Распространение волн в периодических структурах

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sepea

m

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Hb

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v= B KOTOpOM

P-

d~

r

V

e m

= ~

r

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cpe.nHSISI I1JIOTHOCTb.

1) New ton. Principia. livre II. 1686.

ed p

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1822

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+ ot

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m

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I sin (ltd/A) I ltd/A

m

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m

m

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XIX aexa; Konra, Banea-Ilayann,

r. KOWH,

= Vco

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pa60TaM JIar­

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§ 2.

A

ct> Ii r. 2.

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peineuae

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o--d---O---d-o-d---O

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4) 5)

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r.

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B

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2).

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1

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cncreu, 'a

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=

aaase, eCJIH ysecrs, 'ITO '1aCTOTa TaK>Ke BOJlHOBoro '1HCJla, na 'ITO H 06paTHJI BH

MawMR B 1881

r, nOJlHOe peurenae

paccxo

1) I< e I v i n, Popular Lectures, livre I, p, 18

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x=nd.

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=

ax) = A

A cos '2Ifrf.'1t ­

rne a - BOJlHOBoe qHCJlO, 1i3BOJlbHaH nOCTOHHHaH.

'I -

V=

(2.2)

Tor.aa

Yn

r ne B - HeKOTopaH nOCTOHHHaH, 3aBHCHll.I,aH weTKH. Y'IHTbIBaH reneps (2.5), nonysae epaaoBot! CKOpOCTH

cos

qaCTOTa,

27t ('It ­ and),

t-

BpeMH H

A-

(a)

Ia I

= B I sin

-;;ad

Ia I

I_ V

-

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I CD

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Vco=7tdB.

(2.3) npo­

'I

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v

am

/

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0

2

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m

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em

x

~

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r, 4.

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(2.3)

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r ae m ­ ueaoe qHCJlO, TO BeJlHqHHa caeiuenns Yn He H3Me­ HHTCH, T. e. lLDHJKeHHe OCTaHeTCSI TeM JKe caMhlM H 6YlLeT xa­

11

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e

a

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lid.

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V=":: a .

1

2iJ

(2.5)
TaKHM

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a,

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MOlt,

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=

Y

(a),

'I

(a)

= B Isin «ad [,

Ii

r. 5.

v OT BOJl­

(2.6)

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2

3sK, 201.

---~,-

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(2.4),

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1

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a = -+- 2d '

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nOJIHOCTblO

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nOJlO>KeHHe.

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3aHHMaeMoe

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113

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scex

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xoropue

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d

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nOJIOC .np03pa'lHOCTH" H .Henp03pa'lHOCTH" crpyxrypu Onpe./leIlHCTCH nepHO./lH'IHOCTblO crpyxrypu, 06YCIlOBIIHBalOl1\eH HaKanJII1BaHHe clla6LIX OTpSlKeHHH aa KalKJJ.OM 3BeHe npa npOXOlK1leHlIH BOIIHbl B./lOllb see. {eM. n. E, K pac H y III K H H, )KT
n pe11CTaBI1Tb

ce6e

peurerxy,

paaJII1'1HbIX Macc, pacnOJIOmeHHbIX

xa

C

nO

:lla apyroa BJ/.OJlb OCI1 Ox; 3JIeMeHTapHa51 .TJ}H CTeneHI1 CBo60J/.bI, a Kpl1BaSI 'i = 'i (a)

rpex

K

paaJlI1'1HbIX .9TI1M

06bI'IHO

1) nOHBJIeHHe npeneasnsrx 'laCTOT H CBH3aHHblY C HHMH '1aCTOTHbIX

/13"

t ~

Vz

HHme

6blJII1

nepH

a

TOTax MaJIOe conpOTHBJIeHl1e. CHODa OKa3bIBaeTCSI. 'ITO HaJII1'1l1e

1). B 1912 r.

npeJ/

"

3aMbIKaTbC5I '1epe3 I1HJ/.YKTHBHOCTI1. HMelOllI,l1e

CMaTpl1BaeMbIX

B

3JIeMeHTapHOt!


'1H

CB060J/.bI 3JIeMeH llIeTKI1.

2d

+0 d 2

rODOpSI.

'1HCJIO '1aCTOT.

'1aCTOTaMI1. nOCKOJIbKY HHnyK­

npenCTaBJI5IIOT J/.JI5I HI1X 60JIblllOe

'1HCJlO

paCCM

BOJlHOBOrO

MaJIOeCOnpOTl1BJIeHl1e eMKOCTI1. 3Ha'leHl1e

TI1BHOCTI1

cay-rae

l1Ha'le

/I

3aMbIKaTbC5I '1epe3 HMelOllI,l1e

'1aC

napaxerpauu

oopaayiouuo;

'1HCJlY TImOB '1aCTHLI"

'1I1C

n06aBJleHI1eM

xapaKTepI13ylOLl.(I1XC5I HOBblMI1

TaKI1M 06pa30M,

CHrHaJIbi C nOCTaTO'lHO HH3KI1MI1 '1aCTOTaMI1 6ynyT HSITbC5I BnOJIb

aea,

pOCTOM

BeTBb,

llBH>I<eHI1IO,

npn

BbIXOn5llll,aSI

SI'Ie~KY,

KOTOpOM

noxaaano

KaK 3TO

MbI TaKme

aKycmu'l.eClwii

Ha3BaHI1e TapHYIO

BeTBe~,

aonpocaa

euie BepH 113 nasaaa

BemBU;

ace

OHa

CO

'1aCTI1UbI, 0

nepeMelll,alOTC5I

B

.1laHHbl

B O.!lHOM 11 TOM me HanpaBJleHI1I1. llpyra51 B

onrnU'l.ecKOU BemBU;

OHa COOTBeTcTByeT .1lBl1

rAa!3a

30

J

fJlaoa

PAcnpOCTPAHEHHE BOJIH B O.ll.HOMEP KAIJECTBEHHOE PACCMOTPEHHE H 0

B AaHHbltt MOMeHT BpeMeHH 4aCTHu,bl onuoro HJlH HeCKOJlb­ KHX COpTOB ~nepeMew.aIOTcH B OAHOM uanpaaaenan. rorna KaK 4aCTHu,bl Apymx COpTOB nepeMew.alOTcH B npoTHBOnOJlOmHOM HanpaBJleHHH. TaK. nanpaxep. B CJlY4ae, paccxorpeanos Bopuox,

L,

Lz

L,

Lz

L,

2

Lz

§ 4. ct> Ii r. 21. OnTH4eCKaH BeTBb cooTBeTcTByeT ABHmeHHIO. npn KOTOpOM TCH 8 Aa1lHbitt MOMeHT BpeMeHH 4aCTHu,bl M H m nepeMew.alO B npOTHBOnOJlOmHblX HanpaBJleHHSlX. 3aBHCHMOCTb nOCTOHHHott aaryxauas OT4aCTOTbi AJIH pacCMOTpeHHott 60pHOM peurenot npHBelleHa aa qlHr. 20,6. r ne MomHO 3aMeTHTb nOJlOCY nenponycsaHHH H ABe nOJlOCbI nponycsanas. 9Ta Z KpHBaH HBJlHeTCH 0606w.eHHeM 60Jlee 4aCTHoro CJly4aH, npeACTaBJleHHOrO Ha ' Ii r. 22. BOJIH B KpHCTaJlJlH4ecKott peuierxe XJlO­ pacroro HaTpHH, UK KaK CTpyKTypa peweTKH KaMeHHott COJIH npHBOAHT K coBepmeHHO OAHHaKoBoMy pacnOJlOmeHHIO HOHOO: HOHbl Na" H Cl" 4epeAylOTcH, KaK 9TO noxaaauo aa qmr. 22. TaK, uanpaaep. BAOJlb HanpaBJleHHH OCH x CTpyKTypa peiuerxa Hr• 19. nOJlHOCTblO aHaJlOrH4Ha erpyarype, npe.n.CTaBJleHHOfi ua
06w.He aaaesaaa

Flpexcne lJeM npncryrnrn, K crporov o pacnpocrpanenaa ynpymx BOJIH B o He06xoAHMO CAe JI an, pHA 06W.HX aaMClJ 06Cy.n.HTb nexoropue oc06eHHocTH paccxar Bepneucs caosa K paCCMOTpeHHoMy B crenuieay cnyxaio OAHOMepHott petnerxn, cn8MH paanoa xaccu m, pacnoaoxennuaa BeCHH BAOJlb uexoropon npaxoa ua paBHOM OT .n.pyra. ECJlH BbJ6paTb nanpanneuue OCH x BAO xoopnanary n-tt lJaCTHu,bl MomHO npencraa X= nd+~n'

1

a = T' y ~

saerora,

t-

k

= 27tad.

BpeMH. a -

d ~nepHoA pemerxa,

Ul-

Ul

=

BOJlHOBOe 411CJ xpyrosaa lJaCT

,at.tnJIHTYAa. BeJlH4HHa k = 27tad npencraanser c0 ~PH nepexcne OT n-tt lJaCTHu,bl K (n+ 1)-t

BaR

; \

~n+l

= ~ne-ik.

3TOT

¢asoBbllt

2TC;

KpaTHOIt

PacnpOCTpUlieliue 80AIi 8 OOIiOAle

yroJi onpenensercs C T04HOCTblO no BCJlH4HHbI,

.ll.eIlCTBHTeJIbHO,

m-

eCJlH

uenoe nOJIOmHTeJIbHOe

l'aKHM 06paaOM, MOJl<.!O rOBopHTb paBHott Y.lI,BOeHHOMY paCCTOllll11

HJlH orpnuareasnoe 4HCJlO, TO pCUEHHe sanasa H, B 4aCTHOCTIi,

w

SHa4eHlie 4aCTOT

H ~ He MeHlllOTClI npH saMeHe

k

80.llHbI.,

Ha

qaCTHuaMH. 3TOtt BOJlHe COOTBeTcTByeT K

(4.4)

k'=k+2rr:m.

TaKHM 06paaOM, 4aCTOTa ~ (TaK me KaK H KpyrOBall 4aCTOTa

k

.lI,OJlmHa 6bITb nepHO.Il.H4eCKOIl ¢YHKUHelt

w=

=

'I

KaK

yxe

f (k),

nepaoa

27t

F (a),

nepnon

d

onucaepaoa B Bbl60pe

o/n

uaxa. B

w)

OTIlOCHTCJlbHO

a.

CIiCTeMa BelleT ce611, KaK ¢HllbTP HflJl
3TOT peay.n.rar lIBJHleTClI

k

BeJIlitlHHbl

3aMeTHM, 6blJla

6bI

LJTO

eCJlH 6bI ¢YHK­

CHCTCMbI

O.ll,HaKO,

JlHlUb

ljJliJlbTPbl BepXHI1X 11!1Ii HI1Jl
(4.5)

3TH 06l.Ulie pe3YJlbTaTbl, .a.orrOJlHeHHble .lI.eJlbHblX CJlyqaeB A. co (a 0) H A.

T04Kax

C

B caay

nocxonsxy

3Ta

xoopnauaroa,

¢YJlKUHlI

nd,

paBHOIt

onpene.teua

yCJIORHe

BH.ua

paccxorpe-n,

(4.5)

csoacraa

JlHUlb

B

f (k)

¢yHKUHIl

H

F (a)

=

nOCKOJIbl
7t

C006paJl<eHHlI

-

1 2d

<» «:

(4.6)

1 2d'

pacnpocrpaamorca

BOJlHbl

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O.ll.HHM H3 BamHblX CBOItCTR ¢yHKUHIt OOJlOmHTeJIbHblM

pacnpocrpaamoumecs

npocrpaamouraecs 4HCJlO, eMY,

na.reao;

3aKJllOlIeHHOe

TaK

me KaK

H

B

f (k)

3HalleHHIIM

k

F (a)

TaK

4TO eCJIH

ko-

ko

6erYl.UaSl HanpaBO, a BCJIHtIHHe k o 113 HepaBeHCTBa (4.6) cnenyer

+ 27t, 2TC -

1 I),j = T/iT >- 2d.

CTPYI
§ 5. UenOQKa

1f3

r.ma o

C806o~HWX

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JJ.JHl npOCTOTbI 6y.a.eM npe.ll.rrOJlaraTb, LJT B3al1MO.ll,ellCTByeT TOlIbKO co CMeJl
cooraercrnyior pac­

nOJlOmHTeJlbHOe

(-TC,

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IfCnOJlb30BaHbi R nOCJle

B O.ll.HOMeplJOIl u eno4Ke, o6pasOBaHHolt uaMH, CYl.1..leCTBYIOT TOJlbKO CI1Jlbl B3aI1Mo.a.el

nartpaso, OTpHUaTeJIbHblM -

OCHOBHOM HHTepBaJIe

ReJIH4HHe

H

=

06paaOM

B 060HX HanpaBJIeHHlIx.

semuocms,

6Y.ll.YT

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<. k <. TC,

CTpYKTYPOIl

=

Mepe I
HHTepBaJIe HX nepHO.ll.HtIHOCTH; HaH60Jlee

-

C nepHO.ll.li'leCKOIl

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.ll.OCTaTOliHO

IlPOCTblM Bbl60pOM HHTepBaJIa asnaerca CJIe.ll.YIOl.UHIt:

BOJIHbl,

B Ta

HMeJIa ¢HSHLJeCKHIt CMblCJI BO acex T04Kax Mem.ll.Y lIaCTH­

IIBJllleTCII HeHs6emHblM.

HX

~e.

YCIIOBfle (4.8) He lIBlIlIeTClI, Boo6w,e BOaMOJl
neonpeneneuaocn,

ycrpaueua,

xa

~<;"'c'

(4.5)

CJIe.ll.CTBHeM .uHcKpeTHoCTH H rrepHO.ll,H4HOCTH

peurerxa.

lIBJllIlOr.uallClI

'/e'

= Zixad,

k

1.

'tacmoma

CTpYKTypbI. Bo MHorlix BeCbMa BaJl
HJlH BOJIHOBOrO LJHCJla

OTHOCflTeJIbHO

6blJIO OTMe4eHO B rJl.

HenOCpeJl,CTBeHHbIM

UHlI

1

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+ 7t),

TO

COOTBeTcTByer, HanpHMep, CIIYLJall ynpyrnn paBHoMepHo

pacnOJlOJl<eHbl

PaCCMOTpl1M

BHa4alIe CJlYlJatl npo.a.OJlbHbI

=

3HalJeHHe a 0 peweTKa CMel.1..laerclI

(4.7)

c

Mac

.a.etlcTBHlI MeJl<.a.y LJaCTrlUaMfi onpe.a.elIlllOTCH HOlt BO sce x rouxax ynpyrocTbIO CTpyHbI.

coorsercrayer BOJlHa, 6erYl.1..lall HaJIeBO.

O.a.HHaKoDble

COOTBeTcTByeT 6eCKOHe KaK ueJloe, fI paCCTOlI

u.aMH He Mel/lleTCll, TaK 'ITO CHlIbI BaaHMO.ll BaIOTClI. l..facToTa np.r 3TOM paBHa HYJlIO. HYJllI, we

HO OLJeHb MaJlOM 3HalJeHlilf pacCTOllHfllI

3

3aK. 201.

Mem.a.y

a

.lI.JlflHa BO

qaCTHuaMIi.

B

9TO

r saea

34

2

,.,- -n.""':Tpallellue

pacnpocrpausercs B.lI.OJJb peurerxn TaK )J{e, KaK BJlOJJb nenpepuauo­ ro CTep)l{HSI, a CKOpOCTb pacnpocrpaneuaa He aaBIfCHT OT ,l\JJHHbI 1l0JJHbl,

T.

e.

qaCTOTa

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I I.

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C

tIaCTHlJ,aMH,

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BOJJH OKa3blBaeTCSI

pacnpocrpaaeaaa

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B

6blTb

npOTHl3HOM

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pacnpocrpanenaa

npa

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B npllMOM

H

a.

a xpasaa 'I

ECJlH )J{e crpyna aCHMMeTpli'lHa. HanpHMep cesenae, TO 9TO Bblp0)l{l\,eHlfe CHIfMaeTC

(a)

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CKOpOCTH

aanpaaaennn

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v I

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B

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cnenyer 'I

(a)

JIIfTHtIeCKOe

B nae

npenunyutero,

a = l/2d.

TO'lKe

xpanas

1f3

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rOCTIf, TO BeTBb. COOTBeTCTBYIOw.all nOne HarpY)J{eHHotl CTpyHbI, Jle)J{IfT, OlJeBlfllHO, .lI.OJIbHblX KOJJe6aHHtl.

CIfJlY

cHMMeTpHIf

CHCTeMbl,

§ 6. npO~onbHMe Kone6aHHH ~enO paBHooTCTOH~HX

OHa .lI.OJl)J{Ha IfMeTb MaKcHMyM

noxaaars,

tITO

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B

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sa
5.

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4aCTHlJ,a. BOaBpaw.aeMali K nOJIO)J{eHHIO CHJlOtl. 06pa3yeT rapMOHlflJeCKlflt OCUIfJJJllIT

Ee ana­

BeTcTByeT C06cTBeHHali lJaCTOTa '10 ' 3aBH UHeHTa ynpyrocTH H OT MaCCbI tIaCTHUhI. ynpyrocTH HMeeT paaJIHIIHble aHalJeHHlI B Ha

nPOaOAbHblX KOJle6aHHtl, OCTaIOTCSI,

nacroru

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C TeM

llJJHHHblX H

OZ.

JIHWb OTJJH'lHeM, 'lTO

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cnpaaennaauaa,

C

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none pewcux KOJle6aHlitl KatIeCTBeHHoe paccaorpe­

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r. 23.

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I

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8

nOCKOJJbKY nepneH.lI.HKYJlllpHO peure.n urarscs B D.ByX HeaaBIfCHMLIX HanpaBJle nonepellHhlX KOJle6aHHtl l\,OJl)J{Ha D 06w. l\,BYX pa3J1H'lHbIX BeTBelt

a KpOMe roro, crporue Bbl'IHCJIeHHSI npOnOplJ,HOHaJIbHOA noxasusaior, tITO ,l\JISI ,l\JIHH BOJIH, cpasnaxux C paCCTOSlHHeM Me)J{.lI.Y

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§

I 1=

.

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uymo H ace qaCTHUbl

CHCreMa' npe.nCTaBJISleT

M

\{a>K1I.aSl LJaCTtIl.I.a

r oopaayer

noaouin

cocenuae MaCChi. B CIIY4ae 6eCKOHe4Hotl paBHO

r.

npY>KiiHKaMii

npy>KHHKotl

npa

PacnpocrpaHeHue OOAH 8 OaflOM

0603Ha4eHbi MaCCbl

paauoaecaa

nOll0>KeH1IlI

2

TaKHM 06pa30M, paCCMaTpllBaeM311

. CHcTeMa.

onacaaaan

B rrpe.nbl.nYllJ,eM

IfCUyccTBeHHa;

.n.ettcTBHTeJIbHO,

ce6e

lIaCT1IUbT.

Yll.ep)K1IB3eMbie B Hef
IfKK

npY>l
r.

8

flPiU.epbI aHaJIOrH4HbiX 'IlKaH'lecKYIO npHpo.ny.

.nOBOJIbHO

.naJIbHej;!llleM (CM CTpyKTyp.

HMelOl

. 'BoJiee eCTecTBeHHaSl MeXaHH'leCKaSl Mo.n TYpt.1 rrOJlY'laeTCll 8Ta aa.n.a'la

npa

CJiO>l
paccMOTpelI1IH nOCKOJIbKY

'CTlfll. npH'leM aTOMbI Ka)K.noro

.nB

aaecs

copra

B3

• •eIfHO KaK Me)K.n.y cocoa, TaK H C aTO • pellleTKa H3 .nByxaToMHbIX MO

"'~TaKaSl

~lJa>l<eHa aa

~JI

c>j"accott

'ft~OIt ' . ~

M,

cPHr. 26. rrpH'leM Kpy)KKaMH a TO'lKaMH C Maccott

MOlleKyJIa '1Q •

m

(M, m)

06J1a.naeT

CooTBeTCTBYIOllJ,ef\

HeK

npol\OJ/bl

Pacnpocrpaneuue

BOAH

a

OOHOJlepH

fAaBa 2

38 9JIeMeHTOB.

CMeLUeHHSI

3HaKH,

4TO

TaK

M

Macc

ueurp

06LUHtI

m

H

npa

HMelOT

TSimeCTH Macc

OCTaeTCSI

.nBl1mHbI M.

Cnenas

9TH

.uByxaToMHbiX

aa

BeCHSI

3aMe'laHHSI,

MOJleKyJI,

paCCMOTpHM .uBHmeHHe

paCnOJIOmeHHblX

d

O.uHHaKOBbIX paCCTOSlHHSlX

B

paanue neno­

9TOM

nenoxxu

H3

COCTOllHHH

paBHO­

B.uOJIb OCH XX',

npH4eM

6JIHmal:\wHMH COCe.uSlMH.

Mm

Mm

~ ••

t:=d-s.::::::= d



Mm

Mm

d

.1.·

Hr.

26.

S.·

d

Mm

d

=:.r­

'I

(a)

.uOJlmHa

HMeTb,

TaKHM

aaaunaexas HHmHeJ;! nyns, a npyr as,

KOTOpbIX, 'lepe3

'lepe3 T0'lKY '1~ •

BeTBblO, -

6y.ueM C4HTaTb, 4TO B3aHMO.uetlcTBHe HMeeT MeCTO TOJlbKO Mem.uy

Mm

KpHBaSi

onua H3 npoxonnr

OpH YMeHbweHHH .uJIHHbI BOJIHbI HHmH HHMaTbClI, T. MOJIeKyJIa

npn

e. BeCTH ce6S1 no.u06HO TOMy

aaxenena onnoa !a I = 0

6bIJIa

'lacTHIle

y.uaJIeHHH OT T0'lKH

B 3aBHC

CHCTeMbI MomeT KaK

no.uHHMaTbCSI,

TaK

OTHOCHTeJIbHO

H

onYCKaTbCSI

'1~.

T0'lKH

Ha epHr. 27

npenCTaBJIeHO H3­

MeHe}m~ 4aCTOTbI 'I KaK epYHKIlHH

a,

aepxass BeTBb asoopascena anyx cnyxaes: UlTpHXOBaSi KpH­

npH'leM

Bseneu OBe KOMnJleKCHble 9KcnOHeHUHaJlbHble epyHKUHH, onna H3 KOTOpbIX onncuaaer .uBHmeHHe 4aCTHU C Maccotl M, npy­ r as - C MaccoJ;! m,

•

~M=AMe21ti(vt-a.x),

~m=Ame2d(vt-a.x).

(7.1)

3TH epyHKU HH xoryr HMeTb pa3JIH'lHble aMnJIl1TY.u bI, HO xapax­ TepH3YlOTCSI o.uHHaKOBbIMH BeJlH'lHHaMH 'I H a, npH'leM 'laCTOTa "

napaserpoa CHCTeMbI H BOJIH.OBoro 'lHCJIa a. " crporo noxaaars, 'lTO 6JIaro.uapSi ynnoenuoay 4HCJIy CTeneHeJ;! CB060.ubl B TaKoJ;! cHcTeMe 'lHCJIO BeTBetl KpHBOt! v(a) Tome y.uBaHBaeTCSI. Ka4eCTBeHHO 9TO MomHO nOSlCHHTb cnenyio­

.uJISI BaSI

HaSi -

pOCTe

Ia I

y6bIBaHHIO

Pa3JIH'lHOe

cenr)

CyLUecTBOBaTb CJIe.uYIOLUHe naa aana .uBHmeHHSI: a) KOJIe6aHHSI scex aTOMOB npoacxonsr ace BpeMSI B epa3e,

HHe

'1~

T.

.uJIHHbl

BOJIHbl

e.

(7.2)

AM=Am• H .uBH>KeHHe

CHCTeMbI

6)

Uen04KH,

CBO.uHTCSI K nepeMeLUeHHIO acea uen04KH

nepeMeLUalOTcSI HHSlX,

TaK

ECJlH

6bI

4TO

KaK

H3

M

aTOMOB

H

aTOMOB

UeHTp

TlImeCTH

CHCTeMbI

OCTaeTCSI

nOKO~.

B

Mem.uy MOJIeKyJIaMH OTCyTCTBOBaJlO.

TO 'laCTOTa TaKHX KOJIe6aHHJ;! 6bIJIa 6bI pasua '10' npH4eM

Il pH

HaJlH'lHH

OT JIH4Ha OT '10,

m,

ueaoe, HO B npoTHBOnOJIOmHbIX nanpaane­

B3aHMo.uetlcTBHe

B3aHMo.uetlcTBHSI

B03MO-

CHCTeMbI 3aBH­

BeTBH KpHBOtl 'I

no OTHOllleHHIO K

(a)

H,

MaKCHMaJlbHOJ;! 'l

BOJIHbl, 'laCTOTbI KOTOpbIX JIemaT BHe .uOJImHbI

TaK

me,

KaI{

H B

paccaorpenn

'laCTOTa

LUHe 9THM panna

'laCTOTaM, KOMnJIeKCHhI, npH'le

nocmonuuoa samyxauust

na naano

Ha epHr. 28 npasenena 3aBHCHMOCTb nO

panna HyJlIO.

COCTaBJIeHHble

CHCTeMe

9KCnOHeHI.lHaJlbHO aaryxars. BOJlHOBble 'lH

KaK uenoro C coxpaHeHHeM B3aHMHoro paCCTOSlHHSI Mem.uy aTO­ MaMH. 4aCTOTa npu 9TOM

Kam.uotl

napaxerpoa

xon nepxnea

6eCKoHe4Hotl

CaMO

B

H epHr. 20,(BopH). OTMeTHM,

CHT

CJlY4ae

tlTO

nosenenae nepxuen - 2d 13 (BHH­

xoryr

B

crmour­

BeTBH HJIJIlOCTpHpylOT epHr. OT

06pa30M.

a

'laCTOTbI,

meH TOJIbKO O.uHH H3 9THX cJlY'laeB.;

4TO

LUHM

COOTBeTCTByeT

4aCTOTbI,

corion paayxeercs, paccaarpaaaexoa

IIBJISleTCSI epyHKUHeJ;! MomHO

npH

B03paCTaHHIO

'1~,

Bo06LUe

'10~0.

rOBOpSl,

OT

'laCTOTbI 'I, npH'leM

lleTcTBYIOT

.uByM

BJIeHIIbIM na epHr. TaKHM JIaTb

THnaM' aep

27.

06pa30M,

CJIe.uyIOLUHe

cnnoumue H llITP

pa3JIH'lHbIM

paccaorpeauue npu

BbIBO.ubI,

.ueTaJlbHbltI

a

nposenen B .uaJIbHeRllIeM:

1. Hacrora

'I SlBJISleTCSI nepnonauecxoa ep

F'naea 2

40

3

rJlaBa

2.

06J1aCTb

H3MeHeHlUl

k

BeJlHqHH

(HJlH

a)

MO)KHO CBeCTH

MATEMATHqECKOE HCC~Ell OJlHOMEPHhlX PEIUETO COCTOSlmHX H3 OJlHHAKOBhlX

uarepaaay (4.6).

K OCHOBHOMy

3. Ecna 9J1eMeHTapHaH Hqe~Ka, o6pa3ylOlUaH 9J1eMeHT nepuo­ .llHqHOCTH O.llHOMepHo~ peurersn, HMeeT N crenenea caooonu, TO .llJlH Ka)K.llorO 3HaqeHHH k (HJlH a) cyuiecrayer N THnOB BOJIH

p

§ 8.

YpaBHeHHe .lI.BH)I(eHHSI O.ll.HOM cocromuef H3 O.ll.HHaHOBblX

CTporH~ BHBO.ll pe3yJ

3.lleCb 6Y.lleT .llaH

npenunyurax

no.ryuenuux HaMH B Mbl 6y.lleM

paCCMaTpHBaTb

r naaax.

neorpanau

CTOHlUYIO H3 O.llHHaKOBbIX qaCTHU C xacco

npHMO~

B.llOJIb

I

I

o

_I

I

, •••• ! •

H r.

11

Bynex

aayuars

nyxepyex

28.

2

Ilpuxepu, B § 5-7.

cooraercrayioiuae

N =1,

3, 6blllH npaaenenu . 4. 4HCJlO BeTBe~ KPHBO~ ~ = F(a) paBHO qHCJlY creneuen csooonu 9IleMeHTapHO~ Hqe~KH peurerxu. a TaK)Ke qHCIlY nOJlOC nponycxanns, nexoropue H3 KOTOpblX MoryT nepeKpblBaTbcH. H

5.

BOJlHbl

He xoryr

C

qaCTOTO~,

pacnpOCTpaHHTbCH

lle)KalUe~

aaryxaior no

npOXO)K.lleHHH qepe3 cncrexy. TaKOBbI 06lUHe CBO~CTBa nepHO.llHqeCKHX

uiae r II as bl.

H3yqeHHIO

KOTOpblX

ECJlH

nepeon H xepea Yn

.llBH)KeHHH

6y.llyT

OCH

OT

TaK

KaK

crpyxryp.

nOCBHlUeHbl

MaTe­

caenyio­

Mbl

OT

HenOCpe.llCTBeHHO 0603HaqHTb

caoero

paBHOB

B naHHbl~ MOMeHT Bpe

xn=nd+Yn' npennoaoxora», 'ITO O.llHH

cMelUeHHH

Me)K.llY

aa

OTKIlOHeH

nOJlO>KeHHH

OTHOCHTellbHO

B3aHMO.lle~CTBHH Me)K.llY HHMH HaM HHe

Jl106o~

nasuaan, HY

T. .ll.

xoopnaaaru X n

TbIBaTb

xepe

MaTHqeCKOMY

cnenyiouiyio

nasuaan,

.llJlH ee

KOTOpylO 6y.lleM

BHe nonoc nponycxauaa,

H 9KcnOHeHUHaJlbHO

no

paanux paCCTO npononsuue KOJle6aHHH T

qaCTHUbi----iiaqItliaH

HaqaJlbHYlO, CTHUy,

C pa31lHqHblMH qaCTOTaMH.

(HanpaiJJJeHHO~

B COCTOHHHH paBHOBeCHH aa

.llByMH

JlI06blMH

CTOHHHH Me)K.llY n-~ H

(n

q

np yrn X

neooxo

qaCTHuaMH.

+ m)-~

qaCTHua

+

Yn i m- X n = md nOCKOllbKY CMelUeHHe Yk npennoaar aerc HHIO C paccrosnaex d, TO BHpa)KeHHe ( rem.no HIlH orpauarensuo B 3aBHCHMOCTH O

rn, n+m =

Xn

1) B 3TOM rraparpa¢e paCCMaTpHBaeTC~ np MO.lleHCTBHR MeJK.llY tJaCTHI.\aMH ueno-ncn, 3Ta KYJlOHOBCI
npUM. pea.

MareMarU'IeCKOe uCCJle80BaHue oiinc DOTeHUHaJIbHYIO 9HeprHIO, npa nonoum KOTOpOtl BbI'lHCJIHIOTCH

HJIH

CHJIbl B3aHMO./leltcTBHH MelK./lY ./lBYMSl ./laHHbIMH 'laCTHuaMH, MOlKHO

paccaarpanan,

KaK epYHKUHIO TOJIbKO HX B3aHMHoro

V (r) = V

paccrosaua

[(Yn-tm-Yn)VI(md)+

n m>o

(I x n + m - x n I).

+ ~ (Yn+m -

(8.3)

aupascenne .llJIH no.moa nOTeHUHaJIbHolt snep­ acea peurerxu MOlKHO sarmcars B anne

TaKHM 06pa30M, rHH

~ ~

V=C+

rne

nOCTOHHHaH

c=~ ~ V (md) n m>O

V=~ ~ V(!Xn+m-X n]) n m >0

npn

YCJlOBHH

CyMMHpoBaHHH

m (sro

'leHHHM

yCJIOBHe

.llJIH

ycnexov

3aMeTHM, 'ITO C TaKHM lKe

acex

roro, 'lT06bl napu 'laCTHU).

He

MOlKHO 6b1J1o 6b1 npo­

m, no

3Ha'lCHHHM

B 9TOM

cnysae

He06xo.llHMO nOJlY'leHHbllt pe3YJlbTaT pa3.lleJlHTb Ha ./lBa, TaK KaK

unaue

KalK./laH napa 'laCTHH Y'lHTbIBaJlaCb 6bI .llBalK.llbI. Y./l06Hee.

O./lHaKO, OrpaHH'lHTbCSI cyMMHpoBaHHeM TOJIbKO no nOJIOlKHTeJIb­ HbIM

3Ha'leHHSlM

MO.llYJlH y ECJIH C

nOCKOJlbKY

MelK.llY

pa3J10lKHTb B

V (x n + m -

'laCTHuaMH,

psn Teanopa,

x n) = V (md)

r ne V' (md) H V" (md) HbIX

9TO n03BOJlSleT

apryxeura epyHKUHH V. npenno rarars caeurenue Yk

paccTOHHHeM

MOlKHO

m,

dilldr

H

d 2V/dr2

MaJIblM

TO

onycrars no

epyHKUHIO

cpasrremno

V

+

3Ha'leHIIH npOH3BO./l­

r=md. supaaceune (8.4) H (Yn+m - Yn)' 60JlbWHMH

B TO'lKe

CTeneHHMH BeJlH'lHHbl

noaysaex

aupasceune

.llJlH

noauoa

noreaunaasson

npene­ .llBYX.

9HeprHH

peurerxa V =

~ ~ n m

>0

'lJIeHbI He

n

no

KOTOpblX

nrpajor

TaK KaK

OTJIH'IHbllt

m

nee,

BeJlH'lHIta cyw.eCTBeHHO nO

a 'lJleH.

./lJlH

.lleItCTBYlOmyIO aa 9TY nOJlOlKeHHblX cneaa. TaKHM 06pa30M,

aau

Pp = -

Yp

+ ~ (Yn+m + +

Y'lHTbIBaeT 'laCTHU. p

+ m=

xoroporo n lKe xacrauy

co

DblpalKeliHe .llJlH

C

cnn

aa ~ ~

.-

Yn)2V"(md)] = -

a~ ~ [

= -

Yp n m>

°[(Yn+m

p m>O

+ +

= -

+m

pOJIH, TaK KaK He 3aBH

xoroporo n = p, na P-IO 'laCTHUY co CTOpOHbI .OT

n

JIH60

CYMMbl, ./lJlH

(Yp+m - Y p)2V" (md)

+ (YP- Yp-m

(YP - Y p _m)2 V" (md)] =

~

[ - V' (ntd) -- (Yp+m -

Y p ) V"

m>O

+ V' (md) + (Y Pp=

Yn)2 V" (md)]

OT HyJIH B

n = p,

JIH60

P -

Yp_ m) V" (md) J

HJIn, eCJlH 0603:1a'lHTb

[V (md) +(Yn+m - Yn) V' (md) +

+ ~ (Yn+m -

MHpOBaHHH ./lJIH

acer.aa

+ (Yn+m - Yn) V' (md) + + (Yn+m - Yn)2V" (md) + ' ...

DO.llCTaBJIHH 9TO paSJIOlKeHHe B 6peraH

3HaK

TaKHM 06pa30M, nOJIy'lHM

COOTBeTCTBeHHO

CHJIa P p' ./leItCTBYlOmaH na p-IO 'la 06paTHbl~ 3HaKOM npOH3BO.llHolt OT n rio cMew.eHHIO Y p 9TOit sacruuu. O./lHaK crynars K Bbl'lHCJIeHHIO CHJIbl, caenyer Y

,C

TOJlbKO no nOJlOlKHTeJlbHbIM 3Ha­

He06xo.llHMO

Y'lHTbIBaTb .llBalK.llbl B3aHMO./leItCTBHe KalK.llolt BeCTH CyMMHpoBaHHe no

(8.4)

~

V" (md)

"

'"'" V m(Ypm

m>O

'lepe3

V:Z

+ Yp-m -

3TH epOpMyJlbl n03BOJlHIOT cnenan, pan

HH

rAa8a

a

MareAtarU'leClwe uCC:IIe(}o8a1IUe 08ltOM

Hanpaxep, B nocnennen CTpOKe BblpalKCHIi51 (8.6 a) qmry­ papyer '1J1eH -----:U' (md) , npencraanmoumn co6ott CIiJlY, C KO­ ropon (m p)-5I xacraua .nettCTByeT na P-IO '1aCTIiUY; B cnysae neozpauuueuuot: uenoiucu 3Ta cana xoanencnpyerca nporn­ BOrrOJlOlKHO nanpaaneanoa CIiJlOtt V' (md), CKOTOpOtt (p-m)-5I nacrnua .nettcTByeT na P-IO sacrnuy. B cay-rae IWHe'tHol1 uenowcu ne ao 6y.neT 06CT05lTb aaase . ITpe.LlrrOJlOlKIiM. 'ITO uenosxa '1aCTIiU npocrapaerca OT n = - CX) .LlO n 0 (T. e. 'I aCTIiUbI C ti 1, 2•... OTCYTCTByIOT), Ii uaa­ .neM CIiJlbl. xoropue cnenyer BBeCTIi, '1To6bl crpyxrypa peiuerxa B6J1li31i KOHua (n = 0) OCTaBaJlaCb He1i3MeHHott. Bneunrae CIiJlbI

+

+

i IaKKM 06pa30M, .LlJl51 rrOJlY'leHIi51 Orpa peurerxa C o.nIiHaKOBbIM paCCT05lHlieM d M 06XOOUMO. '1To6bI no nuan CUAa, .LlettcTB WeTKI1, paBH5IJlaCb . Ft. O.LlHaKO 3TO ycao He 5IBJl5IeTC5I .LlOCTaT04HbIM. B 3aBliCliMOC cana F t pacnpeneaserca MelK.LlY rpaHli4H urencn, MOlKHO paCCMOTpeTb naa cJlY4a5l:

7

1. B03MOlKHO. 'ITO XOTH B6J1li3H rpaa B03Myw,eHIi5l. O.LlHaKO .LlOCTaT04HO .naJleKO MelK.ny xacrauaan d OCTaeTC5I HeIi3MeHH lKeHl1e IiMeeT MeCTO 06bl'lHO npa CI1J1ax C paCCT05lHlieM, 'ITO, B 4aCTHOCTIi, BCTpe4a aaunux C KpliCTaJlJlli4eCKliMI1 peweTKaMIi. CTBIi51 CIiJl nopsnxa Ld, TO B CYMMe (8.7 '1J1eHbl C 1 L. ITPIi 3TOM paccrosu rrpOCTpaH5IeTC5I B03Myw,eHlie, rro nopaax paBHO Ld,

.LlOJllKHbl 6blTb T04HO TaKIiMIi lKe, KaK CIiJlbl, neacraoaasume CO CTOpOHbl oropoureuuux '1aCTIiU. TaK, aanpaaep, .LlJl51 LJaCTIiUbI C p 2 nano .Ll06aBIiTb CIiJlbr (eplir. 29)

2. Boaxyuteune MOlKeT HOCIiTb rrepli0 B 3aRliCliMOCTIi OT paCCT05lHIf51 Ii MOlKeT p seen peurerxe. B 3TOM cJly'lae Mbl rroJlY4ae T. e. nepaonnsecxyio crpyxrypy C HOBblM WIiM '1eM d. TaKI1M 06pa30M, .LlJl51 roa CIiJlbI F t , rrpliJlOlKellHott K rpaHIi'lllblM sac rYT IiMeTb MeCTO pa3J1114Hble paCCT05lHIi51 4aCTliuaMIi.

=

=

u.~ 5

~ I

I I I

I

I I

-7 -6 -5 -4 -3 -2 -1

1

0

I I

I I I

2

3

-<. m-<.

4

5

6

5U;'~

= -

U~, V~. V~, ... I1TaK, ecnn Mbl XOTIiM, '1To6bl KOHeu uenosnca OCTaBaJlC5I HeIiCKalKeHHbIM, T. e. lKeJlaeM coxpauars Heli3MeHeHHbIM pac­ CT05lHlie d MelK.ny KpattHIiMIi 'I acrnuaaa, TO aeooxonnao BBeCTIi CJlOlKHYIO CIiCTeMY CIiJl, .LlettcTBYIOW,IiX ua nocnennne '1aCTIiUbI uenouxu. TaKIiM 06pa30M. rrOJlHa51 cana, nencrnyiomaa na CO­ BOKyrrHocTb rrOCJle.LlHIiX '1aCTIiU p5l.Lla, paBHa

,

"

co

~,

F t=V 1+2U2 + 3 V a+ ... =L..JmV m ,

(8.7)

m~l

rrOCKOJlbKY cyurecrayer m rrap B3aIiMO.LlettcTBYIOW,IiX MelK.LlY co­ 60tt '1aCTIiU, pacnonozcemnrx C ooeax CTOpOH OT KOHua 11 na­ XO.Ll5lW,IiXC5I na paCCT05lHlil1 md npyr OT npyr a,
Hanpanep, 4T06bl II cnysae CB060.LlHO T. e. uenoxkn, K KOTOpOtt He rrpliJlOlKeHb CIiJlbl, OTCyTcTBoBaJlIi B03Myw,eHIi5l. He06

,

paBeHCTBO: paccrosuna

,

VI = V 2 = MelK.LlY

...

= Vi

4aCTliuaMIi .

=

O.

peurerxa

ITP C

.Ll0 ee rpauauu. Hanporaa, eCJlI1 ace np Jlli4Hbl OT Hy Jl5I, TO 6Y.LlYT IiMeTb MeCTO Jlli60 TOJlbKO B6J11131i rpanauu (cJly4att 1 .LlJlIiHe (cnyuan 2).

Bee BblWeli3J10lKeHHOe OTIIOCliTC5I TOJlbK CJlY4alO, KOTOpOMy cooTBeTcTByeT, nanpa XOlK.neHIiIi epOPMbl rroBepXHOCTI1 Ii rroBepxH lKli.LlKOCTIi IiJlIi TBep.LlOrO re.ra. B cnyuae xoropue BCTpe'laIOTC5I B epli3liKe, B3a1iMO.L crauaan 04eHb 6blCTpO y6bIBaeT C YBeJlli MelK.ny HIiMIi a, TaKI1M 06pa30M, acerna ocyure

3

tllaBa

46

OTCIO.n.a BH.n.HO, 'ITO ypasnenae (8.8) nonyc

I'pauaua nOBepXHOCTHoro CJIOSI. xoropua orpaHH'IHBaeT lKH.ll.lwe HJIH rsepnoe TeJIO, 6y.n.eT onpe.n.eJISlTbCSI L-M aTOMOM KalK.n.oro \ pana '1aCTHU. HCKalKeHHSI peurerxa, HMelOUJ.He MeCTO BHyTpH

noannenne

\ noBepXHOCTHoro CJIOSl, Bbl3blBalOT

.ll06aBO'lHblX

aaunoro rana, .

M",2

=

-4-

a

eCJIH v H ~

Mrc: 2 v2 =

CHJI,

~

f/

•

V m sm2 «amd =

m>O

\ xoropae

H HOCSIT Ha3BaHHe nOBepXHOCTHoro HaTSIlKeHHSI. 3TOT THn B03MYUJ.eHHIt peuienca H pasaepu OXBaTblBaeMol\

HMH

06JIacTH

6y.n.yT

npenaanen-

MOlKHO

CTpyKTyp,

HCCJIeAOBaHbl BCIO

'1aCTHUbl B KOTOpblX

paCCTOSlHHSlX.

3TH

JISlTbCSI

TOJIbKO

CTHUbl;

KpOMe

B §

CJIOlKHOCTb

10,

yxe

HO

ua

CHJI, .n.el\CTBylOUJ.HX

roro. snecs

cel\'1ac

na

onpene­ va­

rpl!HH'IHble

He06xo.n.HMO 3HaTb

xapaxrep pac­

r.n.e

U "m = V " (md). H3 COOTHorneHHSI (8.10)

nepaonnsecxoa

S1BJISleTCSI

v2 (a:

HH'IHOl\ 06JIaCTH. 06bl'lHble MaTeMaTH'IeCKHe MeTo.n.bl, 6a3HpylO­

na

nOJIOlKeHHe

peuieumo raxoro pona aaaax. AHaJIOrH'IHOe MeCTO npu paCCMOTpeHHH BonpOCOB, OTHO­

K

H,

BO-BTOpbIX.

a

raioxe

nsyx orpasceuuu BOJIH aa r panaue npn paccuorpeuaa nnnpocoa.

§ 9.

(CM.§

B 3a.n.a'laX 06

24)

H Bo06UJ.e

nporexaiouiax (axeiouiero nopsnox Ld).

Tpe6ylOUJ.Hx .lleTaJIbHoro aHaJIH3a S1BJIeHHl\,

nose pxuocruoro BepHeMcSI pHM

CJIOSI

reneps

KOJIe6aHHSI

HHSI '1aCTHUbl

nOJIy'lHM,

K nsrpazceuaio (8.66)

peuierua.

npHpaBHHBaSi CHJIy

p-l\

'1aCTHUbl

H

TaK KaK

V:

sin2 [rc:md (a

BeJIH'IHHa cyUJ.ecTBeHlw

V -

peuieuae _

Yp_

a = 11)... -

B

e

3TOM

Fp

CHJIe HHepUHH;

rne

ynoocrea

.n.JISI

3TO H eCTb (8.8)

A 21ti (>t-apd)

e

'1aCTOTa.

•

nO.n.cTaBJISISI

=

V"

=-

4yp sin2 «amd . ..-v-

+

e-21timda

-2]=

sin2

HCnOJIb30BaHO

tiad,

0603Ha

onpenemrn, nnrepecyxnuae

pnnaeaoa

CHCTeMbl,

Hac

noayvenaue B npen OCHOBaHHH '1HCTO Ka'leCTBeHHbIX cooripasceu ¢la30BaSi CKOpOCTb pacnpocrpanenas BO _V _ _

- Ia I-

H K

npa

~

r U"

V

M

HeOrpaHH'IeHHOM

I sin nad I _ d~

I na I

TaKHM

06pa30M,

Mbl

r

V

-

YBellH'IeHHH

npeneay Vco=d

Ae 21ti(vt-apd) [e21timda

n

1) H Mbl

TO HCKOMoe .n.HcnepCHOHHoe CO

(8.9) B (8.8), nOJIy'laeM

+ Yp_m- 2yp=

m=

n03BOJISIeT

(8 9)

'

aup

CYMMa, Bxo.n.S1UJ.aSl B

YpaBHeHHe .nBHlKe­

nonysaea

'1HCJIO, a. 'I -

cnynae

Mrc: 2v2

V-

-

BOJIHOBoe

B

.n.HTCSI K O.n.HOMy '1JIeHy (c

anne

A 21ti (>t-ax p) _

CJiyqaH B3aHMo.neHcTBHH TOJ

paccxor­

=F p = ~ Vm(yp+m+Yp_m-2yp)' m>o

HCKaTb

lp+m

a

CMe*HblMH qaCTlIIJ,aMH. nOJiHOe

BHyTpH

~"

d 2 yp Mdt2 .

r ne

CHOBa

HeOrpaHH'IeHHol\

TaKHM 06pa30M, .n.JISI

13 y.n.eM

'ITO,

BOJIHO

HMeeT

CSlUJ.HXCSI K pacnpOCTpaneHHIO BOJIH B 06JIaCTH COe.ll,HHeHHSI

cpenu

cnenyer,

YHKUHel\

M~2 ~

+ -})=

npe.n.CTaBJIeHHH 0 nOBepXHOCTHblX CHJIaX, COBeprneHHO

HenpHMeHHMbl

peurerox,

lId,

pHO.n.OM, paBHblM

I1pe.n.eJIeHHSI 3THX CHJI MelK.n.y '1aCTHuaMH. Haxo.n.S1UJ.HMHCSI B rpa­ UJ.HecSi

c

m>O

60JIbrnHX

rpaHH'IHble yCJIOBHSI YlKe He MoryT

CyMMOl\

~ V "m (1 21 ~

=

rpaHH'IHblX yCJIOBHl\ .n.JISI

B3aHMo.n.el\CTBYIOT

y.n.OBJIeTBOpSiIOT

.n.llHH

.ri»: V AT;

npHllIJIH

K

pesym

6a.n.eH-naY3JIJIa [CM. OpMyJIy (2.1)].

48

FAa8a 4T06bI

YCTaHoBlfTb

CBHab

3

Me)f(.ll,Y

TaMH 11 nponenanuuun HblOTOHOM CIWPOCTI1

aayxa

ynpyrocrn

TaKOBO,

6bITb

naornocrea COBnaJlaJIO

B soanyxe.

.ll,HCKpeTHoti 4T06bl

4aCTlfU

(T. e . .ll,JUI

4aCTlfU,aMH

nenpepuanoa

O'leHb

~OJIbWHX

crpyrcrypu) OHO

MO.ll,YJIH ynpyrocru KaK

crpyxrypu HaTHlKeHHe Me)f(JlY nByMH

paCCMaTpHBaTb

KaK

CHJlY,

Me)f(JlY HlfMH; TaKlfM 06paaOM, HaTHlK~HHe MelKJlY

I

IVlSI

HenpepblBHyro

li

(p

+ 1)- fl

cpeny,

CKOpOCTb

\

CKOpOCTb

pacnpOCTpaHeHHll

K npe.ll,eJIbHOMY ropoe

OHa

3Ha'leHlilO,

npliHHMaeT

BJIeHHli

(p

+ 1)- ti

H,

1)- fl

(p -

U" (d) [Ypil

C

npyroa CTOpOHbl,

MelKJlY

p-fl

li

(p

p-IO

I1CXO.ll,H aa

4acTlfuy co CTO­

(9.1);

CO CKOpOCTblO,

7t 2V2M

B

""ljiI

Iloaaras a =

COOTHOUleHlffl,

npene.m.aox

nOJlY4aeM aupaacenae nJiH

sin read

(9.3)

CJlY4ae

60JlbUlHX

nJIOTHOCTefl

4aCTlfU ace HaUlli onpeneaeaas COBnaJlalOT C 06b14Hb1MH. Y4HTbIBaH

supaxeaae

.l(JlH MOJlYJlH ynpyrocrn (9.3), a raxsce

Y4HTbIBaH, 'ITO CpeJlHHH nJIOTIlOCTb JlJlH JlaHHOfl CHCTeMbl

aupaxeaae

(9.2a) JlJlH <pa30Bofl

ynpyrocrn €

ar paer

ClKHMaeMOCTH,

a

K03tjJ
sin 2 tuid,

p = M/d,

POJIb

BeJlH4HHbI, 06paTHofl

e.

U"

aHaJIOrH4Ha

=

k

(9.3a)

BeJlli4HHJ

ex

+ l~,

nOJIy4aeM

sin nad ch re~d

+ i cos re

TaKHM 06paaOM, .llJ151 4aCTOT, COOTBeTCT nponycKaHlill, .ll,eflCTBliTeJlbHaH '1aCTb a pasna rare nOlIY43eM

+

2read= ---+- rr 2rei~d sin read = --t- ch re~d.

=

I ~ IUtI ~ = -; V M ch re~d.

(9.3a) nOJlHOCTblO coananaer C epOpMyJlO:t Hsio­

HaOTepMH4eCKOfl

=

IloCKOJlbKy 3TO BblpalKeHlie .llOJllKHO 6 He06xo.ll,IiMO nOJIOlKHTb cos nod = O. O

OTKY.ll,a

Vw=V~ opMyJla

TO

CKOpOCTH MOlKHO npencraaa'rs

B Bline

TOHa; MOJlyJlb

r UtiM'

H JIeBall 4aCTb 3Toro paBeHcTBa llBJlHeTCH .ll, 'lHHOfl, 60JIbIUetl eJlHHHUbI.

Yp

s =dU" (d). 4TO

1~

KOMnJIeKCHbI, TaK KaK

HaMeHeHlie paccroanns

MO.ll,YJlH ynpyr ocrn

3aMeTlfM,

npe.lleJlbHofl. -

+ l)-fl 4aCTlfuaMli onpe.aenaerca COOTHoweHHeM

3THX

0,635 V00

pasnoa

JlJISI n peJleJlbHOfl 4aCTOTbi nonaras a = 1/2d, HaXO.ll,HM

'uJlll sacro-, MeHbwliX npe.ll,e.lbHotl, CO qeHIiH BOJIHOBblX '1liCelI a JletiCTBHTelIbHb IUHX

2y).

OTHOCHTeJlbHOe

(2V A= 2d. B 3

ve=-;V

'1aCTliU, panna

Yp+l d

npa

,

+ Yp-l -

YMeH

pasnouy

Bu pa)f(eHHe

U" (d) [Yp+l - Yp). nOCKOJIbKY cYMMa CHJl, .ll,eacr BYlOlU,lf X aa

YMeHb

TaKlKe

6blJIO yxaaaso, HMeeT MeCTO HeOnHo3Ha paccMaTpliBaTb Jlli60 KaK 6erylU,YlO B TOM

4acTHU,aMli 6yJleT paBHo BeJIli4HHe

POHbl

pacnpO

C

H He aaaucar OT .ll,JIHHbl BOJIHbl.

JlCflCTBYIOWYIO

p-fl

.ll,JIHH BOJIH, MHoro 60JIbWHX T. e. xorna peweTKy MOlKH

xacrnnaaa.

JlOJI)f(HO

HaTSlll<eHliH K OTHOCHTCJIbHOMY ynJIli­

nacxpernoa

MOlKHO

onpene arm, Mony Jib

onpeneaenne

CJIy4ae

onpeneaeuaea

HOpM a.n.noro

nemno. B CJIy4ae

3TO

npene.n.aox

C 06bl4HbiM

oraouienas

He06xo.ll,HMO

CHCTeMbI.

B

no,ly'leHHbIMlf peay.n.r a­

pacseravn no onpeneaeaaio

MaTeMaTU
~

BeJIH4liH3 HOCHT H33B3Hlie nOCTOHHHOfl npe.llCTaBJllllOUI,lie H3MeHeHlie ~ KaK
ssre, 4

3aK. 201.

50 'laCTelt

B0J1110BOrO

a= ~

'lHCJ1a

+ i~

B

TOTbl 'I npHBe,lleHO na ~Hr. 8. llJ1l1 'laCTOT, Jle)){alUHX B aurepaaae ,lleJlbHOlt 'laCToTolt 66J1bWHX

3Ha'lCHHe

'IC'

npe,lleJlbHolt,

nocTOllHHoe

a

Me)){,lly HyJ1eM

-+- 1/2d.

a

'laCTb

rorna

KaK

'laCTOTbI ,llOBOJlbHO 6blCTpO B03paCTaeT.

30M,

nOJ1y'laeM,

'lTO

,llJ111

H npe­

'laCTOT,

66J1bWHX

a ocpa­

'laCTb

TaKHM

~

1

2 _

'I

1t~M

-

r.ae

nocnenasa

HHlIM

m,

H3 HHX K0J1e6J1IOTClI B

npOTHBo~a3e.

a

+ i~

=

-+- (1 /2d)

nO,llCTaBHTb B Bblpa)){eHHlI ,llJ111 YP H Yp+l' YKa3aHHbllt auuie xapakrep K0J1e6aHHj;\ ,llJ111 cnynaa 'I

HJlJ1IOCTpHpyeTclI pue

B311Tbl

H3

KPHBbIMH, H306pa)){eHlIblMH I1a pa60TbI

+

>

'Ic

~Hr. 11, KOTO~

KeJ1bBHHa, H3y'lHDWerO 6YI{BaJ1bHO ace

to.

Kax

. nm {f 0, -= 2 l -+- 1,

,llJ1HHbl H,

BOJ1HbI

cooTBeTcTByeT

Ha060poT,

Ka)){,llolt

TOJ1bKO

'laCTOTe

O,llHO

3Ha'leHHe

COOTBeTcTByeT

,lleltcTBylOT CHJlbl

Tenepb,

'lTO

na

co CTOpOHbl scex

ua paCCTOllHHe, He 60J1bWeC 'leM

~" 1 ~ Vmsin21Camd= 2'

=

o<m
3aMeTHTb,

a

HHIO

V2

=

CD

,llJ1HH

,,~

a»

=

TaKHM

_1_

1t~M

cTpeMlIlUHXClI

~

~ ~

m

a2

o<m
06pa30M,

K

V" sin~ 1tamd

~

2d.

'lTO

,llJlHHbl

BOJ1HbI

m

a

06lUeMy COOTHoweHH 'laCl'OTbI

Ka)){,llOMY

OT BOJIHO

'I

snaxe-

COOTBeTCTByeTl'OJlbKO

O,llHO ananenae BHCHMO

OT

'laCTOTbI 'I neaa­

BeJlH'lHHbl

06J1aCTH

npyroa CTOpOHbl, TaK KaK

2rr:amd MO)){eT

BJleH

B

aane

neuu

m

no cos

nosarensno, xacroru sane

" Vm(l--cos2rr:amd)

(10. t)

6eCKOHe'lHOCTH, nOJ1y'laeM

=

d~ M

~

~

V"

2

mm .

(10 2) .

CTe­

2rr:ad. TO, BbI paxenne

cne­

MO)){HO

npencraaart, CTeneHH

I

2d

,llJlll

L

B no

cos

crayer

MbI

nOJlY'lHJlH.

-CTaBJIlIeT

anasenne-

xpasaa 'I

nyto,

CKOpOCTH

paBHO

Ilpn

pacnpOCTpaHeHHlI

yMeHb­

MeHlIeTClI.

a,

BOJIHOBOe

'lH

~YHKUHIO

MHOr03Ha'lHYIO

H306pa)){eHHolt na ~H

crpnpyercs KpHBOlt,

Ilpn

'lTO, B TO BpeM5I

~YHKUHelt

cocoa

9TOM 'laCTOTa He 06113aTeJlbHO no 3Ha'leHHlI

(a)

H

aa

HMeeT

xonuax

OCHOBH

na r panauax rOpH3

B 'leM MO)){HO HenOCpe,llCTBeHHO y6e,l

'leHHe npoaasonnon

Kpoxe

xoropaa 3aBHCHT OT napaMeTpOB, onpe­ CKOpOCTb

6blTb npencra­

nOJlHHOMa

nOJlHHOMa

0,llH03Ha'lHolt

,lleJ1l1eMbIX CTpyl{Typolt pacCMaTpHBaeMolt CHCTeMbI. weHHH

eCJlH

I1TaK,

xacrnuy

O<m
npe,lleJ1bHOe

HeKOTopOj;\ nOCTOllHHOj;\,

s

Me)){,lly' sacroroa H

naanyio

o<m
BOJ1H.

K

3aBHCHMOCTb

MaJlbHOrO H ,llJ111

m

y,llaJ1eHHbIX OT nee

Kml{,llYIO

L 'laCTHU, Ld. Torna

O,llHO ee

BOJ1HOBbIM 'lHCJ10M HMeeT MeCTO COOTHoweHHe

rr:2'12M

eCJlH

2rr:ad. Orciona BbITeKaeT. 'lTO ,llJl51 ,l L 3Ha'leHHlt cos 2rr:ad H, KaK CJle,ll HHlt a. 3aKJlIO'leHHbIX B OCHOBHOM anrepaa

BnOJ1He onpe,lleJ1ellHOe 3Ha'leHHe ,llJ1HHbl BOJ1Hbl, npeBblwalOlU npe,lln0J10)){HM

Boaapauiaacs uieuy

C

Mbl BH,lleJ1H, B cJ1y'lae B3aHMO,lleltcTBHlI, HMeIOlUero MeCTO

'laCTOTbl

TOJlbKO

510

cos

YqeT JJ.aJIeKHX B3aHMoJJ.eHCTBHH

TOJ1bKO Me)){,lly CMe)){HblMH 'laCTHuaMH, Ka)I{,llOMY 3a,llaHHoMY 3Ha­ 'leHHIO

6epeTClI

B3aHMO,lleltcTBHlI.

acneKTbl ,llaHHolt np06J1eMbl.

§

1 1t 2M

O<

cynxa

npe,lleJ1bHolt,

llJlll roro 'lT06bI y6e­

3

TaK KaK

npH'leM xapaKTep ,llBH)KeHHlI 'laCTHU TaKOB, 'lTO mooue nne cocen­ ,llHTbClI B 9TOM, ,llOCTaTO'lHO 3Ha'leHHe BeJlH'lHHbl

V"· 2 «m m 510 2 -

~

aMnJlHTY,lla K0J1e6aHHlt 9KcnOHeHUHaJ1bHO saryxaer C paCCTOllHHeM. HHe

npenensnoay

onpeneasercs COOTH

O<m
npHHHMaeT

MHHMall

C pOCTOM Mbl

A= 2d (a = 1/2d).

,lleltcTBHTeJ1bHO; ,llJ111 'laCTOT,

,lleltcTBHTeJlbHall

3Ha'leHHe

4aCTOTa, cooraercrayiomas

uac­

3aBHCHMOCTH OT

TeJlbHo.

roro,

'lT06bl

,HeKoTopolt

dvida

cnenyer ace

3a.uaHHolt

B TO'lKaX

a=

yxaaa n., 'lTO C

3Ha'leHHlI 'laCTOTe,

BeJlH'lHHbI 6blJlH

,lle

H3 HHX MoryT 6bITb KOMnJleKCHbIMH, npyr

TaKHe 3Ha'leHHlI BOJIHOBOrO 'lHCJla ,llOJl)){HW

4*

52 KaK perneHlHl, cooTBeTcTBYIOIu,He BOJlHaM, TyxalOI1.I,HM

cnyuaio 1, o6cym.naBweMycSI B (8.7). l1MeeTcSI, onuaxo. npyras B03MomH v = 'I (a) «acaercs OCH a ('I = 0) npa nex

HOCHTCSI K

3KcnoHeHUHaJlbHO sa­

CTBOM

ocofenno saaoioe suase­ peuierox. aHaJlorH'IHblX TeM, KOTO­ § 8 (CM.
C paccToSlHHeM. aTO HMeeT

HHe B cJlyqae orpaHHqeHHblX

pue

6blJlH paccMOTpeHbl B

'ITO K rpaHH'IHolt qaCTHue

nacroroa

CHJla C

HeCKOJlbKO

npHJlOmeHa rapMOHHqeCKaSl B CHCTeMe 6y.n eT cYI1.l,eCTBOBaTb

'I. DPH 3TOM

THnOB

BOJIH

C

pa3Hol\

HHTeHCHBHOCTblO.

aaons peurenoi.

.nel\cTBHTeJlbHbl. 6y.nyT pacnpOCTpaHSlTbCSI KaK BOJlHbl. MHHMbl, OT

.nJlSl KOTOpblX

6y.nyT

rpauauu.

3HaqeHHSI

a

nanpaxe p, na
COOTBeT­

CTBYIOI1.I,HX 3TOl\ qaCTOTe. Te H3 BOJlH, .nJlSl KOTOpblX 3HaqeHHSI

a

Tor.na

KOMnJleKCHbl HJlH '1HCTO

HhlM

aaryxars no xepe y.naJleHHSI me B nenosxe. orpaHHqeHHol\ C oauoro B036y.nHTb TOJlbKO onny H3 3THX BOJlH, TO

3KcnOHeHUHaJlbHO ECJlH

KOHua, .He06xo.nHMO

ysce '1al\

B

ennaauax d), paauua, KaK yxasano B § 8 (cay­

6blJlO

2),

Al d =

cnenyer COOTBeTcTBylOI1.I,HM 06pa30M Bbl6paTb 3aKOH .nBHmeHHSI .nJlSl L nepBblX OT r panauu qaCTHU. B cnysae CMemHoro B3aHMO­ .nel\cTBHSI

sanars

rpaHHqHble

YCJlOBHSI

.nBHmeHHe TOJlbKO

HaH60Jlee

O.nHOl\

ECJlH me B3aHMo.nel\cTBHe npOCTHpaeTCSI paCCTOSlHHSI,

TO

3a.na'la

ua

YCJlomHSleTCSI

rpaamnue

ua

xapaxrepe


30

nOJly'leHa

.nel\cTBYIOI1.I,HX

ypaBHeHHSI .nJlSl

a

HMeeT .nJlSl MaJlblX

snaseuae:

nJleKCHbl

cooTBeTcTBylOT

OCTaJlbHble

(na

paccToSlHHe,

ua

o.nHOMepHol\

peuierxa

H

xae

TpexMepHol\

perneTKH 1).

naauon,

0

xoropoe

(L

~ 1)

.nH'IHa no

peureuae

paccmoenue »

paccMOTpeHHoe

i

OT­

1) nOBepXHOCTHbill CIIOll, B KOTOpOM 3aTylCalO T B036YlKneHHbie HlI C nOBepXHOCTH BOIIHbl, MOlKeT B03HHKaTb B TO'lKaX HapY1lleH nepHO­ .D.M'IHOCTM B CTpyKTypax. 3TO lIBJleHHe 6b1J1o H3yQeHO B pa60Tax n. E. KpacHy1llKHHa H np, [CM. n. E. K pac H y 1ll K H H, B. B. n 0­ T e M K H H, BeCTHHK MfY, N2 10, 33 (1949)]. OHO lIBJllIeTClI aHaJlOrOM JlOKaJlbHblX ypoBHell B KBaHTOBOll MexaHHKe, MCCJle.D.OBaHHbIX VI. E. TaM­ MOM [VI. E. TaM M, Phys. Zs. d. Sow. Union, N2 6, 734 (1932)], a 3aTeM A. A. COKOJlOBblM [A. A. Co K 0 JI 0 B, ZS. f. Phys., 90, N2 7-8, 520 (1934)] B CB1I3H C HaJlMQHeM Hapy1lleHHlI nepMO.D.H'IHOCTH

t,

pe1lleTKM TBep.D.0ro

Te~a. - npUM. pea

a

l/d,

C nepHo.nOM

nOJly'laeM

V" (md) = -

41t 2Md

f

F2 (a)

c

-1/2d

3TO

B cay­

anecs


+1/2d

perneHHl\ KOM-,

pacnpocTpaHSleTcSI

C'IHTaSl

xapaxrepuayioi

CTBHSI Mem.ny '1aCTHuaMH; y'lHTblDaSl, 'ITO

TaKOM

noeepxnocmnw» CJlOeM B cny­

Bee

HaltTH BeJlH'IHHbl,

qaCTOT e.nHHCTBeHHoe .nel\­ me

B03MYI1.l,eHHe, 6y.neM Ha3blBaTb npeiJeJlbflbtM sae

TeopHIO PSl.nOB '-l>ypbe, MomHO,

pasuas HyJIIO.

npe.nnOJlOmeHHH

B03MYlueHHSlM, 3KcnoHeHUHaJlbHO 3aTyxalOI1.I,HM no xepc y.naJleHHSI OT rpaHHu bl• BeJlHqHHY, xapax­ TepH3YlOI1.I,YlO

'1aCTOTbl

YCJlOBHSI

Mem.ny qaCTHuaMH CHJI, 'ITO

CTBHTeJlbHoe H

B

I

I

Cooruoureuuea (10.1) xaanpar onpenenaerca B 3aBHCHMO­ CTH OT a B BH.ne xonewnozo pasJIOmeHHSI B pan '-l>ypbe. Hcnonsays

qaCTHUbI.

OT rpaHHUbl qaCTHU

paCCTOSlHHH Ld). DOKOSlI1.l,HMCSI '1aCTlluaM COOTBeTcTByeT '1aCTOTa, KpHBaSl

.nOCTaTO'lHO

(nepaoa)

1 aldl'

Cfcj

.nOCTaTOqHO 60JlbrnHe

H

ncex L nepsux

He06xo.nHMO CTaBHTb .nJlSl

npOCTJ>I:

rpaHHqHol\

a = -+-

HyJlSl 3Ha'leHHSlX BOJlHOBOrO '1HCJla

perneTl
B xauecrse npaaepa B3aHMo.nel\cTBHSI,

nonuraexcs

npaaonaurero

K

nocros

nporrycsanaa CKOpOCTH pacnpocrpanenaa

'1=W\aj, '12

=

F2 (a)

=

W2a2

rne, paayaeercs, 3Ha'leHHSI BOJlHOBOrO '1HC

nor-o nnrepaana. COOTBeTCTBYIOUJ,He xpaaue

npnseneau

nO.ll,CTaBJHHI HHe

(10.5),

Bblpa)KeHHe

tPyHKUHH

.lI,JHl

P

B COOTHowe­

(a)

HaXO.ll,HM

J

a 2 COS (2'lta md) da

-1/2d

= (_l)m-1 2M W 2 2 2

m

v:

BeJll1lJHHa

'lI1Tb

npencraanse r

cocoa

BTOpylO

d

(10.6)

"

MO)KHO

CTpyKTypOlt,

CKOH

xoropan

x

'ITO paCCMOTpeHHblMH CHJlaMH B3a

TOJlbKO

=

V"(x) ,

tPYHKUlHO

.lI,I1CKpeTHo~

C +1/2<1

V"m =-4'lt2MdW2

3Hall

TaKHM

06pa30M

MeXaHH'leCKHlt

tP

C nOCTOllHHOU CKOpOCTblO pacnpOCTpaHeH COOTBeTcTByIO!lJ.HX nOJlOce nporrycKaHHSI.

AHaJlOrH'lHbIU MeTO.ll, MO)KeT 6bITb HCn

np0l13BO.ll,HYIO

v

HHSI tPHJlbTPOB BepXHHX 'laCTOT, a TaK)Ke COBblX tP l1J1bTPOB, I1MeIO!lJ.I1X O.ll,HY I1J1H H nycKaHHSI.

[j

B

paccxorpenuoa asnue

nocr

cJly'lae

npocTpaHeHHSI JlerKO nOJlY'lI1Tb Bblpa)KeHHe BOlt

lJaCTOTbl

ypbe.

"\V

'V=": _

I

I

I

-(j

.I !i

2

- 2d

w = 4'1t

r ne

w

=

2'1t'l,

YlJHTblBaJI COOTHoweHHe

a

\V •

w2 ,

2 2 '1

4 = M

~

U"m 2 (1 ­

B

BH.l

(10.6),

cos

H

2'ltamd) =

m

-

~

4 (_1)"'-1 W2 (1­ cos 2'ltamd) = m2d2.

~ HI

4W

2

[

= (j2 (1 ­

a

·uI
Hr.

32.

MOH OT

<1HeprHH

KOTOpblMH

md,

B .lI,HCKpeTHOM

pane

pasuux

paCCTOllHl1l1X

PblBHYIO

tPYHKUI1lO

2

.lI,ByX

T. e.

lJaCTHu,.

paCCTOSlHl1e Me)K.lI,y

onpeneaeaayio

TO'leK, paCnOJlQ)KeHHbIX

npyra.

OT

V " (X ) = ­ 2M W2 x2

B.lI,OJlb

TOJlbKO

OCI1

PaCCMOTpl1M

X

na

4~~[n2 d2 12 _

--

nx

(10.7)

4'1

(1.-

_ 4 + cos 4nad

2 'lta d+cos4nad _ 4

cos.

COs

9f

nocKoJlbKY

n2

nenpe­

cosT'

~

cos

1 1 1 ) 1 -4+9-16+'" -cos

V.

tPYHKU,I1IO,

d npyr

4W [ ( =-(j2

npa­

Kosq,qmu.HeHT

OB paneu

B3aHMO.ll,eUCTBHJI paBHO

1 4 (1 -­

2'ltad) ­

+

1

2d

YrJlOBOH

cos

1

12=1- 4 3aMeHSllI

reneps

+ 91 - 161 +

B pa3J10)KeHHI1

(10.8)

a = 'IjW, no.nyxaea Bblpa)KeHl1e .lI,J k 2 B pan ypbe na I1HTepBaJle ( - 'It,

BaSI, 'ITO

xoropas npauaaaer onpeneaeua V" (md). B3aHMO.ll,eUCTBHlI, I1HTerpl1pOBaTb,

TO'lIlO

tPYHKUI1IO lJTO

TaKl1e )Ke 3IJa'leHIHI B TO'lKaX,

4T06bl nOJlY'II1Tb

V" (x)

B03MomHO

HHTerpHpOBaHl1e MomeT

supaxenae

Heo6xo.ll,HMO B

CHJlY

ee

r ae

.lI,Jlll <1HeprHH

.lI,Bam.ll,b1

npo­

HenpepblBHOCTI1;

6bITb npose.aeno npH nouounr TaOJ1HU.

UI1I1

4n 2\12d 2 k 2 = 4'1t 2a2d2 = - - ­ = W2

n2 cos 2k cos 3k = 4 [12-cosk+-4---g

OrpaHH4HBUlICb

B

paccxorpenaux suuie pa3J10>KeHHlIX KO­ no.rysaex rrpH6J1H>KeHHOe peurenne .llJlll rrpH6J1HlKeHHall KpHBa51 6YlleT acnurusari,

He4HbIM 4HCJlOM 4J1CHOB, !PYHKUHH 'I

(a);

TaKa51

KOJle6aHH5I OKOJlO T04HOR KpHBOR.

§ 11.

paanocrn rrOTeHUHaJlOB BO scex ee T04K ypasneaae (11.3) rrOJlHOCTblO aHaJlOrH4HO

H

UJ,eMy

.llBH>KeHHe

B3aHMO.lleRCTBHlI, sacrauax» (CM.

3JIeKTpH'IeCKHH $HJIbTP HH>KHHX '1aCTOT

d 2Yn

npencraaaena cxexa 9J1eKTpHlJeCKOrO !pHJlbTpa HHlKHHX 4aCTOT, oopaaoaauuor-o H3 coe.annennux nocne.nona­ TeJlbHO annyxraanocren L H eMKOCTeR C. KaK YlKe YKaabIBaJlOCb, 6JIarOllapll HaJlHlJHIO eMKOCTeR C nonooaas CHCTeMa aanepxcaaaer

M dt 2

Ha !pHr. 33

~

L

1..

L

1..

l..

L

1000000 \ L

n

n-t

ollHoMepHo

xoropoe nepexoaar

"

= VI (Yn-l + Yn+l

B (11.3), eCJlH OTllOW

Vn .. z

ama+ip

,

Qn"2

n+Z

OK

/

PI'

2d11--­

..

n-l

B

sa I/LC. a Yn aa in.

\!O/II\ISq;«""l\\\Wi~~4\\"">Z\\t:~~~~,A A_"""...~

.....

4aCTHU

HMelOllJ,erO MeCTO TOJlb § 2 H 9), a HMeHHO

/~

A.

/'

~

t'

E:=~I"

na KaK

,

HH3K04aCTOTHble CHrHaJlbI

pacnpocrpaasn.ca. HaR.n:eM ypaBHCHH5I, onacuaaiouure noeenenae 9TOR CHCTeMbI. 0603HalJa51 sepea Qn H V n COOTBeTCTBeHHO aapsn eMKOCTH H rrpHJlOlKeHHYIO K HeR paanocrs rrOTeHUHaJIOB, a sepea in BeJlH4HHY TOKa, nporexaiouiero 4epe3 HHllYKTHBHOCTb, BKJlI04eH­ HYIO B uern, MelKllY noR H (n 1)-R eMKOCTblO, noaysaex. MorYT

YlJHTblBa5l, 4TO

Vn=Qn/C,

npyron

(111)

•


MexaHH4eCKoro CTBeHHO 06pa30M,

CTOPOl/bI,

-.

",}

see

rrOJlY4aeM,

' ..

r. 34.

peayas rarsr,

!pHJlbTpa

nepenecrn na

H

HHlKHHX

no

lJaCTO

paCCMaTpHBaeMblR

4aCTOTa

4TO

lIBJIl

!PYHKuHeR BOJlHOBOrO lJHCJla, a CKOpOCTb

dQn . tn -In+! = /it.

(11.2)

OTCIO,lI.a, .llH!pepepeHUHpYll (11.1), HaXO.llHM

( + in+l ­

1 (d Qn ­ 1 d Qn ) 1 L d?l.. dt 2 = C d-t­ - ----cJI = C i n _ 1 8TO ypanaenne noaaoaser onpeneaars icarourero B JlHHHH, a raioxe BeJIH4HHbl

"

"

CJIe.llOBaTeJlbHO,

Ld1n_V - Vn =Q"-1_g~ dt n-l C C . C

~

- 2d I

2 ) in·

(11.3)

BeJIH4HHY TOKa,

npore­

9JIeKTpH4ecKoro

aapaaa

llJlHHe

BOJIHbr,

CTpeMSlllJ,eRcH

K

6ecKoHe4

panna d/V LC, r-ae d ­ paccroauae Me>K.n caropaun. 3.lleCb TaK>Ke HMeeTClI npeae.ns n BOJlHbl. 4aCTOTbl KOTOpblX npessnuan­ 3H UHaJlbHO aaryxaior C paCCTOHHl1eM. Ee au \f3 BhIpalKeHIUI (9.4); OHO pasno 1

'1c

=

1t

Y LC

•

Cnenyer yxaaars, 'ITO paccaorpeauun anecs qJHJlbTp, OnH­ CbIBaeMbIl't ypaanennea (11.3), He conepxorr conpornaaenaa: ase.neuae HX HeCKOJlbKO HaMeHHeT csoncraa JlHHHH: nOHBJlSIeTCH cnaooe saryxaaae BOJlH, lJaCTOTbl KOTOpbIX JlemaT BHyTpH nOJlOCbI nponycKaHHH. 'ITO CBHaaHO C HaJlHlJHeM aKTHBHWX nOTepb 9Hep­ rHH. H ncveaaer pesxaa rpauuua Mem.ny nonocon HenponYCKaHHH H nonocon nponycxanas, T. e. nOHBJlHeTCH HeKOTopaH aeonpe­ .nCJleHHOCTb B Bw60pe npe.neJlbHol't lJaCTOTW. B 06J1aCTH lJaCTOT,

npeauuraiourax xparanecxyio. aaryxanne peaxo aoapacraer

C

in-2~

in ­

i n +2­

~in

~in+l

'la­

CTOTOl't. Ha qJHr. 34 npnaeaeuu xpaaue, aaoopaacaiounre HaMe­ HeHHe .nel'tcTBHTeJlbHol't H MHHMOl't lJaCTel't BOJlHOBOrO lJHCJla KaK qJyHKUHH lJaCTOTb\. Bonee nonpoeuo 9TH sonpocu 6y.nyT pac­ CMOTpeHbI B rn. 9. Haoopaxenuas na qJHr. 33 9J1eKTpHlJeCKaH JlHHHH nOJlHOCTblO 9KBI1BaJleHTHa nsoaaon JlHHHI1. cxeua KOTOpOl't npasenena aa

-­ i tr- 2

~in_l



If

r. 36.

COCTOHT aa HeCKOJlbKHX paanauuux serse np0JJ,OJlbHbIM H none pelJHbIM KOJle6aHHHM. meHHe HMeeT MeCTO, eCJlH paccaorpers 6e

L

---E--E- -1-- -I-­ L

qJHJlbTpaMI1. S § 5 B CBHaH C HaylJeHHeM COCTOHllJ,elt aa caooonnux ~acTHu. 6bIJlO npencraanmouian lJaCToTy B ~aaBHCHMOCTH

L

L

II n



If

r. 35.

qJHr. 35 11 KOTopaH MomeT 6bITb nonysena .no6aBJleHl1eM K yace paccaorpeuuon npocrenuren JlHHHH ee oroepaxeana, CHM MeT­ pHlJHOrO OTHOCHTeJlbHO aeMJlH. Ilsonnaa JlI1HHH casra MomeT 6bITb cseneua K 60Jlee npocron CHCTeMe, COCTOHw,ett aa I1H.nyKTHB­ aocren, I1MelOw,HX ry me BeJlHlJHHY, 11 ansoe MeHbWHX no BeJlH­ lJHHe euxocrea (cfJHr. 36)

L' = L.

~

C' =

(11.6)

.

Eha JlHHI1H o6J1a.naeT TOlJHO TaKHMH me cBottcTBaMH. KaK H npocremuaa JlHHHH, H xapaxrepnayercs, KaK yxce 6blJlO yxaaauo B § 3, BeJlHlJHHaMH d

V oo

= -y LC

d -Y2L'C' •

1

Vo

=

1t

-y LC

1t

-Y2L'C'

(11. 7)

B aaxmoueuae OTMeTHM. 'ITO MomHO npOBeCTH euie 60Jlee r Jly60KylO aaanormo Mem.ny 9J1eKTpHlJeCKI1MH H MeXaHHlJeCKHMI1

-

iZ

n-I

-------. i3 n

13 /I-I



If

r. 37.

npe.nCTaBJlHIOllJ,YIO co6ott rpexqiaaayio JlI1HH npHBe.neHa na qJHr. 37. B 3TOM cnysae, Ka meM, cHMMeTpHlJHbie COCTaBJllIlOllJ,He nex'ropa aHaJlOrHlJHYIO • ynpyrux npo.nOJlbHbIM H no xoropue MoryT pacnpocrpanarsca ano.n, ue I1TaK, nycn, 6eCKOHelJHaH rpexoaauas npe.nnOJlaraeTCH orcyrcrsue aKTHBHblX no ~TO noxaaauo aa qJHr. 37, aa o.nHHaK06b

epase KOTOphlX BKRIO'leHa

HeKOTOpa51 HHllYKTHBHOCTh

cnenyiouiea npa no aoum

5I'IettKa CB5ISaHa co

CHCTeMhi

L. Kaxcnaa

Ilepsoe

ypaaueaan

HS

C'. r ae C -eMKOCTh Me)/(llY 01lHOtt HS epas H SeMJIett, C' - eMKOCTh Me)/(llY llByM5I epasaMH 01lHOtt H roa )/(e 5I'IettKH. 060SHa'lHM xepea iI, n. i 2 • n H ia, n BeJIH'IHHhI TOKa. npore­ xarouiero B paSJIH'IHhIX epasax n-tt 5I'lettKH. Torna COOTBeT­

M

'1 h

a

creyiouiee ypasnenae

-

1

C' d 2

•

L C iit 2 (£2. n

'1a.

+ ia, n) -

pue

Moryr

1

Bsenex

1. 2. 3.

cHMMeTpH'IHhIC COCTaBJl5IIOIHHe TOKa lIyJIeBott.

npsxoa

H 06parHott nOCJIC1l0BarCJIhHocTett

1, a

M

BeTBett.-

KapTHHa.

rrOJIHOCThIO

rr01l06;1a51

uenovxa,

+ ai2, n + a a, n' Ii, n = ii, n + a2i n + ai3, n'

CB0601lHhlX

a pannu 2

coorBCTCTBelIHO

rpex

epHr.

Ilepexons reneps K HOBhIM nepexennua, ypaaueuan rana (11.8) cacreuy

23).

ny nesoa

Ky6H'IlIh1M

KOPll5lM

2

-Ch.n-1+C Ih,n +L

d 2/ h , n _ ([['J

1

(C') d

2

-7: la. n-l+ C la,n +L 1 + 3 C

(fl2-

COCTaBJI5IIOll(He

- eli, n-l Onpenennx.

2

1

1

Ch

n+l = O.

Ae21ti (.t-kn)

" (r.ne

' 1 - '1aCTOTa.

a

k-

BeJIH'IHlla,

o

nonepenaux.

'Ii) -

KpOMe

yBeJIH'IeHHH

B

roro,

llJIHHhI

saBHCHMOCTH

nyJIeBott

'ITO

BOJlHhI

OT

Ha51 AJIHHe BOJIHhI) MoryT pacnpOCTpall5lThC5I B CHCTeMe,

paCCMaTpHBaeM

3TH CKOpOCTH 5IBJl5IIOTC5I

a

pacrrpOCTpalleHH5I nponoasnux H nonepenn cpe.ll.e. • PaCCMaTpHBa51 5I'IeeK

oriparuo npolIopUHOHaJIh­

roro,

rrOCJIe.ll.OBaTeJIhHOCTH HJIH npsxoa

ormcanaoa,

r

nps­

MOtt H oopamoa nocaenosarens­

nocren ('1a.

.lI.OBaTeJIhHOcTett;

rr pH KaKMX YCJIOBH5IX BOJlHhI anna

t

('1 h )

np 01l0JIhHhI M

CKOpOCTh pacnpocrpaneuna crpe­

-C la, n+l =0.

dd/~f. n + eli, n + L (C') 1+ 3 c 2

0

TOKa

MMTC5I K rrOCT05lHHOMY npeneny, HMeIOll(eMY

(11.9) 1

HS

§ 5.

(CM.

rrOCJle1l0BaTeJlhHOCTH

cooTBeTCTByIOT

npa

7; Ih, n+l = O.

2 / a, n

nonyseua

cocrosurejt

'1aCTHU

YKaiKeM,

1

6hIJIa

BOJIHaM. COCTaBJI5IIOll(I1e lKe noaysaex HS rpex

v

CJIH5I­

paSJIM'lHhIX

llJl51

MS enanauu.

1

peayasrare

JIy'lHBwettc5I B HH5I llBYX

Ih,n=i1, n+ i2. n+ia. n· 2,

fAe

=

KOTOpa51

2i

C

HM)/(HHX '1aCTOT. l{pHBa51 'I 'I (k) COCTOHT H M cnaoenuoa BerBH ('Ia. 'Ii). rro­

ron,

la, n = iI, n

CL 1 +3

Ha epHr. 38 aaocpaxena saBHCHMOCTh Paccxorpennaa suure CHCTeMa otinanaer

C ts, n+l = O. (11.8)

eute .lI.ByM5I ypaDHeHH5IMH. xoro­ noaysenu U3 (11.8) UHKJIH'IeCKott nepecra­

6hITh

1 (C'

= -1 {

'Ii

11;

0110 1l0JI)/(1I0 6hITh 1l0nOJIHeHO HOBKOtt HlIlleKCOB

1 ~/-1-. k CL SIn 1

nBa llpyrMx ypaDHeHH5I rrpHB01l5lT K aHaJIO

(C')d?ll. n

2

(11.9)

= -;tV

llJI5I TOKOB 6YlleT HMeTh BHll

7;-i 1• n- 1 +Ci2, n+ L 1 +2 C { l ] 2 -

-

CHCTeMhl

exxocren C

llBYX

HO

THnOB,

HOCr5lMH

L

CTeMaMH

exxocren

6hlJIO

6hl

rpexrpaanyio

COCT05lll(yIO

H

L'

HS

JIHHHIO.

rr0

rrepHOllH'leC

xapaxrepaayexux paS

H

CB5ISallllhIX

MelK.lI.Y co

(OllHHaKOBhIMH

nOJlY'lHTb

B1l0Jlh

3JIeKTpH'IecKyIO

cac

anaaorox peiuerxn rana Bopna, cocroauien Bo nee neransnsre CBe.lleHHSI MO>KHO aa J1bHOlt paoore Ilaponn 1).

H3

COpTOB.

4aCTHU .llBYX

HaltTH

B

OpHrH­

npyrolt nYTb YCTaHOBJleHHSI aHaJlOrHH K conOCTaBJleHH~ BeJlH4HH

1

U" AHa~orHH Mem~y 9~eKTpHqeCKHMH

§ 12. H

B

MeXaHHqeCKHMH CHCTeMaMH

B npenunyiuea naparparpe onacuaaioume pacnpocrpanenne

MbI

BH.lleJlH,

4TO

ypaBHeHHSI,

3J1eKTpH4ecKHx BOJIH B
raxon >Ke BH.ll, 4TO H ypaane­ HHSI, onacuaaiourae pacnpocrpaaenne ynpyrux BOJIH B MexaHH­ 4eCKOM Ke runa. 3TO naer OCHOBaHHe npen­

HH>KHHX

4aCTOT, HMelor

noaar-ari.,

4TO

Me>K.llY

T04HO

3J1eKTpH4ecKHMH

C­ Lp

4aCTHOCTH,

HMeHHO

eCJlH

MbI

nOCTpoHTb

uiyio

TaKHMH

XOTHM >Ke

npocrpaneaaa KOBblX,

Ka>K.llaSl

3TOT

nyrs

C

c

(B

OTHow

peurerxa, na pasnux paccro

4TO

H3

H

KOTOpbIX

B3aHMO

L2

L2

..... C

......... C

L1

C

3J1CKTPH4C

caortcraaaa

BOJlH),

HaXO.llSllllHXCSI

4aCTHU,

M

H

H MexaHH4eCKHMH CH­

CTeMaMH HMeeTCSI 60Jlee r J1y6oKaSl aHaJlOrHSI. neTaJlbHoMY

csauiena HOCTb

H3Y4eHI110

9;

r J1.

TeM

He

paCCMOTpeTb

nocrarounon aHaJlOrHH

Me>Kny

4T06bI

C

3J1eKTpH4eCKHMH

MbI

orxexan«. J1HHHfi TaKY~

y)Ke

3J1eKTpH4ecKHx

no­

B03MO>K­

T04HOl;TblO,

BnOJlHe

c

IVLC

1

po.n.,

arpaer

KaK neJlH4HHa

.

»m-M-IJ'J'\""""''''Ik\-I0',~;%\.w,,(L~

sepru

OCHOBIIble

J1HHHSlMH H MexamI4eCKHMH

BeJlI14HHa >Ke

6y.lleT

HMeeTCSI

yCTaHoBHTb

peurerxaau. KaK

J1HHHlt

cennac

y>Ke

sonpocu

3TH

roro,

.llJlSl

3J1CKTpHLJeCKHX MeHee

.llJlSl

11-2

n-I

n

t

Vn- z

Vn-I

Vn

V

Qn-2

£In-I

Qn

Q

V U" I M


39.

Hr.

.llJlSl MexaHH4eCKHX peweToK. KJlaCCH4ecKHlt Me>K.llY

n03BOJlSl~llIHlt yCTaHoBHTb ananormo

MeTO.ll,

3J1eKTpOMarHHTHbIMI1

H

MexaHH4eCKHMH

COCTOHT B YCTaIIoBJleHHH B3aHMHoro HHTHOlt

H

3J1eKTpH4ecKolt

KHHeTH4ecKotl

H

snepruea,

norenunansnon

npHBO.llHT K TOMy,

4TO

COOTBeTCTBHSI C

onnoa

snepruen ­

C

S1BJleHHSlMH, Me>K.llY

Mar­

OCTaJlbHblMH 4aCTHLIaMH peurerxa. PaCCMaT

uiyio

3J1eKTpH4eCKY~ J1HHH~, nocrpoeun

J1erKO y6e.llHTbCSI,

CTOPOHbI,~'

HeHHe

npyr on: /OH

HaKoBbltl

BH.ll.

Ka>K.llbltl

H3

He06xo.llHMO conOCTaBHTb BeJlH4HHW

BOJIH

B

OnHaKO

3TO

C

U"

COOTBeTCTBHe

L

H

He

C

(12.1)

M.

S1BJlSlcTCSI

e.llHHCTBeHHO B03­

MO>KHbIM, H MO>KHO npencramm, ce6e .llpyry~, 60Jlee Y.ll06HY~ B pane nnoro

cnyvaen

B03MO>KHOCTb.

COOTBeTCTBHSI

.llHKTyeTcSI

06bI4HO

>Ke

BbI60p roro HJlH

co06pa>KeHHSlMH

crpyxrypoa paccaarpnsaeaon CHCTeMbI.

1) CM, M, Parodi, Camp!' Rend" 216,175(1948),

Y.ll06CTBa

H

J1HHHSI

KOH.lleHCaTOpoB

L[

KOHDeHcaTopaMH,

CJle.llY~IllHMH

onHCbI

ypaBHeHHSI,

.llBYX cncrexax,

Don06HaSl

HH.llYKTHBHocTetl

C1

4TO

3THX

HMe

npencr

C

@MKOCTbIO

C .llByMSI CMe>KHbIMH (.

npa

noxoum

(. BTOPblMH

HH.llYKTHB

COCe.llSlMH"),

pa

H cnpasa OT Hero, H Boo6111,e, npa noxoun C .llByMSI npOCTOTbI C(npaHSISI

.p-MH

COCenSlMH"

H306pa>KeHbI .llJlSl

cnpana

TOJlbKO

KOH.lleHCaTOpoB

B npe.llbI.llYlllHX

aanasax,

TOKa,

+

nporexarouiern

TaKY~

0603Ha'lHM

4epe3

.lleHCaropa K (k p)-My, noM KOH.lleHCaTopOM.

H cae

HH.llyKT

a

>Ke

sepe

HH.llYKTHIlHOC sepea

Qn

-

Torna

d~n =

6yneM HMeTb

+in _ P I I , n + ... + in_I, nl­

[i n_ p, n

-

=~ (in-p. n p C npyron lJ,HaJlOB

[in,

Me)){ny

:t

LI

+ in, n+pl=

JlHH

CTaBHTb

nep

B

sane

aaxxnyroa

4epe3

CTeMbl· 6ynYT

(12.3)

06KJlanKaMH

4epe3

n-ro

pa3HOCTb nOTeH­

xounencaropa,

JI~rKO

Qn-l-Qn C

Vn

Qn-2- Qn C

• V n_ 2 - V n =

i n_ 2, n

Vn

i

JlHHHH,

eMKOCTb

na 3eMJlIO.

npacoenanen

K T04Ke,

(12.4),

H Y4HTbiBaSl

1

JlHHHIO, B TO spexs KaK POJlb unepmoro

9J1eMeHTa

CHCTeMax

nrparor T04e4Hble

B HHX

B 9THX COOTBeT­

MaCCbI

H

c

CJlY4ae
HblX

HaxonHM

L$)A

9J1eKTpH4eCKaSl

nOCJlenOBaTeJlbHO

OKOH4aTeJlbHO

neacaropon

~( d d . ) C----cJi2 = C ~ (H i n _ p, n -(j[ln, n+p =

Jletl

d 2 Qn

=

xoropue

~ Qn-p+ ~n+p-2Qn.

ypaaneuae,

eCJlH

Y4eCTb

(12.2),

COOTBeTCTBHe

(8.8),

non­

nOJlY4eHHbIM npn H3Y4e­

COCTOSlUl,etl H3 4aCTHlJ, paauoa

MaCCbl

M

C npo­

H3BOJlbHbiM 3aKOHOM B3aHMonetlcTBHSl. TaKHM 06pa30M, H306pa­

)){eHHaSl

aa

CBotlcTBaMH 4TO B


39

JlHHHSl

nOJl)){Ha

(B OTHoweHHH xapaxrepa

peiuerxa

H

aaKOHOM

H3- onHHaKOBblX

B3aHMonetlCTBHSl.

CooTBeTcTBHIO

3TH

06J1anaTb

TaKHMH

)){e

•

4aCTHlJ,

csortcraa

C

(12.2)

MO)){HO nars

H3

40)

OnHHM K

KOHlJ,OM

npyron,

npHcoenHHeHa

npH4eM

xoropue aar JlSlnHO

Kaxcnaa

MaCCbl

aa

raxaa npyacamca

K

onHotl


40

Macce, npe.a-

onpeaeaseuoay

HMeeM

nOJlOCOBotl

urynrapyrourae

HHX

ncerna

MO)){HO

CHCTeMa

HH

TaK KaK

C4HTaTb

COnep)){HT

C


o

3a3eMJleH

euie

H

npy

xpoxe TOJlbKO 4TO paCCMOTpeHHbIX, TO nJlS HJlH naoro COOTBeTCTBHSl Heo6xonHM cHcTeMbI I). 3aMeTHM,

reoxerpauecxyio HHTep­

9J1eMeHTaMH,

MO)){HO npencrasan, B BHne npy)){HHOK.

reneps

KOHneHcaTopbl,

CKOJlbKY

paccaorpeuu

MexaHH4ecKHtl
.)){e

paccMaTpHBaTbcSl KaK 4aCTHlJ,bI,

npOH3BOJlbHbIM

6bIJlH

nYCTb

Torna

pacnpocrpanemra BOJlH),

nperaumo.

a npyrHM -

pac­

xonneucaropu

4ecKoMy COOTBeTCTBHlO,

§ 8.

(
¢ H r. 40. aHH}J{HHX 4aCT }J{HHKa; (j HH}

HrpalOT POJlb 9J1eMeHTOB CBSl3H. 3Ta KapTH

HOCTblO coanaaaer C ypasneuuea 'HHH uen04KH,

CBSl3aHa C 3eM­

HHnyKTHBHOCTH,

MoryT reneps

crenu,

(12.5)

p

Eho

I(

4epe3

KOH­

CMaTpHBaTbCSl KaK T04KH CH­

p

6

ynpyrnx CHJI, CBSl3bIBalOUl,HX peurerxy, H 9J1eKTpH4ecKylO 3bIBalOUl,HX

JlHHHSl COCTOHT H3 coenHHeH­

(12.3)

nH
a

KOHneHCaTOpbi.

d. -V V _ Qn-p-Qn L Plitln-p, n - n-p- n - - - - C - - · OTClOna,

T

KOHneHcaTopbI,

llpyrHMH CJlOBaMH, 9Ta T04Ka 3peHHSl CBOnH

CTBeHHO

(12,4)

sac

onHHo4Hotl

nrpars

KOHelJ, KOTOpbIX 0603Ha4aSl

i n __ I , n=Vn- I ­

:t

L2

+ in. n+2 +

in. n-t-p)'

CTOpOHbl,

nOJlY4HTb

n+l

craanenu T04KaMH. 3J1eKTpH4ecKYIO xoaneacaropu WyHTHpylOT BblCOKYIO

I

neaennue

4TO

npoaoaaasre

aaecs

rpanauu: HeB03MO)){HO,

rpnsecxyio JlHHHIO, C onpeneJleHHotl,

aHa

nanpuxe

non4HHSlIOUl,YIOCSl 3anaHHotl B

60Jlee

ypaBH

06J1aCTH npo

HaJlbHotl aaBHCHMOCTblO Me)){ny xacroroa

vH

1) 06 aHaJlOrHH H KJlaCcH$HKaU.fJH CJlO}J{HbIX rpasecxax uerren CM. paoory: E. K pac H II

N~

12 (1949). - Itpu». pea.

5

3aK. 2CU.

n.

y

rorna KaK .nJlSl MeXaHH'IeCKotl peurerxn, KaK 6hlJlO noxasaao B § 10. raxas BOaMOJKHOCTh B npnnunne HMeeTCli. D pH'IHHa sroro aaxmosaerca B TOM, 'ITO, B TO BpeMli KaK C TO'lKH apenas MeXaHHKH DnOJlHe nonycrnxo, '1T06hl nexoropue aa npHHHMaJlH orpauareasnue aHa'leHHSI, HeBOaMOJKHO BeJlH'IHH

U;

nOCTpOHTh 9J1eKTpH'IecKylO JlHHHIO, COCTOlllUYIO aa HH.nyKTHB­ nocrea C orpnuarensnux K09epepHl.lHeHTOM CaMOHH.nyKl.lHH. O.nHaKO 9TH sonpocu MoryT 6hITh paapeuienu, eCJlH ysecrs BaaHMOHH.nYKl.lHIO MeJK.ny KOHTypaMH. Bnepaae 9TO 6hlJlO cnenano DHpCOM H aarex 60Jlee neraxsno paspaooraao JI. 5pHJlJlI09HOM I), paCCMOTpeBWHM BOnpOC 0 BOaMOJKHOCTH nOJlY'leHHlI 9J1eKTpWleCKOrO epHJlhTpa, xapaxrepnayexoro no­ CTOllHHOtl CKOpOCTblO paCnpOCTpaHeHHlI B nOJloce nponycKaHHSI epHJlhTpa. C npyron CTOpOHbl, cnenyer yxaaari., '1TO l TOJlbKO nO.n6Hpali Ha.nJleJKaIUHM 06paaOM K09epepHl.lHeHThl BaaHMOHH­ nyxuaa MeJK.ny KOHTypaMH, MOJKHO nOJlY'lHTb epHJlbTP HHJK­ HHX '1aCTOT, .netlCTBYIOIUHtl B JlHHHH aanepaoca BnJlOTb no '1aCTOT, oxens 6J1HaKHX K npenensnon. Do.no6HhIMH JlHHHlIMH cHa6JKeHbi npHMeHlIeMhle B nonaonnon axycraxe 2) coapexeanue KOMneH­

HOCTb

r-asa,

naxonauierocs

suyrpenuee naanenne raaa

B CHCTeMe, H

noM pesona

B

.nBHJKeHHe MaCCbl U rasa, aaKJlIO'IeHHorO rpycxax, npennoaaras, 'ITO B KaJK.nbltl M B peaOHaTopax Haxo.nHTCli B COCTOllHHH pa DPH 9THX YCJlOBHlIX HMeeM

Up

d;/ =(Pn -

dV n-l

Up ------cIt

=

(

Pnl-I)

Pn-I -

Pn

)

r.ne V n H Vn_1 - CKOpOCTH MaCCbl raaa, a xax, coe.nHHlIIOIUHX n-tl peaonarop C (n aonaropaua, a S - cexenne coe.nHHHTeJlbH

+

::~~:: n-J

n

'1

caropu B).


ITPI1MELJAHl1fl A. AKycTHQeCHHA aHaJlOr 9J1eHTpHQeCHHX H MeXaHHQeCHHX epHJlhTpOB HH}I{HHX QaCTOT MOJKHO npnsecrn aKyCTH'IecKHtl aaanor TOJlbKO 'ITO pac­ CMOTpeHHblX HaMH CHCTeM. Dpe.nnOJlOJKHM, tJTO HMeeTCli 6eCKOHetJHbitl pll.n onunaxoaux pesoaaropoa (epHr. 41) 061>eMOM V. 060aHatJHM xepes U 061>eM Tpy6KH, coenanmoutea nna cocenaax pesonaropa, tJepea p nJlOT­

I1aMeHeHHe KOJlHtJeCTBa raaa, conepaca HaTope, 3a spexs dt paBHO •

p (Vn ECJlH xepea

X 0603HatJHTb

Vn_l) S dt.

CJKHMaeMOCTh r

1 (v n - v n - l ) 5

X=-V

dPIi

H3MeHeHHe .naBJleHHlI B noM pea Teneps c nOMOlUhlO sroro COOTHoweHHlI

rne dPn -

1) L. B rill 0 u in, Proc. of Symposium on Large-scale Digital calculating machinery, Harvard Univ., 1948, p. 110. 2) CM., aanpavep, M. Par 0 d i, Principes de reception sous­ marine dlrtgee, Paris, 1946. 3) J1HHHH 3a.l\ep}KKH c xovneacauaca anepaue npnxeaeau Kans­ MaHOM B 1946 r. (CM. H. E. K a II man, Proc, I. R. E., No 9, 1946; CM. TaK}Ke "OplieMHHKH pa.l\HOJlOKaIlHOHHbIX cTaHIlHH", nepeson c anra., BbIll. 2, CTp. 74, 1I\3.1\-BO "COB. panao" , 1949). B CCCP nO.l\p06HbIH ananas TaKHX JIHHHH nan A. M. Tpaxruaaou (A. M. T P a x T Ma H, Pacxer 9JIeKTpH'IeCKHX ¢HJIbTPOB Ii JlHHHH aanepncxn c yserox no­ reps B 3JIeMeHTax, KaH.l\H.l\aTCKaH .l\HCCepTaIlHlI, HII\II\ MPTO, 1951).­ ITpUM. pea.

HeHHlI .nBHJKeHHlI CKOpOCTh, noayuaex

d 2Pn -----;[i2 =

52

XVup [Pn+l

+ Pn-l -

aTO ypaanenue HMeeT TOtJHO raxon JKe BH nonyxennoe npn paCCMOTpeHHH pacnpocrpa B 6eCKOHetJHotl uenoxxe, cocrosurea H3 KaJK.nali H3 KOTOpbIX saaaaonencruyer TO tJaCTHl.laMH.

5*

JlO)f(HM

Pn

=

nO.llBHlKHO 3aKpenJleHHOrO uearpa cHMMeTp .llJlHHy KalK.llOro H3 MarHHTOB, a '1epe3 .llByMlI COCe.llHHMH MarHHTaMH, npH'IeM BeJl rro narart, MaJlot!: no cpaBHeHHJO C d.

Pe2~i (,t-and),

2[

rae d - paccrosaae Me)f(.llY COOTBeTCTBeHHblMH TO'lKaMH .llBYX nocaenoaare.n.uo pacnoaoxennux pe30HaTopOB. Iloacraaasa 3TO supaxenae .llJlll Pn B nonysennoe ypaaaeuae, HaXO.llHM HCKOMoe COOTHOllleHHe, onpenenmouiee 3aBHCHMOCTb '1aCTOTbl v OT BOJlHOBorO '1HCJla:

Bynex H3Y'laTb pacnpocTpaHeHHe BO rtpennoaaraa, 'ITO MarHHTHOe B3aHMO.llet! TOJlbKO Me)f(.llY nOJlJOcaMH CMelKHbIX MarHHT

Sy-1-. d v=- V sm n cc. 7t

x·

I.. up

Sn_1

Nn_1

-1------------

On-I I-

Orcioaa cnenyer HaJlH'IHe MaKCHMaJlbHot!: '1aCTOTbI pacnpocr­ paneuas, H, TaKHM o6pa30M, paccxorpeaaas CHCTeMa npencra­ BJllIeT cocoa aKycTH'IecKHt!:
Can a c, Journ. de Phys, 7, 161 (1926); CM. TaKlKe art, Phys. Rev., 20, 528 (1922).

1) CM.

St e

\V

Nn

---

, I•

Nn

32'----~!!-.!!:!--------~et.!J Sn-I s,

II

r. 42.

0603Ha'lHM reneps xepea --+- m BeJlH'I psna nOJlJOCOBH xepea an_i, an, an+ i - y ropue 6Y.lleM npennoaarars MaJlbIMH) COOTBe n-ro H (n l j-ro MarHHTOB OT nOJlO Tor-na, eCJlH 1- MOMeHT HHepUHH MarHHT OCH epaurenas. TO ypaaneaae .llBH)f(eHHlI n HMeTb saa

uenoxsn CBH3aHHbiX OCU,HJlJlHTOpOB, otlJlatl.alOw.eA cBoAcTBaMH paCCMOTpeHHoA B § 6 MeXaHH'IeCKOA CHCTeMbl. Pacnpocrpaaeuae MeXaHH'IeCKOrO B03MYW.eHHH B uenosxe, otlpa30BaHHoA H3 MarHHTOB

BbIX MarHHTOB Sn_iNn_i' SnNn' Sn+iNn+l' ... , pacnoaoxen­ HbIX, KaK nOKa3aHO na
I

On

Nn _1

E. Ilpaxep

Bsnne 6bfJIO yxaaaso, 'ITO paccaorpennaa B § 6 MexaHH­ '1eCKall CHCTeMa, npencraaamouias cocoa uenoxxy ynpyro CBlI­ 3aHHblX OCUHJlJlllTOpOB, HeCKOJlbKO HCKyccTBeHHa, H, '1To6bI npen­ CTaBHTb ce6e 60Jlee eCTecTBeHHyJO MO.lleJlb nO.l106Hot!: crpyxrypu, MbI paccuorpena, HCXO.llll H3 Ka'leCTBeHHblX cooopaxenna, CBOt!:­ CTBa uenoxxn nayxaroxnux MOJleKyJl. B 3aKJlJO'IeHHe 3TOt!: rJlaBbI MbI paccxorpax cacrexy, co­ CTOlllllYJO H3 OCUHJlJllITOpOB C neynpyron CB1I3bJO, CBot!:CTBa KO­ TOpOt!: aHaJlOrH'IHbI CBot!:CTBaM yxaaaunoa crpyxrypu H MoryT npe.llCTaBJllITb nexoropun aurepec. 'PaccMoTpHM ueno-ncy, COCTOlllllYJO H3 MaJleHbKHX O.llHHaKO­

s,

d

+

2m21~

"

Ian = - ~(an+i -2a n +an_ 1) -

I "

2

m [2 [ (d

- 2m KaK 06bl'lHO, uieacs BOJlHbJ

2 [ [ (d

­

~ 21)3 + (d ~ 21)3] (a

~ 21)~ -

(d

n l i

~ 21)~] an-

6Y.lleM IICKaTb peureuae B

an =

Ae2~i (vt-andl,

BHl

MATEMATHlJECKOE H3YlJE 60JIEE CJIOlKHblX O,UHOMEPHhlX

HUSI .nJIlI an_I' an U CX n+l B ypaBHeHue .nBHIKeHUSI, nonysaea 2 'I

=

m 2l 2 1t 2/

{

1(1 + ((d ~ 2l)3

2[ (d - 2l)2 -

(d

+1) 2l)2 +

~ 2l)3 ) cos- ad - ~3 sin ad}. 2

(d

7t

7t

OTCIO.na JIerKO aaaers. 'ITO B paCCMOTpeHHOtl CUCTeMe MoryT pacnpOCTpaHSlTbCSI TOJIbKO BOJIHbl, '1aCTOTbl KOTOpblX 3aKJIIO'IeHbl MeIK.ny .nByMSI npe.neJIbHblMU 3Ha'leHUSIMU

ml _

r 1{1(1 2[ (d - 2l)2 -

ml _

r 1{1(1 2[ (d - 2l)2 -

'10

= ---:;- V 7

'11

= ---:;- V 7

(d

1 1} +1) 2l)2 + (d - 2l)3 + (d+2l)3 '

(d

+1) 2l)2 -

2}

d3 •

T. e. 9Ta CUCTeMa seaer ce6S1 KaK nOJIOCOBotl tPUJIbTp. Kpuaaa 'I = 'I (a) aHaJIOrU'IHa KpUBOtl tPu r. 25. Cnenyer ysasars. 'ITO, eCJIU MarHUTHoe nOJIe H napaa­ JIeJIbHO HanpaBJIeHUIO uenoxsa. TO ypaaueaae .nBUIKeHHSI OCTaeTCSI TaKUM IKe C TOtl JIHlllb pa3Huu etl, 'ITO B npaayto '1aCTb .n06aBJISIeTClI '1JIeH - 2mlHcxn- B 9TOM cJIy'lae CHCTeMa TaKIKe npe.ncTaBJISIeT cooon nOJIOCOBotl tPUJIbTP, npenem.aue '1aCTOTbi xoroporo MOIKHO JIerKO HatlTH. B 3aKJIIO'IeHue 3aMeTUM. 'ITO HUIKe TO'lKH Ktop« BrrOJIHe aa­ KOHHO npe.nnOJIOIKeHUe 0 TOM, 'ITO pa3JIU'IHble 9JIeMeHTapHbie MarHUTUKU, Haxo.nllIII.UeCSI B tPeppOMarHUTHOM KpUCTaJIJIe. rro Mepe npU6JIUIKeHUSI K a6COJIIOTHOMy HyJIIO CTpeMSlTCSI ycrano­ BUTbCSI napaJIJIe,1bHO npyr npyry, PaCCMOTpeHHali anecs aanasa COOTBeTCTByeT KaK paa 9TOtl 06JIaCTU resmepa'ryp; 3HaHHe me 3aBUCUMOCTH 'I 'I (a), KaK MOIKHO noxaaars 1), rr03BOJISIeT onpe.neJIUTb nexoropue KJIaCCU'IeCKUe CBotlCTBa paCCMaTpUBae-

=

MblX 06beKTOB.

1) M. Par a d i, Journ. de Physique, 10, 399 (1939).

§ 13.

YpaBHeHHJI .lI.BH>KeHHJI pemerxa X O.ll.HOMepHMH cJlyqaH

Peurerxy XJIOpHCTOro HaTpHSI MOIKHO '1aCTHbltl cJIy'latl onnoxepaux peurero«, K pennux HaMH B § 3 H 7. 31'01' r nn peu tPHr. 43. OH rrpe.nCTaBJISIeT c060ft peurerxy, ceMetlCTBaMH '1aCTHU, MaCCbl KOTOpblX paaa d,

0

M, n-2

• d. 0 M, n-I n-I

M2

d,

• d, 0

Ma n

M, n

d,

• dz 0

M2 n-l

M,

n-t