a circular cross-section and intensity is distributed over the beam cross-section according to the Gaussian law, transverse waves polarized in the boundary plane are absent because the symmetry of the problem. Therefore, as we have mentioned above, the dynamic mechanism of sound generation can be ignored and we may restrict ourselves to the thermal mechanism only. A sound field in a solid halfspace consists of a field of longitudinal waves and a field of transverse waves polarized in the plane perpendicular to the boundary plane. Since the field of longitudinal waves and the field of transverse waves can be separated into independent components in a unique way because of the difference in their polarization, let us consider only the field of longitudinal waves bearing in mind that expressions for the field of transverse waves can be obtained using an analogous technique. Equations (6.17) with the function Q (i.e., the energy released in unit volume per unit time) are initial. In the case of a liquid layer with the thickness H, density ρ, and sound velocity c (0 ≤ z ≤ H), the function Q has the form Q( x, y, z ) = µAmI 0

exp(− µz ) + (1 − A1 ) exp[− µ (2 H − z )] × 1 − (1 − A)(1 − A1 ) exp(−2 µH ) (6.29)  x2 + y2 exp −  a2 

 ,  

where A and A1 are the coefficients of radiation transmission through the free boundary of a layer and the boundary “layer – half-space”, respectively, I0 is the intensity of radiation beam in the center of the spot at the free boundary of a liquid, a is the radius of radiation beam, µ is the coefficient of radiation absorption in a liquid, and m is the modulation index. The energy Q released per unit time in unit volume in the solid halfspace with density ρ1 and sound velocities cl and ct (z ≥ H) is calculated as Q( x, y, z ) = µ1 AA1mI 0 ×

158 GENERATION OF SOUND IN SOLIDS: MODULATED RADIATION

exp[− µH − µ1 ( z − H )] exp[−( x 2 + y 2 ) / a 2 ] . 1 − (1 − A)(1 − A1 ) exp( −2 µH )

(6.30)

Here µ1 is the coefficient of radiation absorption in the solid. The solutions of equations (6.17) must be consistent with the boundary conditions of zero sound pressure at the free boundary of the liquid layer, parity of sound pressure in the liquid layer and normal stress in the solid half-space at the boundary of the layer and the half-space, parity of normal displacements at the boundary of the layer and the half-space, and the condition of radiation decaying at infinity. The desired solution describing a sound field in a solid half-space can be obtained using the technique for solution of boundary problems, which was developed above. In the case of the far wave field and such points of the half-space which are not too close to the boundary “layer – half-space”, we obtain

σ RR = I 0 a 2 mA

  cos θ exp iω  γ − cl  

 2 2 2 exp(iωR / cl )  ω a sin θ exp − 4R  4cl2 

  αρ1µω − ρC ×  p 

  G 2  H  × V i H A A1 ) exp( −2 µH ) ωγ − − − − 1 exp( 2 ) 1 ( 1 )( 1  

 µA exp( − µH ) sin ωγH − ωγ [1 + (1 − A ) exp( −2 µH )] 1 +  1 2+ 2 2 µ ω γ 

(6.31)

2 ωγ ( 2 − A1 ) exp( − µH ) cos ωγH  (3 − 4 / n 2 )αµ1 1 − iωρ / cl ωA1 × + cε 1 + ω 2 ρ 2 / cl2 µ 2 + ω 2γ 2 

exp[−( µ + i (ω / cl ) cos θ ) H ] (1 − V1 )(ω / cl ) cos θ + iµ1 (1 + V1 )  , 1 − (1 − A)(1 − A1 ) exp( −2 µH ) µ12 + (ω 2 / cl2 ) cos2 θ  where γ = (1/c2 – sin2θ /cl2)1/2; θ is the angle between the axis z and the direction to the observation point; R = (x2 + y2 + z2)1/2 is the distance from

RADIATION ACOUSTICS

159

the origin of coordinates to the observation point; and G, V, and V1 are the coefficients [40]:

G=

2 ρc l cos θ

 2 sin 2 θ 1 −  n2 

ρ c  1 l   cos θ 

 2 sin 2 θ 1 −  n2 

 +  

 ρc 2 sin θ  2 + sin θ 1 −  2 2 2 2 2 n n 1 − c sin θ / cl  

4 ρ1ct

 ρ c V = 1 l cos θ  

  ρ1cl  cos θ 

 2 sin 2 θ 1 −  n2 

 2 sin 2 θ 1 −  n2 

−1

2

;

 2  4ρ c ρc   + 1 t sin 2 θ 1 − sin θ − × 2 2  n n  1 − c 2 sin 2 θ / cl2   −1

  ,   (6.32)

2  4ρ c ρc  + 1 t sin 2 θ 1 − sin θ + 2 2  n n 1 − c 2 sin 2 θ / cl2 

V1 = V '− DG

exp(2iωγH ) ; 1 + V exp(2iωγH )

 ρ c  2 sin 2 θ 4ρ c sin 2 θ ρc  + 1 t sin 2 θ 1 − − 1 l 1 − V '=  cos θ  n2 n2 n2   1 − c 2 sin 2 θ / cl2    ρ1cl  cos θ 

 2 sin 2 θ 1 −  n2 

2  4ρ c ρc  + 1 t sin 2 θ 1 − sin θ + 2 2  n n 1 − c 2 sin 2 θ / cl2 

  D =  cl 1 − c 2 sin 2 θ / cl2 (1 − 2 sin 2 θ / n 2 ) 2   

−1

.

   

   ×    −1

,

160 GENERATION OF SOUND IN SOLIDS: MODULATED RADIATION

Expression (6.31) determines the sound field in a solid half-space in the Fraunhofer zone. Using this expression we can calculate the directivity patterns of sound sources for various situations. As usual the component of stress tensor σRR is the analog of sound pressure in a liquid. And in the case of water-like solids, i.e., solids with small transverse sound velocity ct, the directivity pattern of sound sources is similar to the directivity pattern of sound pressure in a liquid half-space, other conditions being equal. It is necessary to note that, in both cases of liquid and solid half-spaces, the directivity pattern may change significantly in the presence of a layer with different properties at the half-space surface. This fact must be taken into account in the process of calculation of sound fields produced in targets by various types of penetrating radiation. A more detailed analysis of sound fields in a solid half-space with a liquid layer at its boundary in some limiting cases when absorption of penetrating radiation occurs mainly either in the liquid layer or in the solid half-space, is given in the next chapter where sound generation by pulses of penetrating radiation is considered.

7. EFFICIENCY OF THERMORADIATION GENERATION OF SOUND Efficiency of sound generation η is assumed to be determined by the ratio of the power of radiated sound Pac to the power of penetrating radiation P,

η = Pac / P .

(6.33)

In the case of sound generation in a liquid, we obtain an expression for the power of radiated sound, Pac =

1 2 ρc

∫ pp

*

dS ,

(6.34)

S

where p is the pressure amplitude in the far wave field, the sign “*” means a complex conjugate value, ρ is the density of a liquid, c is the sound velocity in it, and S is the area of the section, through which sound is transmitted. In the case of solids, we distinguish the acoustic power carried by longitudinal waves Pl, transverse waves Pt, and surface acoustic waves (SAW) PSAW:

RADIATION ACOUSTICS

Pl =

1 2πcl

∫ σ l σ l dS *

, Pt =

S

1 2 ρct

161

∫ σ t σ t dS , *

S

(6.35) PSAW =

ρv SAW 4

∞

∫∫

^ ^*

u u dLdz ,

L 0

here σl and σt are the amplitudes of the components of stress tensor in the far wave field, which are caused by longitudinal and transverse sound waves, respectively; û is the amplitude of displacement velocity in SAW; L is the contour of the SAW propagation front; and vSAW is the velocity of SAW propagation; integration with respect to z takes account of the depth of SAW penetration into the solid. In the case of vertical incidence of radiation at the free surface of a liquid, an expression for sound pressure in the far wave field has the form,

p=−

 k 2 a 2 sin 2 θ µk 2 cos θ mαc exp − P  2πC p R µ 2 + k 2 cos 2 θ 4 

 ,  

(6.36)

where k = ω/c , P = πa2I0, and I0 is the radiation intensity (i.e., the radiation power per unit area) in the center of the spot. If the width of radiation beam is large in comparison with the sound wavelength, i.e., ka >> 1, then the differential dS = 2πR2sinθ dθ in expression (6.34) is approximately equal to dS = 2πR2θ dθ, and the limits of integration from 0 to π /2 may be changed approximately for the limits from 0 to ∞. In this case the efficiency of sound generation

η=

µk c  mα p  2C p µ 2 + k 2

2

  I .  0 

(6.37)

The maximum efficiency is attained under the condition µ = k and is equal to the value c  mα η max =  ρ  4C p

2

  I .  0 

(6.38)

162 GENERATION OF SOUND IN SOLIDS: MODULATED RADIATION

For example, in the case of water ηmax ≈ 5⋅10−12 I0, where I0 is measured in W/cm2 [29]. In the case of a solid half-space, using expression (6.20) we obtain for kla >> 1 an expression for the efficiency of conversion of penetrating radiation into longitudinal sound waves ηl, which is analogous to expression (6.37):  2  m(3 − 4 / n )α µkl ηl = ρ 1 + ω 2 k 2 / c 4  2cε µ 2 + k l2 l  cl

1

2

   I 0 . 

(6.39)

The efficiency of conversion of penetrating radiation into transverse sound waves in this case has the form 2

 m(3 − 4 / n 2 )α   I k 2a 2 ×  ηt =  0 t ρ 1 + ω 2 k 2 / c 4  4cε  l ct

1

(6.40) ∞

exp( − kt2 a 2θ 2 / 2) V22 (θ )dθ , 2 2 2 2 0 1 + ( kt / µ )(1 / n − sin θ )

∫

and as in the case of small angles θ, the reflection coefficient V2(θ ) → − 4θ (expression (6.21)), ηt is smaller than ηl by kt2a2/2 times (ηt ⇒ 2ηl/kt2a2). Thus, in the case of a wide radiation beam (kla >> 1), acoustic energy goes mainly into longitudinal waves. In the opposite case (kla << 1), the values of acoustic energy of longitudinal and transverse waves are comparable. In the case of the Rayleigh waves, the efficiency ηSAW has the form,

η SAW =

v R (3 − 4 / n 2 )α 2 m 2 I 0

1

2 2 2 + 2 2 4  64πcε ρ 1 − v R / cl (1 ω k / cl ) 1 + (ω / µv) 1 − v 2 / c 2    R l

×

2



  ω 2 a 2   *   exp − iV1 ( χ 0 ) + 1 − 2 2    2v R  2v R    

a 2ω 2



2

 2 vR V2* ( χ 0 ) +  cl2 

(6.41)

RADIATION ACOUSTICS

  2 V2* ( χ 0 ) + i 1 − v R V1* ( χ 0 )   cl2   *

163

2

 ,  

*

where V1 (χ0) and V2 (χ0) are taken from expression (6.26) and vR is the velocity of the Rayleigh waves. The maximum efficiency ηSAWmax is expressed as follows:

ηSAWmax =

v R (3 − 4 / n 2 )α 2 m 2 I 0

×

2

2 / 2 (1 + 2 2 / 4 ) 64πcε ρ 1 − v R cl ω k cl

(6.42)   *  iV1 ( χ 0 ) + 1 −   

2

  2 vR V2* ( χ 0 ) + V2* ( χ 0 ) + i 1 −   cl2  

2   2  vR *  V1 ( χ 0 )  . 2   cl  

It is attained at a very small depth of radiation penetration into a substance and for dimensions of radiation spot of the order of magnitude of the generated wavelength. This can be explained by the following: the deeper under the surface of a solid the sound sources are located (i.e., the larger the depth of radiation penetration into a substance is), the more the inhomogeneous wave decays on its route to the surface, where it is transformed into the Rayleigh wave. As one can see from the comparison of expressions (6.38), (6.39), and (6.42), the maximum efficiency of generation of Rayleigh waves is approximately one order of magnitude less than the maximum efficiency of generation of longitudinal waves, but the conditions for the maximum efficiency for longitudinal and Rayleigh waves are different. In some particular case the efficiency of generation of Rayleigh waves may be larger than the efficiency of generation of longitudinal waves (for µ → ∞). Figure 6.5 presents the dependence of the generation efficiency of longitudinal and Rayleigh waves as a function of the coefficient of radiation absorption µ. One can see from the figure that the efficiency of generation of longitudinal waves increases with the increase of the absorption coefficient µ, attains its maximum at the value of the absorption coefficient µ equal in the order of magnitude to the wave number of generated longitudinal wave kl, and then decreases. At the same time, the generation efficiency of Rayleigh waves increases monotonically all the time with the increase of the coefficient of radiation absorption µ, which can be explained by the fact

164 GENERATION OF SOUND IN SOLIDS: MODULATED RADIATION

that sound sources come closer to the solid surface. The asymptotic value of the efficiency of generation of Rayleigh waves is approximately one order of magnitude smaller than the maximum efficiency of generation of longitudinal waves. In the case of small values of the absorption coefficient µ, the efficiency of generation of longitudinal waves is larger than the efficiency of generation of Rayleigh waves, and if µ tends to infinity, the efficiency of generation of Rayleigh waves becomes larger than the efficiency of generation of longitudinal waves.

Figure 6.5 Schematic dependence of efficiency of generation of (1) longitudinal waves and (2) Rayleigh waves on the absorption coefficient.

8. INFLUENCE OF PARTICULAR FEATURES OF ABSORPTION OF PENETRATING RADIATION ON SOUND GENERATION In preceding paragraphs we have considered sound generation in liquids and solids by penetrating radiation when this radiation is absorbed exponentially in the medium. Exponential absorption is characteristic of optical radiation, ultra-relativistic electrons in the energy range where losses are caused basically by their radiation of photons, X-rays, and synchrotron radiation. At the same time, absorption of many types of penetrating radiation varies (see Chapter 1, Section 3). The difference in absorption of different kinds of penetrating radiation must cause the difference of sound fields generated by them. Let us treat here the problem of sound generation by penetrating radiation with an arbitrary law of absorption using a solid half-space as an example. Analogously to expression (6.20), we can write down corresponding expressions for the components of stress tensor in the far wave field under the condition that the law of absorption of penetrating radiation is arbitrary,

RADIATION ACOUSTICS

165

2 m(3 − 4 / n 2 )αa 2 I 0 1 − iωχ / cl exp(iωR / cl ) × σ RR = R 4cε 1 + ω 2 χ 2 / cl4

 ω 2 a 2 sin 2 θ (sin 2 ϕ + cos2 ϕ / cos2 ϑ )  ω × exp  − 2   4 c l  

(6.43)

  ω  ω  K (θ )Φ1  l cos θ cosϑ  + iM (θ )Φ 2  l cosθ cosϑ  , c c   l   l  2 2 m(3 − 4 / n 2 )αa 2 I 0 1 − iωa / cl exp(iωR / ct ) × σ Rθ = R 4cε 1 + ω 2 χ 2 / cl4

 ω 2 a 2 sin 2 θ (sin 2 ϕ + cos2 ϕ / cos2 ϑ  exp  − ωV2 (θ ) ×   4ct2

(6.44)

 ω  ω  1 1 2 2 iΦ 2  l 2 − sin θ cosϑ  − Φ1  l 2 − sin θ cosϑ  , n  ct   ct n  

σ Rϕ = −

ima 2 I 0 sin ϕ sin ϑ exp(iωR / ct ) × 2cct R (6.45)

 ω 2 a 2 sin 2 θ (sin 2 ϕ + cos2 ϕ / cos2 ϑ   ω exp  − ωΦ 3  l cos θ cosϑ  , 2 4ct   ct   where ∞

Q ω  ω Φ1  l cos θ cos ϑ  = ∫ sin  z cos θ  dz ,  I0  0  cl  cl

166 GENERATION OF SOUND IN SOLIDS: MODULATED RADIATION

∞

Q ω  ω Φ 2  l cos θ cos ϑ  = ∫ cos z cos θ  dz ,  I0  0  cl  cl ∞

 Q* ω  ω Φ 3  l cos θ cos ϑ  = ∫ cos z cos θ  dz ,  I0  0  ct  ct are the functions connected directly with the function Q, i.e., with the law of radiation absorption in a substance; z − / cos θ ∞  1 dz  dz l = ∫ 1 − ∫ I 0 cos ϑ  cos ϑ  0 0 

is the depth of radiation penetration into a substance; all other notations * correspond to that introduced earlier except for Q , which is a certain effective energy density for determination of the dynamic force from the next expression:       * 2 4 Q m u  F= , Q * = 1 −   u  z / cos ϑ  1 dz  2  E 1 − mu + 0k  ∫ I 0 cos ϑ     0    

−1 / 2

;

(6.46)

E0k is the initial kinetic energy of quanta of penetrating radiation; and u is the light velocity. Expression (6.46) is true in the case of arbitrary kind of penetrating radiation when its quanta do not necessarily have ultrarelativistic energy (as has been presumed in the case of quanta of penetrating radiation with an exponential law of absorption). In this case the dynamic force is determined as the product of the energy released in the medium by the factor u–1(1 + m2u2/p2)1/2 (instead of a simple division of energy released in a medium by the light velocity as has been done before), where E, p, and m are the energy, momentum, and rest mass of a quantum of penetrating radiation, respectively, which are connected by the known relativistic relationship E2 = p2u2 + m2u4. We should note that the identical notation m used in expressions (6.45) and (6.46) to denote the modulation

RADIATION ACOUSTICS

167

index and the rest mass of a quantum of penetrating radiation, respectively, must not lead to misunderstanding. Analysis of expression (6.45) shows that stress in a solid (pressure in a liquid) can differ essentially from each other in amplitude in the case of different laws of absorption of penetrating radiation, other conditions being equal, only under not too small observation angles θ. Indeed, under large observation angles θ, the basic angular dependence in expressions (6.45) is contained in exponential factors, which do not depend on the law of radiation absorption in a substance. At the same time, in the case of small observation angles, the exponential factors in expressions (6.45) may be taken to be equal to one, and the angular dependence is determined by the functions Φ1, Φ2, and Φ3, which are connected directly with the law of radiation absorption in a substance. A detailed analysis of analogous expressions for the case of sound generation by pulses of penetrating radiation is given in the next chapter.

CHAPTER 7

Pulsed Thermoradiation Sources of Sound in Solids In this chapter we consider sound generation by pulses of penetrating radiation for the same cases that have been treated earlier as applied to harmonically modulated radiation. It is expedient to consider these problems since in practice it is simpler frequently to obtain a pulse of penetrating radiation than a harmonically modulated radiation. Moreover, it is possible to attain much larger power in pulsed penetrating radiation than in a harmonically modulated one.

1. SOUND GENERATION BY RADIATION PULSES IN A SOLID HALF-SPACE Let a pulse of penetrating radiation be incident along the axis z upon the boundary of a solid half-space z > 0. Let us write down the equations describing the sound field in a solid:   ∂2 1 ∂ 1 ∂3  (3 − 4 / n 2 )αQ  2 1 , ∆ − ∆ − ∆ + Φ=−   2 2 χ ∂t 2 3 χcε ρ ∂ ∂ c t c t χ l l   169

170

PULSED THERMORADIATION SOURCES OF SOUND IN SOLIDS

 2 1 ∂2  ∆ −  Ψ = 0, u = grad Φ + rot Ψ, 2 2  c ∂ t t  

(7.1)

Q( x, y, z , t ) = µI ( x, y ) exp(− µz ) f (t ) .

(7.2)

Here f(t) is the function of time, which determines the shape of radiation pulse in such a way that max f(t) = 1. Moreover a very natural restriction is imposed on this function, that is the existence of integrals and convergence of series obtained in the process of problem solution. Assuming for precision that I = I0 exp [−(x2 + y2)/a2], where a is the characteristic radius of the radiation beam, we represent by virtue of problem symmetry the vector potential Ψ by just a single component, which we denote further just Ψ. Using the Fourier transformation with respect to time, we obtain from equations (7.1) and (7.2)   2 iω iω 3   2 ω ∆ + ∆ + ∆ +  Φ ω = χ cl2 χcl2   −

 x2 + y2  (3 − 4 / n 2 )α  exp( − µz ) F (ω ) , µI 0 exp − 2   χcε ρ a   2  ∆ + ω  ct2 

(7.3)

 Ψ = 0 ,  ω 

where ω is the circular frequency and ∞

F (ω ) =

∫ f (t ) exp(iωt )dt

−∞

is the spectrum of the radiation pulse. The solution of equations (7.3) can be written down on the basis of the reciprocity principle as follows:

RADIATION ACOUSTICS

Φ ω ( x, y , z ) =

171

∫ g ( x, y, z; x1, y1 , z1 ) ×

Ω1

   x2 + y2  2 1  exp( − µz ) F (ω ) dΩ ,  − (3 − 4 / n )α µI 0 exp − 1 1    1 χcε ρ a 2     (7.4) Ψω ( x, y, z ) =

∫ h( x, y, z; x1, y1 , z1 ) ×

Ω1

   x2 + y2  2 1  exp( − µz ) F (ω ) dΩ ,  − (3 − 4 / n )α µI 0 exp − 1 1    1 χcε ρ a 2     where g and h are the solutions of the diffraction problem on the field of the point source located at the point (x, y, z), where it is necessary to determine the sound field produced by radiation; and Ω1 is the volume occupied by the thermal sound sources produced by the action of penetrating radiation. Assuming the point (x, y, z) to be located in the far wave field, we can write down expressions for g and h in the form  2  iω  ω − g ( x, y, z; x1 , y1 , z1 ) = − χ   c2  l 

−1

exp(iωR / cl ) × 4πR

[exp( −iklx x1 − ikly y1 − iklz z1 ) + V1 (θ ) exp( −iklx x1 − ikly y1 + iklz z1 )] ,

(7.5)  2  iω  ω − h( x, y, z; x1 , y1 , z1 ) = − χ   c2  l 

−1

exp(iωR / ct ) × 4πR

V2 (θ ) exp[−iktx x1 − ikty y1 + ikt (1 / n 2 − sin 2 θ )1 / 2 z1 )] , where R = (x2 + y2 + z2)1/2 is the distance from the origin of coordinates to the observation point (x, y, z); klx, kly, and klz are the components of the

172

PULSED THERMORADIATION SOURCES OF SOUND IN SOLIDS

wave vector of a plane longitudinal wave in the axes x, y, and z, respectively, the components being directed from the origin of coordinates to the point (x, y, z); V1(θ) is the coefficient of reflection of a plane longitudinal wave; ktx and kty are the components of the wave vector of a plane transverse wave in the axes x and y, respectively, the components being directed from the origin of coordinates to the observation point (x, y, z); V2(θ) is the coefficient of reflection of a plane transverse wave in the case of incidence of a plane longitudinal wave on the boundary; and θ is the angle between the axis z and the direction to the point (x, y, z). Substituting expression (7.5) in expression (7.4) and integrating, we obtain

Φω = −

(3 − 4 / n 2 )αa 2 I 0 1 − iωτ χ exp(iωR / cl ) exp(−ω 2τ a2 / 4) × R 4cε ρ ω (1 + ω 2τ µ2 ) 1 + ω 2τ χ2 (7.6) [ωτ µ (1 − V1 (θ )) + i (1 + V1 (θ ))]F (ω ) ,

Ψω = −

(3 − 4 / n 2 )αa 2 I 0 1 − iωτ χ exp(iωR / ct ) exp(−ω 2 n 2τ a2 / 4) × R 4cε ρ 1 + ω 2τ χ2 ω (1 + ω 2τν2 ) (7.7) (i − ωτν ) F (ω )V2 (θ ) ,

where τµ = cos θ/µcl , τν = (1/n2 – sin2θ)1/2µct , and τa = (a/cl) sin θ are the characteristic delay times of sound waves from elementary thermal (thermoradiation) sources in the vertical and horizontal cross-sections of the region of sound generation; τχ = χ/cl2 is the characteristic time composed from the thermal diffusivity and the propagation velocity of longitudinal sound waves and determining the additional delay of sound waves in the case of “swelling” of thermoacoustic array because of heat conductivity. It is convenient to treat sound fields in solids in terms of components of the stress tensor. We can demonstrate that in the case under consideration only the components σRR = ρcl2∂2Φ/∂R2 and σRθ = ρct2∂2ψ/∂R2 (R and θ are the spherical coordinates), due to longitudinal and transverse waves, respectively, are really independent. Proceeding from the potentials Φ and Ψ to the components of stress tensor σRR and σRθ and using expressions (6.21) for the coefficients V1(θ) and V2(θ) in the case of the free boundary [121], we obtain the next expressions for the components of stress tensor σRR and σRθ with the help of the Fourier transformation:

RADIATION ACOUSTICS

σ RR =

(3 − 4 / n 2 )αa 2 I 0 8πcε R

∞

∫

ω exp(−ω 2τ a2 / 4) exp[iω ( R / cl − t )]

−∞

(1 + ω 2τ χ2 )(1 + ω 2τ µ2 )

173

×

(1 − iωτ χ )[ωτ µ K (θ ) + iM (θ ) F (ω )dω ,

(7.8) ∞ (3 − 4 / n 2 )αa 2 I 0 ω exp(−ωn 2τ a2 / 4) exp[iω ( R / ct − t )] σ Rθ = × 8πcε R (1 + ω 2τ χ2 )(1 + ω 2τν2 ) −∞

∫

(1 − iωτ χ )(i − ωτν ) F (ω )dω ,

where K(θ ) = 1 – V1(θ ) and M(θ ) = 1 + V1(θ ). Expressions (7.8) are the starting point for further analysis. Let us determine now the characteristic features of an acoustic field without setting a specific shape of penetrating radiation pulse or its spectrum. Let τ be the length of radiation pulse, which is determined in some way, e.g., according to the roll-off of the pulse envelope or the portion of energy contained in it. We present integrals in expression (7.8) in the form of two integrals,

σ RR =

(3 − 4 / n 2 )αa 2 I 0 8πcε R

∞  ω 2τ 2  a − exp ∫  4 − ∞ 

  F (ω ) exp iω  R − t  ×       cl 

  τµ 1 i  M (θ ) + K (θ ) ω − 2 2 2 2   τχ τχ τ χ −τ µ 1 + τ χω   

τ χ2

1

  τ  M (θ ) + µ K (θ ) dω +   τχ  

∞

2  ω 2τ 2  1 a  F (ω ) exp iω  R − t  τ µ − × exp   ∫  4   2 2 2 2 c   1 τ τ τ ω − + l     µ χ µ   −∞

 τχ  1 i  M (θ ) + K (θ ) ω − τµ τµ   

   τ  K (θ ) + χ M (θ )  dω  ,   τµ    (7.9)

174

PULSED THERMORADIATION SOURCES OF SOUND IN SOLIDS

σ Rθ =

∞  ω 2 n 2τ 2 (3 − 4 / n 2 )αa 2 I 0  a V2 (θ ) ∫ exp −  8πcε R 4 − ∞ 

2  R  τ χ 1   exp iω  − t  2 2   ct  τ χ − τν 1 + τ χ2 ω 2

 τ i 1 − ν   τ χ

 ω − 1  τχ 

  F (ω ) ×  

 τν 1 −  τχ 

   dω +  

∞

 ω 2 n 2τ 2  1 a  F (ω ) exp iω  R − t  τν × exp −    2 2 2 2   c 4   τ τ τ ω − t     ν χ ν   −∞

∫

  τχ i 1 −   τν

 1 ω −  τν 

  τ χ   dω  .  − 1  τ    ν

The spectral width of functions in the integration elements in expressions (7.9) can be determined as follows: the upper frequency limit for the spectrum of radiation pulse F(ω) is ω = cl/τ; The upper limiting frequencies ω for the exponential functions exp(−ω2τa2/4) and exp(−ω2n2τa2/4) are equal to c2/τa and c2/nτa, respectively; the upper limiting frequencies ω for the rational functions (1 + τχ2ω2)−1, (1 + τµ2ω2)−1, and (1 + τν2ω2)−1 are equal to c3/τχ , c3/τµ, and c3/τν, respectively, where c1, c2, and c3 are the constants depending on the way of determination of the spectrum width. Thus, depending on the relation of the characteristic times of the problem (τ, τa, τµ, τν, τχ), this or that integration element determines the character of the roll-off of the spectral density. We analyze the expressions obtained above for the component of stress tensor σRR at first. This component is determined by longitudinal waves. We consider the limiting relationships τ >> τa and τ << τµ, which mean that the delay time of longitudinal sound waves from elementary thermoradiation sources in the cross-section of the region of sound generation is very small or very large as against the length of laser radiation pulse. Let τ >> τa. In this case we can take exp(−ω2τa2/4) in the first expression (7.9) equal to one in the section of the frequency axis ω, where the spectrum of radiation pulse is essential, and present σRR in the next approximate form:

RADIATION ACOUSTICS

σ RR =

175

2 ∞  R  τ χ (3 − 4 / n 2 )αa 2 I 0  1   ( ) exp F i t ω ω × −   ∫  2 2 8πcε R c  τ χ − τ µ 1 + τ χ2 ω 2 − ∞   l

  τµ 1 i  M (θ + K (θ ) ω −   τχ τχ   

  τ  M (θ ) + µ K (θ )  dω +   τχ   (7.10)

∞

2  R  τ µ 1   ω ω × − ( ) exp t F i   ∫  2 2 2 2 c   1 τ τ τ ω − + l     µ χ µ −∞

  τχ 1 i  M (θ ) + K (θ ) ω −   τµ τµ   

   τ  K (θ ) + χ M (θ )  dω  .   τµ   

We can evaluate the integrals in expression (7.10) for the cases (a) τ >> τµ, τχ; (b) τ << τµ, τχ; (c) τχ << τ << τµ; and (d) τµ << τ << τχ. (a) If τ >> τµ, τχ, the roll-off rate of spectral density in both integrals (7.10) determines the spectrum of radiation pulse F(ω). In this case it is possible to expand rational functions into a series and integrate. Assuming as above that the series converges and restricting ourselves to two first terms of the series we obtain an expression,

σ RR =

(3 − 4 / n 2 )αa 2 I 0 × 8cε R (7.11)

   R  R  M (θ ) f '  t −  + τ µ K (θ ) f ' '  t −  . c c l l     We can see from here that the shape of a sound pulse depends on the envelope of radiation pulse and the parameter τµ. We have to note here that it is necessary to take into account both items in expression (7.11), not only the first one, since the quantities K(θ) and M(θ) depend on both the observation angle θ and the parameter n and change in this case from zero to two, their ratio attaining large values under certain angles θ (e.g., under small angles θ the quantity M(θ) tends to zero and K(θ) tends to two).

176

PULSED THERMORADIATION SOURCES OF SOUND IN SOLIDS

(b) If τ << τµ, τχ, the roll-off rate of the spectral density in integrals (7.10) is determined by rational functions. The spectrum of radiation pulse F(ω) in this case may be substituted approximately by the constant s (a socalled “area” of a radiation pulse): τ

F (ω ) = ∫ f (t ) exp(iωt ) dt ≈ ∫ f (θ ) dt = s .

(7.12)

0

Then the expression for the component of stress tensor σRR takes on the form,

σ RR =

   τµ (3 − 4 / n 2 )αa 2 I 0 s  1 K (θ ) ×  M (θ ) +  2 2 τχ 8cε R  τ χ − τ µ 

 t − R / cl   R  sgn t −  − 1 exp − τχ   cl   

  1 +  M (θ ) +  τ 2 − τ 2   µ χ 

  t − R / cl  τχ  R M (θ )  exp − K (θ ) sgn t −  − K (θ ) −  τµ τµ τµ   cl   

τχ

(7.13)

   .   

One can see from expression (7.13) that in this case the shape of sound pulse depends on the relation of characteristic times τµ and τχ and does not depend on the envelope of radiation pulse. If the condition τµ >> τχ is satisfied also, expression (7.13) is reduced,

σ RR =

 (3 − 4 / n 2 )αa 2 I 0 s  R  M (θ ) sgn t − 2 c l 8cε Rτ µ  

   K (θ ) ×    (7.13a)

 t − R / cl exp −  τµ 

 .  

And if the condition τχ >> τµ is satisfied, expression (7.13) is reduced as follows:

RADIATION ACOUSTICS

σ RR =

177

 τµ (3 − 4 / n 2 )αa 2 I 0 s  K (θ ) ×  M (θ ) + τχ   8cε Rτ χ2 (7.13b)

 t − R / cl   R  sgn t −  − 1 exp − τµ   cl   

 .  

(c) If τχ << τ << τµ, the roll-off rate of the spectral density in the first integral (7.10) is determined by the spectrum of radiation pulse and in the second, by a rational function. Expanding the rational function into a series in the first integral and substituting the radiation spectrum by a constant in the second, we obtain after integration

σ RR =

(3 − 4 / n 2 )αa 2 I 0 8cε Rτ µ sup 2

  R 2 K (θ ) f  t −   cl

 +   (7.14)

 t − R / cl    s  R  M (θ ) sgn t −  − K (θ )  exp − τ µ  τµ  cl    

 .  

In this case a sound pulse is practically determined by the shape of radiation pulse with a certain small addition proportional to the small parameter τ /τµ. (d) If τµ << τ << τχ, the roll-off rate of the spectral density in the first integral (7.10) is determined by a rational function, while in the second it is determined by the spectrum of radiation pulse. Substituting in this case the spectrum of radiation pulse in the first integral by a constant, and expanding the rational function in the second integral into a series, we obtain as the result of integration an expression

σ RR =

  τµ (3 − 4 / n 2 )αa 2 I 0   R s  M (θ ) + K (θ ) sgn t −  2 τ c    l χ 8cε Rτ χ  

   − 1 ×    (7.15)

  t − R / cl   − 2[τ µ K (θ ) + τ χ M (θ )] f  t − R   . exp −  c    τχ l     

178

PULSED THERMORADIATION SOURCES OF SOUND IN SOLIDS

In this case the shape of a sound pulse is determined by both the shape of radiation pulse and the ratio of the characteristic parameters τµ and τχ and also the relation of the quantities K(θ) and M(θ). Thus, in the considered case of pulses of penetrating radiation long in comparison with the characteristic time τa, the shape of the sound pulse due to longitudinal waves does not depend on the characteristic time τa and is determined either by the shape of radiation pulse or by the ratio of the characteristic times τµ and τχ or by both of them. Now let us consider the opposite case τ << τa. The spectral density is limited by the frequencies ω < 1/τa and therefore, ωτ < τ/τa << 1 in this case. Therefore, expression (7.12) is true and an expression for the component of stress tensor takes on the form

σ RR =

∞  ω 2τ 2 (3 − 4 / n 2 )αa 2 I 0 s  a − exp ∫  8πcε R 4 − ∞ 

   τµ 1 K (θ )ω − i  M (θ ) + 2 2 τχ τχ 1 + τ χ ω    1

 τ2  χ  exp iω  R − t   ×   2 c 2    l τ τ −      µ   χ    τµ K (θ ) dω +  M (θ ) + τχ    (7.16)

∞

 ω 2τ 2  1 a  exp iω  R − t  τ µ exp − ×    2 2 2ω 2  4  c   1 τ τ τ − + l     µ χ µ   −∞ 2

∫

   τχ 1 K (θ )ω − i  M (θ ) + τ τ µ µ   

   τχ M (θ ) dω .  K (θ ) + τµ   

It is possible to integrate expression (7.16):

σ RR =

2 2  τµ (3 − 4 / n 2 )αa 2 I 0 s  exp(τ a / 4τ χ )  K (θ ) ×  M (θ ) +  2 2 8πcε R τχ    τ µ − τ χ

 t − R / cl exp  τχ 

   Erfc τ a + t − R / cl   2τ χ τa  

 exp(τ a2 / 4τ µ2 ) + ×  τ µ2 − τ χ2  (7.17)

RADIATION ACOUSTICS

  τχ V1 (θ )1 −  τµ    τχ 1 +  τµ 

   exp − t − R / cl   τµ  

   exp t − R / cl   τµ  

179

   Erfc τ a − t − R / cl   2τ µ τa  

   Erfc τ a + t − R / cl   2τ µ τa  

 −  

    ,   

where Erfc (x) is the complementary error function. Thus we can see from expression (7.17) that in the case τ << τa, the shape of the sound pulse due to longitudinal waves does not depend on the shape of radiation pulse and is determined only by the ratios of the characteristic times τa, τµ, and τχ. Expressions (7.17) are reduced considerably in four limiting cases. (a) If τa >> τµ, τχ, the roll-off rate of the spectral density in expression (7.16) is determined by an exponential function. In this case rational functions can be expanded into a series, and we may restrict ourselves to the first two terms of the expansion. We obtain in the result

σ RR =

(3 − 4 / n 2 )αa 2 I 0 s 2 π cε Rτ a3

 (t − R / c 2 )  l × exp − 2   τa   (7.18)

   (t − R / c ) 2  R l τ K (θ ) .  t −  M (θ ) + 1 −   µ  cl   τ a2     In this case the shape of a sound pulse is determined by both the ratio of the characteristic times τa and τµ and the ratio of the quantities K(θ) and M(θ). (b) If τa << τµ, τχ, the roll-off rate of spectral density in expression (7.16) is determined by rational functions. In this case the exponential function can be taken to be equal to one. Finally, this case becomes analogous to the one considered above and the result of integration is given by formula (7.13). (c) If τχ << τa << τµ, the roll-off rate of spectral density in the first integral of expression (7.16) is determined by the exponential function and in the second integral, by a rational function. Taking correspondingly the denominator of the rational function in the first integral to be equal to one, and assuming that the exponential function in the second integral is also equal to one, we obtain as the result of integration

180

PULSED THERMORADIATION SOURCES OF SOUND IN SOLIDS

σ RR =

 (t − R / c ) 2 τ (3 − 4 / n 2 )αa 2 I 0 s  l 2 π K (θ ) µ exp − 2 2  τ  a τa 8πcε Rτ µ  

 +   (7.19)

 t − R / cl π (M (θ ) sgn(t − R / cl ) − K (θ ) ) exp −  τµ 

  .  

In this case the pulse shape is determined by both the relation of the characteristic times τa and τµ and the relation of the quantities K(θ) and M(θ). (d) If τµ << τa << τχ, the roll-off rate of the spectral density in the first integral of expression (7.16) is determined by a rational function and in the second integral, it is determined by an exponential function. Taking correspondingly the exponential function in the first integral to be equal to one, and equating the denominator of the rational function in the second integral to one also, we obtain the result

σ RR =

  τµ (3 − 4 / n 2 )αa 2 I 0 s   R K (θ )  sgn t − π  M (θ ) + 2  τχ 8πcε Rτ χ      cl

   − 1 ×    (7.20)

 exp  

 (t − R / c ) 2   t − R / cl  2 π l  . + τ µ K (θ ) + τ χ M (θ ) exp − 2  τa    τχ τa   

[

]

In this case the shape of sound pulse is determined by both the relationship of the characteristic times τa, τµ, and τχ and the relationship of the quantities K(θ) and M(θ). Let us return to the component of stress tensor σRθ due to transverse waves. All expressions obtained above for the component of stress tensor σRR can be rewritten for the component σRθ by simple changing of cl for ct, τa for nτa, τµ for τν, M(θ) for V2(θ), and K(θ) for M(θ). We have to remember in this case that under the observation angles θ > arcsin (1/n) the quantity τν becomes imaginary and the reflection coefficient V2(θ) becomes complex. Only real parts of expressions obtained in such a way have physical sense. Therefore, in the case of the observation angles θ > arcsin (1/n), it is necessary to apply the operation of separation of the real part to the expressions obtained. If the quantity τ is complex, it is necessary to compare the absolute value of the parameter τν with other characteristic

RADIATION ACOUSTICS

181

parameters of the problem in order to determine which of the formulae given above corresponds to this situation. We should note also that in situations analogous to the ones described by expressions (7.13), (7.14), and (7.19), the component of stress tensor σRθ has an oscillating term in the case of the observation angles θ > arcsin (1/n). According to expressions (7.13), (7.14), and (7.19), this term is not decreasing in time. This is certainly the result of the approximations made in the process of derivation of these expressions from complete formulae (7.9). Decreasing is determined by the parameter τa, which is ignored in expressions (7.13), (7.14), and (7.19). We can demonstrate that the expressions obtained above may be extended easily to the case of oblique incidence of a pulse of penetrating radiation at a solid boundary. This generalization is attained by introduction of new parameters, a sin θ τa = cl

1/ 2

2   2  sin ϕ + cos ϕ   cos 2 ϑ  

τν =

, τµ =

cosθ cos ϑ , µcl

(1 / n 2 − sin 2 θ ) cos ϑ , µcl

where ϑ is the angle of inclination of the axis of the thermoacoustic array with respect to the axis z, which does not coincide, generally speaking, with the angle of radiation incidence at the boundary because of possible refraction; and ϕ is the angle in the boundary plane between the projections of the axis of thermoacoustic array and the direction to the observation point onto the boundary. Thus, the conducted study allows us to make the following conclusions. Resulting from action of penetrating radiation on a homogeneous and isotropic solid half-space, longitudinal and transverse sound waves arise in its depth, the transverse waves being due only to reflection of longitudinal waves from the solid boundary if the thermal mechanism of sound generation is dominant. Heat conductivity affects essentially the shape of generated sound pulses only in the case when the conditions τχ > τa, τµ, τ (or τχ > nτa, τν, τ for transverse waves) are satisfied. In order to understand better the meaning of these conditions, let us consider the characteristic dimensions of the thermoacoustic array a and 1/µ and also the characteristic dimension clτχ = χ/cl instead of the characteristic times τa, τµ, and τν. The value of χ/cl in the case of good heat conductors does not exceed 10 µm, while it is even smaller in the case of other media. As for the characteristic dimensions of a

182

PULSED THERMORADIATION SOURCES OF SOUND IN SOLIDS

thermoradiation source, the radii of radiation beams (i.e., the parameter a) are usually not smaller than 1 mm and the values of the depth of radiation penetration into a substance (i.e., the parameter 1/µ) can change within a very wide range depending on the radiation wavelength or the radiation length and the substance in which radiation is absorbed. In this case satisfaction of the condition τχ > τa, τµ or τχ > nτa is possible only when the depth of radiation penetration into a substance 1/µ is smaller than the characteristic dimension χ/cl, and the observation angle θ is small. Apparently, the condition 1/µ < χ/cl is realized only in the case of the skin effect, i.e., in the case of absorption of optical radiation for example in good conductors of electricity (e.g., metals). In the case of long radiation pulses τ >> τa, τµ (but τ < τχ still), the shape of sound pulses is determined by the envelope of radiation pulse and the ratio of the characteristic times τa and τχ and also the relation between the quantities K(θ) and M(θ) (see expression (7.15)). In the case of short radiation pulses (τ << τa), the shape of sound pulses is determined by the relationship between the characteristic times τa, τµ, and τχ and also the relationship between the quantities K(θ) and M(θ) (see expression (7.20)). In all other cases the influence of heat conductivity on the shape of sound pulses is negligible and may be ignored. In this case sound pulses produced by longitudinal waves are governed by the following laws. If the length of radiation pulse is large in comparison with the characteristic time τa (τ >> τa), the shape of sound pulses does not depend on the parameter τa and is determined by the envelope of radiation pulse, the parameter τµ, and also the relation between the quantities K(θ) and M(θ) (see expressions (7.18) and (7.19)). Everything said above with respect to sound pulses produced by longitudinal waves is applicable to sound pulses produced by transverse waves with one addition. In the case of the observation angles θ > arcsin (1/n) in situations analogous to those described by expressions (7.14) and (7.19), oscillating components appear in sound pulses, the rolloff of the envelope of these oscillations not being taken into account by expressions (7.14) and (7.19). This roll-off must be determined from complete expressions (7.9). The conditions of pairwise equality of the characteristic times τ, τa, τµ, and τχ determine pairwise the characteristic angles of the problem

θ 0 = arctan( µa ) , τ a = τ µ ; θ1 = arccos( µclτ ) , τ = τ µ ; θ 2 = arcsin( clτ / a ) , τ = τ a ; θ 3 = arcsin( χ / acl ) , τ χ = τ a .

RADIATION ACOUSTICS

183

The next six cases are possible: (a) τχ > τ > τ0; (b) τχ > τ = τ0; (c) τχ < τ < τ0; (d) τχ < τ = τ0; (e) τ0 < τχ < τ; and (f) τ < τχ < τ0; where τ0 = (a/cl)(1 + µ2a2)−1/2. In this case the ranges of application of the expressions obtained above for each of the six considered cases and different observation angles θ are given in Fig. 7.1.

Figure 7.1 Ranges of applicability of expressions (7.11) — (7.20) for different observation angles. Digits in parentheses correspond to the expression numbers.

In the case of irradiation of a metal surface by laser pulses, i.e., when the depth of radiation penetration into a substance is small and determined by the skin effect, sound pulses produced by such radiation in both the cases, when the aggregate state of the metal changes and when it does not, have been given by Hutchins, Dewhurst, and Palmer [222]. In conclusion we should note that if we take M(θ) = 0, K(θ) = 2, and τχ = 0 in expressions obtained for sound pulses produced by longitudinal

184

PULSED THERMORADIATION SOURCES OF SOUND IN SOLIDS

waves and take into account necessary changes in notations, these expressions coincide completely with the analogous expressions obtained above (see Chapter 3, Section 3 and a paper by Kasoev and Lyamshev [110]) while treating the problem of sound generation in a liquid by radiation (laser) pulses not taking into account the effect of heat conductivity.

2. EXCITATION OF RAYLEIGH WAVE BY RADIATION PULSES As was done in Chapter 6, we can write down corresponding expressions for radial and vertical displacements in the far field of the Rayleigh wave excited by pulses of penetrating radiation,

uR =

(3 − 4 / n 2 )αa 2 I 0 1 − i 16π π cε ρv R v R

R

∞ 1 − iωτ χ

∫

2 2 −∞ 1 + ω τ χ

F (ω ) ×

 ω 2τ 2  a  exp iω  R − t  × exp −     4    v R     1  1 + ω τ µ

 ω ∗ i ωV1 ( χ 0 ) + ω 

  2 2 ∗ 1 − vR / cl V2 ( χ 0 ) dω ,   (7.21)

∞ 1 − iωτ (3 − 4 / n )αa I 0 1 − i χ F (ω ) × uz = 2 2 16π π cε ρv R v R R −∞ 1 + ω τ χ 2

2

∫

 ω 2τ 2  a  exp iω  R − t  × exp −     4     v R    1  1 + ω τµ 

 ω ∗ i ωV2 ( χ 0 ) − ω 

  2 2 ∗ 1 − vR / cl V1 ( χ 0 ) dω .  

RADIATION ACOUSTICS

185

Here τa = a/vR , τµ = (µ/vR) (1 – vR2/cl2)1/2, vR is the propagation velocity of Rayleigh wave, and all other notations correspond to those introduced earlier. If the length of radiation pulse τ is small as against the characteristic times τa and τχ, expressions for displacements take on the form

uR =

(3 − 4 / n 2 )αa 2 I 0 2 AS

∞

ω exp(−ω 2τ a2 / 4)

∫ R (1 + ωτ

8π π cε ρv R v R

0

2 2 µ )(1 + ω τ χ )

×

  π π    R   R − t  dω , − t  − ωτ χ sin  − ω  cos − ω    4    vR   vR  4 (7.22) uz =

(3 − 4 / n 2 )αa 2 I 0 2 BS 8π π cε ρv R v R

R

∞

ω exp( −ω 2τ a2 / 4)

∫ (1 + ωτ 0

2 2 µ )(1 + ω τ χ )

×

  π π    R   R − t  dω , − t  + ωτ χ cos − ω  sin  − ω    4  4    vR   vR where 2 2 2 2 ∗ / cl , B = V2∗ ( χ 0 ) + i 1 − v R / cl V1 ( χ 0 ) ; A = iV1∗ ( χ 0 ) + V2∗ ( χ 0 ) 1 − v R

(7.23) ∞

s=

∫ f (t )dt 0

is the “area” of radiation pulse. If n = 2 then A and B are approximately equal to 2. One can see that in the case of short radiation pulses, the shape of sound pulse does not depend on the shape of radiation pulse. If we ignore the effect of heat conductivity, i.e., take τχ = 0 which is always possible when the dimension of a beam of penetrating radiation exceeds 10−5 m, the shapes of sound pulses described by expressions (7.22) are determined only by the quantities τa and τµ. Figure 7.2 gives the shapes of sound pulses for vertical displacements in the case of short radiation pulses and with absence of heat conductivity. One can see that as the depth

186

PULSED THERMORADIATION SOURCES OF SOUND IN SOLIDS

of radiation penetration into a substance increases, the amplitude of a sound pulse decreases and its length grows. If the dimension of radiation beam is much smaller than 10−5 m, i.e., τa << τµ, we can take τa = 0 in expressions (7.22). In this case the shape of a sound pulse depends on heat conductivity (Fig. 7.3) and differs considerably from the shapes of sound pulses in the case when it is possible to ignore heat conductivity.

Figure 7.2 Shapes of sound pulses for vertical displacements in the Rayleigh wave in the case of short radiation pulses and when it is possible to ignore heat conductivity. (1) τa > τµ and (2) τµ = 5τa.

Figure 7.3 Shapes of sound pulses for radial (curve 1) and vertical (curve 2) displacements in the Rayleigh wave in the case of short radiation pulses and strong influence of heat conductivity on the process of sound generation.

3. SOUND GENERATION IN A SOLID HALF-SPACE IN THE PRESENCE OF A LIQUID LAYER AT ITS SURFACE Instead of expression (6.31) characterizing harmonically modulated penetrating radiation, we obtain an analogous expression for a sound pulse generated by a pulse of penetrating radiation in a solid half-space

RADIATION ACOUSTICS

I a2 σ RR ( x, y, z, t ) = 0 8πR

 2 2 2  R   ω a sin θ   exp i t exp ω − −   ∫   cl   4cl2 −∞  ∞

 αρ µω   cos θ 1 exp iω  γ − − cl  ρC p  

187

   AF (ω ) × 

  G  H  ×   1 − V exp( 2iωγH )

 µA exp( − µH ) sin(ωγH ) 2  1 − 1 − (1 − A)(1 − A1 ) exp( −2 µH )  µ 2 + ω 2γ 2 (7.24)

ωγ ( 2 − A1 ) exp( − µH ) cos(ωγH ) + ωγ (1 − A1 ) exp( −2 µH )  −  µ 2 + ω 2γ 2  2 (3 − 4 / n 2 )αµ1 1 − iωχ / cl ωA1 exp{−[ µ + (iω / cl ) cos θ ]H } × cε 1 + ω 2 χ 2 / c 4 1 − (1 − A)(1 − A1 ) exp( −2 µH ) l

(1 − V1 )(ω / cl ) cos θ + iµ1 (1 + V1 )  dω , µ12 + (ω 2 / cl2 ) cos2 θ  where all notations correspond to the ones introduced in Chapter 6. We have to remember that here, as in Chapter 6, we consider for definiteness only the field of longitudinal waves, and a beam of penetrating radiation is incident vertically at the free surface of a liquid layer. The field of longitudinal waves can be separated unambiguously from the field of transverse waves not only according to the character of polarization but also according to the difference of arrival times of sound pulses. Expression (7.24) is rather complex, and therefore it is interesting to analyze it for different limiting cases. Let exp(−µH) ≈ 1, i.e., absorption of penetrating radiation occurs mainly in the solid half-space. Then, expression (7.24) can be rearranged to the form

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PULSED THERMORADIATION SOURCES OF SOUND IN SOLIDS

 2 2 2 I 2 ∞  R   ω a sin θ σ RR ( x, y, z, t ) = 0a ∫ exp iω  − t  exp − 8πR    cl 4cl2  −∞   αρ µω   cosθ 1 exp iω  γ − − cl  ρC p  

   AF (ω ) × 

   H  ×  

2G[ µA1 sin(ωγH ) + 2ωγ ( A1 − 2) sin 2 (ωγH / 2) [1 − V exp(2iωH )]( A + A1 − AA1 )( µ 2 + ω 2γ 2 )

+

(7.25)

2 (3 − 4 / n 2 )αµ1 1 − iωχ / cl × cε 1 + ω 2 χ 2 / c4 l

ωA1 exp[−(iω / cl ) cosθ H ] (1 − V1 )(ω / cl ) cos θ + iµ1 (1 + V1 )  dω . A + A1 − AA1 µ12 + (ω 2 / cl2 ) cos2 θ  In this case the layer affects the sound field in the solid half-space mainly because of its wave properties. If the layer thickness H is much smaller than the radius of radiation beam a in this case, the expression for the sound field in the solid half-space gets reduced to a greater extent for not very small observation angles θ (since the product of frequency by the layer thickness can be treated as a small quantity and the exponents containing this product are equal to one),

σ RR ( x, y, z, t ) = ∞

(3 − 4 / n 2 )αµ1 AA1 I0a 2 × 8πcε R A + A1 + AA1

 ω 2 a 2 sin 2 θ  exp ∫  − 4c 2 l −∞ 

  exp iω  R − t  F (ω ) ×        cl 

(7.26)

RADIATION ACOUSTICS

1 − iωχ / cl2 ω (1 − V1 )(ω / cl ) cos θ + iµ (1 + V1 ) 1 + ω 2 χ 2 / cl4

µ12 + (ω 2 / cl2 ) cos2 θ

189

dω ,

where the reflection coefficient is reduced now to the form V1 = V′ – DG/(1 + V). If we take A = 1 and the value of ρc for the liquid layer equal to zero in expression (7.26), this expression coincides with the analogous expression (7.8) for the sound field in a solid half-space, and the reflection coefficient V1 is the coefficient of reflection of a plane longitudinal wave from a free boundary of a solid half-space. Let us now consider the opposite limiting case exp (−µH) << 1 when absorption of penetrating radiation occurs mainly in the liquid layer. Then, expression (7.24) can be rearranged to the form  2 2 2  ω a sin θ exp − ∫  4cl2 −∞ 

αρ1µ I 0 a 2γA σ RR ( x, y, z, t ) = 4πC p R

∞

   ×  (7.27)

 R  ω 2 F (ω ) G exp[iω (γ − cos θ / cl ) H ] dω . exp iω  − t  1 − V exp(2iωγH )  µ 2 + ω 2γ 2   cl Moreover, if the layer thickness H is much smaller than the radius of radiation beam a, then in the case of not too small observation angles θ, expression (7.27) is reduced (the exponents containing the product of frequency by the layer thickness can be taken equal to one and the quantity ω2γ2 in the denominator of expression (7.27) may be ignored as against µ2):

σ RR ( x, y, z, t ) =

ρ1 G αγ I 0a 2 4πC p µR ρ 1−V

 2 2 2  ω a sin θ exp − ∫  4cl2 −∞  ∞

 R  exp iω  − t  ω 2 F (ω )dω ,    cl where

   ×  (7.28)

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PULSED THERMORADIATION SOURCES OF SOUND IN SOLIDS

cl (1 − sin 2 θ / n 2 ) 1 − c 2 sin 2 θ / cl2 G . = 1−V cos θ Now let the length of the penetrating radiation pulse be greater than the values of a sin θ/cl or H/cl. In this case the roll-off rate of the spectral density in expression (7.28) is determined by the spectrum of radiation pulse F(ω) and the expression for sound field in a solid half-space takes on the form

σ RR ( x, y, z, t ) = −

ρ1

αγ

2C p µR ρ

I 0a 2

 G R f ' '  t − 1−V c l 

 .  

(7.29)

The shape of a sound pulse is determined by the second derivative of the envelope of radiation pulse with respect to time and the angular dependence is determined by the quantity γG/(1 – V). On the contrary, if the length of radiation pulse is much smaller than the quantity a sin θ/cl, the roll-off of the spectral density in expression (7.28) is determined by the quantity exp(−ω2a2 sin2θ/4cl2), and the pulse spectrum can be changed approximately for a constant ∞

F (ω ) ≈

∫ f (t )dt = s . 0

In this case the shape of a sound pulse is described by an expression 2 2 2 ρ1 αγs 2 G a sin θ / cl − 2( R / cl − t ) × I0a σ RR ( x, y, z, t ) = 1−V π C p µR ρ a 5 sin 5 θ / c 5 l

(7.30) 2

2

2

exp{−[( R / cl ) − 1] /[(a sin θ ) / cl2 ]} , and does not depend on the shape of radiation pulse but is determined by the radius of the radiation beam a. Figure 7.4 presents the shape of a sound pulse in a solid half-space when radiation is absorbed mainly in the liquid layer and the length of radiation pulse is small (expression (7.30)).

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191

As one can see from comparison of expressions (7.27) and (7.25), the presence of a liquid layer at the surface of a solid half-space may lead in some situations to essential changes in sound field as compared with the case when a liquid layer is absent. This is true also for high-energy single particles when the “length of radiation pulse” is always small in comparison with other characteristic times of the process of sound generation.

Figure 7.4 Shape of sound pulse originating from longitudinal waves in a solid halfspace for the case, when a pulse of penetrating radiation is absorbed in a liquid layer and its length is small.

The conclusion on the fact that a layer of another substance at the surface of a solid half-space may substantially affect the characteristics of a sound field generated in the half-space by penetrating radiation, is true also for other possible situations and not only for the case of the presence of a liquid layer at the surface of a solid. For example, this fact must be taken into account when a layer of another substance may be found occasionally on a sample under test. This condition may be used also deliberately if one covers a sample with some kind of coating with preliminary set radiation and acoustic properties in order to obtain the necessary acoustic fields in the sample in the case of a specific type of penetrating radiation.

4. EFFICIENCY OF SOUND GENERATION All notations used in this section correspond to the notations introduced in Chapter 6, Section 8. In the case of sound generation by pulses of penetrating radiation we should write down the next expression instead of expression (6.34) (and expression (6.35)): P=

1 p 2 dS . ρc ∫ S

(7.31)

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PULSED THERMORADIATION SOURCES OF SOUND IN SOLIDS

The difference between expressions (6.34) and (7.24) is connected with the fact that the value average with respect to time during the period of harmonic oscillations is considered in expression (6.34). An expression for sound pressure p in the case of “long” radiation pulses (the length of radiation pulse τ is greater than other characteristic times of the process of sound generation) has the form

p ( x, y , z , t ) = −

αI 0 a 2 cos θ R  f ''t −  . c 2C p R µc 

(7.32)

In this case the conversion efficiency is expressed as follows:

η=

αa 2 I 0

f ' ' (t ) a 2 I 0 c a2 . ≈ 6C p ρ µ 2 c 4τ 4 6 ρc 3C 2p µ 2 f (t )

(7.33)

As compared with the maximum conversion efficiency, the conversion efficiency in a pulse at the same radiation intensity I0 is smaller in this case a2/(µ2c4τ4) << 1 times. This may be explained by the fact that a radiation pulse excites a broad spectrum of sound harmonics. Some of these harmonics do not satisfy the condition of maximum efficiency of sound generation. However, we have to note that the intensity I0 in a radiation pulse can be greater than the intensity of harmonically modulated radiation. In this sense utilization of a pulsed operation mode is more efficient. It is possible to obtain analogous expressions for ηl, ηt, and ηSAW for the case of a solid half-space and “long” radiation pulses. In this case acoustic energies of longitudinal and transverse waves are comparable and the acoustic energy of Rayleigh waves depends on the depth of radiation penetration into a solid for any kind of relationships between a and µ (i.e., between the beam width and the penetration depth of radiation into a solid). In the case of “short” radiation pulses (the length τ is much smaller than other characteristic times of the process of sound generation), a sound pulse is proportional to the length of radiation pulse τ (or the total pulse energy E) as follows from expression (7.9). In the case of “long” radiation pulses the efficiency of sound generation is inversely proportional to τ4 (see expression (7.33)), while in the case of “short” radiation pulses the generation efficiency is proportional to τ2:

η ~ I 0τ 2 .

(7.34)

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193

If radiation pulses are very “short”, the intensity I0 can be very large also. As for energy distribution between longitudinal, transverse, and surface waves, in the case of “short” radiation pulses the spectrum of an acoustic pulse is determined by the dimensions of the radiation beam and the penetration depth of radiation into a solid. If a >> µ−1, energy is utilized mainly for excitation of longitudinal and Rayleigh waves. And if a << µ−1, the energies of longitudinal and transverse waves are comparable and the energy of Rayleigh waves is small.

5. INFLUENCE OF PARTICULAR FEATURES OF ABSORPTION OF PENETRATING RADIATION Let a pulse of penetrating radiation be incident on the surface of a solid and not a beam of harmonically modulated radiation as has been described in Chapter 6, Section 4. We consider a sound pulse taking as an example a sound pulse of longitudinal waves described by the component of stress tensor σRR. Other components of stress tensor can be treated in the same way as has been done before (all notations correspond to the notations of Chapter 6, Section 4). In this case a sound pulse is described by the following expression:

σ RR ( x, y, z, t ) =

(3 − 4 / n 2 )αa 2 I 0 8πcε R

∞ 1 − iωτ χ

 ω 2τ 2 a exp − 2 2  4  −∞ 1 + ω τ χ

∫

 ×   (7.35)

 R  exp iω  − t  ω [ K (θ )Φ1 (ωτ µ ) + iM (θ )Φ 2 (ωτ µ )]F (ω ) dω ,   cl  where τa = (a/cl) sin θ, τµ = (l/cl) cos θ cos ϑ, and τχ = χ/cl2 are the characteristic delay times of sound from elementary thermal sources in the horizontal and vertical sections of the region of sound generation. We analyze expression (7.35) in the same way as has been done in Section 1 of this chapter. We have to note first of all that situations are possible when the shape of a sound pulse does not depend on particular features of absorption of penetrating radiation in a medium, i.e., on the specific form of the functions Φ1(ωτµ) and Φ2(ωτµ), but is determined by other characteristic parameters of the problem. Indeed, if the length of penetrating radiation pulse τ is greater than other characteristic times of the

194

PULSED THERMORADIATION SOURCES OF SOUND IN SOLIDS

process of sound generation, i.e., τ >> τa, τl, τχ, the roll-off of the spectral density in the integration element in expression (7.35) is determined by the spectrum of radiation pulse F(ω). In this case we may take the exponent in expressions (7.35) to be equal to zero, the parameter τχ equal to one, and the functions Φ1(ωτµ) and Φ2(ωτµ) may be expanded into a series. We restrict ourselves to the first terms of the expansion Φ1(ωτµ) ≈ ωτµΦ1′(0) and Φ2(ωτµ) ≈ Φ2(0). The shape of a sound pulse is determined in this case by the first and second derivatives of the envelope of the penetrating radiation pulse with respect to time,

σ RR ( x, y, z, t ) = −

(3 − 4 / n 2 )αa 2 I 0 4cε R

  R  M (θ )Φ 2 (0) f '  t − c l  

 +   (7.36)

 R  τ l K (θ )Φ1′ (0) f ' '  t −  .  cl  If we consider short radiation pulses, observation is conducted under large angles θ, and the value of heat conductivity is small, i.e., τa >> τ, τµ, τχ, then the roll-off of the spectral density in the integration element in expression (7.35) is determined by the exponential function (−ω2τa2/4). The spectrum of the penetrating radiation pulse can be changed in this case for a constant equal approximately to τ and the functions Φ1(ωτµ) and Φ2(ωτµ) can be expanded into a series as before. In this case we also restrict ourselves to the first terms of the expansion. The parameter τχ is taken to be equal to zero again. An expression for a sound pulse takes on the form

σ RR ( x, y, z, t ) =

(3 − 4 / n 2 )αa 2 I 0 2 π cε Rτ a3

 (t − R / c ) 2 l exp − 2  τ a 

 ×   (7.37)

   2( t − R / c ) 2  R l τ Φ ′ (0) K (θ ) .  t − Φ 2 (0) M (θ ) + 1 − 1 µ    cl   τ a2   In this case the shape of a sound pulse is also determined by the characteristic time τa and does not depend on the particular features of radiation absorption in a substance. In other cases, i.e., when the length of radiation pulse is very small, observation is performed under small angles θ, and under the considerable influence of heat conductivity, the roll-off of the spectral density in the

RADIATION ACOUSTICS

195

integration element in expression (7.35) is determined by the functions Φ1(ωτµ), Φ2(ωτµ), and (1 − iωτχ)/(1 + ω2τχ2). In other words, it is necessary to know a specific form of the function of energy release Q in order to determine the shape of a sound pulse. Some dependences of rather complex functions of energy release are given in Chapter 1, Section 1. In the case when absorption of penetrating radiation in a medium occurs exponentially, the functions Φ1(ωτµ) and Φ2(ωτµ) can be written down in an explicit form Φ1(ωτµ) = ωτµ /(1 + ω2τµ2) and Φ2(ωτµ) = 1/(1 + ω2τµ2). The main features of generated sound fields in the case of such forms of the functions Φ1(ωτµ) and Φ2(ωτµ) have been considered in Section 1 of this chapter.

6. THERMORADIATION GENERATION OF SOUND BY PULSES OF NON-RELATIVISTIC PROTONS We consider sound generation by a non-relativistic proton beam as an example. Lifshits and Pitaevskii considered this case for a liquid [123]. Energy losses by protons for ionization are described by formula (1.1). We should note that the function z / cos ϑ

1−

∫

0

Q dz I 0 cos ϑ

differs from the energy of a proton E only in normalization with respect to the initial energy of a proton E0. Formula (1.1) is true only up to the value of the energy E∗ when a proton captures an electron. If E ≤ E∗, the rate of energy loss decreases sharply and formula (1.1) becomes inapplicable. For example, the value E∗ for water (or ice) is approximate 1 MeV. Thus, if protons with energy 100 MeV are emitted, approximately 99% of energy is released according to formula (1.1). Analysis of this formula shows that energy loss by a non-relativistic proton increases with the decrease of its energy or with distance traveled by it in a substance. If we approximate the dependence of energy loss by an increasing exponential function and break it at the value E = E∗, we can show that Φ1 (ωτ µ ) ≈

2ωτ µ E0 sin(ωτ µ ) − ωτ µ cos(ωτ µ ) + , 2 E∗ 1 + ω 2τ 2 1 + ω 2τ 2 µ

µ

196

PULSED THERMORADIATION SOURCES OF SOUND IN SOLIDS

4E m   M 4 µ −1 = E02 4π e NZ ln 0  m IM  

−1

,

where M and e are the mass and charge of a proton, m is the mass of an electron, N is the number of substance atoms per unit volume, Z is the nuclear charge of substance atoms, and I is the average energy of excitation of substance atoms. Thus, µ−1 ≈ 25 cm at E = 100 MeV in the case of water (and ice). If such a proton beam is incident vertically on the surface of a solid, the pulse length of the proton beam is small, and the observation angle θ is small also, i.e., at τl >> τa, τ, τχ, then a sound pulse represents basically two pulses of positive excessive pressure of length τ in contrast to the pulse of negative excessive pressure in the case of absorption of penetrating radiation under the exponential absorption law in the analogous situation (formula (7.14)):

σ RR ≈ −

 (3 − 4 / n 2 )αa 2 I 0 E0   R  f  t − + τ µ  + 8cε Rτ µ E∗   cl  (7.38)   R f  t − − τ µ  .   cl

Figure 7.5 Shapes of sound pulses generated by (a) a short pulse of a proton beam and (b) a short pulse of laser radiation in the case of small observation angle.

Figure 7.5a presents shapes of sound pulses for short pulses of a proton beam and in the case of small observation angles θ. The shape of a sound pulse generated by penetrating radiation with an exponential law of absorption (e.g., laser radiation) for the same beam parameters (length of

RADIATION ACOUSTICS

197

radiation pulse, beam radius, depth of radiation penetration into a substance, and radiation intensity) as in the case of the proton beam are given in Fig. 7.5b for comparison. We see that for the same small observation angle θ, the shapes of generated sound pulses differ drastically. Only the particular features of different types of penetrating radiation can explain this difference. Thus, the conducted analysis demonstrates that the shape of generated pulses in the far wave field depends essentially on the particular features of absorption of penetrating radiation in a substance only in the case of very short pulses of penetrating radiation and observation of sound pulses under small angles θ. We should note also that when penetrating radiation leads to considerable ionization of a liquid, the velocity of sound propagation in the ionization region can increase essentially [123]. In this case the parameters τa and τl decrease, respectively. Moreover, in the case of substance ionization, the transition of the energy lost by radiation into thermal energy occurs with a certain delay [123]. Then, the time dependence of thermal sources may differ to some extent from the time dependence of a pulse of penetrating radiation, and therefore the envelope of a sound pulse may change also. We have considered sound generation by penetrating radiation in the far wave field. At the same time an analogous situation in the one-dimensional case or in the case of near wave field is very interesting also. In this case the transverse dimensions of a beam of penetrating radiation are large as against the observation distance, and furthermore, this distance is larger than the depth of radiation penetration into a substance. Then, if we consider normal incidence of radiation on the surface of a liquid half-space and the function Q(z, t) = Q(z)f(t), we can write down an expression for sound pressure in the following form (we use the Green function for a one-dimensional reduced wave equation [51]):

p( z, t ) =

αc 4πC p

∞

  z  exp iω  − t  F (ω )dω ×   c  −∞

∫

(7.39) ∞

  z  z  Q ( z ) exp − iω  exp iω  dz . c  c    −∞

∫

Here c is the sound velocity in a liquid. If the pulse length of penetrating radiation is small compared with the time of wave propagation to the

198

PULSED THERMORADIATION SOURCES OF SOUND IN SOLIDS

observation point, the spectrum F(ω) can be changed for the length of radiation pulse τ and expression (7.39) takes on the form, p ( z, t ) =

αc 2 τ [Q ( z − ct ) − Q ( z + ct )] . 2C p

(7.40)

If we do not take into account the reflected wave −Q(z + ct), the shape of sound pulse corresponds to the shape of the function of energy evolution. This is the basis for the technique of acoustic dosimetry of penetrating radiation (see [26] for example).

Figure 7.6 Dependence of distribution of absorbed energy of electrons in cellophane, which was obtained by an acoustic method according to formula (7.41).

In the case of a solid we can write down in the same way an expression for the component of stress tensor σzz for the normal incidence of short radiation pulses on its surface, small heat conductivity, and approximation of near wave field:

σ zz = −

(3 − 4 / n 2 )αcl2 2cε

τ [Q( z − cl t ) − Q( z + cl t )]

(7.41)

(other components of stress tensor are expressed through the component σzz in the next way: σzx = σzy = σxy = 0 and σxx = σyy = (1 − 2/n2) σzz). Figure 7.6 [25] gives the dependence of distribution of absorbed energy of electrons in cellophane, which has been obtained on the basis of analysis of an acoustic signal according to formula (7.41). In conclusion of this section we should note the following. We have considered here the influence of the particular features of absorption of penetrating radiation on sound generation in a liquid or solid half-space (in the case of a solid half-space we treated only waves in the bulk of the medium). While considering other specific problems of sound generation by

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199

penetrating radiation (e.g., surface waves, normal modes in waveguides, waves in layered media, etc.), one may use the same solution technique that has been described here. However, if we have already some specific solution for a certain law of penetrating radiation absorption, and the characteristic scale of a problem is greater than the depth of radiation penetration into a substance l = µ−1, we can obtain a certain “zero” approximation of solution of a new problem using a new value of l = µ−1 in the old solution.

CHAPTER 8

Moving Thermoradiation Sources of Sound A moving thermoradiation source of sound arises in the process of motion of a beam of penetrating radiation along the surface of a condensed medium. Theoretical studies of sound field of such sources were performed apparently for the first time by Bozhkov, Bunkin, and Kolomenskii [31, 32, 34, 112, 113], Esipov [95], Lugovoi and Strel’tsov [124], and Lyamshev and Sedov [136, 148]. These authors considered sound excitation by a moving laser beam with modulated light intensity. Important features of thermoradiation generation of sound (as compared with traditional techniques) are simple realization of a sound source moving with almost any velocity and acceleration due to the absence of medium resistance, an opportunity for smooth and continuous tuning of modulation frequency of intensity of penetrating radiation (in particular, by changing of the velocity of beam motion, i.e., on account of the Doppler effect), and the absence of side-lobes in the directivity pattern. This chapter considers particular features of generation of a sound field created by an intensity-modulated beam of penetrating radiation scanning the surface of a condensed medium.

201

202

MOVING THERMORADIATION SOURCES OF SOUND

1. SOUND GENERATION BY A MOVING THERMORADIATION PULSED SOURCE IN A LIQUID We consider here the particular features of sound generation in a liquid by a beam of penetrating radiation with intensity modulated by pulses of arbitrary shape, which moves along the liquid surface. We impose almost no restrictions upon the velocity of beam motion along the liquid surface and the shape of its trajectory. We assume only that the trajectory of beam motion is located in a finite area of liquid surface and the sound field is treated in the far wave field with respect to the dimensions of this area. We will obtain general expressions characterizing the spectrum of the sound field of a moving thermoradiation source of sound. The next cases are studied in detail: (a) uniform motion of a beam along a finite trajectory; (b) oscillatory motion of a beam; and (c) uniform motion of a beam along a circle. An interesting particular feature is revealed: in the case of uniform and rectilinear motion of a beam of penetrating radiation, sound generation occurs in the same way as in the case of a motionless beam but at a certain effective shape of a radiation pulse. In this case the envelope of the sound signal in the Cherenkov direction does not depend on the shape of sound pulse and is determined by the geometrical parameters of the region of effective heat release and the observation angle, while the amplitude of the acoustic signal is directly proportional to the energy of the pulse of penetrating radiation. Let us analyze the particular features of the spectrum of generated acoustic signal for the cases of an oscillating beam of penetrating radiation and a beam moving along a circle, investigate in detail the quasi-monochromatic mode of sound generation by such sources, and compare the characteristics of these sources to the characteristics of a motionless quasi-monochromatic radiation-acoustic radiator. Let a beam of penetrating radiation be incident in the positive direction of the axis z at the surface of a half-space filled with a liquid. The function f(t) describes the shape of a radiation pulse (we assume that this function is bounded and continuous) and the transverse distribution of energy in the beam has the form I(x, y) = I0 exp [− (x2 + y2)/a], where a is the effective radius of the beam section at the liquid surface. We assume that the radiation spot moves along the liquid surface along a trajectory with the coordinates x0(t), y0(t). Then the power density of thermal sound sources in the liquid is determined by the expression, Q ( x, y, z, t ) = AµI 0 f (t ) exp[−( x − x0 (t )) 2 / a 2 − ( y − y0 (t )) / a 2 − µz ] .

RADIATION ACOUSTICS

203

An equation for the spectrum of sound pressure generated by a moving beam of penetrating radiation in the liquid has the form

∆pω + k 0 pω = −

AµI 0α exp(− µz ) Cp

∞

2   iωt − [ x − x0 (t )] − exp ∫  a2  −∞

(

[ y − y0 (t )]2   f (t )  f ′(t ) + 2 2 xx0′ (t ) − x0 (t ) x0′ (t ) + 2  a a 

(8.1)

yy0′ (t ) − y0 (t ) y0′ (t )   ,  where k0 is the complex wave number of sound and f ′0(t), x′0(t), and y′0(t) are the derivatives of the functions f0(t), x0(t), and y0(t). Let the trajectory of motion of a penetrating radiation beam be located in a limited area of the liquid surface. We are interested in the sound field in the far wave zone1. We obtain an expression for the spectrum of sound pressure in the far wave zone according to Kasoev and Lyamshev [109, 110] (see Chapter 5, Section 2):

pω = −

2 exp(ikr0 ) ω τ µ α × AI 0 a 2 2r0 1 + ω 2τ 2 Cp µ

(8.2)  ω 2τ 2 a exp −  4 

  − q(ω )r F ∗ (ω ) , 0  

where F ∗ (ω ) =

∞

∫ f (t ) exp(iωt + ik sin θ [ x0 (t ) cos ϕ + y0 (t ) sin ϕ ])dt ,

(8.3)

−∞

τµ = (cos θ)/µc, τa = (a sin θ)/c , c is the sound velocity, θ is the angle between the axis z and the radius-vector of the observation point r0, ϕ is the 1

We mean the Fraunhofer zone with respect to the area of motion of radiation spot along the liquid surface and the upper limiting frequency of the spectrum of sound signal taking into account sound attenuation in a liquid.

MOVING THERMORADIATION SOURCES OF SOUND

204

angle between the axis x and the projection of r0 onto the plane x, y; q(ω) = Im k0 is the coefficient of sound attenuation in the liquid; k = Re k0. If we compare formula (8.2) to the analogous expression for the spectrum of sound pressure of a motionless pulsed radiation-acoustic source, i.e., formula (5.2) (see Chapter 5 and also [88, 151]), we can see that these formulae almost coincide and differ only in the fact that the spectral density of laser pulse F(ω) in formula (8.2) is changed for the ∗ function F (ω). The last depends not only on the shape and length of a radiation pulse but also on the characteristics of motion of a radiation spot (a beam of penetrating radiation) along the liquid surface. Expression (8.2) is obtained on the grounds of rather general assumptions indicated above. It follows from this expression that the spectrum of sound pressure in a liquid is determined by the geometrical parameters of the region of absorption of penetrating radiation in a liquid as ∗ in the case of a motionless beam on the one hand and by the function F (ω), which depends on the spectrum of radiation pulse and motion parameters of penetrating radiation beam on the other hand. An analytic representation of the spectrum of sound field in a liquid (expression (8.2)) is very convenient for consideration of various particular cases of sound radiation by a moving pulsed thermoradiation sound source. We consider some of them below. Let a radiation spot be moving uniformly and rectilinearly with velocity ∗ V along the axis x, i.e., x0(t) = Vt, y0(t) = 0. Then the function F (ω) can be expressed with the help of the spectrum of radiation pulse in a simple way: F ∗ (ω ) = F [(1 − β ∗ )ω ] , ∗

(8.4)

where β = (V/c) sin θ cos ϕ. One can see from formulae (8.2) and (8.4) that sound generation by a moving uniformly and rectilinearly pulsed radiation-acoustic source occurs ∗ in the same way as by a motionless but “compressed” by |1 − β | times ∗ ∗ radiation pulse, which is described by the function f[t/(1 − β )]/|1 − β | and ∗ has the effective length |1 − β |τ, where τ is the length of a penetrating ∗ radiation pulse. We should note that if 1 − β < 0, the “effective” pulse ∗ ∗ f[t/(1 − β )]/|1 − β | is not only compressed but also inverted in time with respect to the radiation pulse. This is connected with the fact that in the case of supersonic motion of a radiation-acoustic source, sound disturbances, which have been produced by the source later, arrive at the observation points in certain directions sooner. Thus, almost all discussion and results of Chapter 5, Section 3 (see also [110]) can be transferred directly to the case of a moving source if we ∗ ∗ consider the effective radiation pulse f[t/(1 − β )]/|1 − β | with length |1 − ∗ β |τ. Therefore, we do not consider the details discussed in Chapter 5 but

RADIATION ACOUSTICS

205

only the particular features of sound radiation in the Cherenkov direction. We mean the direction where sound disturbances from various points get added in-phase. The Cherenkov direction is determined by the equation

β ∗ = (V / c) sin θ cos ϕ = 1 . If this condition is satisfied then, as it follows from expression (8.4), the * function F (ω) takes on an especially simple form F ∗ (ω ) = F (0) =

∞

∫ f (t )dt = σ ,

−∞ *

i.e., F (ω) does not depend on frequency and is equal to the “area” of the * radiation pulse σ. Substituting F (ω) = σ into expression (8.2) and performing the inverse Fourier transformation, we obtain an expression describing a sound field in the Cherenkov direction: p=−

 γ2 AI 0 a 2ασ  4 exp −   s2 8C p r0τ µ2  π s 

2   − exp s ×  4 

(8.5) 

(exp(−γ ) Erfc ( s / 2 − γ / s ) + exp(γ ) Erfc ( s / 2 + γ / s )) , 

where γ = (r0/c − t)/τµ, s = (τa2 +4Cr0)1/2/τµ, and Erfc z is the complementary error function. In order to simplify calculation, we take here q(ω) = Cω2, where C is a certain constant. We note that expression (8.5) coincides with the formula obtained in the process of consideration of sound generation by a very short radiation (laser) pulse [109]. As follows from analysis of expression (8.5), the shape of the envelope of a sound pulse in the Cherenkov direction depends neither on the shape nor on the length of penetrating radiation pulse. The envelope of a sound pulse is determined by the geometrical parameters of the region of effective heat release a, 1/µ, the parameter Cr0, and the observation angle θ, while the pulse amplitude is directly proportional to the energy of penetrating radiation pulse πa2I0σ. The shape of the envelope of a sound pulse in the Cherenkov direction calculated according to formula (8.5) is given by curve 1 in Fig. 8.1. Here the vertical axis presents the amplitude of a sound pulse normalized to the

206

MOVING THERMORADIATION SOURCES OF SOUND

value Aa2I0τα/(4r0τµ2Cp) and the horizontal axis presents the quantity γ = (r0/c − t) / τ, i.e., the dimensionless time. We have assumed in the process of calculation that s = 2. It is necessary to note that the shape of a sound pulse in these coordinates is determined by the single parameter s = (τa2 + 4Cr0)1/2/τµ . If the parameter s decreases, the pulse width (in the coordinate γ) decreases and the amplitude grows.

Figure 8.1 (1) Envelope of a sound pulse in the Cherenkov direction and (2, 3) shapes of pulses of “switching-on” and “switching-off”.

Some particular features of sound generation by radiation pulses of arbitrary shape in the process of uniform and rectilinear motion of a beam of penetrating radiation along a liquid surface have been considered above. It would be interesting to consider in more detail the case of intra-pulse quasi-monochromatic modulation of penetrating radiation in intensity. For example, let 1 − cos ω 0 t , | t | ≤ τ , f (t ) =  | t | >τ . 0 , Let us take ω0τ = 2πn (where n is a certain natural number) in order for the function f(t) to be continuous. The spectrum of the function f(t) consists of three narrow (of width of the order of magnitude of 1/τ) bands at zero and at the frequencies ±ω0:

RADIATION ACOUSTICS

F (ω ) = 2

sin(ωt )

ω

−

207

sin[(ω + ω 0 )τ ] sin[(ω − ω 0 )τ ] . − ω + ω0 ω − ω0

*

We obtain an expression for F (ω), F ∗ (ω ) = 2

sin[(1 − β ∗ )ωt ] (1 − β ∗ )ω

−

sin{[(1 − β ∗ )ω + ω 0 ]τ } (1 − β ∗ )ω + ω 0

− (8.6)

sin{[(1 − β ∗ )ω − ω 0 ]τ } . (1 − β ∗ )ω − ω 0 Proceeding from expression (8.2) we can determine that the form of the spectrum of sound signal pω generated by penetrating radiation is * determined not only by the function F (ω) but also by the product of the 2 2 2 rational ω τµ/(1 + ω τµ ) and exponential exp [−ω2(τa2/4 + Cr0] functions. For example, let the spectrum of this product be determined mainly by the exponential function and therefore, be limited by the frequency of the order of magnitude of 2(τa2 + 4Cr0)−1/2. In this case the spectrum of the functions * F (ω) represents three bands at zero and at the Doppler frequencies * * ±ω0/|1 − β | . Let us consider two cases: ω0 > 2|1 − β |(τa2 + 4Cr0)−1/2 and * −1/2 2 ω0 < 2|1 − β |(τa + 4Cr0) . * In the first case, when ω0 > 2|1 − β |(τa2 + 4Cr0)−1/2 and in particular in * observation directions close to the Cherenkov direction, where |1 − β | << 1, a considerable contribution to the expression for sound pressure is made only by the first term in the spectrum given by expression (8.6). Substituting it into expression (8.2) and performing the inverse Fourier transformation, we obtain an expression for sound pressure in the form

p=−

Aa 2 I 0 s2 α exp {exp[−(d 0τ + γ )] × C p 8r τ (1 − β ∗ ) 4 0 µ

Erfc [ s / 2 − ( d 0τ + γ ) / s ] − exp(d 0τ + γ ) Erfc [ s / 2 + ( d 0τ + γ ) / s ] + (8.7) exp[−( d 0τ − γ )] Erfc [ s / 2 − ( d 0τ − γ ) / s ] − exp( d 0τ − γ ) Erfc [ s / 2 + ( d 0τ − γ ) / s ]} ,

208

MOVING THERMORADIATION SOURCES OF SOUND

*

where d0 = (1 − β )/τµ . It is necessary to note that naturally, in the limit at * (1 − β ) → 0 expression (8.7) transforms into expression (8.5) if we take into account the fact that in the considered case σ = 2τ. Expression (8.7) was obtained by Bozhkov, Bunkin, and Kolomenskii [32] in the process of consideration of the sound field of a uniformly and rectilinearly moving optoacoustic source in the case of the rectangular shape of a laser pulse. 2 * −1/2 In the second case when ω0 < 2|1 − β |(τa + 4Cr0) , it is necessary to take into account all three terms in the spectrum given by expression (8.6). * Substituting F (ω) in the form of expression (8.6) into expression (8.2) and performing the inverse Fourier transformation of the first term, we obtain expression (8.7). The inverse transformations of the second and third terms cannot be expressed in elementary functions but they can be calculated approximately assuming the spectrum of a pulse of penetrating radiation at the frequencies ±ω to be infinitely narrow. We can write down pd in the form

pd =

 k 2 a 2 sin 2 θ  µk d cosθ α Aa 2 I 0ω d   exp − d − Cr0ω d2  × C p 2r (1 − β ∗ ) µ 2 + k 2 cos 2 θ 4   0 d   (8.8) cos(ω d t − k d r0 ) , *

where ωd = ω0/|1 − β |, kd = ωd/c. Expression (8.8) describes a * monochromatic sound wave at the Doppler frequency ω0/|1 − β | . It is necessary to note that an analogous expression describes sound generated by a continuous monochromatically intensity-modulated penetrating radiation in the case of a motionless beam with the only difference that the motionless beam generates sound at the modulation frequency ω0. Analysis of expressions (8.7) and (8.8), taking into account expression (8.5) given above, provides an opportunity to note the following features of a sound field generated by a moving uniformly and rectilinearly radiationacoustic source in the case of intra-pulse monochromatic modulation of penetrating radiation in intensity. The shape of the envelope of generated sound pulse in the Cherenkov direction does not depend on the shape of penetrating radiation pulse and therefore, is described by general expression (8.5). The envelope of the sound signal is determined by the geometrical parameters of the region of effective heat release a and 1/µ, the parameter Cr0, and the observation angle θ, while the amplitude is directly proportional to the energy of radiation pulse. In the observation directions close to the Cherenkov direction, where * * |1 − β | << 1 and the condition ω0 > 2|1 − β |(τa2 + 4Cr0)−1/2, the sound field is a pair of pulses with the shape described by expression (8.7). These

RADIATION ACOUSTICS

209

pulses are the responses to “switching-on” and “switching-off” the pulse of penetrating radiation. If we move away from the Cherenkov direction, the * value of |1 − β | increases, and the time interval between the pulses of switching-on and switching-off increases too (this interval is of the order of * * magnitude of 2|1 − β |τ). If the condition ω0 < 2|1 − β |(τa2 + 4Cr0)−1/2 is * satisfied, an almost sinusoidal filling at the Doppler frequency ω0|1 − β | appears between the pulses. The amplitude of this filling is determined by expression (8.8). The aforesaid is illustrated by Fig. 8.1. Curves 2 and 3 present the shape of the envelope of a sound signal in observation directions close to the Cherenkov direction and are calculated according to formula (8.7) at the values of the parameters s = 2 and d0τ = 5 (curve 2) and d0τ = 10 (curve 3). Each curve consists of two pulses of “switching-on” and “switching-off” located symmetrically with respect to the vertical axis, each of these pulses consisting of pulses of compression and rarefaction with identical shapes. As the parameter d0τ increases, i.e., if we move away from the Cherenkov direction, the time interval between the pulses grows, tending towards the length of the radiation pulse, which is equal to 2τ in this case, and the amplitudes of the pulses decrease. Let us consider one more particular case of motion of a radiation spot at the liquid surface. Let a beam perform an oscillatory motion along the axis x according to the law x0(t) = b sin Ωt, y0(t) = 0. We obtain an expression * for F (ω), F ∗ (ω ) =

∞

∑ J n (kb sin θ cos ϕ ) F (ω + nΩ) ,

(8.9)

n = −∞

where Jn(x) is the Bessel function. If the length of a radiation pulse is larger than the period of beam motion (τ > 2π/Ω), then, as follows from expressions (8.2) and (8.9), the spectrum of sound signal consists of two bands with width of the order of magnitude of 1/τ at the frequencies nΩ multiple to the frequency of beam motion, where n is an integer number. In the opposite case when the pulse length is smaller than the motion period, the spectrum of sound signal is continuous. Finally, in the limiting case of a very long pulse of penetrating radiation (τ >> 2π/Ω), the width of the bands at the frequencies nΩ becomes very small and considering them approximately very narrow, we obtain an expression determining sound pressure in a liquid

(

)

2 α AI 0 a 2 ∞ n ω nτ µ p=− ∑ (−1) 1 + ω 2τ 2 exp − ω n2τ a2 / 4 − q(ω n )r0 × C p r0 n µ n =1

210

MOVING THERMORADIATION SOURCES OF SOUND

× J n (k n b sin θ cos ϕ ) cos[ω n (t − r0 / c)] ,

(8.10)

where ωn = nΩ and kn = ωn/c. It is interesting to compare the characteristics of sound radiation by an “oscillating” radiation-acoustic source at the lower frequency ωl = Ω for example with those of a motionless monochromatic radiation-acoustic source. One can see that the directivity pattern of the source performing oscillatory motion with frequency Ω differs from the directivity pattern of the motionless source by the factor 2J1(k1b sin θ cos ϕ); and, in the selected direction in the case of the corresponding length of the beam track at the surface (so that k1b sin θ ≈ 1.8), the pressure amplitude in the plane of motion exceeds the sound amplitude from a motionless monochromatic radiation-acoustic source approximately by 20%. It is necessary to note also that k1b = Ωb/c = Vm/c, where Vm is the maximum speed of motion of an oscillating source, and since k1b > 1, it radiates sound effectively when the maximum speed of motion is larger than the sound velocity in the liquid. Moreover, differing from a motionless monochromatic radiation-acoustic source, the sound field of a source performing oscillatory motion is concentrated close to the motion plane and it is absent in the plane perpendicular to the motion direction. Figure 8.2 presents the surfaces of the functions

p1 (θ , ϕ ) =

2   Ω 2τ 2 α AI 0 a 2 Ω τ µ a − q (Ω) r  × exp − 0   C p r0 1 + Ω 2τ 2 4 µ  

 Ω J1  b sin θ cos ϕ  , c  which describe the angular dependence of the amplitude of sound pressure produced by a beam of penetrating radiation performing oscillatory motion at the frequency Ω. The amplitude of sound pressure measured in relative units is plotted in the vertical axis, while the horizontal axes present the values of observation angles θ and ϕ from 0 to 90°. Calculation was performed for the next values of the problem parameters: k1a = 2, k1/µ = 21/2, q(Ω) = 0, i.e., not taking into account sound attenuation in a liquid that is unimportant in this case, k1b = 3 (Fig. 8.2a), k1b = 6 (Fig. 8.2b), and k1b = 9 (Fig. 8.2c). The relief of the surface of the function p1(θ, ϕ) is plotted by the curves representing sections of the p1(θ, ϕ) by the planes θ = const and ϕ = const. The sections are made per three degrees.

RADIATION ACOUSTICS

211

Figure 8.2. Angular dependences of sound pressure.

One can see in the figure the major features of sound radiation by an oscillating radiation-acoustic source. The maximum of the directivity pattern is located in the plane of beam motion ϕ = 0. There is no sound field in the plane perpendicular to the direction of motion ϕ = 90°. Changing the amplitude of beam motion b, one can change the direction where sound radiation is maximal or create “gaps” in the directivity pattern. In conclusion let us consider the particular features of sound radiation by a radiation-acoustic source in the case of motion of a radiation spot along a circle: x0(t) = B sin Ωt, y0(t) = B cos Ωt. In this case F ∗ (ω ) =

∞

∑ J n (kb sin θ ) exp(inϕ ) F (ω + nΩ) .

n = −∞

(8.11)

In the case of a very long radiation pulse τ >> 2π/Ω, its spectrum can be written down approximately in the form F (ω + nΩ) = 2πδ (ω + nΩ) . (8.12) Substituting this expression into formula (8.11) and then into expression (8.2) and performing the inverse Fourier transformation, we obtain an expression for sound pressure

p=−

2 2 2   α AI 0 a 2 ∞ n ω n τ µ exp − ω n τ a − q(ω )r  × ( ) 1 − n 0 ∑   C p r0 4 1 + ω n2τ µ2 n =1  

212

MOVING THERMORADIATION SOURCES OF SOUND

× J n (k n B sin θ ) cos[ω n (t − r0 / c) + nϕ ] .

(8.13)

As one would expect by proceeding from general physical concepts, the directivity pattern of a radiation-acoustic source performing motion along a circle at a certain frequency is the product of expressions describing the directivity pattern of a motionless monochromatic radiation-acoustic source and the directivity pattern of sound radiation by a body moving along a circle [88]. In this case as in the previous one, the amplitude of sound pressure at the frequency Ω can be larger than the amplitude of sound pressure in the case of a motionless radiation-acoustic source if a certain value of circle radius B is selected. In this case estimations show (we do not give them here) that the efficiency of sound generation by a beam of penetrating radiation moving along a circle, which is determined by the ratio of the power of generated sound oscillations to the power of penetrating radiation, is almost the same as in the case of a motionless monochromatic radiation-acoustic source. The results obtained above can be extended readily to the case when a beam performs motion along a circle and the intensity of penetrating radiation is modulated monochromatically with sound frequency ω0. If we assume that the length of radiation pulse is large as against both the modulation period (τ >> 2π/ω0) and the period of beam rotation (τ >> 2π/Ω), i.e., if we consider the quasi-monochromatic mode of sound generation, it is possible to demonstrate that sound waves with frequencies ωn± = ω0 ± nΩ, where n = 0, 1, 2, …, are emitted apart from waves at the frequencies multiple to the rotation frequency nΩ with the characteristics of acoustic radiation described by expression (8.13). Directivity patterns of sound sources at each of these frequencies are described by the expression

Pn ± =

2   ω2 τ 2 α AI 0 a 2 m ω n ±τ µ exp − n ± a − q (ω n ± )r0  ×   Cp 2r0 1 + ω 2 τ 2 4 n± µ  

(8.14) J n ( k n ± B sin θ ) , where m is the modulation index. Summarizing, we note the next particular features of sound generation in a liquid by a beam of penetrating radiation moving at its surface. In the case of a uniform rectilinear motion of a beam operating in the pulse mode, the characteristics of generated sound signal are the same as in the case of a motionless beam but for a certain effective shape of radiation pulse. A sound signal with the shape of envelope independent of the shape

RADIATION ACOUSTICS

213

of radiation pulse, and determined by the parameters of the region of effective heat release and the observation angle, is generated in the Cherenkov direction. The amplitude of this pulse is directly proportional to the energy of penetrating radiation pulse. In the particular case of intrapulse quasi-monochromatic modulation of penetrating radiation, the presence of modulation does not influence the characteristics of the sound field in the Cherenkov direction and the directions of observation close to it. The quasi-monochromatic component at the Doppler frequency arises in the sound field only at a certain angular distance from the Cherenkov direction. In the case of oscillatory motion or rotation of a beam of penetrating radiation, and in the case of a long enough radiation pulse, only sound disturbances at frequencies multiple to the frequency of oscillatory motion or rotation frequency are generated. In the case of a certain set of the parameters of beam motion, the efficiency of sound generation by an oscillating or rotating radiation beam is not smaller than the efficiency of sound generation by a motionless quasi-monochromatic radiation-acoustic source, but the sound beam is very narrow. We should note also that the analytical presentation of the spectrum of sound field in a liquid (8.2) can be useful for consideration of the particular features of thermoradiation generation of sound in a liquid in the case of other forms of the trajectories of motion of a beam of penetrating radiation at the surface of a liquid.

2. SOUND EXCITATION BY A MOVING THERMORADIATION PULSED SOURCE IN SOLIDS Now let us consider sound excitation by a pulsed moving thermoradiation source in solids. Let a beam of penetrating radiation be incident in the positive direction of the axis z on the boundary of a solid homogeneous and isotropic halfspace z > 0. We take the Gaussian intensity distribution in the beam as usual. Let us assume that the radiation spot moves at the solid surface along a trajectory with the coordinates x0(t), y0(t). We assume also that the power density of sound sources Q in a solid is determined by the expression Q ( x, y, z, t ) = µI 0 f (t ) exp[−( x − x0 (t )) 2 / a 2 − (8.15) ( y − y0 (t )) 2 / a 2 − µz ] ,

MOVING THERMORADIATION SOURCES OF SOUND

214

where f(t) describes the time shape of the pulse of penetrating radiation, and the coefficient of transmission through the solid boundary is taken into account directly in the expression for the radiation intensity I(x, y). Let us write down an expression for the Fourier transforms of the scalar potential Φω and the only (because of the axial symmetry of the problem) component of the vector potential Ψω of the displacement vector in the sound field:   2 (3 − 4 / n 2 )α iω iω 3   2 ω ∆ + ∆ + ∆ + Φω = − µI 0 exp(− µz ) F ∗ (ω ) ,   2 2 χ χ c ρ  ε cl χcl   (8.16) 2   ∆ + ω Ψ = 0 , ω  ct2   where F ∗ (ω ) =

∞

2 2   iωt − ( x − x0 (t )) − ( y − y 0 (t )) ( ) exp f t ∫  a2 a2  −∞

 dt .  

We are interested in determining the sound field in the far wave zone, i.e., the Fraunhofer zone with respect to the dimensions of the area of motion of penetrating radiation beam and the upper frequency limit of the spectrum of sound signal. We obtain expressions for the components of stress tensor σRR and σRθ originating from longitudinal and transverse waves, respectively and for vertical displacements uz in the Rayleigh wave at the boundary of a solid just in the same way as above (see previous sections and also [151, 159]):

σ RR =

(3 − 4 / n 2 )αa 2 I 0 8πcε R

∞

∫

−∞

ω exp(−ω 2τ a2 / 4) exp[iω ( R / cl − t )] (1 + ω 2τ χ2 )(1 + ω 2τ µ2 )

×

(1 − iωτ χ )[ωτ µ K (θ ) + iM (θ )]F∗l (ω )dω ,

σ Rθ =

(3 − 4 / n 2 )αa 2 I 0 8πcε R

∞

∫

−∞

ω exp(−ω 2 n 2τ a2 / 4) exp[iω ( R / ct − t )] (1 + ω 2τ χ2 )(1 + ω 2τν2 )

×

RADIATION ACOUSTICS

215

(1 − iωτ χ )(i − ωτν )V2 (θ ) F∗t (ω )dω ,

uz =

(3 − 4 / n 2 )αa 2 I 0 1 − i 16cε ρ (πv R ) 3 / 2

R

∞

∫

2 exp(−ω 2 a 2 / 4v R ) exp[iω ( R / v R − t )]

2 2 2 2 1/ 2 − ∞ (1 + ω τ χ )[1 + ( ω / µv R )(1 − v R / cl )]

   ∗ − ω 1 − ω i V 2  ω    

τa =

(8.17)

1/ 2

2  vR   cl2 

×



 V1∗  F∗R (ω )dω ,  

a sin θ cosθ (1 / n 2 − sin 2 θ )1 / 2 χ , τµ = , τν = , τχ = , cl µcl µct cl2

R is the distance from the region of sound generation to the observation point, K(θ) = 1 − V1(θ), M(θ) = 1 + V1(θ), θ is the angle between the axis z and the direction to the observation point, ϕ is the angle between the axis x and the projection of R onto the plane (x, y), and vR is the velocity of Rayleigh wave, ∞

F∗l =

  ω f (t ) exp iωt + i sin θ [ x0 (t ) cos ϕ + y 0 (t ) sin ϕ ] dt , c l   −∞

∫

∞

F∗t =

  ω f (t ) exp iωt + i sin θ [ x0 (t ) cos ϕ + y 0 (t ) sin ϕ ] dt , ct   −∞

∫

(8.18)

∞

F∗R =

  ω f (t ) exp iωt + i sin θ [ x0 (t ) cos ϕ + y0 (t ) sin ϕ ] dt . vR   −∞

∫

The coefficients of reflection of longitudinal and transverse waves V1(θ) and V2(θ) from the free boundary of a solid in the case of incidence of a longitudinal wave at the boundary are expressed as follows: V1 (θ ) =

2 sin θ sin 2θ (n 2 − sin 2 θ )1 / 2 − (n 2 − 2 sin 2 θ ) 2 2 sin θ sin 2θ (n 2 − sin 2 θ )1 / 2 + (n 2 − 2 sin 2 θ ) 2

,

216

MOVING THERMORADIATION SOURCES OF SOUND

V2 (θ ) = −

4n sin θ cos 2θ (1 − n 2 sin 2 θ )1 / 2 2

2

1/ 2

2 sin θ sin 2θ (1 − n sin θ )

2

+ n cos 2θ

,

(8.19)

*

and the analogs of reflection coefficients in the case of Rayleigh wave V1 * and V2 have the form

i vR V1∗ = − 2 L cl

2   2 2cl  n − 2 vR 

1/ 2

 c2    L =  l − n2   v2   R 

2 1/ 2  2 2     n  cl    2 2cl  ∗  , V2 = L  2 − 1  n − 2  , v R   v R     

2 2  3c 2 (c 2 / v R − 1)cl2 / v R + 2 − l − l 2 2 2 2   vR cl / v R − n  

(8.20)

 c2   2c 2  2 l − 1   l − n 2  .  2   2   vR   vR  If we compare expressions (8.17) with analogous expressions (7.3) (see Chapter 7), we can see that these expressions differ only in the fact that in expressions (8.17) the spectral density of the pulse of penetrating radiation ∞

F (ω) =

∫ f (t ) exp(iωt )dt

−∞

is changed for corresponding expressions (8.18). The last depend not only on the shape and length of a pulse of penetrating radiation but also on the characteristics of motion of a radiation beam at the solid surface. Expressions (8.17) are obtained on the basis of rather general assumptions. It follows from them that in the general case, sound fields in a solid are determined by the geometrical parameters of the region of radiation absorption, as in the case of a motionless source on the one hand, and by the functions F∗l, F∗t, and F∗R depending on the spectrum of a pulse of penetrating radiation and the parameters of beam motion on the other hand. Analytical expressions obtained for sound fields are convenient for consideration of various particular cases of motion of radiation-acoustic sources of sound. Let us consider some of them.

RADIATION ACOUSTICS

217

Let a radiation spot be moving uniformly and rectilinearly with the velocity v along the axis x, i.e., x0(t) = vt, y0(t) = 0. In this case the functions F∗ are expressed in a simple way with the help of the spectrum of an optical pulse F(ω): F∗l (ω ) = F [(1 − β l∗ )ω ] , F∗t (ω ) = F [(1 − β t2 )ω ] , (8.21) ∗ )ω ] , F∗ R (ω ) = F [(1 − β R ∗

∗

∗

where βl = (v/cl) sin θ cos ϕ, βt = (v/ct) sin θ cos ϕ , and βR = (v/cR) sin θ cos ϕ . It follows from expressions (8.17) and (8.21) that sound generation by a moving uniformly and rectilinearly pulsed radiation-acoustic source occurs ∗ in the same way as by a motionless source, which is compressed by |1 − β | ∗ ∗ times, described by the function f[t/(1 − β )]/|1 − β |, and has the effective ∗ ∗ length |1 − β |τ. Here τ is the length of penetrating radiation pulse and β are the corresponding Mach numbers for longitudinal, transverse, and ∗ Rayleigh waves. It is necessary to note that if 1 − β < 0 for some type of ∗ ∗ waves, the “effective” pulse f[t/(1 − β )]/|1 − β | is not only compressed but also inverted in time with respect to the pulse of optical radiation. As has been noted above, this is connected with the fact that in the process of supersonic motion of a source, sound disturbances produced by a radiationacoustic source later arrive at the observation points in certain directions earlier. In this case a situation may arise that motion may be subsonic for some types of waves, e.g., longitudinal waves, but supersonic for other waves, e.g., Rayleigh waves. Thus, almost all considerations in Section 1 of this chapter and the results by Lyamshev and Chelnokov [151, 159] can be applied directly to the case of a moving thermoradiation source if we consider an effective ∗ ∗ ∗ pulse of penetrating radiation f[t/(1 − β )]/|1 − β | with length |1 − β |τ. Therefore, we do not give all details here. They are covered in the section devoted to sound generation by radiation pulses in a solid. Here we consider only the particular features of sound generation in the Cherenkov directions, meaning the directions where sound disturbances from different points of motion trajectory of a radiation-acoustic source are added inphase. The Cherenkov directions for different types of waves are determined by equations v v v sin θ cos ϕ = 1 , sin θ cos ϕ = 1 , cos ϕ = 1 . cl ct vR

218

MOVING THERMORADIATION SOURCES OF SOUND

If these conditions are satisfied, then it follows from expressions (8.21) that the functions F∗l, F∗t, and F∗R acquire an especially simple form ∞

F∗l (ω ) = F∗t (ω ) = F∗ R (ω ) = F (0) =

∫ f (t )dt = s ,

−∞

i.e., F∗l, F∗t, and F∗R do not depend on frequency and are equal to the “area” s of the radiation pulse. Substituting F∗(ω) in expressions (8.17), we obtain expressions describing sound fields of longitudinal and Rayleigh waves in the Cherenkov directions,

σ RR =

2 2  τµ (3 − 4 / n 2 )αa 2 I 0 s  exp(τ a / 4τ χ )  K (θ ) ×  M (θ ) +  2 2 8cε R τχ    τ µ − τ χ

 t − R / cl exp  τχ 

   Erfc  τ a + t − R / cl   2τ χ τa  

 τχ exp(τ a2 / 4τ µ2 )  V1 (θ )1 − 2 2  τµ τ µ − τ χ    τ t − R / cl Erfc  a −  2τ µ τa 

  τχ  − 1 +   τµ  

   exp − t − R / cl   τµ  

   exp t − R / cl   τµ  

0

    ,    (8.22)

1/ 2

   

ω 1 / 2 exp( −ω 2 a 2 / 4v R2 )

∫ [1 + (ω / µv

 ×  

   Erfc  τ a + t − R / cl   2τ µ τa  

 2  (3 − 4 / n 2 )αa 2 I 0 s 2  ∗  v R uz = V1 + i1 −   8cε ρ (πv R ) 3 / 2 R1 / 2  cl2   ∞

 +  

2 2 1/ 2 ](1 + ω 2τ χ2 ) R )(1 − v R / cl )



 V1∗  ×  

×

  π π    R   R − t  dω , − t  + ωτ χ cos − ω  sin  − ω    4    vR   vR  4

RADIATION ACOUSTICS

219

where ∞

2

Erfc ( x) = (2 / π ) ∫ e − t dt x

is the complementary error function. It follows from the analysis of expressions (8.22) that the shapes of the envelopes of sound signals in the Cherenkov directions depend on neither the shape, nor the length of a radiation pulse. Envelopes of sound pulses are determined by the geometrical parameters of the region of effective heat release a and 1/µ and the observation angle θ, while the amplitudes of pulses are directly proportional to the energy of a pulse of penetrating radiation πa2I0s. If a pulse of penetrating radiation is modulated quasimonochromatically in intensity, for example in the form 1 − cos ω 0τ , | t | ≤ τ , f (t ) =  | t | >τ , 0 , where ω0τ = 2πk and k is a natural number, then analogously to the way it has been done in the previous section of this chapter for the case of sound generation in a liquid, we can obtain similar expressions for a solid. Analysis of these expressions allows us to note the following features of sound fields generated by moving uniformly and rectilinearly thermoradiation sources of sound in the case of intra-pulse quasimonochromatic modulation of penetrating radiation in intensity. The shapes of the envelopes of generated sound signals in the Cherenkov directions do not depend on the shape of penetrating radiation pulse. The envelopes of sound signals are determined by the geometrical parameters of the region of effective heat release a and 1/µ and the observation angle θ, while the amplitudes are directly proportional to the energy of radiation pulse. In the observation directions close to the Cherenkov directions, where ∗ ∗ |1 − β | << 1 and the conditions ω0 > 2|1 − β /τa|, sound fields represent a pair of pulses. These pulses are the responses to “switching-on” and “switching-off” a thermoradiation source. If we move away from the Cherenkov directions, the time interval between the pulses of “switchingon” and “switching-off” increases (this interval is of the order of magnitude ∗ ∗ of 2|1 − β |τ). If the conditions ω0 < 2|1 − β |/τa are satisfied, an almost ∗ sinusoidal filling at the Doppler frequency ω0/|1 − β | arises between these pulses.

220

MOVING THERMORADIATION SOURCES OF SOUND

Now let us consider the case of oscillatory motion of a radiation spot at the surface of a solid. Let a beam of penetrating radiation perform oscillatory motion along the axis x according to the law x0(t) = b sin Ωt, y0(t) = 0. We obtain for F∗(ω): F∗l (ω ) =

F∗t (ω ) =

∞

 ω J m  b cos ϕ sin θ  F (ω + mΩ) ,   cl m = −∞

∑ ∞

 ω J m  b cos ϕ sin θ  F (ω + mΩ) ,   ct m = −∞

F∗ R (ω ) =

∑

(8.23)

∞

 ω J m  b cos ϕ  F (ω + mΩ) ,   vR m = −∞

∑

where Jm is the Bessel function. If the length of radiation pulse is larger than the period of beam motion τ > 2π/Ω, then as follows from expressions (8.17) and (8.23), the spectrum of sound signals consists of bands with width of the order of magnitude of 1/τ at frequencies multiple to the frequency of beam motion mΩ, where m is an integer number. In the opposite case when the length of radiation pulse is smaller than the motion period, the spectra of sound signals are continuous. Finally, in the limiting case of a very long radiation pulse, i.e., when τ >> 2π/Ω, the width of bands at the frequencies mΩ becomes very small and assuming them infinitely narrow, we obtain expressions determining sound fields in a solid,

σ RR =

2 τ 2 / 4) (3 − 4 / n 2 )αa 2 I 0 ∞ exp(−ω m a (−1) m × ∑ 2 2 2τ 2 ) 2cε R ( 1 )( 1 ω τ ω + + m χ m µ m =1

 2 ω τ µ K (θ ) cos[ω m (t − R / cl )] − J m  m b sin θ cos ϕ {ω m c   l 3 2 ωm τ χ τ µ K (θ ) sin[ω m (t − R / cl )] + ω m τ χ M (θ ) cos[ω m (t − R / cl )] +

ω m M (θ ) sin[ω m (t − R / cl )]} ,

RADIATION ACOUSTICS

σ Rθ =

221

2 2τ 2 n a / 4) (3 − 4 / n 2 )αa 2 I 0 ∞ exp(−ω m × (−1) m ∑ 2 2 2 2 2cε R (1 + ω mτ χ )(1 + ω m τν ) m =1

 2 ω J m  m b sin θ cos ϕ {ω m cos[ω m (t − R / ct )]V2 (θ )(τ χ − τν ) +   ct 2 ω m sin[ω m (t − R / ct )]V2 (θ )(1 + ω m τ χ τν )} ,

(8.24) 2 2 1/ 2 ∗ ∗ (3 − 4 / n 2 )αa 2 I 0 V2 + i (1 − v R / cl ) V1 uz = × R π 3 / 2 cε ρ (2v R ) 3 / 2 ∞

∑

m =1

( −1) m

2 2 2 ω m exp( −ω m a / 4v R ) 2 2 (1 + ω m τ χ )[1 + ω m (1 − v R2 / cl2 )]1 / 2 / µv R

×

  π π    R   R − t   × − t  + ω mτ χ cos − ω m  sin  − ω m    4  4    vR   vR  ω J m  m b cos ϕ  ,   vR where ω = mΩ. We can see that the directivity patterns of a radiation-acoustic source performing oscillatory motion at the frequency Ω differ from the directivity patterns of a motionless source by the factors 2Jm[(ω/cl)b sin θ cos ϕ], 2Jm[(ω/ct)b sin θ cos ϕ], and 2Jm[(ω/vR)b cos ϕ], respectively, and in the chosen direction in the case of a certain selection of the track of radiation beam at the surface (so that either b(ω1/cl) sin θ ≈ 1.8, or b(ω1/ct) sin θ ≈ 1.8, or b(ω1/vR) sin θ ≈ 1.8) the amplitudes of corresponding sound fields in the motion plane exceed the amplitudes of sound fields from a motionless monochromatic radiation source by 20% approximately. We should note also that (ω1/cl)b = vmax/cl , where vmax is the maximum velocity of motion of an oscillating source; and since ω1b/cl > 1, the oscillating source emits sound effectively in the case where its maximum motion velocity is larger

222

MOVING THERMORADIATION SOURCES OF SOUND

than the corresponding velocity of sound in a medium. Moreover, differing from the case of a motionless monochromatic thermoradiation source, sound fields of a source performing an oscillatory motion are concentrated near the motion plane and they are absent in the plane perpendicular to the motion direction. In conclusion let us consider the particular features of sound generation by a thermoradiation source in the process of motion of a radiation spot along a circle x0(t) = B sin Ωt, y0(t) = B cos Ωt. In this case

F∗l (ω ) =

F∗t =

∞

 ω J m  m B sin θ  exp(imϕ ) F (ω + mΩ) , c   l m = −∞

∑

∞

 ω J m  m B sin θ  exp(imϕ ) F (ω + mΩ) , c   t m = −∞

∑

F∗ R (ω ) =

(8.25)

∞

 ω J m  m B  exp(imϕ ) F (ω + mΩ) . v  R  m = −∞

∑

In the case of a very long pulse of penetrating radiation τ >> 2π/Ω, an expression for the spectrum F(ω + mΩ) can be written down approximately in the form F (ω + mΩ) = 2πδ (ω + mΩ) .

(8.26)

Substituting this expression into expression (8.25) and then into expression (8.17), we obtain an expression for sound fields,

σ RR =

2 τ 2 / 4) (3 − 4 / n 2 )αa 2 I 0 ∞ exp(−ω m m a − × ( 1 ) ∑ 2 2 2τ 2 ) 2cε R (1 + ω mτ χ )(1 + ω m µ m =1

{

 2 ω J m  m B sin θ  ω m (τ µ K (θ ) + τ χ M (θ )) cos[ω m (t − R / cl ) + mϕ ] +   cl

(

)

}

2 ω m M (θ ) − ω m τ χ τ µ sin[ω m (t − R / cl ) + mϕ ] ,

RADIATION ACOUSTICS

σ RR =

223

2 2τ 2 n a / 4) (3 − 4 / n 2 )αa 2 I 0 ∞ exp(−ω m (−1) m × ∑ 2 2 2 2 2cε R (1 + ω mτ χ )(1 + ω m τν ) m =1

{

 ω 2 τ −τ J m  m B sin θ  V2 (θ ) ω m ( χ ν ) cos[ω m (t − R / ct ) + mϕ ] + c   t (8.27)

}

2 ω m (1 + ω m τ χ τν ) sin[ω m (t − R / ct ) + mϕ ] ,

 2  ∗  v R + − uz = V i 1 2   cε ρ (2v R ) 3 / 2 (πR)1 / 2  cl2   (3 − 4 / n 2 )αa 2 I 0

∞

∑

m =1

( −1) m

1/ 2

   

2 2 2 ω m exp( −ω m ) a / 4v R 2 2 (1 + ω m τ χ )[1 + (ω m / µv R )(1 − v R2 / cl2 )]1 / 2



 V1∗  ×  

 ω J 0  m B  × v  R 

  π π      R  R   + mϕ   .  + mϕ  + ω mτ χ cos + ω m  t − sin  + ω m  t −   4  4  vR   vR     Naturally, proceeding from general physical concepts, the directivity patterns of a thermoradiation sound source performing motion along a circle at a certain frequency are the product of expressions describing the directivity patterns of a motionless monochromatic thermoradiation source and the directivity patterns of sound radiation by a body moving along a circle (see Section 2 of this chapter and [148]). In this case as in the previous ones, the amplitudes of sound fields at the frequency Ω at a certain circle radius B may be larger than the amplitudes of sound fields from a motionless monochromatic radiation-acoustic source. The results obtained above can be extended readily to the case in which a beam of penetrating radiation moves along a circle and radiation intensity is modulated monochromatically with sound frequency ω0. If we assume that the length of radiation pulse is large compared with both the modulation period τ >> 2π/ω0 and the period of beam rotation τ >> 2π/Ω, i.e., if we take the quasi-monochromatic mode of sound radiation, then

224

MOVING THERMORADIATION SOURCES OF SOUND

sound waves with frequencies of the form ωm = ω0 ± mΩ, where m = 0, 1, 2, …, are emitted, apart from the waves at the frequencies multiple to the rotation frequency mΩ with the characteristics described by expressions (8.27). Summarizing, we can indicate the following particular features of sound generation in a solid by a beam of penetrating radiation moving at its surface. In the case of uniform and rectilinear motion of the beam and pulsed operation mode of a source of penetrating radiation, the characteristics of generated sound fields are the same as in the case of a motionless beam but with a certain effective shape of radiation pulse. In the Cherenkov directions (each kind of sound field has its own Cherenkov direction), a sound pulse is generated with the shape of envelope independent of the shape of the pulse of penetrating radiation and dependent on the parameters of the region of effective heat release and the observation angle. The amplitude of this pulse is proportional to the energy of radiation pulse. In the case of a very long pulse, the amplitude of the acoustic oscillations generated in the Cherenkov direction increases up to the rise of shock waves. Thus, for example, the increase of the amplitude of Rayleigh wave is determined by the expression  2  (3 − 4 / n 2 )αa 2 I 0 t  ∗  v R uz = V2 + i1 − 2  c R 8cε ρ (πv R ) 3 / 2 l   ∞

ω exp( −ω 2 a 2 / 4v R2 )

∫ [1 + (ω / µv 0

1/ 2

   

2 2 1/ 2 ](1 + ω 2τ χ2 ) R )(1 − v R / cl )



 V1∗  ×  

×

(8.28)

  π π    R   R − t  dω . − t  + ωτ χ cos − ω  sin  − ω    4  4    vR   vR This effect was predicted theoretically by Dykhne and Rysev [92] and discovered experimentally by Velikhov, Dan’shchikov, and Dymshakov [48]. In the particular case of intra-pulse quasi-monochromatic modulation, its presence does not influence the characteristics of sound fields in the Cherenkov directions and the observation directions close to them. A quasimonochromatic component at the Doppler frequency arises in sound fields only at certain angular distances from the Cherenkov directions.

RADIATION ACOUSTICS

225

In the case of oscillatory motion or rotation of a beam of penetrating radiation and a long enough radiation pulse, sound disturbances are generated at frequencies multiple to the frequency of oscillatory movements or rotation frequency, respectively.

CHAPTER 9

Sound Generation by Single High-Energy Particles Theoretical studies of sound generation by single high-energy particles have become important now in connection with possible practical applications (e.g., the DUMAND Project). The history of this problem has been considered already partly in the Introduction. This chapter treats sound generation by single elementary particles in various model situations, i.e., in an infinite space, in a solid half-space (bulk waves), at a solid surface (the Rayleigh wave), and efficiency of sound generation in an infinite space. We consider only the particles which give birth to cascade showers in the process of absorption in a substance, since single particles with energy insufficient for production of showers generate very weak acoustic fields. As geometrical dimensions of a cascade shower are greater than 10−5 m, we may ignore the effect of heat conductivity on sound generation under these conditions.

1. SOUND GENERATION BY A PARTICLE IN INFINITE SPACE Formulation of the problem of sound generation by single high-energy particles in infinite space (i.e., when the distance from the region of sound generation to the free boundaries of a body is large and we may ignore waves reflected from the boundaries) is possible for example in the case of 227

228

SOUND GENERATION BY SINGLE HIGH-ENERGY PARTICLES

high-energy neutrino and muons, which have large penetrating capability and can produce nuclear-electromagnetic cascades deep within a substance. Figure 9.1 gives a scheme of rise of the cascade in a substance in the process of absorption of a single high-energy particle. We take equations (6.9) as initial in this case, i.e.,   1 ∂2  (3 − 4 / n 2 )α div F  div u = , ∆ − ∆Qdt − ∫   2 2 2 c ρ ε c ∂ t c ρ l l   2    ∆ − 1 ∂  rot u = − rot F ,  ct2 ∂t 2  cl2 ρ 

where F is the dynamic force, F = Q/c; Q is the energy released per unit time within unit volume; and c is the velocity of light in vacuum. We consider a homogeneous and isotropic solid space bearing in mind the fact that the result for a liquid space may be obtained readily as a particular case.

Figure 9.1 Rise of a nuclear-electromagnetic cascade in a substance as the result of absorption of a high-energy particle.

We take the dependence of the function of energy evolution Q on time in the form of a delta-function since the time of energy evolution is much smaller than other characteristic times, and the spatial dependence of the function Q is approximated by the expression

Q ( x, y , z ) =

 x2 + y2 exp −  a2 πa 2 

µE

  exp(− µz ) Θ( z ) ,  

RADIATION ACOUSTICS

229

where E is the cascade energy, 1/µ is the effective cascade length, a is the effective cascade radius, and Θ(z) is the Heaviside function. The origin of coordinates is selected in this case in the point of cascade rise and the axis z is directed along the cascade axis in the direction of its development. Figure 9.2 dives an idea of the geometry of the problem. We may expect that in the process of problem solution, such approximation for the function of energy evolution Q(x, y, z) would provide an opportunity to obtain basic features of a sound field generated by a particle. This approximation is very rough because of the fact that it does not take into account the increase of the cascade radius in the process of its development.

Figure 9.2 Geometry of the problem of sound generation in an infinite space by a single high-energy particle.

First, let us consider the far sound field generated in an infinite medium by a nuclear-electromagnetic cascade. Using the technique for solution of boundary problems, which has been developed in Chapter 6, we can write down expressions for the components of a stress tensor in a sound field in a solid elastic medium,

σ RR = −

(3 − 4 / n 2 )αE 8πcε Rτ µ2

exp

   R − cl t   Erfc  τ a − R − cl t  , exp −  2τ µ  cl τ µ  cl τ a  4τ µ2   

τ a2

(9.1)

σ Rθ =

   R − ct t  τ2  Erfc  τ a − R − ct t  , exp a exp −  2τ µ ct nτ a   ct nτ µ  8πcRct n 2τ µ2 τ µ2     E sin θ

230

SOUND GENERATION BY SINGLE HIGH-ENERGY PARTICLES

where σRR and σRθ are the components of stress tensor, which originate from longitudinal and transverse waves, respectively; R and θ are the spherical coordinates of the distance from the cascade to the observation point and the angle between the cascade axis and the direction to the observation point; τa = a sinθ/cl and τµ = cosθ/µcl ; and other notations correspond to the ones introduced earlier. The near wave field of longitudinal waves in the case of observation in the direction perpendicular to the cascade axis is described by the following expression:

σ RR = −

c (3 − 4 / n 2 )α µE l 4πcε πR

 2 2  ω a  exp − × ω ∫  2  4 c 0 l  

∞

(9.2) π  R cos − ω  − t  dω .  4   cl Parameters of a cascade produced by a high-energy particle are approximately equal for various liquid or solid media if the values of density and charges of elements constituting these media are close. Thus, according to the data by Askarijan et al. [194], the parameters of a cascade 15 from a neutrino with energy of the order of magnitude of 10 eV in water are 1/µ ≈ 4 m and a ≈ 2 cm. In this case the estimation of effective sound pressure (in Pa) in the near wave field (f ≈ 30 kHz) is determined according to expression (8.2) as peff ≈ 0.1

E 1 , E0 R

(9.3)

where and below R is the distance (in meters) and E0 = 1016 eV. Formula (9.3) corresponds to the analogous expression for estimation of the level of sound signal in the near wave field given by Berezinskii and Zatsepin [24]. If we take Antarctic ice for example as the medium, where a cascade arises in the process of absorption of a high-energy neutrino, the parameters of a cascade are approximately the same there as in water. In this case if we take the next numeric values: cl = 4⋅103 m/s, n = 2, α = 5⋅10−5 s−1, and cε = 2⋅103 J/(kg⋅C) that correspond to the ambient temperature −20°C [178], then the estimation of effective sound pressure in the near wave field (f ≈ 90 kHz) is given according to expression (9.2) by the next relationship:

RADIATION ACOUSTICS

eff σ RR ≈

231

E 1 . E0 R

(9.4)

According to formulae (9.3) and (9.4), the effective sound pressure in ice is approximately one order of magnitude higher than the effective sound pressure in water, other conditions being equal. This difference is caused by the fact that the Gruneisen parameter for ice is approximately one order of magnitude larger than that for water (Γ = αc2/Cp). In the case of the far wave field at τa >> τµ, i.e., in the case of observation almost perpendicularly to the cascade axis, we obtain the next expression for longitudinal waves:

σ RR = −

 2 E (3 − 4 / n 2 )α 1 R − cl t  ( R − cl t ) exp − cε  cl2τ a2 2π π τ a2 R cl τ a 

   . 

(9.5)

Correspondingly, the estimation of effective sound pressure (in Pa) in ice under the conditions considered above is eff σ RR ≈ 0.1

E 1 . E0 R

(9.6)

An analogous estimation for effective sound pressure in water is approximately one order of magnitude smaller as in the case of the near wave field. The boundary between the far and near wave fields for the parameters of the cascade given above and the effective frequency f = 30 kHz for water and f = 90 kHz for ice lies at the distance R ≈ 100 m. In this case the level of sound pressure in ice at a distance 100 m in the observation direction perpendicular to the cascade axis is 10−3 Pa approximately, and the level of sound pressure in water is 10−4 Pa for a particle with the energy of the order of magnitude of 1016 eV. The far wave field at τa << τµ, i.e., in the case of observation under any angle except for those close to the perpendicular to the cascade axis, is characterized, as calculation shows, by the next expression (for longitudinal waves):

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SOUND GENERATION BY SINGLE HIGH-ENERGY PARTICLES

 τµ R − cl t R − cl t exp − R < cl t , exp , c c | | | | τ τ  µ l µ l |τµ | 2 (3 − 4 / n )α E  (9.7) σ RR =  R 8πcε τ µ2 exp - R − cl t  − τ µ exp - R − cl t  , R > c t . l   c |τ | |τ |  cl | τ µ |  µ     l µ  Effective sound pressure in ice for the cascade parameters given above and observation directions close to the cascade axis is (f ≈ 1 kHz) eff σ RR ≈ 10 − 5

E 1 . E0 R

(9.8)

Correspondingly, sound pressure in water under analogous conditions is approximately one order of magnitude smaller than effective sound pressure in ice. It is interesting to note that the level of effective sound pressure in the far wave field in the observation direction perpendicular to the cascade axis (θ = 90°) is approximately four orders of magnitude higher than the level of effective sound pressure in the case of observation along the cascade axis (θ = 0°). The same may be said with respect to sound pressure in water. As for the level of shear stress caused by transverse waves and originating from the dynamic mechanism of sound generation, according to expression (9.1) it is approximately five orders of magnitude smaller than the corresponding level of pressure caused by longitudinal waves for the same observation angles. And moreover, the angular factor sinθ is present in the expression for shear stress in the far wave field, which exists due to transverse waves. It is very interesting in this case that in the far wave field, the level of sound pressure produced by longitudinal waves at small θ is of approximately of the same order of magnitude as the level of shear stress produced by transverse waves at θ = 90°. A schematic shape of a sound pulse in the far wave field, which is produced by longitudinal waves and described by expression (9.5), is given in Fig. 9.3a. Analogously, the shape of a sound pulse in the near wave field, which is described by expression (9.2), is given in Fig. 9.3b. Figure 9.4 presents the maximum values of the relative level of the component of stress tensor σRR as a function of the observation angle θ according to general expression (9.1). As for the component of stress tensor σRθ , according to expression (9.1) its relative levels differ from the relative

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levels of the component of stress tensor σRR only in the additional factor sinθ.

Figure 9.3 Shapes of sound pulses produced by longitudinal waves in the (a) far and (b) near wave fields in the case of observation in the direction perpendicular to the cascade axis.

Figure 9.4 Maximum values of the component σRR as a function of observation angle.

Berezinskii and Zatsepin [24], Askarijan et al. [194], and others considered the near wave field with cylindrical symmetry, which arises in an infinite liquid space from a nuclear-electromagnetic cascade. It has been noted that using a set of receivers of sound pressure, it is possible to determine the direction of the cascade axis in space but it is impossible to say where a particle producing this cascade has come from, from “above” or from “below”. Observing the far wave field under different angles, it is possible to determine the direction where a particle has come from. Naturally, in the case of a single receiver, the situation when a particle comes from “above” and the receiver is positioned “below” a cascade, and the situation when a particle comes from “below”, while the receiver is positioned “above” a cascade, are indistinguishable in principle. However, in the case of two receivers such situations become distinguishable. The presence of transverse waves in a solid apart from longitudinal waves can provide additional information on the distances from a cascade to the reception point. However, the level of transverse waves produced due to

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SOUND GENERATION BY SINGLE HIGH-ENERGY PARTICLES

the dynamic mechanism of sound generation is much smaller than the corresponding level of longitudinal waves. But if it is possible to detect them, this is an additional information channel. We have to keep in mind that various inhomogeneities may exist in real solids. Scattering of longitudinal waves into transverse ones at these inhomogeneities masks the dynamic mechanism of sound generation.

2. SOUND EXCITATION BY A SINGLE PARTICLE IN A SOLID HALF-SPACE Let a high-energy particle be incident upon the surface of a solid half-space. It is assumed that a cascade arises directly at the half-space boundaries. The function of energy release Q in this case is expressed in the following way:

Q ( x, y , z , t ) =

 y2 + x2 exp −  a2 πa 2 

µE

  exp(− µz )δ (t ) ,  

(9.9)

where E is the energy of the cascade or the initial particle (under the condition that the total energy of the initial particle is transferred to the cascade and not taken away by some particles which do not take part in the rise of the cascade, e.g., muons). Analogously to the way it has been done in the case of sound generation by short pulses of penetrating radiation, we can write down expressions for the components of stress tensor in the far wave field,

σ RR =

(3 − 4 / n 2 )αE 8πcε Rτ µ2

 2  τ exp a  4τ µ2 

  R − cl t     V1 (θ ) exp c τ  × l µ    

 τ  R − cl t    R − cl t   Erfc  τ a − R − cl t   , − exp − Erfc  a +  2τ µ  clτ µ   2τ µ clτ a  clτ a       

σ Rθ =

(3 − 4 / n 2 )αE 8πcε Rτν2

 n 2τ 2 a V2 (θ ) exp  4τ 2  ν

  exp R − ct t  ×  cτ    t ν  

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 nτ R − ct t  , × Erfc  a + 2 ct nτ a  τ  ν (9.10)

σ Rϕ =

 2  τ exp a  4τ µ2 8πcRct n 2τ µ2  E sin ϕ

   τa R − ct t    R − ct t   exp − c nτ  Erfc  2τ − c nτ  − t a   µ   t µ 

 R − ct t     Erfc  τ a + R − ct t   . exp  ct nτ µ   2τ µ ct nτ a       As it has been done in the case of sound generation by short pulses of penetrating radiation, we can write down expressions for the components of stress tensor in the case when τa >> τµ and nτa >> |τν|,

σ RR = −

  2 (3 − 4 / n 2 )αE  R − cl t  ( R − cl t ) exp −    2π π cε Rτ a2  cl τ a cl2τ a2  

   M (θ ) + 

  2( R − c t ) 2 τ  2   µ exp − ( R − cl t )  K (θ )  , l − 1    τ   2 2 cl2τ a2    cl τ a  a  (9.11)  (R − c t) 2 (3 − 4 / n 2 )αE R − ct t t σ Rθ = − exp − 2 2 2 2 2  c n τ 2π π cε Rn τ a t a  ct n τ a

σ Rϕ =

 V (θ ) ,  2 

 (R − c t)2 R − ct t t exp −  c 2 n 2τ 2 π π cRct n 2τ a2 ct nτ a t a  E sin ϕ

 ,  

and in the case when τa << τµ and nτa << |τν|:

σ RR =

R − ct t (3 − 4 / n 2 )αE  V1 (θ ) exp Θ ( cl t − R ) −  cl τ µ 4πcε Rτ µ2 

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  R − cl t  Θ( R − c t ) , − exp − l  cl τ µ     (9.12) R − ct t (3 − 4 / n 2 )αE V2 (θ ) exp Θ ( ct t − R ) , σ Rθ = 2 ct τν 4πcε Rτν

σ Rϕ =

 R − ct t exp −  ct nτ µ 4πcRct n 2τ µ2  E sin ϕ

  sgn( R − c t ) , t  

where Θ(χ) is the Heaviside function.

Figure 9.5 Shape of sound pulses for the components of stress tensor σRϕ and σRR in the cases τa > τµ and nτa > |τν|.

Figure 9.6 Shape of dependent pulses in the case of observation under small angles: (a, b) τa < τµ and (c) nτa < |τν|; (1) θ < arcsin (1/n) and (2) θ ≥ arcsin (1/n).

Shapes of sound pulses for the components of stress tensor σRR, σRθ, and σRϕ in the case τa >> τµ and nτa >> |τν|, i.e., the case described by expressions (9.11), are shown in Fig. 9.5. Figure 9.6 presents shapes of sound pulses for the components σRR and σRϕ in the case τa << τµ , which is described by expressions (9.12), and the shape of a sound pulse for the component of stress tensor σRθ . Schematic dependences given in Fig. 9.5

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and 9.6 demonstrate that the shapes of sound pulses change strongly if the conditions of observation change (the same may be said about their absolute values also).

3. PARTICULAR FEATURES OF EXCITATION OF RAYLEIGH WAVES In a way analogous to how the expressions for displacements in the Rayleigh wave in the case of short pulses of penetrating radiation have been obtained in Chapter 7, Section 2, we can write down corresponding expressions for the case of generation of Rayleigh waves by a single highenergy particle under the conditions of the rise of a cascade directly at the surface of a solid half-space. They differ from expressions (7.14) only in the absence of the parameter τχ, which is taken to be equal to zero, and normalization to the total cascade energy E,

uR =

(3 − 4 / n 2 )α 2 E

A

8π 2 π cε ρv R v R

R

∞

∫

0

ω exp(−ω 2τ a2 / 4) × 1 + ωτ µ

π   R cos − ω  − t  dω ,   vR  4 (9.13) (3 − 4 / n 2 )α 2 E uz = 8π 2 π cε ρv R v R

B R

∞

∫

0

ω exp(−ω 2τ a2 / 4) × 1 + ωτ µ

π   R sin  − ω  − t  dω ,   vR  4 where all notations correspond to Chapter 7, Section 2. As the cascade length is greater than its transverse dimensions, then, according to Chapter 7, Section 2, the absolute values of displacements in the Rayleigh wave excited by a high-energy particle are very small.

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4. EFFICIENCY OF SOUND GENERATION In the case of sound generation by single high-energy particles (which are in some sense analogs of “short” pulses of penetrating radiation), we determine the conversion efficiency as follows:

η = Eac / E ,

(9.14)

where Eac is the total energy in the acoustic pulse and E is the particle energy. We consider sound generation only in an infinite space. Using expression (9.1) we can obtain the following expression for η:

η≈

(3 − 4 / n 2 ) 2 α 2 cl2 Eµ

ρcε2

a2

=

Γ 2 Eµ

ρcl2 a 2

,

(9.15)

where Γ is the Gruneisen parameter. For example in the case of a neutrino with energy E = 1016 eV and the parameters of a cascade in water a ~ 2 cm and 1/µ ~ 4 m, the efficiency η is of the order of magnitude of 10−11 for sound generation in water and 10−10 for sound generation in ice.

CHAPTER 10

Experimental Study of Thermoradiation Excitation of Sound This chapter presents results of numerous experimental studies of thermoradiation excitation of sound by penetrating radiation in condensed media, which were published earlier (see [33, 46, 173, 186] for example). At the beginning we will consider the particular features of the field of a laser thermooptical source in the case of quasi-harmonic generation of sound in a liquid by intensity-modulated laser radiation. We will give also the results of experiments on sound generation by laser pulses and moving laser beam. We will discuss the characteristics of acoustic signals generated by laser pulses and excitation of surface acoustic waves in solids. After that we will consider sound excitation by X-rays (synchrotron radiation), proton and ion beams, beams of electrons, positrons, and γ-quanta in liquids and solids. It will be demonstrated that comparison of experimental results with calculations performed on the basis of theoretical concepts exposed in preceding chapters indicates conclusively the fact that the theory of thermoradiation generation of sound has been confirmed experimentally quite well. This provides grounds for utilization of this theory for selection of sources of penetrating radiation (their types, parameters, etc.) for generation of acoustic fields with various preset and remotely controlled characteristics in condensed media, determination of radiation parameters by methods of radiation-acoustic dosimetry, and other applications.

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240 EXPERIMENTAL STUDY OF THERMORADIATION EXCITATION

1. LASER THERMOOPTICAL (THERMORADIATION) SOURCES OF HARMONIC SOUND OSCILLATIONS IN WATER Let us recall some theoretical results and calculations. The formula characterizing the field in the far wave zone (the Fraunhofer zone) of a laser thermooptical source of sound as well as that of a thermoradiation source of sound in a liquid with a free surface has the form p(r ) = −

exp(ikr ) ωmαA Popt Ψ (θ ) , 2πC p r

(10.1)

where Ψ (θ ) =

 (ak ) 2  exp − sin 2 θ    4 µ 2 + k 2 cos 2 θ   kµ cos θ

(10.2)

is the function describing the directivity pattern of a source and cPopt = I0πa2 is the power of penetrating (optical) radiation in a beam. It is possible to demonstrate that the directivity pattern determined by expression (10.2) has a single peak corresponding to the direction along the normal into a liquid at (aµ)2 > 6 − 4⋅21/2 ≈ 0.35. If the region of heat evolution satisfies the conditions (µ/k)2 << 1 << (aµ)2/2, i.e., if it is a sufficiently broad and thin surface layer of a liquid satisfying the conditions of efficient generation, then an expression for the directivity pattern has the form

Ψ (θ ) ≈

  ak  2  exp −  θ   .   2   1 + (k / µ ) 2   k/µ

(10.3)

Its angular half-width (Fig. 10.1a) is equal to ∆θ ≈ 2 / ak .

(10.4)

It is necessary to note that as the frequency ω increases, the amplitude of sound pressure in the case of a free liquid boundary acquires an almost stationary level at k ≥ µ .

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Figure 10.1 Directivity patterns of laser thermooptical sources of sound directed (a) along the laser beam and (b) along the surface of a liquid.

Let us give numerical estimations for the case of water and laser radiation. Here and below we use the next values of parameters: α = 3⋅10−1 K−1, Cp = 4.2 J/(g⋅K), and c = 1.5⋅105 cm⋅s−1. These values correspond to the temperature 18°C. We assume that sound is generated by a modulated pulse of a neodymium laser with pulse energy E = 1.5⋅103 J and length τ = 10−3 s. Let us suppose that the laser beam radius is a = 2.5 cm, 100% intra-pulse modulation of light intensity m = 1 is performed, and the condition of efficient sound generation µ = k = 4 cm−1 is satisfied (this corresponds to the modulation frequency 100 kHz; water in the region of light absorption may be tinted in order to provide the necessary value of µ). In this case according to expression (10.1), the amplitude of sound pressure at the distance R = 10 m in the direction θ = 0° is equal to p0 ≈ 500 Pa and the half-width of the directivity pattern is ∆θ ≈ 0.2 rad. Let us consider the case of a narrow directivity pattern directed sharply along the surface of a liquid (Fig. 1b). Such a directivity pattern is realized at (aµ)2/2 < (µ/k)2 << 1. It corresponds to radiation of an extended (as against the wavelength of sound) chain of in-phase monopoles located under the free surface of a liquid (a thermooptical source of the “rod” type). There is no sound radiation strictly along the surface by virtue of the boundary conditions. The direction to the maximum constitutes a small angle (µ/k << 1) with respect to the liquid surface, and the total angular width of the peak at the level of half-amplitude is ∆hθ ≈ 2⋅31/2µ/k << 1. In the maximum of the directivity pattern we have Ψ(θm) = 0.5 exp [−(ak/2)2]. Let us estimate sound oscillations excited by a neodymium laser with the same parameters as in the previous numerical example but in clear water

242 EXPERIMENTAL STUDY OF THERMORADIATION EXCITATION

(µ = 0.17 cm−1, λlight = 1.06 µm). If the sound frequency is 100 kHz, k = 4 cm−1, and the radius of the light beam is a = 0.5 cm, then the amplitude of sound wave emitted at the angle µ/k ≈ 4.2⋅10−2 rad to the surface of the liquid at a distance of 10 m is p ≈ 90 Pa. It is quite common to characterize the efficiency of a sound source by the amplitude of sound pressure at a distance of 1 m or, in other words, normalized to the distance R = 1 m. Keeping this in mind, we obtain for estimates given above the values p1 = 5000 Pa⋅m and p2 = 900 Pa⋅m. It is common to characterize the threshold value of sound pressure by the value pthr = 10−5 Pa. Comparing pthr, p1, and p2, one can see that sound oscillations can attain quite impressive values of amplitude in the case of laser excitation. Now let us consider experimental results. Frequencies of the order of magnitude of tens and hundreds of kilohertz are optimal from the point of view of production of sources with various directivity patterns in laboratory conditions. As thermooptical generation of sound is most efficient at µ ~ k, a laser must be selected to provide a value of light absorption coefficient at its wavelength in the range from parts of inverse centimeter to several inverse centimeters. This requirement is well satisfied by neodymium lasers with the light wavelength λlight = 1.06 µm or YAG-lasers with the same radiation wavelength. The value of µ in clear water for this wavelength is 0.17 cm−1 and it may be increased readily by solving an absorbing admixture in water if necessary. Results of experimental studies of generation of quasi-monochromatic sound waves in a liquid by modulated laser radiation were published in many papers including those by Bozhkov, Bunkin, and Kolomenskii [32] and Bunkin, Mikhalevich, and Shipulo [42]. Sound waves were excited in water with the help of radiation of a CW or pulsed YAG-laser with wavelength λlight = 1.06 µm. Radiation intensity was modulated by the Pockels cell positioned beyond the laser cavity. Sinusoidal voltage with frequency ω was fed to the cell. Changing the frequency of the control signal of the cell makes it possible to tune smoothly the modulation frequency of laser radiation and the frequency of the excited acoustic field. Changing the size of the radiation spot at the surface of a liquid and the coefficient of absorption of optical radiation (in the case of water it may be changed in a wide range by solution of CuCl2) makes it possible to realize various types of laser thermooptical sources of sound. Experiments [35, 42] were conducted in a water-filled basin. Its walls were covered by a sound-absorbing material. Utilization of the pulsed operation mode of a laser provided an opportunity to separate in time the direct sound signals and the signals reflected by walls, and the influence of the last was insignificant. The water temperature in the basin was 20 ± 1°C. The laser operated in the modes of pulsed and intra-pulse quasi-

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monochromatic modulation of radiation intensity. Sound oscillations in water were detected by a hydrophone at various distances and depth from the region of operation of a laser thermooptical sound source up to several meters. In the process of study of the directivity patterns of laser thermooptical sound sources in the far wave field, the radius of the light spot at the surface of a liquid was equal to 2 cm, the absorption coefficient was µ = 4 cm−1, and the frequency of excited sound oscillations was 100 kHz. A laser thermooptical source of sound with a relatively narrow directivity pattern with the maximum in the direction of the normal to the surface of a liquid is produced under such conditions. In this case the conditions of the far wave field are satisfied already at a depth of about 1 m. Figure 10.2 gives the directivity pattern of a laser thermooptical sound source, which was measured at a fixed distance of 0.75 m from the radiator. The directivity pattern has a single maximum. Its measured width is 2∆θ ≈ 0.2 rad and the calculated value is 0.23 rad. Measured absolute values of the amplitude of sound field are also in good agreement with the results of theoretical calculation.

Figure 10.2 Directivity pattern measured experimentally in the far wave field: a = 2 cm and k = µ = 4.2 cm−1.

Figure 10.3 Dependence of sound pressure amplitude in the far wave field of a laser thermooptical sound source on the coefficient of absorption of optical radiation in water. (1) Calculated curve; (2) experimental data; a = 2 cm and k = 4.2 cm−1.

The solid line in Fig. 10.3 shows the theoretical dependence of sound pressure amplitude in the far wave field on the value of the coefficient of absorption of optical radiation in water. Experimental data are presented also. One can see that the theoretical and experimental data obtained agree well.

244 EXPERIMENTAL STUDY OF THERMORADIATION EXCITATION

In the process of experimental investigation of a laser thermooptical sound source of a “rod-like” shape, sound was excited in water by radiation of a pulsed YAG-laser with the radiation wavelength 1.06 µm, pulse length τ = 0.5⋅10−3 s, and pulse energy E = 0.2 J. Light intensity (within a pulse) was modulated according to a harmonic law with the frequency 105 Hz and 100% modulation depth. The radius of the light spot at the surface of water was 0.25 cm and the absorption coefficient was µ = 0.17 cm−1.

Figure 10.4 Angular dependences of amplitude of sound oscillations in the (a) far and (b) near wave fields of a quasi-periodic laser thermooptical sound source.

Figure 10.5 Dependence of direction of maximum for the angular distribution of amplitudes of sound fields of a laser thermooptical sound source in the wave field.

The distribution of sound field amplitude in water was measured at various distances x from the source. In this case the size of changing of the 2 parameter xµ /k in experiments was 0.7 – 3.2, and the positions of the observation point covered the near and far wave fields (xµ2/k << 1 and xµ2/k >> 1, respectively). Figure 10.4a and b demonstrate the dependence of sound pressure amplitude on the angle θ = arctan (z/x) at the values of the

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245

parameter xµ2/k equal to 0.7 and 2.3, respectively. The dependence of the direction of the maximum of the angular distribution θ0 on the parameter xµ2/k is given in Fig. 10.5. Circles indicate the results of experimental measurements and the calculated data are given by a solid line. Comparison of experimental and calculated data demonstrates that they agree well. The directivity pattern does not have sidelobes. If we increase the parameter xµ2/k, the direction of the maximum of the directivity pattern approaches the value µ/k, which is characteristic to the far wave field.

Figure 10.6 (a) Dependence of level of sound pressure at the axis of an optoacoustic source on the laser power and (b) angular dependence of sound field in the case of laser excitation of sound in water.

Experiments on laser excitation of sound in water were conducted not only in laboratory conditions. Muir, Culbertson, and Clynch [241] investigated laser generation of sound in a lake. A neodymium laser was used, which operated in a pulsed mode with intra-pulse modulation of optical radiation intensity. The modulation frequency provided a quasimonochromatic mode of sound generation in water. Namely, the length of laser pulse was τopt = 10−3 s and the frequency of intra-pulse modulation (the frequency of sound) was f = 20⋅10−3 Hz. Measurements were conducted at a distance of 10 m from the axis of the thermoacoustic array; ka << 1. Obtained experimental results also agree well with theoretical conclusions. For example, it follows from the theory that the amplitude of sound pressure in the far wave field of a laser thermooptical sound source grows linearly as the power of optical radiation increases (see formula (10.1)). This fact was confirmed experimentally. A solid line in Fig. 10.6a shows a corresponding theoretical dependence, while dots indicate experimental data [241]. The level of sound pressure at the axis of a laser thermooptical sound source,

246 EXPERIMENTAL STUDY OF THERMORADIATION EXCITATION

i.e., in the direction of the maximum of the directivity pattern, is plotted in the vertical axis. The values of pressure are adjusted to a distance of 1 m and normalized with respect to 10−6 Pa. The horizontal axis gives the change of the power of optical radiation in kilowatts. Figure 10.6b presents theoretical (solid line) and experimental (dots) results characterizing the angular dependence of the sound field of the same laser thermooptical sound source. In this case the length of laser pulse was τlight = 10−3 s and the sound frequency was f = 5⋅103 Hz. Measurements were conducted at a distance of 16.8 m, and the coefficient of light absorption in water was µ = 15.7 m−1. In conclusion let us give the calculated data [241] characterizing the properties of directivity of a laser thermooptical sound source and its efficiency. Figure 10.7 presents plots demonstrating how the directivity of a laser thermooptical sound source in the far wave field in sea water changes depending on the radiation wavelength of a laser (light wavelength) and the modulation frequency of laser radiation (sound wavelength). If we characterize the directivity properties of an array by the angle θ1/2, under which the radiated acoustic power decreases two times, then the next formula is true:

θ1 / 2 = (180 / π )( µ / k ) .

Figure 10.7 Width of the directivity pattern of a laser thermooptical sound source (thermoacoustic array) in sea water. Numbers at curves mean sound frequency.

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In Fig. 10.7 the angle θ1/2 is plotted in the vertical axis, while the horizontal axis presents the light wavelength. The parameter is the modulation frequency of a laser beam. In the process of calculation the authors [241] used certain averaged values of the coefficient of light absorption for sea water. One can see that the greatest directivity may be attained using a laser radiating blue-green light and in the range of high frequencies. For example, the aperture of the directivity pattern of a laser thermooptical sound source operating in sea water at a frequency of about one megahertz may attain the value of about several thousandths of degree (!) while its value at a frequency of about one kilohertz is equal to several degrees. This is not surprising. The linear size of the array is about 40 m!

Figure 10.8 Level of sound pressure in the field of a laser thermooptical sound source in water. Numbers at curves mean the frequency of sound.

The data given in Fig. 10.8 characterize the efficiency of a laser thermooptical sound source. Calculations have been conducted according to a formula [24] LS = 20 log( fαWopt / 4C p ) ,

where f is the frequency of sound (in Hz) and LS is the level of sound pressure (in dB) normalized to 10−6 Pa and the distance 1 m. The formula is obtained under the assumption that a laser thermooptical sound source is formed in clear water and generation conditions are in fact optimal, i.e.,

248 EXPERIMENTAL STUDY OF THERMORADIATION EXCITATION

µ = k. Sound pressure is determined at the axis of the array in the far wave field at θ = 0°. The level of sound pressure LS is plotted in the vertical axis and the horizontal axis presents the power of laser source. The parameter is the modulation frequency of laser radiation (sound frequency). One can see that powerful lasers are needed for laser thermooptical excitation of harmonic sound oscillations of considerable amplitude in water. For example, the laser power of 106 W is needed at a frequency of about 100 kHz for production of the sound pressure amplitude of 103 Pa⋅m. Taking in view the state-of-the-art nature of laser technology, apparently such power may be obtained only in the pulsed quasi-periodic mode of laser radiation as yet. Calculation results given in Figs. 10.7 and 10.8 are certainly approximate. More precise calculation can be performed in every specific case using theoretical formulae given above.

2. SOUND EXCITATION IN WATER BY LASER PULSES Pulsed excitation of sound by laser radiation was considered by many researchers (for example, see reviews [41, 140, 142] and a book by Ready [175]). Attention was attracted first of all by the opportunity to produce in such a way sound oscillations of a very big amplitude (up to hundreds of atmospheres) increasing with the increase of power of optical radiation. The efficiency of conversion of the energy of penetrating (laser) radiation into the energy of sound in the case of the thermoradiation (thermooptical) mechanism increases proportionally to the radiation intensity. Therefore, studies of thermooptical excitation of sound in the pulsed mode were conducted many times as laser technology developed. Starting from the first studies [260, 261], a common experiment setting is the following. Pulsed laser radiation is incident perpendicularly upon the free surface of a liquid or the surface adjoining a transparent wall. The radiation is absorbed in the surface layer of the liquid. A sound pulse is excited as the result of heat release. The pulse shape is determined by the boundary conditions, thermal characteristics, and parameters of laser radiation including the envelope of a laser pulse. The authors of one of the first experimental papers [221], who studied the particular features of acoustic signals excited in a liquid by laser pulses, investigated the change of the shape of an acoustic pulse as a function of the coefficient of light absorption in a liquid, the dimensions of the laser spot at its surface, and the length and power of laser pulse. The changes of shape and amplitude of acoustic pulses observed by them did not fit into the

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theoretical one-dimensional model [210, 215], which existed at that time. They explained this discrepancy between experimental and theoretical results by the rise of nonlinear effects. As one can see, in reality the observed changes of acoustic pulse shape could be explained within the framework of the linear (three-dimensional) theory of laser thermooptical excitation of sound, which was developed later [109, 110].

Figure 10.9 Oscillograms of acoustic signals in the far wave field, which are generated by “short” laser pulses in a water solution of copper sulphate.

Figures 5.4 and 5.5 (see Chapter 5) show sound pulses in a liquid, which were predicted theoretically under the assumption that laser pulses have a rectangular shape [109], and the pulses obtained experimentally [221]. The calculations were grounded on the conditions of the experiment [221]. The oscillograms of the acoustic signals generated in the far wave field by short laser pulses are shown in Fig. 10.9 [90]. Sound pulses were generated in a water solution of copper sulphate. Changing the solution concentration, it was possible to change the coefficient of light absorption. The light source was a ruby laser, which emitted pulses with length 4⋅10−8 s and energy 0.08 ÷ 0.1 J. The oscillograms were recorded at the next values of parameters: (a) µ = 0.3 cm−1, a = 23 cm, and θ = 60°; (b) µ = 0.8 cm−1, a = 2.3 cm, and θ = 60°; and (c) µ = 1 cm−1, a = 3 cm, and θ = 40°. One can see that naturally, the shape of the sound pulse agrees with the one calculated according to formula (5.28) for “short” laser pulses (see Fig. 10.9). The shape of the acoustic signal is universal and does not depend on the shape of the envelope of a laser pulse. It is determined by the ratio of the characteristic times of sound travel along the generating lines of laser thermooptical sound source. Indeed, as we can see from the parameters of the thermoacoustic array given above, the following condition is always satisfied: the length of an optical pulse is essentially smaller than the characteristic times of sound travel within the spatial region occupied by a laser thermooptical sound source.

250 EXPERIMENTAL STUDY OF THERMORADIATION EXCITATION

It has been demonstrated in Chapter 5 that the shape of an acoustic signal in the far wave field is determined by the second derivative of radiation (laser) pulse if the length of the laser pulse is greater than the characteristic times of sound travel (i.e., when the laser pulse may be considered “long”). It was also established there that in the case of laser pulses with length small as against the time of sound travel along the length of a laser thermooptical sound source τµ and comparable to the time of travel along its width (diameter) τa, or in other words, when a laser thermooptical sound source has the shape of a long narrow cylinder (a “rodlike” array), the sound signal represents a pulse of rarefaction, which repeats in its shape the envelope of an “overturned” laser pulse with a certain positive addition proportional to a small parameter.

Figure 10.10 Comparison of results of a “numerical experiment” [198] and theoretical conclusions [110]. (a) The case of a “long” laser pulse and (b) a “short” laser pulse. (1) Calculation; (2) I′′(t − r/c); and (3) −I(t − r/c).

Berthelot and Busch-Vishniac [198] conducted a kind of a “numerical experiment”. They conducted calculations for the two mentioned cases and obtained total correspondence of these calculations to the conclusions exposed above. The results are given in Fig. 10.10 [198]. The equation of optical generation of sound was solved numerically. Calculations were performed for a laser pulse with intra-pulse sinusoidal modulation. A laser pulse was set by the expression I (t ) = I 0 (t ) sin 2 (ωt / 4π ) = 0.5 I 0 (t )[1 − cos(ωt / 2π )] , where I0(t) = (10.8/τ) exp(−5t/τ). This expression is a good approximation of the description of the shape of a real laser pulse. The comparison was conducted for the modulation frequencies 5⋅103 and 30⋅103 Hz and the pulse length τ = 10−3 s. Solid lines in Fig. 10.10a and b show the results of

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calculation of the shape of the acoustic signal for “short” and “long” laser pulses: (a) r = 4 m, θ = 60°, f = 5⋅103 Hz, and µ = 50; (b) r = 20 m, θ = 60°, f = 30⋅103 Hz, and µ = 0.33. A line with dots shows the functions I′′(t − r/c), i.e., the second derivative of the optical pulse, and I(t − r/c) (Fig. 10.10b), i.e., an “overturned” laser pulse. The vertical axis shows the normalized amplitude of sound pressure, and the time in milliseconds is plotted in the horizontal axis.

Figure 10.11 Shape of sound pressure wave. (a) Experiment and (b) theory.

Berthelot and Busch-Vishniac [198] conducted also an experimental study of laser excitation of sound in clear water in a basin with dimensions 18.29 × 4.57 × 3.66 m3. A neodymium laser with radiation wavelength 1.06 µ was used. Light was polarized with the help of the Brewster cell and light intensity was modulated by the Pockels cell. Light pulses with the length up to 1.2⋅10−3 s and energy about 1.5 J were emitted. Intra-pulse modulation was performed also. Figure 10.11 demonstrates typical experimental and numerical results. An acoustic signal was received by a hydrophone,

252 EXPERIMENTAL STUDY OF THERMORADIATION EXCITATION

amplified, filtered, and fed to a digital oscilloscope. Data from the digital oscilloscope were fed to a computer, where they were recorded simultaneously with the results of calculation. Experimental results are given in Fig. 10.11a, while Fig. 10.11b presents calculated data. The amplitude of sound pressure in pascals is plotted in the vertical axis and the horizontal axis presents time. Experimental conditions were: r = 0.5 m, θ = 75°, a = 0.5 cm, µ = 13.7, f = 35⋅103 Hz, and τ = 1.2⋅10−3 s. Calculation was conducted on the assumption of Gaussian distribution of light intensity in a laser beam (Fig. 10.1b). One can see that experimental and calculated data agree quite well. This agreement is even better if the shape of the envelope of laser pulse for calculation is taken as constant, i.e., I(t) = I0.

3. SOUND FIELD EXCITED BY A SEQUENCE OF LASER PULSES The low efficiency of thermoradiation generation of sound in continuous mode of harmonic modulation of radiation set a problem to find a more “economical” technique of modulation of intensity of penetrating radiation (light). A suggestion was to excite sound in a liquid by a periodic sequence of short (e.g., nanosecond) laser pulses [138, 139]. Such a mode of operation of lasers is well studied and used frequently in practice. In this case a sound field at the repetition frequency of laser pulses and the frequency of harmonics multiple to this frequency is excited in a liquid. The field components at each of these frequencies have their own directivity patterns. The amplitudes of these harmonics are equal to the doubled amplitudes of the acoustic signals in the case when sound is excited by a continuous harmonically modulated laser radiation with the same power and at the corresponding frequency. Thus this mode of pulse sequence provides a significant general increase of the conversion coefficient. At the same time each harmonic obeys the laws of optical generation of monochromatic sound. Thus, by varying modulation frequency, it is easy to select the optimal conditions of excitation for a chosen group of harmonics while other harmonics are suppressed. Let us consider particular features of sound generation by a periodic sequence of pulses of penetrating radiation (laser pulses). We approximate the shape of a single pulse by the function F(t) = P0 exp(−t2/τ2), where P0 is the peak radiation power in a pulse and τ is the pulse length. In this case the power density of the thermal sound sources arising in a medium has the form

RADIATION ACOUSTICS

Q ( R, t ) =

253

∞   µαP0  r 2  C 0 + ∑ C n cos(nΩt ) exp − µz − ,   a 2  πa 2   n =1

where Cn = 2π1/2τT −1exp(−π2n2τ2/T 2), n = 1, 2, …; T is the period of pulse repetition; and Ω is the cyclic frequency of pulse repetition. An acoustic field with a broad discrete spectrum of frequencies, which is determined by the repetition frequency of radiation pulses, their length, and the absorption coefficient of penetrating (laser) radiation, is excited in a liquid. Calculated envelopes of spectra of excited sound oscillations along the axis of the beam of penetrating (laser) radiation for different values of the absorption coefficient of penetrating radiation (light) are given in Fig. 10.12.

Figure 10.12 (1–3) Envelopes of spectra of sound fields excited in a liquid by a periodic sequence of laser pulses at different values of µ and (4) the envelope of the spectrum of a periodic sequence of laser pulses [139]. µ = (1) 4, (2) 19, and (3) 100 cm−1, and (4) τ = 500 ns.

The maximum amplitude of sound field at a preset absorption coefficient µ is attained at the frequency   1 f max ( µ ) =   2π 

1/ 2

1/ 2   2 2  µ 2c 2   µc µ c + 1  −    16 2 4  τ     

.

The maximum possible amplitude of sound field for this excitation technique is realized at the value of the coefficient of radiation absorption µ = 21/2/cτ at frequency (21/2πτ)−1. The ratio of intensity of a spectral component of the sound field excited by a sequence of pulses of penetrating radiation at a preset frequency to the

254 EXPERIMENTAL STUDY OF THERMORADIATION EXCITATION

intensity of the monochromatic sound excited by radiation with monochromatic modulation of intensity and with the same average power is equal to Is/Im = 4 exp(−2π2n2τ2/T 2). In the case, when the on-off ratio of radiation pulses exceeds significantly the number of an excited harmonic of acoustic field, the ratio of intensities attains its maximum value, equal to four. Results of experimental studies of sound excitation by a sequence of laser pulses in a liquid (water) are given and discussed in detail by Lyamshev [137]. Here we give only some of them. Acoustic oscillations were excited in water by a periodic sequence of Gaussian laser pulses of length 500 ns and repetition frequency 20 kHz. The radius of laser beam was 0.05 and 1 cm. The absorption coefficient of laser radiation was changed from 0.17 to 9 cm−1. This provided an opportunity to realize the cases of the “rod-like” and “disk-like” laser thermooptical sound sources. Changing of the coefficient of optical absorption µ was performed by dissolving copper sulphate in a sound-transparent cuvette of sufficient size, which was positioned in a basin within the region of action of a laser source. The cuvette dimensions were always larger than the dimensions of the thermooptical source produced by laser radiation. The conditions of the experiment provided an opportunity to realize the far wave field. Measurements of directivity patterns of different acoustic harmonics were conducted and the results of calculation were compared to experimental data. Calculations show that in the case of a “disk-like” thermoacoustic array (kna >> 1, µa >> 1), sound oscillations propagate basically in the same direction as laser radiation. The divergence of the acoustic beam in the far wave field is determined by the relationship ∆Q ~ (kna)−1. In the case of a “rod-like” array, sound oscillations propagate mainly at an angle close to 90° to the direction of light propagation. Measurements were conducted for the first six harmonics at frequencies 20, 40, 60, 80, 100, and 120 kHz. Measured directivity patterns for the first, third, and sixth harmonics (in the far wave field) for the case of a “disklike” laser thermooptical sound source are shown in Fig. 10.13. Sound waves excited in a liquid under these experimental conditions (a = 1 cm, µ = 4 cm−1) propagate mainly perpendicularly to the surface of a liquid that corresponds to calculations. The results of measurements agree well with the data of calculation. The angular distributions of the acoustic fields for the first and sixth harmonics in the case of a “rod-like” laser thermooptical sound source are given in Fig. 10.14. In this case µ = 0.17 cm−1 and a = 0.05 cm. Excited sound oscillations propagate under an angle close to 90° with respect to the propagation direction of laser radiation. One can see that the directivity patterns of different harmonics of acoustic spectrum get separated in angle.

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Calculated curves are plotted by solid lines. The condition of the far wave field is satisfied for the component at the frequency 40 kHz. This condition is not satisfied for the component at the frequency 100 kHz, but the condition of the Fresnel zone is satisfied for this component. One can see that the experimental data coincide with the calculation within the measurement error. The observed scatter of experimental data is caused by the conditions of the experiment.

Figure 10.13 Angular distribution of acoustic field in the far wave zone for a “disklike” thermooptical source of sound [137]. (1 – 3) Experiment: (1) the first harmonic f = 20 kHz; (2) the third harmonic f = 60 kHz; and (3) the sixth harmonic f = 120 kHz. Solid, broken, and dotted lines correspond to calculation.

Using the dependence of efficiency of sound generation on the coefficient of light absorption, it is possible to excite a smaller number of components of the acoustic spectrum than the number of components in the spectrum of optical radiation. This is caused by the existence of the optimal coefficient of absorption for each frequency of sound oscillations. Excitation of acoustic oscillations with wave numbers smaller than the absorption coefficient is inefficient. The amplitude of each single sound

256 EXPERIMENTAL STUDY OF THERMORADIATION EXCITATION

harmonic is a maximum if the absorption coefficient µ is equal to its wave number k. This is illustrated by the dependence of the amplitude of sound pressure on the coefficient of light absorption measured for the fifth harmonic (f = 100 kHz) and given in Fig. 10.15.

Figure 10.14 Angular distribution of acoustic field of a “rod-like” laser thermooptical source [137]. (a) The second harmonic, f = 40 kHz and (b) the fifth harmonic, f = 100 kHz. Solid lines present calculation and circles correspond to experiments.

Figure 10.15 Dependence of the amplitude of sound pressure on the coefficient of light absorption in the maximum of directivity pattern in the far wave field of a laser thermooptical sound source [137]. A solid line corresponds to calculation and circles present experimental data.

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Experimental results given above confirm qualitatively and quantitatively the major theoretical conclusion, i.e., an opportunity to generate harmonic acoustical oscillations by a periodic sequence of short radiation (laser) pulses.

4. ACOUSTIC FIELD OF A MOVING LASER THERMORADIATION SOURCE OF SOUND IN WATER As we have mentioned already in Chapter 8, the thermoradiation technique of sound excitation allows us easily to produce sound sources moving with almost arbitrary speed. This opens new opportunities in acoustics. For example, supersonic motion of a beam of penetrating radiation (laser beam) along the surface of a liquid absorbing radiation or the solid surface is accompanied by excitation of the Mach wave in a medium, which propagates at a characteristic angle to the motion axis. Utilization of this effect provides an opportunity to create a narrow-beam pulsed radiator of sound. If the intensity of penetrating radiation in a beam (light in a laser beam) is modulated at a certain frequency, then it is possible to tune the frequency of emitted sound on account of the Doppler effect without changing the modulation frequency. It turns out that on the contrary, it is possible to select the configuration of a thermoradiation or laser sound source in order to reduce as much as possible the effect of motion on the directivity and frequency of oscillations in the emitted acoustic field. Motion on a circle produces a sound field at multiple frequencies. Changing of the radius of a “tubular” radiation (laser) beam at a certain supersonic rate provides an opportunity to focus generated sound under the irradiated part of the surface of a liquid or solid. Opportunities to control sound in this case are determined certainly by the opportunities to control parameters of penetrating radiation. The laser thermooptical and thermoradiation sound sources considered are ideal in the sense that sound is emitted by a certain region of a medium itself, which is affected by penetrating (laser) radiation, and no foreign bodies are introduced into it. Acoustic fields excited by moving thermoradiation and laser thermooptical sources of sound are similar in many features to electromagnetic fields produced by moving charges. In particular, these are the acoustic analogs of the Cherenkov (the Mach wave) and transition radiation and radiation in the process of sharp acceleration or braking of a charge (source), i.e., bremsstrahlung.

258 EXPERIMENTAL STUDY OF THERMORADIATION EXCITATION

It is appropriate to recall some basic theoretical concepts before starting to describe and discuss results of experimental studies of acoustic field of a moving laser thermooptical sound source in water (see Chapter 8, Section 1). Let us consider a moving laser thermooptical sound source in a liquid, which is produced by a pulsed laser beam scanning the liquid surface. We can write down an expression for the power density of sources, Q(R ′, t ) = Q R (R ′) f (t )(1 + m cos Ωt ) .

(10.5)

Here R′ = (x − vt, y, z), f(t) is the time envelope of a laser pulse, and Ω is the frequency of intra-pulse modulation. We assume that τΩ >> 1, where τ is the length of laser pulse, which is determined by the function f(t); QR(R′) = µAI(x − vt, y) exp(−µz). We restrict ourselves to consideration of the particular features of the acoustic field of a laser thermooptical sound source in the far wave zone. In other words, as in Chapter 8 we consider the field at the distances R >> L, where L is the length of the trajectory of laser beam motion. According to previous exposition (see also [31, 112, 147, 148, 240]) we can write down the next space-time representation for sound signals from a moving laser thermooptical sound source, ∞

p0 (R, t ) = −

iωα ∗ (k ) f (ω − β ∗ω ) exp − iω  t − R  dω , QR    4πC p R c    −∞

∫

∞

p D (R, t ) = −

∫

−∞

(10.6)

iωαm ∗ Q R (k ) f ∗ (ω − β ∗ω ) × 4πC p R (10.7)

  R  exp − iω  t −  dω , c    where p0(R, t) is the pulsed contribution and pD(R, t) is the Doppler contribution into the field of a moving laser thermooptical sound source, k=

R ωu ∗ (k ) = Q ( R )e ikR dR , f (ω ) = 1 , u = , QR ∫ R c R 2π

∫ f (t )e

i ωt

dt .

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259

The length of the sound train of the Doppler-compressed envelope is τD = τ |1 − β*|. The characteristic scales due to spatial distribution of a source are τa = c−1a sinθ and τµ = (µc)−1cosθ. Depending on a relation between the time τD determining the spectral width of the functions f(ω(1 − β*)) and the times τa and τµ determining the spectral width of the function QR*, the shape of sound signal is determined either by the Doppler-compressed envelope or by the spatial distribution of the source [147]. First, we assume that τD >> τa, τµ, 2πΩ. Then we can consider the function QR*(k) in the expression for the Doppler signal (10.7) changing smoothly as compared to the function ωf and take it outside the integral at the value ω = Ω(1 − β*)−1. In the case of a supersonic source, if the observation point is located within the angular interval 0 ≤ θ < θm, the major contribution into expression (10.7) belongs to the integral with respect to the negative ω. In this case the projection of the source velocity onto the observation direction is larger than the sound velocity (β* > 1). Consequently, the sound signal arrives at the observation point in the reverse sequence as compared to the sequence of its generation by the source. Therefore, this Doppler signal may be called anomalous. The modulus sign should be substituted for it in the expression for the Doppler frequency. In the case of subsonic motion of a source or in the interval θm < θ ≤ π in the case of supersonic motion, the anomalous Doppler signal paD is absent and there is only the normal Doppler signal pnD. The Doppler signals are described by an expression [33] p(n,a)D ( R , t ) =

mAα |J 4πC p R

     R   R  |  Jf ′ J  t −   − iΩ D f  J  t −   − c   c       (10.8)

∗ ( k ) exp[i ( k R QR D D − Ω D , t )] , ∗

where J = (1 − β )−1 is the Doppler factor, kD = uΩD/c, and f ′(ξ) = df/dξ. The contribution proportional to f′(J(t − R/c)) is caused by tuning of moving sources connected with the fact that the envelope changes with time. This contribution is small as against the second one. In the case of a rectangular envelope in the intermediate time moments f ′(t) ≡ 0, it arises only at the end points of the source trajectory. Formula (10.8) for a rectangular envelope was obtained by Lyamshev and Sedov [148] and Bozhkov, Bunkin, and Kolomenskii [31]. It follows from expression (10.8) that the amplitude of the Doppler signal produced by a moving source can be obtained from a corresponding expression for a motionless source. In this case it is enough to substitute the

260 EXPERIMENTAL STUDY OF THERMORADIATION EXCITATION

components of the Doppler wave vector uΩD/c for the components of the wave vector uΩ/c and multiply the expression obtained by the modulus of the Doppler factor. This agrees with a general rule, which allows us to describe the sound field of a moving volume-distributed source in the far wave field proceeding from the equivalent motionless source [240], which is obtained from the initial one by compression of the time scale t → Jt and multiplication by the modulus of the Doppler factor. Therefore, sound excitation by a scanning laser beam occurs in the same way as by a motionless source but with the time-compressed envelope |J|f(Jt) and modulation frequency equal to the Doppler frequency. This rule as applied to pulsed excitation of sound by a scanning laser beam was formulated by Lyamshev and Sedov [148]. Using this rule, it would be possible to obtain immediately formulae analogous to expression (10.8) proceeding from calculations of sound excitation by a motionless laser thermooptical sound source, as has been stressed above (Chapter 8, Section 1). Let us consider the most typical cases, which are realized at different relations between the Doppler length of a laser pulse τD and times of sound travel along the projections of longitudinal and transverse dimensions of a laser source onto the observation directions τµ and τa. Let τD >> τa, τµ that corresponds to excitation of the sound pulses long in comparison with the times τa and τµ. This case is realized at the velocity of source motion satisfying the conditions |1 − β*| >> τa/τ, τµ/τ almost in the whole half-space z ≥ 0 if long laser pulses are used, and in the region of manifestation of the anomalous Doppler effect if one uses short pulses. The inverse Fourier transformation applied to formula (10.6) gives an expression for pressure in an excited sound pulse under given approximation accurate to the terms of the order of magnitude of O(τµ3/τD3), p0 ≈

  Aαa 2 I 0 R  τ µ | J |3 f ′′ J  t −   , c  2C p R  

(10.9)

i.e., in the case of large values of Doppler length under these conditions, the shape of sound signals in the far wave field is determined by the second derivative of the effective time envelope of a laser pulse, and pressure is proportional to the light intensity and inversely proportional to the absorption coefficient of laser radiation. At τµ >> τD >> τa that may be realized in the case of the “rod-like” shape of a source, the projection of the motion velocity of a laser thermooptical sound source onto the observation direction vR satisfies the condition τµ/τ >> |1 − vR/c| >> τa/τ (this may require supersonic motion of

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the source at the surface of a liquid). Pressure in a sound pulse is determined by the expression

p≈−

 | t ′ |  sf Aαa 2 I 0   . | J | f ( Jt ′) − exp −  τ µ  2C p Rτ µ  2τ µ   

(10.10)

The shape of a sound pulse is determined by the envelope itself in this case:

| J | f ( Jt ′) , t ′ = t −

R , sf = c

∞

∫ f (t )dt .

−∞

Let us consider the case τD << τa, τµ, which is realized for short laser pulses at small motion velocities and for any value of τ at vR ≈ c. Application of the inverse Fourier transformation to formula (10.6) gives in this case an expression for pressure in a sound pulse,    ′ Aαa 2 I 0 s f  4τ µ t p≈− exp −   2  τ 8C p Rτ µ  π τ a   a    t′  τ t′ t ′  exp Erfc a + + exp −  τµ   2τ µ    2τ µ τ a 

Erfc( x) =

2

∞

   

2

  − exp τ a  2τ µ   

2

  ×  

     Erfc τ a − t ′   ,  2τ µ τ a      

(10.11)

2

−t ∫ e dt .

π x

It is necessary to note that expression (10.11) is always true for the observation directions, where vR = c. In the considered case τD << τa, τµ, the shape of emitted sound pulses does not depend on the time envelope of the laser pulse f(t) and is determined by the geometry of the region of heat release. Under these conditions a sound pulse consists of a rarefaction phase with preceding and consequent phases of compression. In this case sound pressure is proportional to the total energy of laser pulse. Special attention will be given further to the comparison of experimental data obtained with theoretical results for the characteristic cases considered here.

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Let us proceed to the results of experimental studies of the acoustic field of a moving laser thermooptical sound source. They are exposed in the most complete form by Bozhkov et al. [33]. We will follow this paper restricting ourselves basically to consideration of pulsed moving laser thermooptical sound sources. Bozhkov et al. considered sources of “rod-like” shape, which move in water rectilinearly, and both uniformly and with acceleration along a finite trajectory. In this case pulses of a YAG-Nd3+-laser (λlight = 1.06 µm), without an additional intra-pulse modulation of light intensity, were used. The pulses had the characteristic length at the half-height τ = 0.5 ms and the energy E = 1.5 J. The source dimensions were a = 0.25 cm and µ−1 = 5.5 cm. The distance from the light track to the observation point was varied from 1 m to 4 m. Subsonic and supersonic velocities of source motion were studied. In the experiment in the case of uniform motion of a laser thermooptical sound source, the length of laser pulse τ = 0.5 ms exceeded significantly the characteristic times of sound traveling along the transverse and longitudinal dimensions of the radiator τa = 1.7 µs and τµ = 40 µs. Utilization of long laser pulses provided an opportunity to increase energy contribution by τv/2a times (as compared to the case of a motionless source) without changing the aggregate state of a medium. In this case the Doppler length of a laser pulse τD could be both greater and much smaller than the times τa and τµ depending on the motion velocity and the direction of observation. The case τD >> τa, τµ at small subsonic velocities of a source was realized in the whole half-space occupied by the liquid, while at velocities close to the sound velocity it was realized beyond the region of the observation directions close to the line of motion, i.e., at vR/R << c(1 − τa/τ). It was demonstrated that the shape of sound pulses of an “effective” motionless source is described in this case by formula (10.9). In the case of the Gaussian shape of f(t), a sound signal consists of a rarefaction pulse with preceding and consequent “splashes” of compression (so called “switchingon” and “switching-off” pulses). The studies conducted by Bozhkov et al. demonstrated that, in the case of motion of a laser thermooptical sound source, such shape of pulses stays the same in a wide range of velocities almost up to the values of v, at which τD ≈ 2 max (τa, τµ). A characteristic oscillogram of a sound pulse detected in the plane of radiator motion (ϕ = 0) at a distance 4 m from the light track in the case of the depth of the observation point equal to 0.6 m (θ = 8.5°) for the velocity of source motion v = 0.980 c, is given in Fig. 10.16. For the selected observation direction τD = 14.4 µs, τµ = 6 µs, and τa = 1.6 µs. In this case R ≈ 4L, where L is the length of the trajectory of motion of a laser beam, R >> a, µ−1, and the observation point is located in the far wave field with respect to the trajectory length and the source dimensions a and µ−1. The

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263

shape of the sound pulse corresponds to the second time derivative of the envelope of the laser pulse given in Fig. 10.17a (the time scale is compressed 34 times approximately). The total length of the compression pulse in the sound signal ∆τ = 16 ± 2 µs is determined by the time interval between the points of inflection at the fronts |J|f(Jt), which coincides to a good precision in the case under consideration with the length of the “effective” laser pulse at the half-height τD = 14.4 µs. Such coincidence of experimental and calculated data was obtained for the whole considered range of values of τD. The amplitude of the rarefaction phase of the sound signal given in Fig. 10.16 was 45 ± 10 Pa and that of the compression phase was 30 ± 10 Pa.

Figure 10.16 Oscillogram of a sound pulse excited at τD >> τa, τµ [33].

Figure 10.17 Shape of envelope of laser pulse (a) without modulation and (b) with harmonic modulation of light intensity at the frequency 100 kHz [33].

If the observation direction was changed, the length of a sound signal changed correspondingly to the change of the Doppler factor J = (1 − β* cosθ ′)−1. Namely, when θ ′ increased from 3 to 8.5°, ∆τ increased from 6 to 16 µs, while the calculated value of τD increased from 7.6 to 14.4 µs and τµ increased from 2.1 to 6 µs at v = 0.985 c. The amplitude of sound pressure, which is determined as |J|3f′′(Jt) according to expression (10.9), changes proportionally to |J|3 ~ ∆τ−3.

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Moreover, sound pressure within the considered range of τD is proportional to τµ, which grows with increase of θ. Figure 10.18 gives the dependence of the amplitude of sound pulses on the parameter τµ/∆τ3 in the case of changing of the observation angle from 26.2 to 19.3° and the constant value ϕ = 17.7°. In this case τµ changed from 14.4 to 6 µs and the measured length of rarefaction pulses changed from 26 to 3.4 µs. A linear dependence of sound pressure on the parameter τµ/∆τ3 was observed in a broad range of angles θ ′ and ϕ, for which τD >> τa, τµ. A certain deviation of experimental data from the theoretical linear dependence at small θ ′ (see Fig. 10.18) is connected with the fact that in this case τD becomes comparable to τµ and the condition τD >> τa, τµ gets violated.

Figure 10.18 Dependence of amplitude of sound pulses on the parameter τµ/∆τ3 in the case of change of observation direction [33].

The value of τD becomes comparable to the values of τµ and τa for the observation directions satisfying the condition |1 − β*| ~ τa/τ, τµ/τ. It is necessary to note that this condition is realized for any selected observation direction R if vR is close enough to c that may require supersonic motion of a source at the liquid surface. Measurements demonstrated that the shape of sound pulses in this case differed from the shape of pulses considered earlier for the case τD >> τa, τµ. The amplitude and length of sound signals under these conditions depend on the source velocity to a less extent. The cubical dependence of the amplitude on J transforms into a linear one. If the motion velocity of a laser thermooptical sound source increases further and the value of τD decreases, a transition to the case τµ >> τD >> τa occurs. This case is realized at τµ/τ >> |1 − β*| >> τa/τ. Sound pulses excited under these conditions are described by approximate expression (10.10). A rod-like source configuration used in experiments provided an

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opportunity to fulfill such observation conditions when τµ ≈ 2τD ≈ 4τa at v = 1.003 c close to ϕ = 0 and θ ′ = 13.3°. An oscillogram of a pulse detected under such conditions is given in Fig. 10.19a. The pressure trend in the rarefaction pulse repeats the shape of the effective envelope |J|f(Jt) as formula (10.10) predicts. The pressure amplitude in the rarefaction pulse is 102 ± 30 Pa. The amplitude of positive splashes described by the second term in formula (10.10), which is proportional to the small quantity τD/τµ, is equal to 20 ± 10 Pa in this case. The length of the rarefaction pulse at its half-height (the pulse is given in Fig. 10.19b) is ∆τ = 5.3 ± 2 µs that corresponds to the calculated value τD = 4.7 µs. Further reduction of the Doppler length leads to the fact that τD becomes much smaller than τa and τµ, i.e., τD << τa, τµ. This case is realized for both short laser pulses (almost at any velocity of radiator motion and direction of observation) and long laser pulses (in the observation directions localized near the directions with vR = c). Sound pulses excited in this case are described by formula (10.11). Figure 10.19b gives an oscillogram of a sound pulse detected at R = 4 m, θ ′ = 9°, ϕ = 0, and v = 0.993 c. Under these conditions τD = 1.2 µs, τµ = 6.3 µs, and τa = 1.7 µs. The sound pulse has the shape of a rarefaction pulse located between compression pulses. The total length of the rarefaction pulse ∆τ = 7.8 µs is close to τµ, i.e., the length of the sound signal is determined by the geometrical dimensions of a source. Conducted experiments demonstrated also that the shape of excited sound signals depended on the shape and length of light pulses at τD << τa, τµ as would be expected from theoretical concepts.

Figure 10.19 Oscillograms of sound pulses excited (a) at τµ > τD > τa and (b) τD < τa, τµ [33].

Now let us consider the particular features of sound fields in the case of uniform motion of a source with velocities equal to or exceeding the sound velocity in a medium. If v > c, three characteristic cases are realized depending on the position of the observation point: vR < c, vR = c, and vR > c. All results given above for the case of radiator motion with subsonic velocity stay the same in the region of observation directions with vR < c for sound fields of a radiator moving with supersonic velocity.

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An important feature of radiator motion with supersonic velocity is the presence of observation directions where vR = c. In-phase accumulation of sound disturbances from all parts of the trajectory of radiator motion occurs in these directions in the far wave field. Due to this fact it is possible to increase energy release and form especially intense and short acoustic signals, which are called the Mach wave in acoustics, increasing the length of the light track L. Within the framework of this description, the length of the light track L is restricted only by the rise of nonlinear acoustic phenomena at large values of L and high intensity of sound. Formation of the Mach wave is demonstrated by the plots given in Fig. 10.20. These plots characterize the dependence of the amplitude and length of sound pulses on vR. They are obtained for a fixed position of observation point R = 4 m, θ ′ = 6°, and ϕ = 0 in the case of the velocity of radiator motion changing in the range 0.9 – 1.15 c. As the value of vR approaches c, 3 the amplitude of sound grows proportionally to |J| within the region τD >> τa, τµ, then it slows down to linear dependence on |J|, and at τD << τa, τµ, the amplitude attains its maximum value p = 1.5⋅102 ± 50 Pa at E = 1.5 J. The length of sound pulses is equal to τD within the accuracy of measurements at τD >> τa, τµ and to a value about max(τa,τµ) at τD << min(τa, τµ). The minimum length ∆τ ≈ 2 µs is attained in the Mach wave at vR = c. It is necessary to note that the pulses with the maximum amplitude were observed under the conditions of this experiment at vR = 0.97 c and not in the Mach wave as one can see from Fig. 10.20. Bozhkov et al. explain this as follows [33].

Figure 10.20 Dependences of (1) amplitude and (2) length of excited sound pulses on the projection of the velocity of source motion onto the observation direction [33].

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In the case of small length of laser pulses, a motionless laser thermooptical sound source emits sound signals of the maximum amplitude in the directions R0, where cotθ ′′ = 1.7/µa. In fact, such a configuration of a laser thermooptical sound source is possible that signals of negligibly small amplitude are emitted in the directions of the Mach wave propagation. The condition cotθ ′ = 1.7/µa must be satisfied also in the direction of Mach wave propagation (vR = c) for generation of maximum sound amplitude on account of motion in the case of fixed energy of a laser pulse. In the case of preset configuration of the source and fixed observation direction (θ ′ = arctan(µa/1.7); ϕ) this is performed by selection of motion velocity vR ≈ c[1 + (µa/1.7)2]1/2(cosϕ)−1. An oscillogram of characteristic sound pulses observed at vR = c, R = 400 cm, θ ′ = 6°, and ϕ = 0 is given in Fig. 10.21a. The shape of these pulses coincides with the shape of the signal given in 10.19b and corresponding to the case τD << τa, τµ. It was assumed in the process of calculation that the laser pulse had a rectangular shape [112]. Linearity of the dependence of sound amplitude in the Mach wave given in Fig. 10.20 demonstrates that the experiments were conducted within the framework of linear acoustics. The transition to the Mach wave corresponds within a theoretical description to τD → 0 and sound pulses are described (as in general at τD << τa, τµ) by formula (10.10), i.e., they do not depend on the envelope of laser pulse and are determined by the region of heat release. On the other hand, the results given in Fig. 10.22 are a good experimental confirmation of the theoretical conclusion that the amplitude of a sound signal generated by a very short laser pulse is proportional to the energy of this laser pulse (and not the power as in the case of a long laser pulse or under the conditions of generation of monochromatic oscillations) and does not depend on the shape of laser pulse.

Figure 10.21. (a) Oscillogram of a sound pulse in the Cherenkov direction and (b) the shapes of sound signal in the Mach wave ((1) experimentally detected pulses and (2) calculated pulses) [33].

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Bozhkov et al. [33] studied also the region of observation directions, where the projection of the motion velocity had supersonic values vR > c. A special feature of sound fields in this region was inversion of excited sound pulses with respect to time, i.e., disturbances emitted from different points of the trajectory were detected in the time sequence reversed with respect to that of their generation. This fact is described by the change of the sign of J in the argument of f(Jt) in the process of transition to the region vR > c. In other respects the description of sound signals under these conditions coincides with the case vR < c.

Figure 10.22 Dependence of sound pressure amplitude in the Cherenkov direction on the energy of laser pulse [33].

Figure 10.23 demonstrates oscillograms of sound pulses obtained at the velocity of radiator motion v = 1.008 c in the motion plane (ϕ = 0) for vR = 0.998 c, θ ′ = 6° (see Fig. 10.23a) and vR = 1.002 c, θ ′ = 9° (see Fig 10.23b). As the shape of utilized laser pulses was close to symmetrical, there were no changes in the shape of generated acoustic pulses connected with signal inversion in time at vR > c. The difference in the amplitude and length of the pulses arises because of the presence of the directivity properties of a rod-like source. Under the conditions corresponding to Fig. 10.23 τD << τa, τµ and the length of sound pulses is determined by τµ. In this case τµ = 4.2 µs and ∆τ = 6.7 µs at θ ′ = 6° and τµ = 6.3 µs and ∆τ = 7.8 µs at θ ′ = 9°.

Figure 10.23 Oscillograms of sound pulses excited at (a) subsonic and (b) supersonic velocities of motion of a laser thermooptical sound source [33].

Now let us present briefly some results of experimental studies of periodic sound fields of moving laser thermooptical sources [33]. Also

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studied experimentally were parameters of sound fields of moving laser thermooptical sources produced in water in the process of absorption of sufficiently long sound pulses with the characteristic energy E = 0.2 J and intensity-modulated according to a harmonic law at the frequency Ω/2π = 105 Hz (see Fig. 10.17b). A laser pulse contained about 50 periods of modulation. In this case 2π/Ωτ = 2⋅10−2 s. The shape of the envelope of sound trains excited in the experiment is described well by the Dopplercompressed time envelope of the laser pulse. Detection of sound trains was performed at the Doppler frequency in the frequency band ∆ω/2π = 9 kHz that provided an opportunity to detect without distortions the trains of the length τD ≥ 0.1 ms, which corresponded to |J|−1 ≥ 0.2 and ΩD/2π ≤ 5⋅105 Hz. Directivity patterns with respect to sound amplitude F(θ ′,ϕ) at θ ′ = const measured in the horizontal plane at two values of velocity of radiator motion v1 = 0.15 c and v2 = 0.47 c are presented in Fig. 10.24. The amplitude of a sound signal was measured in each direction at the Doppler frequency. Solid curves represent results of calculation.

Figure 10.24 Directivity pattern of a moving laser thermooptical source of sound in the horizontal plane at v = (1) 0.15 c and (2) 0.47 c [33].

Motion of a laser thermooptical sound source changes its directivity pattern in the vertical plane too. Figure 10.25 presents the distribution of sound pressure amplitude measured in the wave field in the motion plane of a rod-like laser thermooptical sound source moving with velocity v = 0.3 c and the dependence of the half-width of its directivity pattern in the same plane on the velocity of its motion. Solid curves in this figure represent results of calculation, while experimental data are given by dots. The range of angles θ, where the directivity pattern given in Fig. 10.25b was calculated, was restricted by the domain of applicability of the theory. Results obtained demonstrate basic features introduced by motion into the directivity pattern of a radiator. In the case of a subsonic velocity the directivity pattern gets extended in the horizontal plane in the direction of motion. This effect is accompanied by the pattern narrowing in the direction of radiator motion and its broadening in the opposite direction.

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Figure 10.25 (a) Distribution of sound pressure amplitude in the wave field for a moving “rod-like” laser thermooptical sound source in the vertical plane at v = 0.3 c and ϕ = 0 and (b) half-width of its directivity pattern in the same plane as a function of the velocity of its motion [33].

5. LASER THERMOOPTICAL EXCITATION OF SOUND IN SOLIDS – EXCITATION OF SURFACE WAVES Results of investigation of laser thermooptical excitation of sound in solids were published in many papers, starting from the first papers devoted to this problem [260, 261]. In contrast to experimental studies of laser generation of sound in liquids, the majority of the results were qualitative, since the theoretical basis of these experiments were solutions of one-dimensional initial and boundary problems of thermooptical generation of sound, and experimental data obtained could not be reliably interpreted theoretically. Another reason for this was the fact that investigation of thermooptical acoustic signals in solids required more complex experimental equipment (different types of receivers for detection of longitudinal, shear, and surface waves, dependence of results on reliability of receiver contact with a medium, its dimensions, etc.). Considerable progress was achieved here too at the beginning of 1980s (for example, see [186] and the papers cited there) and especially in investigation of laser excitation of surface waves (see a review [105]). Here we restrict ourselves basically to presentation and discussion of results of studies of laser thermooptical excitation of a surface acoustic wave (SAW). Excitation of these waves aroused interest of the researchers

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rather recently in connection with the problem of research on surface physics and chemistry. It is possible to excite SAW in a broad frequency range and with various time and spatial parameters with the help of laser radiation. Laser generation of SAW was described apparently for the first time by Lee and White [234]. The Rayleigh wave was excited in the process of absorption of a single pulse of a Q-switched ruby laser in an aluminum film deposited upon the studied area. Various materials were used as a substrate: ceramics and fused and crystalline quartz. Laser radiation was focused on a narrow rectangular strip. Therefore, the SAW front was close to the plane one. However, a limited bandwidth of detectors of Rayleigh waves did not allow the researchers to resolve the time shape of SAW adequately. In fact, the amplitude of the spectral component of SAW at the frequency of the principal resonance of the detector was measured. The next paper devoted to laser generation of SAW [232] was published only after 11 years in 1979. Radiation of a Q-switched laser was used. The radiation was focused by a cylindrical lens into a narrow strip on the plane face of a sample near the rib. Velocities of longitudinal, shear, and Rayleigh waves were measured simultaneously. The authors indicate the next advantages of laser excitation of acoustic signals in solids: an opportunity to conduct measurements using samples of small dimensions and simple shapes within a wide range of temperature and pressure, immediacy of data acquisition, and simultaneity of measurements of velocities of all three types of waves. A technique for laser generation of periodic SAW was suggested for the first time by Ash, Dieulesaint, and Rakhouth [192]. A harmonically modulated laser beam was directed through a periodic mask at the studied surface absorbing optical radiation. A sharp peak in efficiency of SAW excitation was observed at the modulation frequency corresponding to the Rayleigh wavelength equal to the mask period (Fig. 10.26a). Measurement of the directivity pattern of a SAW optoacoustic array gave the results coinciding well with the theoretical results for a collinear array (Fig. 10.26b). Estimations of SAW amplitude coincided with experimental data in the order of magnitude. Naturally, utilization of a periodic mask for laser excitation of periodic SAW is optional. It is sufficient to have harmonic modulation of intensity of optical radiation and focus it as one can see, in particular, from theoretical consideration of the problem of laser excitation of SAW. This version of laser excitation of SAW was realized later by Veith and Kowatsch [258], who used a mode-locked dye laser for this purpose. Optical radiation was absorbed in a film deposited upon the studied surface. The laser beam was focused by a spherical or cylindrical lens. SAWs were detected by a transducer tuned to the repetition frequency of periodic laser pulses 76.4

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MHz. In particular, it was determined that the amplitude of acoustic signal was directly proportional to the laser power (see Fig. 10.27) that agreed completely with theoretical conclusions.

Figure 10.26 (a) Dependence of SAW amplitude on the frequency of light modulation and the mask characteristic and (b) directivity pattern of SAW optoacoustic array [192]. Solid lines present calculation and dots correspond to experimental data.

Figure 10.27 Dependence of SAW signal amplitude on the power of laser radiation in CW mode [258]. Radiation was focused by (1) a cylindrical or (2) spherical lens.

The paper by Royer and Dieulesaint [247] also needs special attention. The authors investigated theoretically excitation and propagation of SAW in an anisotropic elastic medium when its surface was subjected to periodic

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(laser) heating. A boundary problem was solved using the reciprocity principle that substantially simplified the procedure of acquisition and analysis of the solution. Incidentally, this fact was indicated by the authors themselves [247]. Calculations were compared to experiments conducted earlier by Veith and Kowatsch [258]. In particular, a good coincidence of theoretical and experimental dependences of SAW amplitude on the power of laser radiation was revealed (Fig. 10.28).

Figure 10.28 Dependence of SAW signal amplitude on the power of optical radiation incident upon the surface of a crystal of lithium niobate [247]. A direct line corresponds to theory [247] and circles show experimental data [258].

Generation of pulsed SAW was studied by Aindow et al. [191], Khodinskii, Korochkin, and Mikhnov [183], and Brueck, Deutsch, and Oates [202]. Aindow et al. [191] used a neodymium laser, which operated in the Q-switched mode at wavelength 1.06 µm. The length of laser pulse was about 17 ns (at the half-width) at pulse energy up to 6⋅10−3 J. Detectorstransducers had bandwidth from 1 to 10 MHz. The laser beam was directed perpendicularly to the sample surface and focused (when this was necessary) with the help of a lens. Slabs of aluminium, steel, copper, etc., were used as samples. Taking in view the theory described in Chapter 7, experimental results by Aindow et al. [191] are noteworthy. First, acoustic pulses of Rayleigh wave were observed in the near wave field and therefore, it was no wonder that the pulse shape had the characteristic form of Nwave. Such form is proper to acoustic pulses in the near wave field of a laser thermooptical sound source. Secondly, it follows from the theory that in the case of the laser pulse length small in comparison with the characteristic times of sound travelling (with respect to the spatial

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dimensions of a laser thermooptical sound source), the amplitude of the acoustic pulse increases proportionally to the energy of optical radiation, and its length grows proportionally to the diameter of the optical spot at the sample surface (the width of a laser thermooptical sound source). These theoretical conclusions are also confirmed by experiments [191] (see Fig. 10.29).

Figure 10.29 Dependence of Rayleigh pulse amplitude on the energy of laser radiation in an aluminium sample [191].

Aindow et al. studied also the dependence of the amplitude of acoustic pulses of longitudinal and shear waves on the power and energy of laser radiation. It was revealed (as was stressed by Khodinskii, Korochkin, and Mikhnov [183]) that the amplitudes of these pulses increased proportionally to the energy of laser radiation (Fig. 10.30). This corresponds completely to the theory of thermoradiation and thermooptical (laser) generation of pulsed acoustic fields in solids. The results of the first experimental studies of SAW excitation in the case of motion of a laser beam at the surface of a solid at various subsonic, transonic, and supersonic velocities of beam motion (in comparison with the propagation velocity of Rayleigh waves) were published apparently by Velikhov et al. [49]. A CO2-laser with power up to 10 kW was used. The power was constant during a generation pulse. Laser radiation was focused into a strip of length l = 10 mm and width a = 1 mm. The strip was moved along the surface of a massive aluminium sample (5×10×30 cm3) with the help of a rotating mirror. The scan velocity was changed by changing the angular velocity of the mirror. The absolute error of velocity measurements did not exceed 1% and the relative error did not exceed 3%. The scan length L (3 ÷ 17 cm) was changed by moveable screens. The power of optical radiation was reduced

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with the help of attenuators. SAWs were detected by a transducer glued to a corner transducer, which was positioned at the end of the scan track. The same detectors were positioned at the adjacent and opposite faces of the sample in order to detect background waves. Experimental results were compared to theoretical conclusions [92]. The last correspond completely to analytical concepts and theoretical results given in Chapter 7.

Figure 10.30 Change of amplitude of acoustic pulses as a function of energy in a laser beam [191] for (a) a pulse of longitudinal waves and (b) a pulse of transverse (shear) waves. A transducer with resonance frequency 3 MHz was used for longitudinal waves and a transducer with the resonance frequency 4 MHz was used for transverse waves. (1) Steel and (2) aluminium.

Experimental and theoretical results agree well with each other as in the case of bulk waves. A sharp (resonance) increase of efficiency of SAW generation was observed when the velocity of beam motion approached the velocity of Rayleigh waves (Fig. 10.31). The width of the resonance is determined by the size of the spot a and the length of the scan track L. If there is an offset δv from the resonance value of cR, the amplitude growth

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continues up to the moment when SAW and the beam would not go away from each other at a distance ~ a. Thus the width of the resonance is estimated by the value of δv ~cRa/L that corresponds to experimental data. As the track length increases, the resonance curve becomes narrower (Fig. 10.31) and the wave amplitude grows. Deviation of the dependence of the wave amplitude from the linear one with distance is connected apparently with SAW diffraction.

Figure 10.31 (1) Dependences of Rayleigh wave amplitude in the resonance and (2) resonance half-width on the length of the scan track [49].

The amplitude of surface displacement in the scan mode may attain large values (it was up to 0.1 µm in the experiment [49]) in the process of very weak heating of the surface. Nevertheless, it stays proportional to the power of laser radiation. The estimates of displacement [92] agree well with the data of measurements [49]. Dykhne and Rysev [92] obtained the solution of the problem of SAW excitation by a moving laser spot shaped as a strip. If the scan velocity coincides precisely with the velocity of the Rayleigh wave, the last grows linearly with time, and the component of particle velocity normal to the surface repeats the distribution of light intensity over the spot section. A relative change of volume in the process of deformation in the case of aluminium has an order of magnitude of vI0L/a (v = 7⋅10−12 cm2/W is the dimensional coefficient and I0 is the light intensity). According to Dykhne and Rysev, an advantage of scan mode is an opportunity to affect surface layers of a material purely mechanically without any significant heating. The problem of efficiency of laser excitation of sound in solids has been considered in Chapter 7. Using the results by Krylov and Pavlov [115], we

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give here estimates of efficiency of laser excitation of Rayleigh, longitudinal, and shear waves. Krylov and Pavlov [115] analyzed laser excitation of SAW in a nonheat-conducting medium paying major attention to the relative efficiency of excitation of different types of waves (longitudinal, shear, and Rayleigh waves) depending on the value of kRa, where kR = ω/cR is the wave number of the Rayleigh wave. It was determined that at kRa ~ 1 the major part of power of all types of acoustic waves (~ 67%) belonged to the Rayleigh waves, while the part of energy contained in bulk waves was essentially smaller (26% for transverse waves and 7% for longitudinal waves). As kRa increased, the part of energy contained in longitudinal waves grew rapidly and finally it became dominant. This can be explained by the fact that in the process of thermal expansion of a medium only a longitudinal wave is excited, while a transverse wave arises in the process of reflection of the longitudinal wave from the boundary. Therefore, sharp focusing of a laser beam is necessary for thermooptical excitation of high-frequency SAW. This condition was satisfied in the experiments by Brueck, Deutsch, and Oates [202] and Veith and Kowatsch [258]. The theoretical consideration exposed above and the experimental results of laser excitation of sound in solids have been restricted by the framework of the thermal and thermooptical mechanism and a solid elastic medium has been assumed to be isotropic. Meanwhile, the problem of investigation of acoustic processes arising in the case of interaction of laser (penetrating) radiation with solids is very complex even under the conditions where the volume density of absorbed energy of light is small, there is no change of aggregate state of a substance, and acoustic effects arising may be described within the framework of a linear theory. We give the results of a recent experimental study of SAW generation in the case of optical radiation affecting a semiconductor (silicon) [1] as an example. If a semiconductor is affected by optical radiation with a quantum energy hν exceeding the band-gap energy Eg, excitation of acoustic waves occurs on account of two different mechanisms: the thermal mechanism connected with heating and cooling down of the crystal lattice (only this mechanism has been considered above) and the deformational mechanism connected with photo-generation of electron-hole plasma. Longitudinal acoustic waves excited on account of the deformational mechanism were studied experimentally by Gauster and Habing [214] and Veselovskii et al. [50]. Pogorel’skii indicated the need to take into account the deformational mechanism in the process of analysis of optical generation of surface waves (SAW) [172]. Gusev and Karabutov developed a theoretical model for description of Rayleigh waves in the case of light absorption in a semiconductor [79]. Avanesyan et al. were the first who detected

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experimentally SAW produced by deformation of the crystal lattice of silicon in the process of optical generation of electron-hole plasma [1]. The most illustrative basic result of experiments was the change of polarity of detected SAW in the process of changing of the wavelength of optical radiation absorbed in silicon. This result follows formally from the theoretical consideration of the problem of laser excitation of sound (including SAW in silicon). One can see this considering for example the right-hand side of the inhomogeneous wave equation describing laser generation of sound waves in semiconductors. It is possible to demonstrate that the terms describing the thermal and deformational mechanisms of optical sound generation have different signs. In fact, the change of SAW polarity in the process of transition from the deformational to thermal mechanism (or vice versa) is connected with the fact that the increase of semiconductor (silicon) temperature leads to expansion of crystal lattice and the increase of the concentration of nonequilibrium carriers leads to compression [214].

Figure 10.32 Scheme of experiment on excitation of surface waves in silicon. (1) A crystal, (2) piezoelectric sensor, (3) cylindrical lens, and (4) light beam [1].

The geometry of the experiment is given in Fig. 10.32. A silicon crystal was shaped as a cylinder with radius R ≈ 1.5 cm and thickness of about 1 cm. The planes (111) were the bases of the cylinder. A piezoelectric sensor with resonance frequency 18.2 MHz was positioned at a polished cut of the cylindrical surface. The sensor provided an opportunity to detect the vertical component of sound particle velocity v in SAW. In order to excite weakly diffracting SAW, optical radiation was focused by a cylindrical lens into a strip of the length H and width a in such way that H2a−1 >> R. The condition λR ~ a was satisfied in the experiments, where λR is the Rayleigh wavelength. The first condition meant that the experimental conditions

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satisfied the requirements of one-dimensionality of the problem of SAW excitation. This simplified comparison of experimental results and estimations. In the case when the second condition was satisfied, the efficiency of SAW excitation was close to optimal. Light exposure was performed by pulses of radiation of a YAG-Nd3+ laser at the fundamental wavelength (λ1 = 1.06 µm and hν1 = 1.17 eV) and the second harmonic wavelength (λ2 = 0.53 µm and hν2 ≈ 2.34 eV). The pulse length was τL ≈ 20 ns. The main experimental result was the change of polarity of detected SAW in the process of changing the wavelength of radiation absorbed in silicon. It is presented in Fig. 10.33. The analysis shows that this effect is connected with the change of the dominant mechanism of SAW excitation.

Figure 10.33 Oscillograms of detected acoustic pulses for different wavelengths of incident optical radiation [1]. (a) λ1 = 1.06 µm, scale 0.2 µs/div.; (b) λ2 = 0.53 µm, scale 0.5 µs/div.

In conclusion let us give some results of measurements of directivity patterns of laser thermooptical sound sources in solids. These results were published in several papers (see [186] and especially [222]). In experiments [222] acoustic signals were excited in aluminium semispherical samples. The experimental scheme is shown in Fig. 10.34. A beam of a pulsed YAG-Nd3+-laser operating in a Q-switched mode at the wavelength 1.06 µm was incident perpendicularly on the plane surface of a solid aluminum semi-sphere. The pulse length was ~50 ns and the pulse energy was ~30 mJ. Measurements of longitudinal and shear waves (the radial and tangential displacements of the semi-sphere surface, respectively) were performed with the help of piezoelectric sensors of corresponding polarization. Piezoelectric sensors were equipped with a special clamping device and a thin oil film was deposited at the sensitive surface of a sensor contacting the semi-spherical surface of a sample in order to improve

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acoustic contact. A high-quality acoustic resonator tuned to a fixed frequency 1 MHz was installed at the sensor output.

Figure 10.34 Scheme of setup for measurement of directivity patterns of laser thermooptical sound sources in solids. (1) An aluminium (solid) semi-sphere with diameter 10 cm; (2) piezoelectric sensor; (3) laser beam; (4) laser beam splitter; (5) Q-switched laser; (6) photodiode; (7) oscilloscope input [222].

Figure 10.35 Experimentally measured directivity patterns of a “linear” laser thermooptical sound source in an aluminium sample [222]. (a) Longitudinal vibrations and (b) transverse vibrations. Points indicate experimental data.

A laser beam was focused by lenses at the sample surface either into a spot with diameter from 1 to 3 mm or into a line with width 0.025 cm and length 8 cm. The condition of the far wave field of a laser thermooptical sound source was satisfied for the diameter of aluminium semi-sphere. Figure 10.35 shows the directivity characteristics of a “linear” laser thermooptical sound source with the dimensions indicated above, which were obtained in the result of measurements at the frequency 1 MHz for longitudinal and transverse waves. Analogous characteristics obtained on

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the basis of calculations are given in Fig. 10.36. One can see that theoretical and experimental results agree quite well at least qualitatively.

Figure 10.36 Theoretical dependences characterizing directivity of a “linear” laser thermooptical sound source in an aluminum sample for (a) longitudinal and (b) shear waves [222].

6. SOUND EXCITATION BY X-RAYS (SYNCHROTRON RADIATION) IN METALS The results of the first experimental studies of sound excitation by X-rays in condensed matter and first of all in metals, were presented by Kim and Sachse [226, 227]. These results are given in the most complete form in the paper by Sachse, Kim, and Pierce [250]. There is nothing surprising in utilization of X-rays for sound excitation in condensed media and especially in metallic targets after studies of laser excitation of sound. In both cases electromagnetic radiation is used to generate sound. However, in contrast to laser radiation, the track length of quanta of X-rays in metallic targets depends on physical parameters of a target (the atomic number of a substance) and the energy of radiation quanta. For example, the penetration depth of X-rays with energy 10 keV in aluminium is about 1.4⋅10−2 cm while it is about 3.76⋅10−7 cm in the case of laser radiation. This difference may be essential from the point of view of applications. The source of X-rays in the experiments [226, 227, 250] was the synchrotron of Cornell University (USA). Synchrotron radiation of X-ray range was used. We should recall that synchrotron radiation arises when particles such as electrons and positrons for example are accelerated in a vacuum chamber of an accelerator up to relativistic velocities in the process

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of their motion along a curved trajectory in a magnetic field. Synchrotron radiation used in the experiment [250] was generated in the process of motion of a “package” of electrons in the storage ring of the synchrotron. Fundamental characteristics of synchrotron radiation are high intensity, broad spectral range, strong polarization, pulsed time structure, and natural collimation. Generated X-rays become homogeneous and wide-band at the energy of about 10 keV. As the energy of accelerated particles grew, these properties of radiation were violated. X-rays used in experiments had the following parameters: the pulse length 0.16⋅10−9 s and the pulse repetition period about 2.56⋅10−6 s. This pulse periodicity was determined by the time of motion of an electron package in the storage ring of the synchrotron. The energy of the beam of photons (per pulse) of X-rays incident directly at a sample-target was 1.12⋅10−6 J per pulse. Figure 10.37 presents the block scheme of the experiment. X-rays (synchrotron radiation) were directed through (1) a collimator at (2) a protecting screen and then at (3) a sample-target. (4) A wide-band piezoelectric transducer was fixed at the opposite side of the target. A signal from the transducer output was fed to the input of (5) a preamplifier with the band 0.01 – 2 MHz and then to (6) an integrating amplifier and (7) data collection system. A synchronization signal was used to improve noise stability of measurements. This signal was taken from (8) an X-ray detector, which had a response time not larger than 10−9 s. The synchronization signal was fed to (6) the integrating amplifier containing an analog-to-digital converter. The minimum dimension of the X-ray beam at the target surface was 3 mm.

Figure 10.37 Block scheme of experimental set-up [250].

Sample-targets had the shape of a disk with diameter 5.72 cm and thickness 1.52 cm. They were made of aluminum, stainless steel, copper, bronze, and titanium. Aluminum cylindrical blocks with length 10.2 mm were used to measure the directivity characteristics of thermoradiation sound source in solids. It is expedient to note that the thickness of the disk

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targets was essentially larger than the radiation length, i.e., the penetration depth of X-rays in the target material (as it was mentioned above, in the case of aluminum µ−1 ≈ 1.4⋅10 cm). The characteristic times of sound propagation along the dimensions of a thermoradiation sound source in targets were larger than the length of X-ray pulses. In other words, the conditions of sound excitation in solids by very short pulses of penetrating radiation were satisfied. Figure 10.38 presents a typical recorded shape of a signal obtained from a wide-band transducer after digital filtration. Sampling time started at the arrival of a pulse of X-rays at the surface of a sample-target. The letter p indicates an ultrasonic radiation-acoustic pulse caused by longitudinal waves, and the letter s indicates the pulse produced by transverse waves; s(−1) corresponds to a pulse of transverse waves caused by the action of a preceding pulse of X-rays incident on a target and p(1) corresponds to a pulse of longitudinal waves cased by the next pulse of X-rays. We should remember that the repetition period of X-ray pulses was 2.56 µs. Check measurements of ultrasonic signals excited in a sample-target were conducted with the help of a piezoelectric transducer fixed at the lateral surface of a target and not at its rear side in the target center on the axis of X-ray beam incident at the target. This was done in order to determine once more that ultrasonic pulses were excited in the target and not in the piezoelectric detector located at the path of the X-ray beam.

Figure 10.38 Recording of signals from a thermoradiation sound source in a stainless steel target at the output of a piezoelectric transducer after wide-band digital filtering and derivation: p – pulses of longitudinal waves, s – pulses of transverse waves [250].

The dependence of amplitude of an ultrasonic pulse on energy in a beam of X-rays incident at a steel target is shown in Fig. 10.39. The vertical axis presents the values of voltage proportional to the amplitude of ultrasonic signal and the directions characterizing energy in the beam of X-rays are

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plotted in the abscissa. The beam energy is directly proportional to the amplitude of current in the particle beam in the synchrotron and electrical voltage is directly proportional to current in the synchrotron beam. The data were obtained as the result of averaging of at least one hundred single measurements (for each point). The analogous dependence of ultrasonic signal amplitude on the energy of photons of X-ray beam is given also in Fig. 10.40. The gap in the data in the area of the electrical voltage 0.7 V is caused by synchrotron operation.

Figure 10.39 Dependence of acoustic signal amplitude (longitudinal waves) generated by a beam of X-rays on the energy of photons in the beam [250].

Figure 10.40. Dependence of amplitude of acoustic signal (longitudinal waves) on energy in a beam of X-rays (notations are the same as in Fig. 10.39; another sampletarget) [250].

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Measurements of the dependence of amplitude values of an ultrasonic signal on the size of the aperture of an X-ray beam were conducted. Experiments were conducted in such a way that the beam was passing in front of a sample-target through a narrow slot. The slot height was constant and constituted 3 mm, while its length could be changed within the range from 3 to 20 mm. The results of measurements for a stainless steel target are given in Fig. 10.41. One can see that at the beginning a proportional dependence of peak value of amplitude of the ultrasonic thermoradiation signal on the slot length (aperture) was observed, and then this dependence was lost. Sachse, Kim, and Pierce [250] determined experimentally that in the case of significant increase of the slot length (the aperture of X-raybeam) the shape of ultrasonic signals (which stays almost constant in the process of initial change of the slot length) is subjected to considerable changes. The change of the shape of acoustic signals explains the loss of direct proportionality between the peak amplitude of ultrasonic signals and the linear dimension of the aperture of X-ray beam in the case of significant change of the aperture (slot) length.

Figure 10.41 Dependence of the maximum value of the amplitude of an acoustic signal of longitudinal waves in a sample-target on the dimensions of the aperture of an X-ray beam.

All presented experimental results agree quite well with theoretical conclusions. Indeed, it follows from theory that in the case of excitation of acoustic signals in a solid elastic half-space by very short pulses of penetrating radiation the amplitude of acoustic signal changes directly proportionally to the energy of a pulse of penetrating radiation. Just these conditions were realized in the experiment. The depth of penetration of Xrays into the target material was essentially smaller than the target

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dimensions and the radiation pulse length was very small as against the time dimensions of the thermoradiation sound source created in the target by the action of pulses of X-rays. It follows from theory also that, in the case of constant radiation intensity in a beam incident upon a target, the increase of the linear dimensions of the beam (slot) aperture must lead to the proportional change of the acoustic signal amplitude. Just such a dependence was observed in the experiments [250]. Sachse, Kim, and Pierce [250] measured also the directivity characteristics (angular dependence of acoustic signal amplitude) of a thermoradiation sound source in a solid. The results are given in Fig. 10.42. In the case corresponding to Fig. 10.42a, the beam dimensions were 3 × 3 mm2, and in the case given in Fig. 10.42b the dimensions were 3 × 15 mm2. These characteristics relate to longitudinal waves in a sample-target. As was noted by Sachse, Kim, and Pierce [250], the polar characteristics are very similar to the angular acoustic characteristics of an optoacoustic source arising in a solid under the effect of short pulses of focused laser radiation [222] (see Figs. 10.34 – 10.36).

Figure 10.42 Polar characteristics of a thermoradiation acoustic source in aluminium at different dimensions of a penetrating radiation beam. Measurements were conducted for longitudinal waves at the repetition frequency of pulses of X-rays 1.2 MHz [250].

7. SOUND EXCITATION BY A PROTON BEAM Experimental studies of sound excitation by a beam of protons in a condensed medium were conducted many times (see, for example, [84, 220, 253 – 255]. Here we will discuss in detail the results published by Danil’chenko et al. [84], Hunter and Jones [220], and Sulac [253].

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Sulac [253] conducted extensive experiments on sound excitation by proton beams in liquids at the accelerators of the Brookhaven National Laboratory (USA) and Harvard University (USA). The proton beam came from an accelerator. It was directed through a collimator into a water basin. The acoustic signal was detected by a hydrophone equipped with a preamplifier and a circuit for amplification and detection. Measurements were conducted in the pulsed mode. Experiments on sound excitation in water by a beam of protons with energy 200 MeV were conducted using the linear accelerator of the Brookhaven National Laboratory. The basin dimensions were considerably larger than the proton track length and spatial dimensions of an acoustical signal in water. This provided an opportunity to perform space-time selection of the direct acoustic signal and signals reflected from the basin walls. The proton track length in water was about 30 cm. The beam reset (pulse length) changed within the limits from 3 to 200 µs and energy release in a pulse varied within the limits from 1010 to 1021 eV. The diameter of the proton beam was constant and equal to 4.5 cm. Analogous experiments were conducted at the cyclotron of Harvard University with a beam of protons with energy 158 MeV. In this case the energy release in a pulse was 1015 eV, the pulse length was 50 µs, and the track length in water was about 16 cm. Cyclotron experiments were conducted not only with water but also with various liquids in different conditions (at different values of liquid temperature and static pressure). The dimensions of the basin in this case were smaller than those of the basin used in the experiments with the linear accelerator but still considerably larger than the characteristic sound wavelength. The third series of experiments was performed using the accelerator of the Brookhaven National Laboratory with a beam of protons with energy 28 GeV (and very small pulse length). Energy release in a pulse was smaller than 1019 eV as in the experiments with the linear accelerator. In a typical experiment of this series, 3⋅1011 protons covered the distance of 20 cm during a pulse, the beam diameter was varied from 5 to 20 mm, and the pulse length was smaller than 2 µs. Opposite to the cyclotron measurements, the length of a pulse of protons was always smaller than the time of sound propagation along the beam diameter, i.e., the conditions of sound excitation by a very short pulse of penetrating radiation were satisfied. The scheme of the experiments is shown in Fig. 10.43. The measurements demonstrated naturally that in the near wave field of a thermoacoustic array, an acoustic signal has the shape of an N-wave. The dependence of the N-wave period on the beam diameter was measured in the experiments with a beam of protons with energy 28 GeV (Fig. 10.44). One can see that the observed dependence is linear. This

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corresponds to the theoretical conclusion that under the condition τ < τa the length of acoustic signal is directly proportional to the beam diameter.

Figure 10.43 Scheme of sound generation by a proton beam. L is the track length of protons, d is the diameter of the region of a proton beam, R is the distance to the observation point, and A is the characteristic distance to the boundary between the near and far wave fields [253].

Figure 10.44 Dependence of acoustic signal length on the diameter of proton beam [253].

Figure 10.45 demonstrates the experimental data characterizing the dependence of acoustic signal amplitude on energy release in a pulse in the case of a proton beam of a very small duration (τ ≤ 10 µs). Theory predicts that the linear dependence of signal amplitude on the energy of penetrating radiation must be observed. This fact was confirmed in the experiments. Analogous data are given in Fig. 10.46 but for a proton beam with smaller energy release in a pulse. The experiments were conducted at the cyclotron of the Harvard University. It follows from theory that in the case of constant energy (power) in a beam the amplitude of acoustic signals must change in inverse proportion to the square of the beam diameter. The experimental data confirming this rule are given in Fig. 10.47. According to the equation of sound generation by penetrating radiation in condensed media, the amplitude of sound signals increases in direct

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proportion to the ratio of the coefficient of thermal volumetric (linear) expansion to the specific heat conductivity of a medium. Sulac confirmed this connection experimentally (see Fig. 10.48) [253].

Figure 10.45 Dependence of acoustic signal amplitude on energy release in a proton beam (H2O, 20°C, d = 4.5 cm, R = 100 cm) [253].

Figure 10.46 Dependence of acoustic signal on energy release in a proton beam (H2O, 20°C, d = 1 cm, R = 8 cm) [253].

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More evidence of the thermoradiation mechanism of sound generation by penetrating radiation under a moderate density of energy released in a medium is the dependence obtained of the amplitude of an acoustic signal of a thermoradiation sound source on temperature [253]. This dependence for a proton beam in water is given in Figs. 10.49 and 10.50. A particular feature of this experimental dependence is the fact that the coefficient of thermal expansion for water must turn into zero at 4°C. Meanwhile, the amplitude of an acoustic signal generated by a proton beam vanishes at approximately 6°C. This fact may be explained by the existence of an additional mechanism of sound generation, namely microstriction compression of a medium under the action of ionizing particles [10, 66]. This leads to the effect of compensation of thermal expansion of a medium caused by the thermoradiation mechanism at the temperature about 6°C [220].

Figure 10.47 Dependence of acoustic signal amplitude on the diameter of a proton beam [253].

Hunter and Jones [220] conducted thorough experiments in order to determine the presence of a non-thermoradiation mechanism of sound generation in water affected by a proton beam. Experiments were conducted at the accelerator of the Brookhaven National Laboratory with a beam of high-energy protons 20 GeV. The pulse length was varied in the range 1.5 ÷ 3 µs that is essentially less than the time of sound travelling along the beam diameter (8 ÷ 10 mm and 5 ÷ 7 µs, respectively). A proton beam was introduced into a special Dewar flask with water. Water temperature could

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be varied, and these changes were carefully monitored. Sound pulses were detected by a miniature high-sensitivity hydrophone made of the piezoelectric ceramics of zirconate-titanate, with the transmission band from 0.1 Hz to 120 kHz, and equipped with a circuit of amplification and detection.

Figure 10.48 Dependence of acoustic signal on the ratio α/Cp [253].

The purpose of the experiments was to determine how the shape of a sound pulse changes in the process of changing of water temperature within the temperature interval close to the critical temperature where the coefficient of thermal expansion of water turns into zero and then changes its sign for the opposite one. A preliminary judgment that the power spectra of signals of thermoradiation and non-thermoradiation “origin” under certain conditions (in their major energy-carrying part) must differ since the shape and length of pulses of different origin is different, was made on the grounds of the idea on a possible non-thermoradiation (non-thermoelastic) mechanisms. These ideas were realized in experiments. It was determined that in the case of sound generation in water by a proton beam with the characteristics described above and broad-band detection of generated acoustic signals, the signal amplitude vanishes at 6°C (Fig. 10.51), but in the case of detection in a relatively narrow band corresponding to the energy-carrying band of N-wave produced due to the thermoradiation mechanism (this band is determined by the time of sound travelling along the transverse dimensions of a beam), the acoustic signal vanishes at a temperature of 4°C (Fig. 10.52)

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which corresponds to the temperature dependence of the coefficient of thermal expansion of water (Fig. 10.53).

Figure 10.49 Dependence of coefficient of thermal expansion α (a solid line) and acoustic signal amplitude (circles) on temperature [253].

Very interesting studies of sound excitation by a proton beam in water were conducted by Danil’chenko et al. [84]. Measurements were performed using a beam from the synchrotron of the Institute of Theoretical and Experimental Physics (Russia). Protons with energy 200 and 190 MeV were used. The collimator diameter was 4 cm. Water temperature was monitored. The pulse length was 100 ns and much smaller than the length of acoustic pulses (τa ≤ 100 µs). This made it possible to consider the process of heating of water volume in the region of beam action to be instant. Total energy release was varied within the range 8⋅1015 ÷ 23⋅1019 eV. Measurements of acoustic signals were performed by a hydrophone and a measuring circuit with the frequency band 0.1 ÷ 80 kHz in the near wave field of a thermoradiation sound source produced in water by the beam. The amplitude of the positive half-wave of N-signal was detected. Figure 10.54 shows the dependence of the acoustic signal on the energy of protons, which was obtained as the result of the measurements. Data by other researchers who conducted analogous measurements using proton, laser,

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and electron beams of various energies [68, 69, 254] are given in this figure for comparison.

Figure 10.50 Dependence of acoustic signal amplitude on temperature [253].

Figure 10.51 Shape of acoustic signal at various values of temperature in the frequency band 7 – 80 kHz [220].

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Figure 10.52 Shape of acoustic signal at various values of temperature in the frequency band 7 – 40 kHz.

Figure 10.53 Dependence of acoustic signal amplitude on temperature in the frequency band 7 – 40 Hz [220].

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Figure 10.54 Dependence of acoustic signal amplitude on particle energy. (1) Data by Danil’chenko et al. [84]; (2) data by Golubnichii, Kalyuzhnyi, and Korchikov [68]; (3) data by Golubnichii et al. [69]; and (4) data by Sulac et al. [254].

Figure 10.55 shows the dependence of acoustic signal amplitude on the hydrophone position at the beam axis [84]. The mechanism of proton absorption corresponding to the so-called Bragg peak is observed clearly at the end of the track of protons. The presence of this mechanism is caused by the known dependence of ionization losses on proton energy. The dependence of losses of α-particles of polonium in air on the residual track is given in Fig. 10.56 for comparison (see [188], p. 207). One can see a good correlation of results given in Figs. 10.55 and 10.56. The experimental results given above, which characterize sound generation by protons in a condensed medium, concern basically the case when the dimensions of medium volume are large in comparison with the track length of protons in a medium. However, the cases are interesting when the ratio of the track length to the target thickness changes within the range of change of proton energy. Such experiments were conducted by Borshkovskii, Volovik, and Lazurik-El’futsin [36, 38]. As it follows from physical concepts and the theory, a peak must be observed in the curve characterizing the dependence of acoustic signal amplitude on the energy of protons incident at a target shaped as a plate. The authors observed the peak and named it the acoustical peak of protons. The physical nature of this peak is essentially the same as that of the Bragg peak at the curve characterizing proton absorption in a medium. The experiments [38] were conducted using the accelerator of the Institute of Theoretical and Experimental Physics. Relatively low-frequency acoustic vibrations (f = 66 kHz) arising in an aluminum plate-target (the plate thickness h = 0.2 cm) were detected. Proton pulses (the pulse length τ = 20 µs) with initial energy Ep = 24.6 MeV in a beam with diameter

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d = 0.75 cm after a collimator were incident at the plate center. The energy of the proton beam was changed by transmitting it through a holder with decelerating plates made of copper foil.

Figure 10.55 Dependence of signal amplitude on hydrophone position along the beam axis [84].

Figure 10.56 Dependence of ionization created by α-particles of polonium on residual track [188].

Figure 10.57 shows the dependence of the displacements x1 (curve 1) and x2 (curve 2) of the front and rear surfaces of the plate, respectively on proton energy. Curves 1 and 2 were obtained by calculation on the basis of the simplest one-dimensional model of thermoelastic (thermoradiation) excitation of sound by protons in a plate. Circles indicate experimental results. Light circles correspond to the experiments by Borshkovskii, Volovik, and Lazurik-El’futsin [38]. The peak in curve 2 corresponds to the energy of protons with a track length in aluminum equal to the plate

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thickness (Ep = 10 MeV). A dashed-line curve corresponds to the half-sum of the displacements x1 and x2, which is proportional to absorbed proton energy according to estimations [38]. Black circles show experimental results published earlier by Borshkovskii and Volovik [36], which may be an illustration demonstrating ionization losses in the process of interaction of a proton beam with a substance and increase of the amplitude of excited sound. These experimental data are given in Fig. 10.58 presenting the dependence of the amplitude of the acoustic signal generated in an aluminum plate by protons with initial energy ~70 MeV on the thickness of a plexiglas plate (radiation absorber). Absorbing plates were installed in the path of a proton beam in the front of the aluminum plate. The experiments were conducted using the linear accelerator of the Institute of Technical Physics of the Academy of Sciences of Ukraine. The maximum value of the amplitude of acoustic signal corresponds to the Bragg peak. Results of calculation corresponding to these experimental data are given in Fig. 10.57. These experiments [38, 175] demonstrated that the maximum corresponds to such energy of protons incident on an aluminium target cell, when the track length of protons in the aluminium plate is equal to its thickness. The results [36, 38] agree with the theoretical conclusions given in Chapter 6: the condition of equality of the radiation length to the plate thickness is optimal for sound generation by penetrating radiation.

Figure 10.57 Dependence of acoustic signal amplitude on proton energy for an aluminium plate [38].

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Figure 10.58. Dependence of signal on the thickness of a plexiglas absorber [38].

8. EXCITATION OF ACOUSTIC WAVES IN METALS BY ELECTRONS, POSITRONS, AND γ-QUANTA Sound excitation by electrons was considered in the first papers on radiation acoustics. At first it was a theory describing generation of the Cherenkov acoustic radiation by an electron moving uniformly with a supersonic velocity in a metal [101, 203]. The first papers devoted to radiation-acoustic experiments also concerned sound excitation by an electron beam in solids. It was characteristic that these studies were conducted almost simultaneously with the first experiments on laser excitation of acoustic waves in solids [260, 261]. Multiple experiments on sound excitation by electrons in condensed media including sound excitation by electron beams in metals were performed after that. These studies were conducted using various accelerators in the broad energy range in conditions, where the thickness of sample-targets was larger than the radiation length, or with thin plates (see [37, 39, 197, 242] for example). Excitation of acoustic waves by electron beams in water was investigated by Golubnichii et al. [69] and Lyamshev and Chelnokov [156]. Papers discussing the nature of acoustic waves generated by electron beams in crystals and natural solids were published recently [3, 15, 45, 116, 170]. We will give some typical results of studies of excitation of acoustic waves by electrons, positrons, and γ-quanta in metals. Borshkovskii et al. [37] experimented with the linear accelerators of the Physical and Technical Institute of the Russian Academy of Sciences. The scheme of experiments was traditional. A beam of electrons (positrons) from a linear accelerator was incident on a plate made of the metal under investigation. A piezoelectric transducer was installed on the plate. The transducer was

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connected to an amplification and detection circuit. A typical length of a particle “package” incident on a sample-target was τ = 1 µs. A flow of bremsstrahlung was produced by an electron beam with energy Ee = 620 MeV in a tantalum target with thickness 5.3⋅10–2 cm. After transmission through a series of collimators, the beam of bremsstrahlung had an average transverse dimension d = 1.5 cm at the surface of the studied target plate. Figure 10.59 shows the dependence of the amplitude of acoustic signal on the diameter of an electron beam at the target, a lead plate with the thickness 0.2 cm. The plate was thin, i.e., its thickness was smaller than the track length of electrons. The experiments were conducted at electron energy Ee = 20 MeV, a constant number of particles in the beam, and radiation pulse length τ = 2 µs. Thus, in the case of variation of the beam diameter at the plate surface, the total energy (power) of the electron beam stayed constant. One can see that the amplitude of acoustic signal changes approximately in inverse proportion to the square of the beam diameter. If we take into account the fact that the efficiency of thermoradiation conversion is directly proportional to the intensity of penetrating radiation, then these experimental data agree well with the results of the theory of thermoradiation sound excitation in solids as in the cases of sound excitation by beams of protons and synchrotron radiation (X-rays) considered earlier.

Figure 10.59 Dependence of acoustic signal amplitude on the diameter of an electron beam at a target [37].

Figure 10.60 demonstrates the results of measurements of the dependence of the amplitude of acoustic signal in the case of a thick aluminium plate (h = 5 cm). The plate thickness in these experiments was of the order of magnitude of the track length of electrons in aluminium, and the length of the radiation pulse was smaller than the propagation time of sound along the beam dimensions in the plate. The observed amplitude of acoustic signal changes directly proportionally to the electron energy in the beam, as follows from theory.

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Figure 10.60 Dependence of acoustic signal amplitude on the energy of electrons for a thick aluminium plate [37].

Figure 10.61 Dependence of acoustic signal amplitude on the number of electrons for a thick lead plate [37].

A dependence of acoustic signal amplitude in a thick lead plate on the number of electrons (positrons) in a pulse of penetrating radiation is shown in Fig. 10.61. The plate thickness was h = 5.0 cm. The particle energy was Ee = 620 MeV. As the total energy in a radiation pulse is directly proportional to the number of particles, the experimental results agree with theory. Figure 10.62 gives the dependence of acoustic signal amplitude on the energy of electrons for a thin aluminium plate (h = 0.2 cm). In the considered case the plate thickness is small as against the radiation length. The energy of electrons in this experiment was smaller than the critical one, i.e., Ecr ≤ 40 MeV, and the main losses in the process of absorption of an electron beam in aluminum were ionization losses. In this case the total number of secondary electrons or δ-electrons in the energy range from 10 to 30 MeV did not change. Just this fact is the explanation of the absence of dependence of acoustic signal amplitude on the energy of electrons in a beam.

Figure 10.62 Dependence of acoustic signal amplitude on the energy of electrons for a thin aluminium plate [37].

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Finally, the dependence of intensity (not amplitude!) of acoustic oscillations excited in a thick lead plate on the total number of equivalent photons of γ-radiation at Ee = 620 MeV is given in Fig. 10.63. One can see that in this case direct proportionality between the amplitude of acoustic signal and the energy of photons of γ-radiation is also observed.

Figure 10.63 Dependence of acoustic signal intensity on the total number of equivalent photons for a lead plate with the thickness 5.0 cm [37].

Malugin and Manukin [161] give data on sound excitation in solids by a beam of low-energy electrons. These data deserve our attention also because of the fact that the target was an aluminium cylinder with the mass 1 t. The cylinder had the first quadrupole mode of vibrations 104 rad/s and the Q-factor 105. Cylinder vibrations were detected by a capacitance transducer. Signal accumulation was performed. The source of electrons was an electron gun positioned near the lateral surface of the cylinder. The gun accelerated electrons to an energy of 0.5 keV. The length of a current pulse was τ = 8 µs. The interest in investigation of the impact of low-energy particles on such massive bodies arose in connection with the problem of detection of gravitational waves with the help of detectors in the form of massive elastic bodies. Figure 10.64 shows the dependence of the amplitude of acoustic vibrations in a cylinder on the energy of electron beam. Considering given data, one can draw the conclusion (as Malugin and Manukin did themselves [161]) of direct proportionality of an acoustic signal to the energy of electrons. Thus, the experimental data given are the evidence in favor of the thermoradiation mechanism of sound generation by beams of electrons (positrons) and γ-quanta and agree with the theory presented in previous sections.

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Figure 10.64 Dependence of amplitude of acoustic vibrations in a cylinder on the energy of an electron beam [161].

9. SOUND GENERATION BY AN ELECTRON BEAM IN WATER In conclusion of this chapter we give the experimental results obtained by Lyamshev and Chelnokov [156], who investigated sound excitation by an electron beam in water. Experiments were conducted at the linear accelerator of electrons of the Nuclear Research Institute of the Russian Academy of Sciences. This accelerator provided an opportunity to obtain a beam of accelerated electrons with energy in the range from 10 to 70 MeV. The length of pulse of the electron beam was 1.4 µs. The modes of a single pulse of electron beam and a sequence of pulses with repetition frequency 50 Hz were possible. The average current of an electron beam in the mode of a sequence of pulses with the repetition frequency 50 Hz could be changed from 0 to 10 µA, i.e., the energy in a single pulse could be changed from 0 to 10 J. Figure 10.65 gives the scheme of the experimental setup used for investigation of sound generation in water by an electron beam. The setup consists of an accelerator of electrons LUE-100, a special basin with acoustically insulated walls, a wide-band hydrophone and a wide-band amplifier, a cable connecting the control room with the physical room, and a system of detection of acoustic signals. The basin was equipped with a coordinate device for hydrophone positioning. An electron beam was set with respect to the current strength, electron energy, and shape of the

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section of an acoustic signal. A layer of a special rubber with reflection coefficient at frequencies over 5 kHz not smaller than 0.2 (with respect to pressure) was fixed at the internal surface of the basin walls. A special window was installed in the place of the beam entry into the basin. The window was covered by a polyethylene disk. The basin itself was installed upon a layer of foam plastic in order to reduce possible acoustic coupling of basin walls with its mounting and the collimator. The wide-band hydrophone of 5-mm diameter had an average sensitivity 10 µV/Pa and a band up to 200 kHz. The amplifier provided amplification of about 1.7⋅103 with nonuniformity not larger than 6 dB in the same band. Detection was performed by photographing oscilloscope scans. Oscilloscope scans were triggered by the noise pulse accompanying the startup of the electron beam. It was determined that the length of the noise pulse triggering the oscilloscope scan was about 40 µs. Therefore, there was no sense in positioning the hydrophone closer than 6 cm from the region of absorption of the electron beam (the region of sound generation). The shape of the thermoacoustic array produced in water, was determined after this with the help of measurements of delay times from the scan start at the oscilloscope screen to the starting time of an acoustic signal at various positions of the hydrophone in the basin. The thermal mechanism of sound generation was confirmed by both proportionality of acoustic signal amplitude to the total energy contained in a pulse of electron beam and significant decrease of acoustic signal in the case of pulldown of water temperature to 4°C.

Figure 10.65 Scheme of experimental set-up. (1) A hydrophone; (2) an amplifier; (3) a cable; (4) an oscilloscope with a camera; (5) a basin; (6) a layer of foam plastic; (7) rubber walls; (8) an input acoustic decoupling; (9) a collimator; (10) a film for monitoring of transverse dimensions of a beam; and (11) accelerator output.

It is necessary to note that significant heating of the region of beam absorption was observed in the mode of sequence of pulses with repetition

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frequency 50 Hz under water temperature close to 4°C (the acoustic signal increased with time). The same effect under room temperature (and more precisely at 16°C) manifested itself to a much less extent (it was almost imperceptible). This is quite natural if we proceed from the fact that the electron beam heats the region of absorption for several kelvins, heating from 4°C for several kelvins giving much larger increase of acoustic signal than heating for the same amount of kelvins from 16°C. This happens because of the fact that the coefficient of thermal expansion of water is close to zero at a temperature of 4°C and increases almost linearly with further increase of temperature. Figure 10.66 gives the sectional view of the basin with a thermoacoustic array formed in it for two values of electron energy in the beam: Ee = 20 and 50 MeV. The shape of this array represents a truncated cone with the opening angle of about 10°. The length of this array is about 10 cm for electrons with energy Ee = 20 eV and about 25 cm for electrons with the energy Ee = 50 eV which corresponds to the energy loss by an electron equal to 2 MeV per 1 cm track in water.

Figure 10.66 Sectional view of basin and thermoradiation array formed in water at electron energy (1) 20 and (2) 50 MeV.

If the hydrophone position with respect to the thermoacoustic array is changed, the shape and amplitude of acoustic signals change too. The shape and amplitude of acoustic pulses in the case of the hydrophone position with respect to the axis of the thermoacoustic array at a distance of 10.5 cm and different distances from the array origin is given in Fig. 10.67. In this case the energy of electrons is equal to 20 MeV and the average current is 2.5 µA, such that the energy in a single pulse is about 1 J. The initial diameter of the electron beam at its entry in the basin window is equal to 2 cm. If we position the hydrophone near the origin of the thermoacoustic array (curve 1) then this corresponds to the approximation of the near wave field from an instantly heated cylinder. Sound pressure in a liquid in this case can be presented in the form

αE c p= L πR

∞

 ω 2 a 2  π −  cos − ω  R − t  dω , ω exp  ∫  4c 2   4 c    0

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where E is the total energy of a pulse of electron beam, L is the length of a thermoacoustic array, R is the distance form the array axis, and a is the beam radius. In this case for curve 1 in Fig. 10.67, the calculated value of amplitude is equal to 40 Pa and the duration of the compression phase is about 20 µs, which agrees well with the experimental values of 30 Pa and 22 µs. If the distance from the hydrophone to the origin of the thermoacoustic array is increased, the approximation of the expression given above ceases to be valid in particular, because of the finiteness of the array size. Moreover, there is a possibility of influence of waves reflected from the polyethylene disk covering the input window of the basin and direct emission of acoustic waves by the disk itself on the shape of acoustic pulses.

Figure 10.67 Shape and amplitude of acoustic signals in the case of the hydrophone position at a distance 10.5 cm from the axis of the thermoradiation source and different distances from it: 3, 8, 10.5, 13, and 18 cm (curves 1 – 5, respectively).

An interesting detail was noted. When the electron beam hit directly a basin wall (not through the special window), the character of the acoustic signal changed sharply: its amplitude increased several times and its initial shape changed strongly. This result may be confirmation of the theoretical conclusion on the possibility of significant change of an acoustic signal in a multi-layer medium. The experiments conducted provide an opportunity to draw a conclusion on the fact that in the cases, where theoretical results (based on the thermal

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mechanism) can be compared to experimental data, good agreement is observed.

10. SOUND EXCITATION BY A BEAM OF IONS IN METALS Papers devoted to sound excitation by ion beams in metals were published recently [189, 251]. The results of experimental studies of generation of + acoustic waves by a beam of Ar ions in aluminium published by Satkewicz et al. [251] may be the most characteristic. An aluminium disk with diameter 1.4 cm and thickness 0.3 cm was used in experiments. A piezoelectric detector made of PZT-ceramics was fixed at one side of the + disk. A beam of Ar ions was incident on the opposite side of the disk. Their energy changed within the range from 1 to 10 keV and current in the circuit “beam – target – piezoelectric detector” could be changed from 0.3 to 14 µA at constant ion energy (in other words, the number of particles in the beam was changed). The modulation frequency of ion intensity in the beam could be changed also from 15 Hz to 20 kHz. The diameter of the + beam of Ar ions at the target surface (an aluminium disk) was equal to 300 µm. Some results of the experiments are given below.

Figure 10.68 Dependence of acoustic signal amplitude in aluminium on current in a + beam of Ar ions at different values of particle energy in the beam [251].

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Figure 10.68 shows the dependence of the amplitude of an acoustic signal at the output of the piezoelectric detector on the current in the beam at the frequency of modulation of its intensity 2 kHz. The parameter is the energy of ions in the beam. The dependence of acoustic signal amplitude S b on the current ip can be approximated by an expression S ≡ aip , where b is the index characterizing the inclination of the straight line and equal to 0.96 approximately, and a is the numerical coefficient. Analogous dependences were observed within the whole range of changing of the modulation frequency of beam intensity from 15 Hz to 20 kHz. The data characterizing the dependence of acoustic signal amplitude at the output of the piezoelectric detector on the voltage Vi characterizing ion energy are given in Fig. 10.69. The observed dependences may be described b by the relationship S = aVi as above. The index value here is also equal to b = 0.96 approximately.

Figure 10.69 Dependence of acoustic signal amplitude in aluminum on the energy of an ion beam at different values of current in the beam [251].

These experiments demonstrate that an almost linear bond between the amplitude of an acoustic signal and the energy (or the number of particles) of ions. The authors explain a certain small difference from linear dependence (the index is not equal to one still) by the fact that the mechanism of direct transfer of momentum of particles in the beam to the metal target takes part in generation of acoustic waves by heavy ions in metals apart from the thermoradiation and thermoelastic mechanisms. The role of the last mechanism becomes less and less significant as the particle energy grows and therefore, in the case of moderate density of energy of

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penetrating radiation absorbed in a substance and large particle (ion) energy, the dominant mechanism is the thermoradiation mechanism.

CHAPTER 11

Some Applications of Radiation Acoustics Investigation of acoustic effects of interaction of penetrating radiation with matter opens new opportunities for investigation of penetrating radiation itself (acoustic detection and radiation-acoustic dosimetry and diagnostics), study of physical characteristics of substances, nondestructive testing (radiation-acoustic microscopy and radiation-acoustic sounding and visualization of inhomogeneous condensed media), and also nontraditional ways of controlled radiation-acoustic action upon physical-mechanical and chemical structure of substances. Accelerators and lasers are being introduced to a larger and larger extent in modern technology. Unification of radiation, laser, and ultrasonic technologies provides basically new opportunities for solution of important applied problems. Some of applications of radiation acoustics have been described in literature already (see [97] for example). Here we give only the examples of its applications concerning monitoring of product quality (radiation-acoustic microscopy and visualization), detection of super-high-energy elementary particles, and also some applications of new-generation super-powerful accelerators, which could seem very unusual.

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1. SCANNING RADIATION-ACOUSTIC MICROSCOPY AND VISUALIZATION Traditional techniques of investigation and visualization of microscopic objects such as optical and electron microscopy have many restrictions. For example, an optical microscope and a scanning electron microscope have high resolution but they are unsuitable for investigation of internal regions of opaque materials. If X-ray TV microscopes are used, difficulties connected with interpretation of the images obtained arise. This is especially true in the case of studying low-contrast objects. Radiation-acoustic microscopes do not have such shortcomings [154, 157]. The action of a radiation-acoustic microscope is based upon the phenomenon of generation and propagation of sound and thermal waves in an object, which are excited by the sounding intensity-modulated penetrating radiation. It should be noted that in the majority of cases, the role of heat conductivity is ignored in the process of consideration of sound generation by penetrating radiation since the dimensions of the region of heat release are always large compared with the thermal wavelength. On the contrary, in the case of a radiation-acoustic microscope, a beam of penetrating radiation is focused, the dimensions of the region of heat release are small, and thermal waves often play a fundamental role. Acoustic vibrations and thermal waves arising in an object are detected by sound detectors most frequently. An acoustic signal depends on local physical properties of an object. Therefore, in the case of scanning by a beam in two mutually orthogonal directions, a radiation-acoustic image of an object is formed. In the general case it results from three processes: variation of absorbed power of penetrating radiation due to change of radiation properties of an object from one point to another, interaction of thermal waves with thermal inhomogeneities of an object, and interaction of acoustic waves with elastic inhomogeneities of an object. The first process provides information on only the radiation-absorption properties of an object. If this process is dominant, the radiation-acoustic image is essentially identical to the optical or scanned electron image. The resolution of the radiation-acoustic microscope in this case is determined by the diameter of the sounding beam, and the depth of visualization of a subsurface structure is determined by the penetration depth of radiation. The second process is characterized by interaction of thermal waves with microscopic inhomogeneities of an object. It gives qualitatively new information and provides an opportunity to expand essentially the knowledge on physical properties of an object. The third process carrying information on mechanical irregularities of an object plays an essential role if the acoustic wavelength is of the same order of magnitude as the dimensions of microscopic inhomogeneities in an object (usually this takes

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place at modulation frequencies of penetrating radiation exceeding 100 MHz). In this case the radiation-acoustic image is identical to the acoustic one (as in an acoustical microscope) and resolution is of the order of magnitude of the sound (hypersound) wavelength.

2. SCANNING LASER-ACOUSTIC MICROSCOPY Historically, the first example of radiation-acoustic microscopy was laseracoustic microscopy (or photoacoustic microscopy as it is often called) [162, 205]. A typical block scheme of a photoacoustic microscope is shown in Fig. 11.1. An intensity-modulated laser beam (of the infrared, ultraviolet, or visible range) scans the surface of an object under investigation. Modulation is performed by mechanical or electrooptical methods. An acoustic signal from a detector is fed to a synchronous detector via a preamplifier. The output of the synchronous detector is connected to a visualization device (a display, plotter, or storage oscilloscope) with the scans synchronized to the system of scanning of a laser beam. Depending on the way of detection of acoustic signals, photoacoustic microscopes are divided into schemes with microphones and schemes with piezoelectric transducers. There are also schemes of photoacoustic microscopes with detection of a useful signal with the help of an auxiliary laser beam or a photodetector.

Figure 11.1 Block scheme of a scanning photoacoustic microscope. (1) A laser, (2) a modulator, (3) a control system of scanning, (4) a focusing system, (5) an object under investigation, (6) an acoustic (thermal) detector (sensor), (7) a preamplifier (in the case of harmonically modulated laser radiation it is an amplitude or phase detector and in the case of pulsed laser radiation it is a spectrum analyzer), (8) a synchronous detector, (9) and (10) scan generators, and (11) a visualization device.

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In the case of a microphone technique a sample is placed into an optoacoustic cell (Fig. 11.2) consisting of the hermetic chamber filled with a gas or air, a microphone, and a sample holder. The chamber has a window transparent to sounding laser radiation. Acoustic oscillations arising in a gas chamber under the effect of a laser beam on an object are detected by a sensitive microphone. In the case of a photoacoustic microscope with a piezoelectric transducer (Fig. 11.3), a studied object is in direct contact with the piezoelectric transducer detecting bulk acoustic waves. In the case of a photoacoustic microscope with detection of an optoacoustic signal by an auxiliary beam (Fig. 11.4), either the change of the optical refraction coefficient in a medium in the layer near an object or sound vibrations of an object are detected. The last version of detection is especially convenient in the case of investigation of surfaces with protrusions and cavities.

Figure 11.2 Block scheme of an acoustic (gas-microphone) cell. (1) An input window, (2) a chamber wall, (3) an object holder, (4) a microphone, and (5) an object.

Figure 11.3 Receiving part of a photoacoustic microscope with a piezoelectric transducer. (1) An object, (2) a transducer, and (3) an object holder.

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Figure 11.4 Method of detection with the help of an auxiliary laser beam. (1) A sounding laser beam, (2) a heated region of an object, (3) an object, (4) a direction of mechanical scanning, (5) an auxiliary laser, and (6) a photodetector.

Figure 11.5 Optoacoustic (photoacoustic) image of an electronic chip (an argon laser with the power 0.1 W; the modulation frequency of light intensity 1 kHz; the resolution of photoacoustic microscope 5 µm).

Some fields of application of photoacoustic microscopy are nondestructive profile analysis, i.e., investigation of structure of layered inhomogeneous materials; study of electronic chips (Fig. 11.5); monitoring of chemical composition of complex chemical compounds; investigation of the crystal structure of semiconductors in the process of ion implantation; an opportunity to visualize volumetric or surface areas with different

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thermal characteristics because of inhomogeneity of crystal structure; direct monitoring of laser annealing; study of phase transitions in crystals; and also measurements of thickness and monitoring of uniformity of anodic deposition of films upon semiconductor substrates. Designers of photoacoustic microscopes expect a lot from applications not only in electronic industry but also in medical sciences and biology. Photoacoustic microscopes are inferior to optical and electron microscopes in resolution but surpass them in the amount of information contained in images as they provide an opportunity to visualize the details of microstructure of objects opaque to photons and electrons, open new fields of microscopy, and may broaden essentially traditional techniques of microscopic analysis.

3. SCANNING ELECTRON-ACOUSTIC MICROSCOPY Schemes of photoacoustic microscopes with electronic excitation, where the role of the laser beam is performed by an electron beam [208], have been developed and used during the last decades. A focused electron beam is used for excitation of acoustic and thermal waves in a sample (solid) and acoustic signals are detected by a piezoelectric detector being in direct contact with a sample. In other words, the same scheme as in a photoacoustic microscope with piezoelectric detection is used. The first papers describing schemes of electron-acoustic microscopy were published more than ten years ago (in 1980) [201, 207]. The first scanning electronacoustic microscope was described in 1983 [201, 207, 211]. The advantage of utilization of electrons instead of photons is first of all the fact that an electron beam can be focused into a spot of smaller dimensions. Moreover, the track length of electrons (le = µ −1) in media opaque to light can be essentially larger than that of photons. Both these factors open opportunities to increase the resolution of microscopes. Estimations show that the value of le in solids can be determined with a precision sufficient for applications according to a formula [243] l e = E e1.43 / 10 ρ ,

where le is in micrometers, Ee is in keV, and ρ is in g/cm3. For example, in the case of the electron energy Ee = 30 keV, we obtain le = 0.7 µm for gold, le = 1.4 µm for copper, and le = 5.5 µm for aluminum. The first scanning electron-acoustic microscopes were designed on the basis of standard scanning electron microscopes. They were a certain

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version of their modification. Essentially, a standard scanning electron microscope was complemented just with several devices: a modulator of electron beam intensity (additional deflector plates in the chamber of a scanning electron microscope) and a sample holder equipped with a piezoelectric detector with electronic circuitry necessary for amplification and visualization of an acoustic signal. Scanning electron-acoustic microscopes turn out to be more universal devices than standard scanning electron microscopes or even scanning photoacoustic microscopes. Moreover, one and the same device can operate in both modes (the modes of a scanning electron-acoustic microscope and scanning electron microscope) as a rule. Comparison of images obtained in different operation modes provides new opportunities in investigation of the structure of a studied sample object.

Figure 11.6 Scheme of electron-acoustic visualization [208]. (1) A focused electron beam (with modulated intensity), (2) a periodically heated area performing periodical expansion, (3) ultrasonic waves generated by the periodically expanding area in the sample, (4) a piezoelectric detector, (5) electron beam scan, (6) the output of a piezoelectric detector used to enlarge images, and (7) changes in brightness of an enlarged image, which indicate conversion of electron energy into an acoustic signal depending on the properties of sample material.

Figures 11.6 and 11.7 show schematically the major processes of image formation and the block scheme of a scanning electron-acoustic microscope designed on the basis of a standard scanning electron microscope. The following data can give some idea on its parameters. In the case of the accelerating voltage 30 kV, the peak value of current in an electron beam with spot size of about 1 µm attains 10 µA. Successful experiments were

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reported with an electron beam with diameter at the sample surface of about 0.1 µm and maximum (peak) current in it 0.1 µA. A piezoelectric detector and a sample holder are combined quite often in a single device. For example in one of the versions of scanning electronacoustic microscopes, a sample was glued to a piezoelectric detector (a disk made of PZT-ceramics with thickness 0.5 mm and diameter 12 mm).

Figure 11.7 Block scheme of a scanning electron-acoustic microscope on the basis of a standard scanning electron microscope [208]. (1) The chamber of a scanning electron microscope, (2) an electron gun; (3) the first electronic lens, (4) the second electronic lens, (5) deflector plates, (6) the terminal lens and yokes, (7) the terminal aperture, (8) a sample and a piezoelectric detector, (9) a generator of pulsed and sinusoidal signals, (10) a rectifier, (11) an amplifier, (12) a detector, (13) an amplifier, and (14) an imaging device (a TV tube).

The optimal rate of electron beam scanning over the sample surface in a scanning electron-acoustic microscope depends on time constants of the electronic devices and the frequency of beam modulation. If the diameter of the element to be resolved at the sample surface is about 1 µm and the scan area at the sample under investigation is 0.1 × 0.1 mm2, there are

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0.1/0.001 = 100 elements to be resolved along only one scan line. If we take 1000 cycles-oscillations to resolve a single element in the case of modulation frequency of beam intensity 1 MHz and 100 lines in the image, visualization of a single electronic (acoustic) image takes 10 s. In the case of slow scanning the signal-to-noise ratio is improved and the resolution of a scanning electron-acoustic microscope increases.

Figure 11.8 Image of a silicon structure with a phosphorescent coating [208].

Scanning electron-acoustic microscopy and visualization are used in microelectronics to monitor the quality of electronic chips, determine

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defects of the crystal structure of metals and alloys, study the character of dislocations and other defects in material characteristics under large loading, visualize vibration modes of surfaces, and monitor nondestructively the presence of defects (cracks) in microscopic samples, etc.

Figure 11.9 Image of structure of borders in polycrystalline copper obtained by a scanning electron-acoustic microscope [208].

Figure 11.8 shows images of a silicon structure with a phosphorescent coating in order to illustrate the opportunities provided by scanning electron-acoustic microscopy. One can see that in the image obtained with the help of a standard scanning electron microscope (Fig. 11.8a), silicon structures under the coating are invisible. On the contrary, Figs. 11.8b and d demonstrate a good quality image of a silicon structure obtained with the help of a scanning electron-acoustic microscope. An image of the structure of crystalline particles in a polycrystalline copper sample obtained by a scanning electron-acoustic microscope is given in Fig. 11.9. Characteristic changes of image brightness are visible directly at the borders.

4. X-RAY – ACOUSTIC SCANNING VISUALIZATION Suggestions to use a beam of X-rays instead of a laser beam in photoacoustic microscopy and visualization were discussed by Kim, Sachse, and Pierce [226, 250]. These authors published the first experimental results of studies of scanning X-ray – acoustic visualization [250]. They used the synchrotron radiation of the X-ray range with energy 10 keV. The source

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was the high-energy synchrotron source of Cornell University (USA). The radiation represents a periodic sequence of pulses with length 0.160 ns, energy 1.12 µJ per pulse, and repetition frequency 390.6 kHz. The studies were conducted using aluminium sample disks with diameter 5.72 cm and thickness 1.57 cm. It is necessary to note that the track length of X-ray quanta with energy 10 keV in aluminium is 0.14 mm. In the case of laser radiation, it equals 3.76 nm.

Figure 11.10 Results of measurements of spatial distribution of mean-square values of amplitudes of acoustic signals excited by a modulated X-ray beam [250].

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A series of experiments was conducted in the case of scanning by the beam in mutually perpendicular directions along the sample surface. The beam dimensions at the sample surface were 2 × 2 mm2 and the dimensions of the scanned area were 12 × 12 mm2 or 18 × 18 mm2 with orientation according to the center of a disk-target. An acoustic signal was detected by a piezoelectric transducer made of PZT-ceramics. Detectors with diameters 18.5 and 1.3 mm were used. The detectors were fixed in the center of a sample disk at the surface opposite to the irradiated surface. Experiments with a detector of 1.3-mm diameter fixed at the lateral side of the disk were conducted also. Detectors were damped in order to secure the necessary frequency band. The studies were conducted using two modes. In one of them a collimated beam of X-rays was directly applied to the surface of a disk sample and moved in mutually perpendicular directions. The mean-square value of acoustic signal amplitude was detected at the output of the detection-amplification circuit. This value was represented at a video monitor synchronously with the beam motion. In the other mode called by the authors the mode of double modulation, a low-frequency intensity modulation of X-rays that already were a periodic sequence of pulses with a very high repetition rate (two or three orders of magnitude higher than the modulation frequency) equal to 390.6 kHz was performed. A mechanical modulator in the form of a disk chopper made of stainless steel was installed after a collimator in front of the sample target. Modulation of the initial flow of X-rays with frequency from 0.5 to 2.5 kHz was performed at certain rotation rates of the disk chopper. The amplitude and phase of the envelope of a low-frequency acoustic signal could be detected at the receiver output. Utilization of the mode of double modulation was based on the understanding of the fact that the thermal wavelength in a sample target corresponding to the low-frequency envelope was larger than that for the repetition rate of the sequence of pulses of initial X-ray flow (λT ≡ f −1/2). This provides an opportunity to increase the depth of the monitored (by a thermal wave) subsurface layer of a sample (thermal wavelength). The last fact is important as acoustic waves of megahertz range decay rather rapidly. One of the basic problems, which the authors tried to solve, was to clarify the opportunities to monitor defects by an X-ray – acoustic technique. Experiments were conducted on visualization of a spatial pattern of distribution of amplitudes of acoustic signals excited by an X-ray beam in a disk sample with an internal inhomogeneity in the form of a cylindrical cavity oriented in the disk plane. Measurements conducted both without low-frequency modulation of X-rays and in the modulation mode did not give a satisfactory image of the cavity in the pattern of spatial distribution of amplitudes of acoustic signals excited in the disk by X-rays. It was suggested in this connection to use the ratio of mean-square values of

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amplitudes of acoustic signals of a low-frequency envelope in the target at two modulation frequencies as an informative signal. Experimental results are shown in Fig. 11.10. Figure 11.10a presents a pattern of spatial distribution of mean-square amplitudes of acoustic signals corresponding to the envelope at frequency 2.5 kHz and Fig. 11.10b corresponds to frequency 0.5 kHz. Figure 11.10c and d show the spatial distribution of the envelope of mean-square values of amplitudes of acoustic signals for the envelopes at frequencies 2.5/0.5 kHz and 1.5/0.5 kHz, respectively. One can see a rather clear image of the defect (cavity) inside the sample disk. The experiments conducted demonstrated a real opportunity to use X-ray – acoustic visualization for nondestructive testing. It is evident that there is a real opportunity to implement scanning X-ray – acoustic microscopy under the condition of sufficient focusing of an X-ray beam.

5. ION-ACOUSTIC MICROSCOPY AND VISUALIZATION Application of an ion beam to scanning radiation-acoustic microscopy and visualization was discussed for the first time apparently in 1983 by Lyamshev and Chelnokov [154, 157]. Design of an ion-acoustic microscope was reported in 1985 [257]. Results of experimental studies of scanning ionacoustic visualization of defects in metals were discussed by Satkewicz et + al. [251]. A beam of Ar ions was formed, directed, and moved at the surface of an aluminium sample with the help of modified ray optics of a standard mass-spectrometer. The beam formed a spot with diameter 300 nm at the sample surface. Accelerating voltage was varied within the range from 1 to 10 kV and the modulation frequency of the beam intensity was varied from 15 Hz to 20 kHz. The experiments pursued two goals. It was necessary to determine experimentally the character of the mechanism of radiation-acoustic conversion and clarify an opportunity to use it for ionacoustic visualization. It was determined experimentally that in the considered range of ion energy, the dominant mechanism of sound excitation by a modulated ion beam in the process of its interaction with a target is the thermoradiation mechanism. It was demonstrated also that an ion beam may be used effectively for scanning radiation-acoustic microscopy and visualization together with beams of photons (laser beams), electrons, and X-rays. Experiments were conducted using samples in the form of aluminium disks with diameter 1.4 cm and thickness 0.3 cm, which had cylindrical cavities with internal diameter of 0.1 cm inside.

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Figure 11.11 shows schematically a device for fixation of a sample and a piezoelectric detector used for detection of acoustic signals excited by ion beams in an aluminum sample.

Figure 11.11 Scheme of a device for fixation of a sample with a piezoelectric + detector [251]. (1) A beam of Ar ions, (2) an aluminium disk sample, (3) a cylindrical cavity simulating a defect, (4) PZT-ceramics, (5) direction of scanning, (6) an acoustic signal, (7) an insulating plate, (8) an earthing plate, and (9) sample holder.

Figure 11.12 Distribution of acoustic signal amplitude for an aluminum sample with two intersecting cavities.

Figure 11.12 demonstrates the change of an acoustic signal in the process of linear movement of an ion beam at the surface of an aluminum disk along a single line of the image and at different values of modulation

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frequency of beam intensity. An aluminum disk was used in experiments, which had two intersecting cylindrical cavities of 1-mm diameter inside at a depth D = 0.5 mm. One can see from the data given in Fig. 11.12 that at a low modulation frequency, when the condition λT/D > 1 (λT is the thermal wavelength) is satisfied, the amplitude of acoustic signal increases, when an ion beam is above the plane. On the contrary, if λT/D < 1, the signal decreases. At frequency 15 Hz the thermal wavelength in aluminium is equal to 1.45 mm and corresponds approximately to the distance between the centers of the cavities. The experiments [251] are also evidence of the fact that in the cases under consideration the resolution of ion-acoustic visualization at a high frequency is determined by the diameter (size) of an ion beam. Figure 11.13 gives the results of the experiments in which an aluminum sample disk with an inclined cylindrical cavity is scanned, i.e., a cavity with diameter 1 mm positioned at a certain angle with respect to the sample surface. The depth of cavity position was different in different places with respect to the sample surface where scanning was performed. As in the previous case, one can see that if the depth of cavity position is small (λT/D > 1), an increase of acoustic signal amplitude is observed. On the contrary, the signal amplitude decreases at λT/D < 1. Results given in Fig. 11.14 demonstrate the situation where a beam was moved at the surface of the same sample as in the previous case but along a fixed line, where the depth of cavity position was equal to D = 0.3 mm, and the modulation frequency of beam intensity was changed. Comparison of the results of measurements of acoustic signal amplitudes given in Figs. 11.13 and 11.14 indicates the existence of frequency dependence of the signal, which corresponds approximately to frequency law f −1/2 in the process of transition from the case λT/D > 1 to the case λT/D < 1. This is more evidence of the existence of two modes of visualization: a “thermal” mode, when thermal waves play the main role in formation of the acoustic signal, and an “elastic” mode, when the main role is played by elastic deformations, i.e., acoustic waves in a sample. Experiments on visualization of the same sample as in the case given in Fig. 11.14 but using scanning electron-acoustic microscopy and scanning laser acoustic microscopy are of interest too. Results of these experiments are shown in Fig. 11.15. The modulation frequency was about 78 kHz. This figure shows an image of a disk with a cavity obtained using a scanning electron-acoustic microscope and a line-by-line record of changes in the acoustic signal close to it (at the right). Corresponding records obtained with the help of a scanning laser-acoustic microscope are shown in Fig. 11.15b. One can see different contrast of images and the dominant roles of the “thermal” mechanism of radiation-acoustic visualization at λT/D > 1 and the “elastic” mechanism at λT/D < 1.

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Figure 11.13 Profile of an inclined cavity in a sample in the case of constant modulation frequency and variable depth [251].

Figure 11.14 Profile of an inclined cavity in the case of constant depth and variable modulation frequency [251].

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Figure 11.15 Image of an inclined cavity in an aluminium disk obtained using a scanning electron-acoustic microscope and laser excitation of sound [251].

The results given here (and in the preceding sections) indicate real opportunities to study various kinds of penetrating radiation in devices for radiation-acoustic visualization and microscopy.

6. ACOUSTIC DETECTION OF SUPER-HIGHENERGY PARTICLES IN COSMIC RAYS – THE DUMAND PROJECT The energy 1017 eV is commonly considered to belong to the range of super-high energy. The range boundary is not connected with a physical phenomenon. It is determined by the energy threshold of detection of extensive air showers by the largest existing installations such as those in Yakutsk (Russia) and Haverah-Park (Great Britain). The areas occupied by these two installations are 18 and 12 km2, respectively. Dimensions of installations are determined by the necessity to detect very rare events, i.e., emergence of particles of super-high energy in the spectrum of cosmic rays. The following data may give an idea of the number of detected events. During the decade of operation at Haverah-Park, they detected 70,000 showers with energy over 6⋅1016 eV, 52,000 showers with energy within the interval 1017 – 1018 eV, 4000 showers with energy over 1018 eV, 144 showers with energy 1019 eV, and only 16 showers with energy 5⋅1019 eV [22].

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Neutrino radiation arouses a lot of interest. This is caused by the enormous penetrating capability of neutrino. This capability provides a real opportunity to “look” into stars. This is especially true about studies at the energy Eν > 1 TeV, when only neutrino can carry information on highenergy processes taking place during the epoch with large red shifts, unique physical objects, i.e., hidden sources, etc. A new branch has arisen in astrophysics: neutrino astronomy. It is in the phase of its experimental development now. The Baksan neutrino telescope in the Northern Caucasus was put into operation. Similar installations are under construction in the USA and Italy. However, the main installation, which will open broad opportunities for experimental neutrino astronomy, is the DUMAND. It is in the design stage yet. We will discuss DUMAND below. Neutrino radiation is divided conventionally into two classes: atmospheric and cosmic neutrino radiation. In the first case neutrino are generated as a result of interaction of accelerated particles with atomic nuclei of matter in the atmosphere. Cosmic neutrino radiation arises in cosmic objects as the result of collision of accelerated particles with atomic nuclei and also due to the interaction of high-energy protons with lowenergy relict photons in space. Unstable particles like pions and kaons (π- and k-mesons) are born in the process of interaction of protons with atomic nuclei or photons. These particles disintegrate giving birth to muons (µ) and neutrinos (ν). High-energy neutrinos are detected by muons, hadrons, and electrons produced by them, which arise in the process of interaction of neutrinos with nucleons and give birth to nuclear-electromagnetic showers in an ambient medium or detector. In the case of high energy, a muon retains the direction of neutrino motion and has a large track length in soil or water. The track length of muons with energy higher than 1 TeV in water exceeds 3 km. Thus, muon detection provides an opportunity to determine “immediately” the direction of the neutrino source. In underground experiments mouns are registered by special detectors that are most frequently scintillation detectors. In underwater experiments the detector is water itself. Bremsstralung photons, electron-positron pairs, and hadrons giving rise to electromagnetic and nuclear-electromagnetic showers are created along a muon trajectory. At energy Eµ ≥ 100 TeV, an electromagnetic shower arises along a muon trajectory. Hadrons generated in neutrino-nucleon collisions give rise to nuclear-electromagnetic showers. The length of such a shower in water or ground is small in contrast to an electromagnetic shower generated by a muon. This is why the showers caused by a neutrino can be detected only if it interacted with nucleons inside a detector. In water the electrons of electromagnetic and nuclearelectromagnetic showers produce the Cherenkov optical radiation, which

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can be detected by a system of optical detectors, i.e., a lattice of photoreceivers. The idea of the possibility of recording cosmic neutrinos was suggested first by Markov in 1960 [239]. A new stage of high-energy neutrino astronomy began with the discussion of the project of the deep underwater experiment DUMAND (Deep Underwater Muon and Neutrino Detection). Initially the DUMAND detector was planned to be a spatial lattice of photodetectors submerged in the ocean at a depth of about 5 km. The distance between the sensitive elements (photodetectors) should not exceed the optical transparency length of water for visible and near-ultraviolet parts of the spectrum. The water layer over the installation serves as a shield from cosmic-ray muons. Askar’yan and Dolgoshein [8, 213] and independently Bowen [199] suggested an acoustic technique for detection of super-highenergy neutrinos (Eν ≥ 107 TeV) with the help of a spatial lattice of hydrophones. The essence of the technique of acoustic detection of super-high-energy neutrinos consists of the following. A nuclear-electromagnetic cascade produced by the interaction of a neutrino with the detector substance is accompanied by a fast (practically instantaneous) heating of water in a narrow channel along the shower axis. This causes the expansion of a volume of liquid in the channel and leads to a pressure pulse propagating in water perpendicularly to the shower axis. As estimates show, at very high energy, a narrow (several centimeters) and long (~10 m) particle beam forms in the vicinity of the shower axis. Heating of the medium within the region of action of the beam occurs due to ionization losses of the shower electrons slowed down to an energy lower than the critical one. Therefore, a cylindrical thermoradiation sound source forms in the region of absorption of shower electrons. Its radius (the radius of the heated part of the channel) is determined by the electron distribution over the channel cross-section. The length is L = [ln( Eh / Ecr )]1 / 2 , where Eh is the shower energy (the energy of hadrons initiating the shower), Ecr = 73 MeV is the electron critical energy, and L is in meters. We have given already the scheme of calculation of the sound field generated by a high-energy particle in a condensed medium and corresponding estimates in Chapter 7. However, it is expedient to give some estimates again. The authors of a large series of papers devoted to detection of high-energy particles (to the DUMAND project) analyze the parameters of the sound field in the near wave zone of an electromagnetic cascade arising in water, i.e., of the thermoradiation source of sound [9, 11, 56, 58,

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193, 194, 200]. They obtain the radiation acoustic pulse form and amplitude using the equation of thermoradiation sound generation in a liquid and calculating the energy release function Q(t, r) by different methods. The time dependence of the function Q(t) is always taken in the form of a δ(t)function as the initiation time of an electromagnetic cascade is much less than the other characteristic “acoustic” times, while the spatial dependence Q(r) is determined on the basis of some approximations and various direct calculations of the energy release density in the cascade. For example, the following expression for calculation of radiation-acoustic signal amplitude in sea water has been given by Askarijan et al. [194], p max =

0.44ϕ (r ) E r

1018

,

where p is the sound pressure (in Pa), E is the particle energy (in eV), and r is the distance from the channel axis (in meters). The factor ϕ(r) takes into account the deviation of the law of sound signal attenuation from the cylindrical one r−1/2 and ϕ(r) = 1.0; 0.95; 0.28; and 0.12 for r = 50, 100, 250, and 500 m, respectively. The acoustic signal amplitude for a particle with energy E = 1017 eV at a distance of 100 m from the cascade axis is equal to pmax = 4⋅10−3 Pa. The shape of an acoustic signal and its maximum amplitude depend, in fact, on the particular features of ionization losses in the nuclear-electromagnetic shower channel, i.e., on the form of the distribution function describing energy release Q(r). Figure 2.2 (Chapter 2) gives the shapes of acoustic pulses for different dependences of the function of energy release Q(r). In order to detect the acoustic signal from a shower in the ocean, it is important to know not only the signal amplitude and shape (its spectrum) but also the value of attenuation of sound during its propagation as well as interference characteristics, i.e., the spectrum of ambient noise in the ocean. All this leads to the necessity of determination of the optimal detection frequency band of the radiation-acoustic signal. The sound attenuation decreases as the frequency decreases (i.e., the sound absorption length increases) but, on the other hand, it is known that the level of ambient noise (interference) in the ocean increases as the frequency decreases. That is why in the majority of papers associated in one way or another with the DUMAND project, the estimates of the value of effective sound pressure are given either for some optimal frequency band or at a fixed (optimal) frequency. The opportunity to use the deposits of rock salt NaCl and Antarctic ice as the working medium of an acoustic neutrino detector was discussed. Basic opportunities for using various condensed media as the working medium of the detector were considered also.

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Effects of sound generation in condensed media by high-energy particles were considered explicitly in Chapter 7. Here we give only some estimates for the near wave field of a cascading particle. Let us recall the expression for the tensor of normal stress (sound pressure) in the case of observation of an acoustic signal in the direction perpendicular to the cascade axis, c 3 − 4 / n2 σ RR = − µE l 4πcε πr

 2 2  ω a exp ω − ∫  cl2 0 

∞

    r π   dω .  cos t ω − −    c 4 l    

The parameters of the cascade formed by a high-energy particle are approximately the same for various liquid and solid media if the densities of these media and the charges of the elements constituting them are close to each other. For example, according to Askarijan et al. [194], the parameters of the cascade from a neutrino with energy of the order of magnitude of 1015 eV in water are L = µ −1 = 4 m and a = 2 cm (a is the radius). In this case according to the above expression, the estimate of the effective sound pressure (in Pa) in the near wave field (f = 30 kHz) is peff ≈ 0.1

E 1 ; E0 = 1016 eV . E0 r

This formula is analogous to the expression for estimation of the level of sound signal in the near wave field of the thermoradiation source produced by a cascading particle given by Berezinskii and Zatsepin [24]. If we take for example Antarctic ice as the medium where a cascade forms due to absorption of a high-energy neutrino in it, the parameters of the cascade are approximately the same as in water. If we take the numerical values cl = 4⋅103 m/s, n = 2, α = 5⋅10-5 K−1, and cε = 2⋅103 J/(kg⋅K) that correspond to the ambient temperature t = −20°C, the estimate of the effective sound pressure in the near wave field according to the above expression for σRR is given by the following expression (f = 90 kHz): eff σ RR ≈

E 1 . E0 r

One can see that the effective sound pressure in ice is approximately one order of magnitude higher than that in water, other conditions being equal. This difference is caused by the fact that the Gruneisen parameter (Γ = αcl2/cε) for ice is approximately one order of magnitude larger than that for water.

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In order to estimate the number of hydrophones necessary for construction of an acoustic neutrino detector, it is important to know the maximum distance from the cascade axis rmax where detection of the acoustic signal is still possible. That is why it is necessary to determine the optimal frequency of the frequency band for reception of the acoustic signal generated by a nuclear-electromagnetic shower. Volovik et al. [56] conducted calculations for different values of energy and substances including water. The analysis of the calculation results taking into account the number of possible events, i.e., emergence of super-high-energy particles in the detector volume and the rate of the number of events per year, have led the authors to the conclusion that the transition from water to other condensed media (detector working media) does not give essential advantages to big installations designed for detection of super-high-energy neutrinos.

7. NEUTRINO FOR GEOACOUSTICS – THE GENIUS PROJECT As construction of more and more powerful accelerators opens new opportunities for further studies of particle physics at larger and larger energies, “old” accelerators designed initially for purely basic research are applied in various fields of science and technology like medicine, biology, etc. One of the newest fields, where the advances of high-energy physics can be used effectively for both basic research and industry, may be an example of future applications of accelerators. We are talking about geophysical applications of high-energy neutrino beams. Figuratively speaking, neutrino may play the same role in this field as that of X-rays in medicine and nondestructive testing. A general scheme of geophysical application of neutrino beams is as follows [184, 248, 249]. A neutrino beam formed by an accelerator is aimed in a specified direction and travels over a considerable distance in the Earth. As it propagates, the beam generates secondary radiation of different types: muon, radio, and acoustic. Neutrinos themselves can be a probing device: neutrino absorption along its path can be a measure of the quantity of matter along it. In this case secondary emissions serve for neutrino beam detection and measurement of its parameters. In other cases (one of them we will discuss below) a neutrino beam can be used only as a source of secondary radiation, which in the course of propagating “selects” information on the characteristics of the Earth. This is the situation if the problem is to investigate surface layers of the Earth’s crust (ocean), for finding minerals for example.

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The opportunities provided by geoacoustic applications of neutrinos, i.e., use of the acoustic radiation produced by a powerful neutrino beam propagating in the Earth, for acoustic sounding of the Earth’s bowels are described briefly in this section. The realization of geoacoustic (as well as geophysical in general) applications of neutrino beams demands contradictory conditions to be satisfied: neutrino penetrating capability must be combined with a strong interaction with matter, as the neutrino beam must perform the role of a sufficiently powerful thermoradiation sound source. As calculations showed [184, 249], these conditions can be satisfied for the neutrino beams of energy Eν ≈ 1 TeV and higher. A neutrino beam of such energy produced by an accelerator is narrow and sharply directed, and this provides the high volume density of the energy released due to its interaction with matter. This creates the conditions of efficient sound generation in ground or water. The minimum energies of accelerators needed to obtain such neutrino beams depend on the accelerator type and lie within the interval of several tens of teraelectronvolts, i.e., just within the energy range of new-generation accelerators being designed now. As for the optimal energy of neutrinos for geoacoustic research, it is certainly essentially higher and may possibly be attained only in the far future. Thus, when we speak about neutrinos for geoacoustics, we speak certainly about ideas that as yet are only theoretical and at present we can speak only about projects and forecasts for accelerators of future generations. Moreover, the phenomena taking place at such an energy can be observed now only in experiments with cosmic rays (they were considered in the preceding section) and to a large extent, they need further investigation. Neutrino is a neutral particle. It is impossible to accelerate a neutrino to the necessary energy (Eν = 1 ÷ 10 TeV) directly in an accelerator, since neutrinos do not interact with an electromagnetic field. Therefore, the production of neutrinos by “elementary reactors” (unstable particles) moving at a very high velocity is used. Such a fast unstable particle can be obtained in two ways: it is possible to obtain a “slow” unstable particle, a pion (π-meson) for example, and then accelerate it before it has time to decay, or to accelerate a proton, which produces a fast unstable particle immediately after its collision with a target. Both cases are discussed in the literature devoted to neutrinos for geophysics [184]. In existing cyclic proton accelerators and accelerators, which are being designed now, proton acceleration occurs in a circular vacuum chamber, where 1013 – 1015 protons during the time ≤ 103 s are accelerated up to the maximum energy. In order to obtain a neutrino beam, these protons are brought out of the circular chamber and directed at a special target. When protons interact with the nucleus of the target, several secondary particles

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emitted under very small angles to the direction of proton motion, mainly ± ± pions (π -mesons) and kaons (k -mesons), are formed. After focusing by a system of magnetic lenses (separation of particles with charges of different sign occurs simultaneously), mesons go into a long straight vacuum + channel, i.e., a so-called decay channel, where a part of them decays: π → + + + µ + νµ and k → µ + νk . Let us take pions. Their lifetime is comparatively small (τ ≈ 2.4⋅10−8 s). However, the distance, at which they decay, is rather large because of the Lorentzian retarding of time and equals approximately lπ = 56 (Eπ/TeV) km. For example, only about 20% of pions decay at energy Eπ = 1 TeV at the length lπ = 1 km. In the process of pion decay, a muon is born apart from a neutrino, and in the process of muon decay, neutrinos and antineutrinos are born in their turn. However, muons make almost no contribution to a neutrino flux, while their lifetime is approximately two orders of magnitude longer than the lifetime of pions. As the result of a chain of transformations p → π → ν, the average energy of a neutrino in a beam is approximately 20 times smaller than the energy of primary protons. Thus, although it is suggested to use first of all cyclic accelerators to attain the maximum energy of accelerated particles, these accelerators have considerable disadvantages as the sources of neutrino beams – degradation of energy of primary particles (protons), a large length of the decay channel, etc. Unique opportunities may be provided by linear accelerators, where particle acceleration occurs during the time of a single flight of a particle. The calculations performed in P. N. Lebedev Physical Institute, Russian Academy of Sciences, [184] showed that pion accelerators could be very effective sources of neutrino beams for geophysical research. Pion accelerators are at least four times more effective than proton accelerators in the ratio of the initial energy of protons and the energy of unstable particles, i.e., pions. In contrast to annular accelerators, a linear accelerator can operate at high rates of pulse repetition (70–100 Hz) that provides an opportunity to obtain a high average intensity of the neutrino beam. As we know now, this is important from the point of view of efficiency of radiation-acoustic conversion. An important advantage of linear pion accelerators in comparison with proton accelerators is the smaller energy of the primary beam that makes its deflection easier. Finally, one can use “packages” of particles with a very high density in a linear accelerator, which provides the opportunity to obtain short pulses with a number of particles sufficient for detection of a radiation-acoustic signal. Speaking about neutrino beams we have always meant their pion “origin”. Meanwhile, as we have noted earlier, kaons are also the source of neutrinos in the process of interaction of accelerated protons with the nuclei of the target. However, estimations show that the role of kaon neutrinos is

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rather insignificant. Their contribution may be noticeable only far from the axis of a neutrino beam [184]. It has been noted also that the necessity of a large-length decay channel (1 ÷ 10 km) is a big disadvantage of a superpowerful proton accelerators serving as a source of a neutrino beam. However, as has been demonstrated by physicists from Lebedev Physical Institute, there may be no need of a decay channel if one uses so-called “direct” neutrinos born in the process of decay of charmed particles (Λ, D, etc.) discovered rather recently. This is still more evidence of the fact that the results of basic research in the field of high-energy physics can change our views on the sources of neutrino beams. Rujula et al., who introduced the idea of using a thermoacoustic signal generated in rocks by a neutrino beam for geological research, were apparently the first who made estimates of proton accelerator parameters needed for the creation of the required neutrino beam and the acoustic signal generated by it [249]. On these estimates they based their proposal for the GENIUS (Geological Exploration by Neutrino Induced Underground Sound) project. They considered the possibility of construction of a circular accelerator for proton energy Ep = 3 ÷ 20 TeV, which was named Geotron in contrast to the proton accelerator for the energy Ep = 1 TeV named Tevatron, since in the latter case a new generation accelerators for teraelectronvolt energy are meant. Thus, the GENIUS project can be the second example of an immense project in the field of high-energy physics and radiation acoustics. We may consider the DUMAND project to be the first. While it is significant that a proton accelerator for energy Ep = 1 TeV will every few minutes “eject” 1014 protons with total kinetic energy 1 MJ, then these parameters are even more impressive for the Geotron. For example, the number of protons per pulse is 1015 and the total energy is ~109 J. If the most modern superconductors are used for the construction of the accelerator magnet system, the radius of its circular vacuum channel will be equal to approximately 6 km for the proton energy Ep = 10 TeV, and for the proton energy Ep = 20 TeV it will be 12 km, respectively. The length of the decay channel is estimated as l = 7.5 km (Ep/10 TeV), i.e., about l = 7.5 km at Ep = 10 TeV and l = 15 km at Ep = 20 TeV. The synchrotron radiation power is a very important parameter of the accelerator. Rujula et al. [249] estimate this power according to the formula Ws = 6⋅10−14 NEp4/R2 ; here N is the number of particles (protons), R is the radius of the circular channel (in kilometers), and Ep is the proton energy (in TeV). Thus, at Ep = 10 TeV, N = 1015, and R = 6 km, we have Ws = 16 kW and at Ep = 20 TeV, Ws = 64 kW, correspondingly. In order to protect people nearby from synchrotron radiation, and taking into account the fact that the problem of controlling a decay channel of length of about 15 km is not the “simplest” one, Rujula et al. [249] consider as one of the

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possibilities the construction of a circular chamber and decay channel of a proton accelerator for energy Ep = 10 ÷ 20 TeV in the sea. A scheme of a neutrino geoacoustical experiment is given in Fig. 11.16. A neutrino beam propagating deep in the Earth is the source of acoustic waves and sound detection has to be performed by a geophone array at the Earth’s surface or a hydrophone array if measurements are conducted at sea. As for the parameters of a neutrino beam, since only several percent of pions have time to decay in the decay channel, according to estimations [249] the beam should contain only 1013 neutrinos if there are 1015 particles in the proton beam, and the neutrino energy should be approximately Eν = 0.3 TeV at Ep = 10 TeV. The next formula is given [249] for the radius a of the neutrino beam: a = 10.8 m (L/1000 km)⋅(10 TeV/Ep). Here L is the distance to the accelerator (in km). If L = 1000 km, then a = 10 m. This means that the neutrino flux in the beam at this distance constitutes ~1010 neutrinos per 1 m2.

Figure 11.16 Scheme of neutrino-geoacoustical sounding [249]. (1) A neutrino beam, (2) the Earth surface, (3) a geophone (hydrophone) array; R is the distance to the geophone array and d is the diameter of the neutrino beam.

The calculation of acoustic radiation from a neutrino beam, which was conducted by Rujula et al. [249], was based on the equation of thermoradiation sound generation in the form 2  2  2  c ∆ − ∂  p (r , t ) = − αc 1 (1 + σ ) ∂ε ,  C p 3 (1 − σ ) ∂t ∂t 2  

where α is the coefficient of volumetric thermal expansion of a medium, Cp is the specific heat capacity, σ is Poisson’s ratio, c is the sound velocity, and ε is the cubic density of energy dissipated in a medium. We should note that the factor (1 + σ)/[3(1 − σ)] is approximately equal to one and the quantity αc2/Cp = Γ is the Gruneisen parameter of a medium. Estimates show that about 100 interactions with the medium substance per 1 cm occur at proton energy Ep = 10 TeV in a neutrino beam. This means that, finally, the ionization losses constitute ~ 100 erg/cm. At distance L = 1000 km from the Geotron, the cubic density of the energy

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released in a medium is approximately equal to ε = 3⋅10−5 erg/cm3. In many cases the Gruneisen parameter is about one and the estimate for the amplitude of the acoustic pulse is p ≈ 3⋅10−5 dyne/cm2. The acoustic signal has the characteristic form of the so-called N-wave. Rujula et al. [249] give also the following estimate for the N-wave amplitude: for Ep = 10 TeV, L = 1000 km, R = 1 km (R is the depth of the neutrino beam in Earth, the distance to the geophone) and rock salt NaCl (for example) p ≈ 1.8⋅10−5 dyne/cm2 and for the same parameters and Ep = 20 TeV a more exact calculation gives p = 4.1⋅10−5 dyne/cm2. A certain increase (more than twice) of the signal amplitude is caused in particular by the fact that as the neutrino energy grows, the beam diameter decreases, and the cubic density of the energy released in a substance increases additionally. An important characteristic of an acoustic signal is its frequency spectrum. The spectrum density of an acoustic signal has the form p(ω ) → ω 1 / 2 exp( −ω / ω 0 ) , where ω0 = c/a and a is the radius of the neutrino beam. The maximum spectral density pmax(ω) corresponds to the frequency ω/2 and 〈ω 〉 = 1.50ω 0 , 〈ω 2 〉 2 = 3.75ω 02 , ∆ω = [〈ω 2 〉 − 〈ω 〉 2 ]1 / 2 = 1.22ω 0 . The characteristic frequency (in Hz) is given by an expression f0 =

ω0 c . = 2π 2πa

Rujula et al. [249] give the next expression for the radius of the neutrino beam a: L    10 TeV  a = 10.8 m  .   1000 km   E p  It follows from the formulae given above that

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L  Ep  c  〈 f 〉 = 23 ,    1 km/s   1000 km  10 TeV where 〈f〉 is expressed in hertz. For L = 1000 km, Ep = 10 TeV, and c = 4.74 km/s (rock salt) we can obtain 〈f〉 = 109 Hz and the frequency band ∆f = 89 Hz. These estimates (and the estimate of the acoustic signal amplitude especially) are very rough as has been stressed by Rujula et al. [249].

Figure 11.17 Scheme of neutrino-geoacoustical mineral prospecting [249]. (1) The Earth (oceanic) surface, (2) horizontal sedimentary, (3) impervious sedimentary, (4) gas and oil, (5) water, (6) a mixture of water, oil, and gas, (7) a rock base rich with organic substances, and (8) the propagation direction of the sound waves excited by a neutrino thermoradiation source of sound.

The signal amplitude is very small. However, the fundamental quantity from the point of view of signal reception is the signal-to-noise ratio. The level of noise (seismic noise or oceanic noise if we mean underwater reception) within the range of the characteristic frequencies (~100 Hz) can be approximately five orders of magnitude higher than the signal level, but nevertheless, signal reception under these conditions may be not at all hopeless. The noise stability may be increased using a lattice (an array) with a large number of receivers. If the noise at the array receivers is uncorrelated while the signal at all receivers of the array is correlated, then the noise 2 stability of reception increases proportionally to n1/ , where n is the number

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of receivers in the array. Furthermore, one can use coherent signal integration to increase noise stability. As the positions of the sound source and receiving array are fixed, one can take a big enough integration time. In particular, utilization of a linear pion accelerator in the Geotron design would contribute to this. It would provide an opportunity to increase the repetition rate of neutrino (acoustic) pulses. In real conditions the amplitude of the acoustic signal generated by a neutrino beam can be several orders of magnitude higher. As was noted by Tsarev and Chechin [184] and Balitskii et al. [15], a natural solid medium can behave as an active medium: the initial acoustic radiation by a neutrino beam can act as a kind of a trigger mechanism initiating the rise of acoustic emission in a medium. The signal from this emission is essentially higher than the signal generated by a neutrino beam and the “useful” signal is the signal of acoustic emission. Figure 11.17 presents one of the possible schemes of neutrino geoacoustical mineral prospecting. A neutrino beam is transmitted through various geological strata with different values of Gruneisen coefficient. Measuring the change of the acoustic signal by a lattice of geophones while moving it along the beam track, one can obtain information on the type of rocks and minerals.

Conclusion We can state that theoretical and experimental studies of thermoradiation mechanism of sound generation by penetrating radiation in a condensed medium have advanced quite far by now. The processes of sound generation in the case of continuous (modulated) and pulsed action of penetrating radiation on a substance are studied. Basic laws of formation of acoustic signals are established and the connection of the characteristics of these signals with the parameters of radiation and thermodynamic, radiation, and acoustic properties of substances are revealed. The optimal conditions and efficiency of thermoradiation generation of sound are determined. The particular features of sound generation by a particle beam moving along the surface of a liquid or solid in the cases of subsonic, transonic, and supersonic velocities of beam motion and the arbitrary form of modulation of radiation intensity in the beam are studied. The possibility of production of sound sources operating in a broad frequency range from sonic to hypersonic frequencies in liquids or solids is demonstrated. It is possible to change the frequency, directivity, and intensity (power) of a radiationacoustic sound source by selecting the type of radiation and controlling the parameters of a radiation beam. The processes of sound generation by single high-energy particles in a substance are studied. The theory of thermoradiation sound generation in condensed media has been confirmed reliably by experiments. This provides grounds for justified selection of sources of penetrating radiation for the solution of practical problems of thermoradiation sound generation such as radiation-acoustic microscopy and visualization, radiation-acoustic dosimetry, application of radiation-acoustic effects to nondestructive testing, etc. 339

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There are the prospects of carrying out immense and maybe fantastic (as it may seem now) projects, i.e., the DUMAND project in neutrino astrophysics and the GENIUS project in neutrino geoacoustics. Radiation excitation of sound can be useful also in other situations, which may seem unusual, e.g., for excitation of sound pulses in cosmic bodies by laser radiation from the Earth or powerful beams of penetrating radiation from space platforms and creation of vertical underwater acoustic arrays in the ocean, with efficient height or length of about several tens of meters (if the sources of laser radiation operating in the blue-green optical range are used) and of thousands of meters (beams of muons and neutrinos). These problems do not have “simple” technological solutions if traditional radiators are used. Summarizing, we have to note however that only linear theory has been treated in this book. As one can see, this theory describes radiation-acoustic effects when the intensity of penetrating radiation is small. And the phenomenon of interaction of penetrating radiation with a substance is nonlinear by its nature. From this point of view both the theory and the conditions of experiments (practical application of the conclusions of the linear theory) are restricted within the framework of the perturbation method. Furthermore, only comparatively simple “model” problems have been considered within the framework of this linear theory. As for thermoradiation generation of sound in solids, only the model of an isotropic solid has been considered. However, in practice one has to deal with solids of complex structure like semiconductors, piezoelectrics, ferroelectrics, ferromagnetics, etc., where it is important to take into account the interaction of penetrating radiation with various subsystems, i.e., the lattice and electron-hole, spin, etc., subsystems, as well as the interaction of these subsystems in complex solid structures. It is possible that this research will lead into new fields of science such as radiation acousto-electronics and radiation magneto-acoustics. Broad prospects for development of new technology may be opened by the studies of nonlinear radiation-acoustic effects, which arise in a substance when there are no changes of the aggregate state of this substance and no phase transitions, but the expansion rate of the volume of the substance absorbing radiation is large enough or change of its thermophysical properties occurs. The heated region of a medium “works” in this case as a source of finite amplitude waves, which can transform in their turn into shock waves. Mechanical, physical, and chemical properties can change under the impact of these shock waves that may be used for new technologies. If a powerful, penetrating radiation affects a substance in the conditions when phase transitions occur and substance evaporation or optical breakdown (under the action of powerful laser radiation) take place, shock and sound waves of huge amplitude with parameters unattainable by

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traditional means can rise. Radiation sources of great power are already developed or being developed now (regretfully they are intended for the use in ray and beam weapons as has been reported [87]). Hopefully, these sources will never be used for their “direct” purpose but applied to peaceful tasks including radiation-acoustic technology.

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ADDENDUM1

Acoustooptics of Penetrating Radiation Studies of optical excitation of sound in gases and condensed media are the substance of optoacoustics, which was developing extensively during the last decades. This is illustrated by the 9th International Conference on Photoacoustics and Photothermal Phenomena [1]. Light beams are beams of particles (photons). In this connection the topic of this book may be considered as optoacoustics (photoacoustics) of penetrating radiation [2]. At the same time, there is another broad field at the interface of acoustics and optics, i.e., acoustooptics. Its topic is investigation of acoustooptic phenomena arising in the process of interaction of light with acoustic oscillations and waves in a substance. Acoustooptics is the basis for development of modern and future devices for microelectronics, signal processing, telecommunications, and informatics. For example, multichannel distributed optical-fiber acoustic receivers are being developed on the basis of optical interferometers and the effects of interaction of light with acoustic waves [3]. Acoustooptic devices using surface acoustic waves are applied widely [4]. Investigation of diffraction, scattering, and interference of waves of penetrating radiation (the de Broglie or matter waves) in the process of their interaction with acoustic oscillations and waves in a substance can be treated as acoustooptics of penetrating radiation, which is also the subject of radiation acoustics. Some fields of acoustooptics of penetrating radiation 1

A considerable time has passed since publication of this book in Russian. New results in the field of radiation acoustics have been published during these years. Some of them concerning acoustooptics of penetrating radiation are given in this section. References to other problems are given as footnotes to the main text. 359

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may be considered traditional now. First of all, this concerns the studies of interaction of X-rays with ultrasound [5]. Other fields are at the initial stage of their development yet. Several years ago the studies were started, which led to the development of acoustic tunneling microscopes [6, 7]. Tunneling microscopy has a very high spatial resolution at the molecular and atomic levels. The first atom interferometer was constructed about five years ago [8]. Probably, it will find its application to acoustic research and measurements at the quantum level. We can give other examples also. Further, we will examine these three sections of acoustooptics of penetrating radiation in some more details.

1. DIFFRACTION OF X-RAYS AND NEUTRONS BY ULTRASOUND IN CRYSTALS The first studies on diffraction of X-rays by acoustic oscillations in solids were performed about 50 years ago [5]. Fundamentally new results were obtained in the 1980s – 1990s (see [9, 10] for example). A common mechanism of acoustooptic interaction based on the modulation of the refraction index of light in the field of sound waves is inefficient in the Xray range. However, in the case of Bragg diffraction the change of the phase differences of the X-ray waves scattered by displaced atoms becomes essential Therefore, the intensity of X-ray reflection turns out to be very sensitive to the level of acoustic disturbances. The regular character of acoustic displacements manifests itself rather weakly at comparatively low frequencies. In the case of high frequency or short waves, an acoustic wave produces in a crystal a macroscopic superlattice with a period equal to the ultrasonic wavelength that leads to strong reflection of X-rays. The amplitude of sound waves can be measured according to the positions of the peaks of reflection intensity. The boundary between low and high frequencies is determined by a relationship between the sound wavelength λs and the extinction length τ of X-rays. The last depends on the energy of X-ray quanta and the material of the sample (crystal). The high-frequency range corresponds to the condition λs ≤ τ and the low-frequency range is determined by the condition λs > τ. The character of the oscillations of intensity of X-ray reflection depends on λs and τ and the sample (crystal) thickness. Probably, the latter prompted Zolotoyabko et al. [10] to study the effect of surface acoustic waves propagating in a thin-film structure at a silicon substrate on diffraction of X-rays. Naturally, they observed a strong change of intensity of X-ray reflection with the change of the amplitude of surface acoustic waves within the high-frequency range. The period and phase of oscillations of the intensity of X-ray reflection are sensitive also to

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weak distortions of the crystal lattice and the parameters of the layered structure. All this opens new opportunities for nondestructive testing of acoustooptic devices for microelectronics. Similar phenomena were observed also in the case of diffraction of thermal neutrons at the spatial lattice formed by acoustic waves in a crystal [11]. As in the case of X-rays in the low frequency range, the increase of the amplitude of acoustic waves (ultrasound) leads to the expansion of the angular interval of the Bragg reflection and the intensity of neutron reflection attains a certain kinematic limit. In the high-frequency range when formation of a superlattice occurs, oscillations of the intensity of neutron reflection are observed. In the case of diffraction of neutrons (in contrast to the case of X-ray diffraction) not only exchange of momentum but also exchange of a very small energy with the ultrasonic wave are essential. Other fundamentally important phenomena are observed too. There is no need here to dwell on these phenomena. They are described by Iolin and Zolotoyabko [11] and in the literature cited by them.

2. SCANNING ACOUSTIC TUNNELING MICROSCOPY Tunneling microscopy and scanning tunneling microscopy are based on the effect of the strong (with respect to the exponential law) dependence of the change of the tunneling electron current between the needle point and the sample surface on the distance between them. Relative scanning of the needle point and the surface provides an opportunity to measure the smallest roughnesses of the surface and perform the topography of the surfaces literally at the molecular or even atomic level. The latter is very important for studies in surface physics. This stimulated a large number of studies on tunneling and scanning tunneling microscopy. Recently, scanning tunneling microscopy was applied to detection of high-frequency (1 – 12.5 MHz) acoustic waves [12]. Acoustic pulses with high-frequency filling applied to the surface of a sample made of an electrically conducting material were detected as the disturbances of the tunnel current. The nonlinearity of the characteristic of tunnel current (the dependence of current on the distance between the needle point and the surface) provided an opportunity to detect the envelope of an acoustic pulse and, therefore, its time and phase parameters and amplitude. It has been determined that the sensitivity of the proposed technique of detection of high-frequency acoustic waves [12] is comparable with the sensitivity of optical methods but its time resolution is much higher. Scanning tunneling

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microscopy proved to be very useful for investigation of the properties of conductor and semiconductor surfaces but less useful in the case of dielectrics. It was revealed that scanning tunneling microscopy using the effects of interaction of tunnel current with an acoustic wave in a sample provides an opportunity to conduct analogous studies of dielectrics with the same high sensitivity. Moreover, it is possible to study and monitor the subsurface structure of materials with the help of scanning acoustic tunneling microscopy. Du Sidan et al. [7] considered various mechanisms of action of acoustic waves on the formation of the “image” of the surface and subsurface structures of various materials in scanning acoustic tunneling microscopy.

3. INTERFEROMETERS USING MATTER WAVES – ATOM INTERFEROMETERS In 1802 Young demonstrated that light could behave like waves at a liquid surface, which propagating and passing through two parallel slots interfere and form a pattern of alternate troughs and crests. Later, in the 19th century Michelson and Fresnel developed optical interferometers on the basis of these ideas and conducted very precise measurements of various physical phenomena using these devices. It was determined that the precision of measurements is limited by the light wavelength. Taking into account the fact that light consists of particles (photons) and the wave nature is inherent also with other particles, physicists just could not avoid thinking about development of interferometers using not photons but other particles and the de Broglie waves corresponding to them. Electrons, protons, and neutrons were used initially as such particles. The last turned to be the most “convenient” also because of the fact that they penetrate into substance relatively well. But the development of an interferometer using atoms, i.e., “heavy” particles, and very short matter waves corresponding to them seemed the most attractive idea. Atoms may be treated as waves only to a certain degree and it is very difficult to reveal their wave nature. Even at relatively small motion velocities, they have a very small wavelength in conformance with the laws of quantum mechanics. Their wavelength is small even in comparison with low-energy electrons and neutrons, for which interferometers were developed earlier [12]. Extremely short matter wavelengths for atoms (matter wavelengths are 1000 times shorter than light wavelengths), and the fact that atoms do not pass through a substance in contrast to neutrons for example, made the development of atom

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interferometers very difficult. Despite big difficulties this development was achieved quite recently, about five years ago. The first two experiments with atom interferometers were conducted in Konstanz (Germany) and Massachusetts Institute of Technology (MIT, USA). The Young experiment with helium atoms was conducted in Konstanz and an interferometer using sodium atoms was developed in MIT. Three diffraction gratings are used in the interferometer with sodium atoms. The first of them splits a collimated beam of sodium atoms into two divergent beams. The second grating is positioned at a large enough distance from the first one (lower at the propagation track of the beams where they have diverged already) and changes the direction of one of the beams in such way that they would converge and interfere. The interference pattern of “troughs” and “peaks” of the type of a standing wave is observed in the region of interference of the beams in the plane perpendicular to the direction of their propagation. In fact, this is the statistical distribution of probability of occurrence of particles in this or that place of the interference pattern. Detection is performed by a running light grating with the period identical to the period of the interference pattern, which moves in the plane of the last. The running grating is created with the help of a He-Ne laser. Material nanostructures were used as diffraction gratings in the experiments discussed above. It was demonstrated in 1986 that it is possible to create diffraction gratings for atomic beams on the basis of standing light waves produced by beams of coherent laser radiation. They can play the role of atomic beam splitters and deflectors like ultrasonic waves serving as optical beam splitters and deflectors. The phenomenon of diffraction of matter waves or particles at light waves was predicted by Kapitsa and Dirac already in the 1930s. Atom interferometers are used for very precise measurements of the effects of rotation, acceleration, gravitation, and other physical phenomena and quantities. Two research teams reported recently successful experiments on development of an atom interferometer analogous to the Sagnac optical interferometer [8]. In this interferometer two atomic beams propagate in the opposite directions along the perimeter of a certain area and meet at the initial place. If the equipment is rotating, the beams have a phase shift when they recombine. The phase shift is directly proportional to the area swept by the beams and the relative rotation around the axis perpendicular to the plane of rotation of the beams (equipment). It was noted for comparison that the Sagnac optical interferometer constructed by Michelson in 1925 for measurement of rotation of the Earth had a perimeter equal to several football fields. Analogous experiments with a neutron interferometer require several square centimeters. The Sagnac effect is the basis of laser ring gyroscopes used widely in aviation. Schmiedmayer et al. [13] reported successful measurements of the so-called refraction index of matter waves

364

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of sodium atoms in gases (helium, neon, and argon) performed with the help of an interferometer using sodium atoms. Ekstron et al. [14] give the data on measurements of electrical polarizability of sodium atoms with a precision six times higher than that of the measurements conducted earlier using other modern methods. All this can be an impressive illustration of the efficiency of atom interferometers. It is a common opinion that atom interferometers have an excellent future. Atomic sources are rather cheap as well as the devices for splitting and deflection of atomic beams. As for application of atom interferometers in acoustics, there are no reports on this topic yet as far as we know. But there are no doubts that in the near future, atom interferometers may become the basis for development of essentially new acoustic measurement techniques and devices as it happened with optical interferometers or scanning tunneling microscopes.

REFERENCES The 9th International Conference on Photoacoustic and Photothermal Phenomena, June 27 – 30, 1996, Nanjing (China), Conference Digest. 2. L. M. Lyamshev “Radiation acoustics: photoacoustics of penetrating radiation.”, The 9th International Conference on Photoacoustic and Photothermal Phenomena, June 27 – 30, 1996, Nanjing (China), Conference Digest, PL-1.3, pp. 4 – 5. 3. L. M. Lyamshev and Yu. Yu. Smirnov “Distributed optical-fibre acoustic detectors”, Akusticheskii Zhurnal, 1995, v. 41, No. 4, pp. 533 – 546. 4. “Acoustic surface waves”, Ed. by A. Oliner, Springer, Heidelberg, 1978. 5. W. Spencer “Investigation of resonance vibrations and structure violations in single crystals by the technique of X-ray diffraction topography”, in “Physical acoustics. Principles and methods.”, v. 5, Mir, Moscow, 1973. 6. A. Morean and J. H. Ketterson “Detection of ultrasound using a tunneling microscopy”, J. Appl. Phys., 1992, v. 72, No. 3, pp. 861 – 864. 7. Du Sidan et al. “Theoretical study on tunneling acoustic microscopy”, The 9th International Conference on Photoacoustic and Photothermal Phenomena, June 27 – 30, 1996, Nanjing (China), Conference Digest, pp. 542 – 543. 8. B. G. Levi “Atoms are the new waves in interferometers”, Physics Today, 1991, v. 44, pp. 17 – 20. 9. I. R. Entin and I. A. Puchkov “Oscillating dependence of intensity of X-ray reflex excited in a crystal by ultrasound”, Fizika Tverdogo Tela, 1984, v. 26, No. 11, pp. 3320 – 3324. 10. E. Zolotoyabko et al. “Acoustic field study in layered structures by means of X-ray diffraction”, J. Appl. Phys., 1992, v. 71, No. 7, pp. 3134 – 3137. 11. E. M. Iolin and E. V. Zolotoyabko “Interference phenomena in the process of dynamic diffraction of neutrons under ultrasonic excitation”, Zhurnal

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Teoreticheskoi i Eksperimental’noi Fiziki, 1986, v. 91, No. 6 (12), pp. 2132 – 2139. 12. “Optics and interferometry with atoms”, Ed. by J. Mlynek, V. Babykin, and P. Meyestre, Appl. Phys. B, Special Issue, 1992, v. 54, p. 321. 13. J. Schmiedmayer et al. “Index of refraction of various gases for sodium matter waves”, Phys. Rev. Lett., 1995, v. 74, No. 7, pp. 1043 – 1047. 14. C. R. Ekstron et al. “Measurement of the electric polarizability of sodium with an atom interferometer”, Phys. Rev. A, 1995, v. 51, No. 5, pp. 3883 – 3888.

SUBJECT INDEX

acoustic tunneling microscopy 361 acoustooptics 359, 360 air shower 18, 30, 325 Airy function 100 atom interferometer 360, 362 – 364 Bessel function 57, 63, 67, 209, 220 boson 8, 11, 12 Bragg diffraction 360 Bragg peak 19, 295, 297 bremsstrahlung 16, 32, 257, 299 bubble mechanism of sound generation 4, 30, 31, 35 Buger-Lambert law 48 cascade 18, 19, 21, 27, 30, 227 – 234, 237, 238, 327 – 330 cascade shower 18, 19, 227 Cherenkov direction 202, 205 – 209, 213, 217 – 219, 224, 267, 268 Cherenkov mechanism of sound generation 31 – 33 Cherenkov (acoustic) radiation 2, 31, 32 Cherenkov wave 33 copper sulphate 249, 254 correlation length 58 – 61, 68 cosmic rays 12, 13, 18, 19, 325, 331

cyclotron 287, 288 deformational mechanism of sound generation 277, 278 directivity pattern 51, 59 – 61, 73, 80 – 85, 92, 101 – 103, 116, 150, 151, 155, 156, 160, 201, 210 – 212, 221, 223, 240 – 243, 245 – 247, 252, 254 – 256, 269 – 272, 279, 280 Doppler effect 201, 257, 260 Doppler frequency 208, 209, 213, 219, 224, 259, 260, 269 Doppler wave 260 DUMAND Project 4, 26, 227, 325, 327, 328, 333, 340 dynamic mechanism of sound generation 35 – 37, 148, 152, 157, 232, 234 efficiency of sound generation 4, 5, 23, 31, 37, 39, 72, 160, 161, 163, 164, 191, 192, 212, 213, 227, 238, 252, 255, 271, 275 – 277, 279, 339 electron-acoustic microscope 314 – 318, 323, 325 extensive air shower 18, 30, 325

367

368

SUBJECT INDEX

far wave field (zone) 49, 51, 52, 54, 61, 62, 72, 73, 80, 89, 91, 94, 105, 117, 118, 123, 141, 143, 146, 148, 154, 158, 160, 161, 165, 171, 197, 202, 203, 214, 231 – 234, 240, 243 – 246, 248 – 250, 254 – 256, 258, 260, 262, 266, 280, 288 fermion 8, 10 Fourier integral 64 Fourier transformation 106, 140, 170, 172, 205, 207, 208, 211, 260, 261 Fraunhofer zone (field) 43, 51, 141, 160, 203, 214, 240 GENIUS Project 5, 330, 333, 340 geoacoustics 5, 330, 331, 340 gluon 10 – 12 Grand Unification Theory 8, 11, 12 Grueneisen parameter 25, 103 hadron 8 – 11, 326, 327 Heaviside function 106, 229, 236 inverse piezoelectric effect 38 kaon (k-meson) Kirchhoff approximation 53 Kirchhoff integral 53 latent energy 35 lepton 8 – 12 linear accelerator 1, 287, 297, 298, 302, 332 Love wave 152 Mach number 217 Mach wave 257, 266, 267 matter (de Broglie) waves 359, 362, 363 meson 2, 9, 10, 326, 331, 332 microshock waves 4, 28 monochromatic sound 34, 39, 47, 64, 73, 208, 252, 254

muon (µ-meson) 3, 5, 9, 19, 30, 227, 234, 326, 327, 330, 332, 340 N-wave 130, 135, 273, 287, 291, 335 near wave field (zone) 27, 50, 51, 124, 125, 128 – 130, 197, 198, 230 – 233, 244, 273, 287, 288, 292, 304, 327, 329 neutrino 3 – 5, 7 – 9, 11, 12, 19, 30, 227, 230, 238, 326 – 337, 340 photoacoustic microscope 311 – 315, 318 pion (π-meson) 10, 326, 331, 332 pulsed radiolysis 35 quark 9 – 12 radiation-acoustic microscopy 3 – 5, 309 – 311, 321, 325, 339 radiation-acoustic visualization 309, 321, 323, 325, 339 Rayleigh parameter 55, 56, 60, 61, 68, 69, 87 Rayleigh wave 151 – 156, 162 – 164, 184 – 186, 192, 193, 214 – 218, 224, 227, 237, 271, 273 – 277 reciprocity relationship 42, 43, 45, 142 reciprocity theorem 42, 49, 93 Sagnac interferometer 363 scintillation detector 326 shear stress 232 shear wave 137, 274, 276, 279, 281 signal-to-noise ratio 317, 336 strain tensor 146, 148, 150, 151 stress tensor 160, 161, 165, 172, 174, 176, 178, 180, 181, 193, 198, 214, 229, 230, 232 – 236 strictional mechanism of sound generation 33, 34 surface acoustic wave (SAW) 155, 161, 239, 270 – 279, 359, 360

SUBJECT INDEX

synchrotron radiation 12, 20, 41, 105, 147, 164, 239, 281, 282, 299, 318, 333 theory of underwater explosions 29 thermal (thermoradiation) mechanism of sound generation 3, 4, 19, 20, 24, 26, 30, 31, 33 – 35, 37 – 39, 42, 50, 94, 123, 145, 148, 152, 157, 181, 277, 278, 290, 292, 302, 303, 306, 308, 321, 339

369

thermal relaxation time 41 thermoacoustic array 128 – 130, 150, 155, 156, 172, 181, 245, 246, 249, 254, 287, 303 – 305 thermodynamic generation of sound 24, 47 thermoradiation excitation of sound 5, 39, 42, 50, 55, 72, 75, 131, 239 transition radiation 33, 257 tunneling microscopy 361