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à<~C>à<g)lhVjºlnChzfåigacÀj#ln>hdfbdÆdeihdw<¹qâahdgifqbQw<yzlnhkjql)<wnCcºydhzg^>h5mydgihdag fCà<hzmalj[<gfydc~Cgifceafyd~>j[lj[f5 eij#l~Cw<hdhdlnà<g P hdgº¹qw1dei~C>f[ljCw<h^lnhml~>¿Vj[QR yzgbd~>ydftbQw<ydffeag3gafjtbdcw<bQw<feagºdwnhdºmlnhzft~Cg·#Æmlnbdj[cg Z ÊbdcgihmlnhkjÅgah iw<¹¡b1j[g qº~Cl3ÛÜw<Cf~Cgafg gj[fdgRiw<yd#Ædgif~>C¹q>jgaf Cf RVydgiydfgifgjj[Ædgic¹q]RVydgifgijÅ~ gt1>fj[gihdagdgq¹¡w dgaf·fyzbxeicCgiydcfÐëgih(bmlncÀj[Ciyd~C>gacRclndIlyBtdÓRdlnhdf3~Cl^al)1j[eqdgiw<ydbd~Clnà<g P Çågijjg¹qw1dei~C>f[ljCw<h5Ê|dCgih;RVydgfw<yd~>gilnhkj3zgafºbdcw<|d~>âa¹qgaf·bdÆk¿1f]RVydgif~CCeifq^~Il^¹¡w dea~>Cflj[>w<hdy(¹qCicw<bdÆzw<hdgqea~Cgiijcw<fÀj#l ij[QR yzgnÊ gafjÅyzhdgt|mlnfgRfydcå~Il$RVydgi~C~CgcgibxwnfgnÊzmlnhdf~>giChFRVyd>âa¹qg#Æmlnbz>j[cgnÊ1~Cg¹qw dâa~>gbQw<ydc~>g#Æml¹¡b dgbdcgiffCw<h¡mlnhdfÏydhdg)l) >j[eÅdgaw<yzbd~IlnàngfydcÏ~CgafÏbmlncw<Cf±dg~Il`R yzga~C~>gfw<hkjåbd~IlnieafÏdgbxgj[>jf±aw<¹qbxwnf[lnhkj[flnawnydfj]RVydgafiÊ1zCfbQw<f>j[Û5l)¿<lnhkjbxwnydc|zyzjdg¹¡gifydcgacålçngabdceiaCfCw<h¡~Cgiydcf)lnclnijeac>fj]RVydgaf5 gahkj[ceag PÑ6gifj[cl)nlyBt§gt1bxeic>¹¡gihVj[lnyBt ¹¡gihdeafbmlnc[l~C~Câi~Cgi¹¡gihVjqigijj[g_ej[yddg"j[Ædeiw<cQR yzgZw<hkj iw<hddydjly
bxgicÛÜgiij[>w<hdhdgi¹¡gihkj/dy|mlhdå5 eij[ln~Cwnhdhmlnà<gdy!Ñ6ÈÌgijWq~Il¹¡>fglyºbxwnChkjDzybdcw agiffyzf dg¹qgafyzcgdgÆmlnyzjgbzceaiCfCw<h"bmlnc~Il¡ce)ln~>Cflj[>w<h 5 ydh_à<clnhd_hdwn¹º|dcg·5 eij[ln~Cw<hzhmlnà<gifdg¹¡>acw<bdÆdw<hzgafaÊEigaa9mlnhdf~Cg )ldcg"dgZ~Clbmlcj[>a>bmlj[>w<h²dyÑ6ÈÌ qydhdg"Chkj[gicawn¹¡bmlc[ln>fw<h Chkjgachdlj[>w<hmln~>g fydc¡~ eij#l~Cw<hdhdlnà<gdg_¹qCicw<bdÆzw<hdgaf¡bmlnc~Cl¹qeijÆdw1zg_dg_ceaa>bdcw aj[e(Ðëawnydc[lnhkj^»¼<¼<ézÊlnhdhdg t1gÎ_Ó P Ú lnc lC~C~>gaydcfaʱydhdgeijyddgqgt1bxeic>¹¡gihVj[ln~Cgqbdcea~C>¹¡>hmln>cg3wnyz cg~Cl<w<>g q^~Il )lnclnijeac>f[ljCw<h mhzgdg¡bxgj[>jf@aw<¹qbxwnf[lnhkj[flnawnydfj]RVydgafÐÜagURVyd5Ûëln>jÅzeiÛëlnyzjq!~ Ædgiydcg@lnj[ydgi~C~>gçÓ P
K 1 1 >b > 1 1
g @p K K ! @"#
ѱ w<| ¾gj dgigbmlnc[là<c[lnbzÆdgågifj±dgclnbdbQga~CgicD~>gaf/e4R ydlj[>w<hdf/iw<ydbd~>eagaf zgaf/j[clnhdfdydijgaydcf±ea~Cgiijcw<fÀj#l ij[QR yzgafaÊ ~Cgifåe4R ydlj[>w<hdfåcga~Clj[<gaf)q3~Cgiydcfdxeacgahkj[f幡w dgafÏdgÛÜwnhdijCw<hdhzga¹qgahkj@ÐÜea¹qgijjgaydciÊ1ceaigabzjgaydc[ÓlnChzfGRVydg~>gafå|dlnfgifådg~Il·jga#Ædhz]RVydgtdg~IlRj[gihdf>w<hChzfeaceagÊ jga#ÆdhdQRVydgtbdceaawnhdCfeagbdlncå~Cl·hdwnc¹qgtÇÌ ËÞ O ¼<¸ Z`i »Kj »<»/lDbxwnydcRdeijgac¹¡>hdgact~Clj[gihdf>w<h q iCcaydjRw<yzngacj·5 ydh¹qCacw<bdÆdwnhdgDRVymlnhdC~9gafÀj#Ædlncà<eea~>gaijcQR yzga¹qgahkj PÇågafc[lnbdbQga~>ffw<hkjgt jc[lnj[f5 ydhZclnbdbQw<cjdg@aw<hkj[cljSj ¸/l P
$&%'$&%($ )+*,.-0/2143657198;:=<?>@5./BA&:?,.<?>6-C57*>6/*D5E:?8F57AG57198&:?,hj[clnhdfdydijgaydcåea~Cgiijcw<fÀj#lj]RVydg!ÐÜceaigabzjgaydcwnyea¹qgijj[gayzc#Ógifjydh iw<hddgihdf[l j[gayzcea~Cgiijc]RVydgÊzdg
fydcÀÛëlnag SmgijºdgbQgac¹q>jj[1j[e!d>ea~Cgiijc]RVydg ε ÊGawnhdfj>j[yzedg dgiyBt lnc¹l j[ydcgaf ÐÜ~ ydhdg BtzgÊ/~ lnyzjcg¹¡wn|dC~>gçÓfeabmlceagif3bmlc·yzhdg!dCfÀj#lnhdig d(t) = d0 + ξ(t) ÐëwIH d0
gafÀj3yzhdg!aw<hdfÀj#lnhkjgbQw<fj[>nggij·wIH~Cgdeabz~Ilniga¹qgahkj ξ gafÀjåiw<¹¡b1j[ebQw<fj[>ÛQiw<¹¡¹qg~Ò >hdd]RVydg~Cl mà<ydcg O?PQO Ó P Çågaw<hzdgahdflj[giydcÏbdceifgahkjgsqRfgif|xwnchdgifåydhdgdxeacgahdigtdgtbQwnj[gihkj[Cgi~ U(t) = U0 + u(t) gjC~QgafÀjbmlncaw<yzcybmlcåydh iw<ydc[lhVjåea~>gaijcQR yzgI(t) = I0 + i(t) Ðëfwn¹¡¹qgafdgRln~>gaydcfiw<hdfÀj#lnhkj[gifgj5 ei)lncÀj[fÅChzfj#lhVj[lnhdeif#Ó P É1lqalnbmlni>j[eRgifjeaicj[g
C =εSm
d=
εSm
d0(1 + ξ/d0)=
C0
1 + ξ/d0,
wIH C0 = εSm/d0cgibdceifgihVjg3~Clº)lnbmla>je@dy aw<hdzgahdflj[giydc mnly^cgabQw<f n P
ѱ lnc¹ljydcg3¹¡w<|zC~CggafÀjyzhdg!¹¡gi¹º|dc[lhdg^Ðëalnc[lnj[eicCfeagbmlcRf[l ¹lnffggjfl fw<ydbz~Cgaffg)Ó P Ì ~C~Cg3gafjfw<yz¹¡>fgq~Òl)lnhkj qyzhdg^ÛÜwncag g t1jeacCgayzcg F Ðëde mhdCg^bQw<fj[>ng fydnlhVjq~Òl/t1g j[clnae^fyzc¡~Cl dà<ydcgO?PVO Ó P Ézl#Æmlncà<g_lniw<ydfÀj[]RVydgKq~ lnccCâicg gifj lnffgZawn¹¡bd~>gt1g²ÐÜ)l) >jenÊÏlnc¹¡ljydcg 5t1gZbQgacÀÛÜw<ceigZgijw<cmag alnbd>~C~IlCcgq5 eaàkl~CCflj[>w<h(dg bdcgaffCwnh§fÀj#lj]RVydgçÓ Ï~Ò >¹¡bQeadlnhdag¹¡ei)lnhdQRVydg dg^agijqgihdfgi¹º|d~Cg¹¡gi¹º|dclnhdg i #Æmlncà<glcc>âacgÐëclnbdbQw<cjdg~Cl^ÛÜw<cag¡gt1gacaeag fydcº~Cl"¹qga¹º|dclnhdgqfydc3~Il^ j[gaffgzg~Il¹¡gi¹º|dclnhdgnÊm~>gafÅdgiyBt"eij#lhVjw<cCgahkjeagafmlnhzf~>g·¹ a¹qg@fgihdf#ÓÖÊxà<eihdeacln~Cgi¹¡gihkjlbdbxgi~Ceig·>¹¡bQeadlnhdag@¹qe i)lnhz]RVydgÅdg~IlR¹¡gi¹º|dclnhdgÅgafÀjåhdwj[eag Zm
P Ñ6l@#Æmlncànglniw<ydfÀj[]RVydg3Ð mÑ6w<lnGndÓWq@~ lçlnhkjÏgafÀjåalnc[lij[eic>feagbmlncÏ~ C¹qbxeimlnhdiglaw<ydfÀj[QR yzgsRVy5 gi~C~>gbdceafgahkj[gUq3~Cl¹qga¹º|zc[lnhdgºÐë>¹¡bQeamlhdagdgtclç¿nw<hdhdgi¹¡gihkjÖÓhdwj[eag
½
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ZLÊnc[lnbdbQw<cÀj/dg~ClbdcgiffCw<h!aceaeigbmlnc/fw<h!¹¡w<y1<ga¹qgahkj fydc ~Cgåbdcw dydj/dg~ClfydcÀÛëlnagzg~Il¹qga¹3|dc[lnhzg
bmlc~Cl3 >j[giffg·jga~C~>gsRVy5 w<c>gahkj[eig·fydc~Il mà<yzcg O?PVO?P
d 6Rf1%<343ifx%'&#$
dAξ
Av
AF
I
I
U+Q −Q
WÏAÂ O?PVO áGc[lhdfdyzij[giydcei~Cgiij[cw<fj[lj[QRVydg pmf#Ædei¹l!dg@bdcChdiCbQg P
$&%($;% >?AF57198;:6, -08&>63 *,<?> ,, 5 *Ñ-gafaw<ydbz~Ilnà<gifei~Cgiij[cwklniw<ydfj]RVydgiftlnffydceaftbmlc~>g@jc[lnhzfdydj[giydcbQgayzngahkj j[cgdeaicj[fbmlncdgayBt
e4R ydlj[>w<hdfÅaw<yzbd~CeigafÅcgi~CIlhVj~CgafÅà<clnhddgiydcfÅei~Cgaj[c]RVydgaf u gij i Ðëde mhdCgifÅbdceiaeazga¹q¹¡gihVj#ÓÅgjÅ~Cgifànc[lnh idgiydcflniw<ydfj]RVydgifaÊEbdcgiffCw<h^lniw<ydfj]RVydg p gijÅ >jgaffg@bmlncjCiyd~Iln>cg v q~ lçlnhkjÅdg@~Clº¹¡gi¹º|dc[lhdg PÑ6l!cga~Clj[>w<hU(t) =
Q(t)
C(t)=
Q0 + q(t)
C0/(1 + ξ/d0),
wIH Q = Q0 + q cgibdceifgahkjg^~Cl"#Æmlcà<gei~Cgiij[c]RVydg¡dy²aw<hzdgahdflj[giydcaÊDfw<jºlny²bzcga¹qCgicwncdcgdg~ClRVymlnhkj>j[e ξ/d0Ê
U0 + u(t) ≈ Q0
C0+
q
C0+U0
d0ξ ,
dwnhdhdg@dCcgaj[ga¹qgahkjÅ~Clºbdcgi¹¡>âacgdgafÅdgiyBt eRVyml j[Cwnhdfdg@aw<yzbd~IlnàngnÊzbQw<ydczgafÅf>à<hmlnyBtÆmlnc¹¡wnhd]RVydgifdg@bdyz~Cf[l j[Cwnh ω gjl)nga·~Ilaw<hkngahkj[>w<h ej ω t Ê
u(t) =1
jωC0
i(t) +U0
jω d0
v(t) , Ð O?PQO Ó
wIH v(t) =∂ ξ
∂tgifj~Il3 j[gaffg·dg@~Ilº¹qga¹º|zc[lnhdg@w<cCgihVjeagRfga~Cwnh"~>gRfgahzf>hdd]RVyde dà<ydcg O?PVO?P
Ñ6lZfgiaw<hdzg eRVymljCw<h²dgaw<ydbz~Ilnà<gqj[clndydj3~Ò e4R yzC~C>|dcgqdgif3ÛÜw<cigafg tzgicieagaf!fydcº~Il^¹¡gi¹º|dclnhdg PÇågifÛÜw<cagif3ÐÜw<c>gahkj[eigafÅmlnhzf~>gR¹i¹¡gRfgahdfRVydg·~CgRdeibd~Ilniga¹qgahkjÖÓåfw<hkj~CgafÅfyd>lnhkj[gifUp ~ClºÛÜw<cigRea~Cgiijcw<fÀj#lj]RVydg
− Q2
2εSm= − Q2
2Cd= −(Q0 + q)2
2C0d0≈ − Q2
0
2C0d0− Q0q
C0d0= − Q2
0
2C0d0− U0q
d0,
~Cl·ÛÜw<caggt j[eicCgiydcg F q·~ l)nlhVjådyjc[lnhdfdydj[gayzc@Ðëbdcw1zyd>jdg~ClfydcÛëlnig Smdg~Cl·¹qga¹3|dc[lnhzg
bdlncÅ~ w<bdbQw<feRdg@~Clºbdcgiff>w<h"lnaw<yzfj[QRVydgUR yz-câaànhdgacln>jÅf-~Il¹¡gi¹º|dclnhdgRhdgR C|dcln>jbmlnf#ÓÖÊ
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~ClÛÜwncagRzgce)lnj[Cwnh fj#l j[]RVydg Q20/(2C0d0)
fydcÅ~Clº¹¡gi¹º|dc[lhdgnÊ ~ClÛÜwncagR~>Ceig@lny ¹¡w<y1<ga¹qgahkjdg@~Il¹qga¹º|zc[lnhdg −(Zm + S2
mZL)v .ÝChzf5~Ò e4R ydlj[>w<h 5 e4RVydC~>C|dcgtdgafÅÛÜw<cagafÅfi eiacj i ga~C~>g
F =U0
jω d0i+ (Zm + S2
mZL)v .
Ñ6g·cgibxwncj)ÊEmlhdfigijj[g3g t1bdcgiff>w<h5ÊQdg·~Il!bdcgiff>w<h_law<ydfÀj[QR yzgqq~ lçlnhkjdy"¹¡>acwnbdÆdw<hdg¡ÐÜdydgDq~Il3ÛÜw<cag@gt jeac>gaydcg@gijlny ¹¡wnyz<gi¹¡gihVjdg@~Cl!¹qga¹3|dc[lnhzgçÓÊ
p = − F
Sm+ ZLSmv ,
aw<hzdyd>jq~Clºfgaiw<hddgeRVymljCw<h dg@iw<ydbd~Clnà<g pF
Sm− ZLSmv = −p =
U0
jω d0Smi+
Zm
S2m
Smv .Ð O?P »<Ó
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P Ñ-gifaw<¹qbxwnf[lnhkj[giflncCln|d~Cgif u gij idgÅ~Iltj[gihdf>w<hqgijÏdgÅ~ Chkj[gihdfj[edgfwncj[>gdy¡jc[lnhzfdydj[giydcfw<hkjlChdf6Ðëiw<¹¡b1j[gj[gih y¡dgafÏw<cCgahkj[lj[>w<hdfbdceia>feagiffydcÅ~Il mànydcg O?PVO Óå~C>eagifbmlnc~Ilcga~Il j[Cwnh
u = −ZeiÐ O?P ékÓ
gijÅbmlcfyd>jgR~Cgife4RVymlj[>w<hdf·Ð O?PQO Óågij3Ð O?P »<ÓåbQgayzngahkj j[cg·eiac>j[gifUp
v =jω d0
U0
1
Ze
[
Ze +1
jωC0
]
u Ð O?P Z Ógij
Smp = −[
(
U0
ωd0
)21
Ze + 1/(jωC0)+ Zm
]
v . Ð O?P ½<Ó
ÇågafÏdgachdCâicgifcga~Il j[Cwnhdf/awnhddyd>fgahkj)q·~Ò g t1bdcgiff>w<hqdg~Ò guq)lni>j[egihbdcgaffCwnh u/p dy!j[clnhdfzydijgaydciÊu
p= −U0Sm
jω d0
Ze
Zm[Ze + 1/(jωC0)] + [U0/(ωd0)]2 ,
Ð O?P ÞkÓgijq!agi~C~Cgdg@~ C¹qbxeimlnhdigRlnawnydfj]RVydg·bdceifgahkjeag·bmlncÅagi~Cyd i aÒÊ
Za = − p
Smv=
1
S2m
[
(
U0
ωd0
)21
Ze + 1/(jωC0)+ Zm
]
. Ð O?P Y Ó
É1~>gjc[lnhdfdydj[gayzcgafjqaCcaydjw<y1<gacÀjÐëw<yf5~Ò >¹¡bQeamlhdag5 gahkj[ceag@dy bdce)ln¹qbd~Cm)l j[gayzcgifjÅfyBu if[ln¹q¹qgahkjÅà<c[lhddgçÓÖÊd~CgRiw<ydclnhkj i gifjhVyd~-gijagifzgayBt^dgichd>âacgafrRVymlnhkj[j[eafÅfi eiac<gihVjSp
Mp =
(
u
p
)
i=0
= − U0Sm
jω d0Zm
Ð O?P[\ Ó
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:;0<J88-L8-K0."<%3@#"<%M0
gj(Za)i=0 =
Zm
S2m
. Ð O$P ¸kÓÑ-gafg tzbzcgaff>w<hdf·Ð O$P \ ÓÏgij@Ð O?P ¸kÓÏÛÜw<hkjlbdbmlncl Cjcg RVydg~ g4uq)lni>jetdy ¹qCacw<bdÆdwnhdg qºiCcaydjåw<y1<gacÀj
MpgafÀj>hk<gacfgi¹¡gihVjbdcw<bxwncj[>w<hdhdgi~C~>gsq~ C¹qbxeimlnhdigRlniw<ydfj]RVydgTq!aCcaydjw<yzngacÀj (Za)i=0
PÑ6l jgahdfCw<hei~Cgaj[c]RVydg u0
q a>ciyd>jRwnyz<gicjgafj·dej[gic¹qChdeigqdlnhdf@~Ilbdclj[QRVydg!bmlncR~Iljga#ÆdhdQRVydgdg@~CljgahdfCw<h >hdfeiceag¡Ð O?PVO?P ½nÓ P
$&%($;% &68&:6-C57198;:6:?* *: 5E*: *D5 57*>6/Ñ-w<cfR5 ydhÛÜw<hdj[Cwnhdhdga¹qgahkj@gihei¹¡gjj[giydcaÊ~IlÛÜw<cag!gt j[eic>gaydcg F gt1gacaeagqfyzc·~Cl¹¡gi¹º|dclnhdggafÀjhVyd~>~Cg±gij9~CgϹ¡wnyz<gi¹¡gihVj9dgÏ~IlŹ¡gi¹º|dclnhdg±Dq~Ilj[gihdf>w<h u lnbdbd~>]RVydeaglny·j[c[lhdfdyzij[giydcDgifjhq~Ò w<cCà<>hdgdg·~IlbdcgaffCwnh p c[l)¿<wnhdhdeag P Çågijjgbzcgaff>w<h_lniw<ydfÀj[]RVydg p gafjcgi~C>eagUq!~Ilº >j[giffg v dg~Clº¹¡gi¹º|dclnhdgbmlc~Ò >¹¡bQeadlnhdag ZL
pp = ZLSmv .
Ð O?PVO ¼kÓÑ-gåcgabQw<cj9dgagijj[gcga~Clj[>w<h·mlnhdf9~CgafGeRVyml j[CwnhdfÐ O?PVO Ó6gjÅÐ O?P »nÓ6iw<hddydj q~ gt1bdcgaffCw<h3dg~Ò >¹¡bQeadlnhdagei~Cgaj[c]RVydg@bdceifgihVjeagbmlnc~Cgtjc[lnhzfdydj[giydcaÊ
u
i=
1
jωC0
+
(
U0
ωd0Sm
)21
ZL + Zm/S2m
, Ð O?PVO?O Óqº~Ò g tzbzcgaff>w<h dgR~Il3bzcgaff>w<h à<eahdeiceigRbmlnc~Ilfw<ydcagRgih ÛÜwnhdijCw<hzy^iw<ydclnhkj i ÛÜw<ydchd5lny jc[lnhdfdyd ijgaydciÊ
p = − U0
jω d0Sm
ZL
ZL + Zm/S2m
i =ZLZm/S
2m
ZL + Zm/S2m
Mpi ,Ð O?PVO »<Ó
gj qº~Ò g t1bdcgiff>w<h^dy dea|djÅà<eahzeace@bmlc~Ilºfwnydcig·gih^ÛÜwnhdijCw<h dy iw<ydc[lhVj i ÛÜw<yzchd-ly jc[lnhzfdydj[giydc
Smv = − Zm/S2m
ZL + Zm/S2m
U0Sm
jω d0Zmi =
Zm/S2m
ZL + Zm/S2m
Mpi .Ð O?PVO ékÓ
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5 ydh"¹¡>acwnbdÆdw<hdg RVymlhd_C~gafj#Æmlncà<e·ea~Cgiijc]RVydgi¹¡gihkj P Ì ~C~>gRgafjdeiacj[g·mlnhdfÅ~Clqhdwnc¹qg·ÇÌ±Ë O ¼<¸ Z`i» j»n»/lÏgij·ga~>~CggafÀj3y1j[C~>Cfeag!lnyÑGÈÌ mlnhzf~>gqalndcg¡dgifej#ln~>w<hdhmlà<gaf@gahbdcgiffCw<hdgaf·¹¡>acw<bdÆdw<hzgaf PÑ w<| ¾gijdg·ag·bmlnc[là<c[lnbzÆdggifjdgclnbdbQga~>gacÅagjj[g·j[gi#Ædhd]RVydggijdg3bzceaiCfgac~Cgaf|mlnfgaffydc~Cgif RVydga~>~Cgifgi~C~CgRcgabQw<fg PÚ w<yzc/¹¡gifydcgac/~Clj[gihdf>w<h|qiCciyd>jDw<yzngacjaÊnydhzgC¹qbxeimlnhdig Z Êbxgj[>jgåbmlc/c[lnbzbxw<cÀjWq~Ò >¹¡bQeadlnhdagdg¡#Æmlncà<g_ÐÜC¹qbxeimlnhdig!5 gihkj[ceig Ze
dybdce)ln¹qbd~>malj[giydc#ÓÖÊ5gafÀj3iw<hdhdgiij[eiggahfeicCg¡l)<gi~Cg!¹qCicw ibdÆzw<hdg ydhzg jgahdfCw<h dj[g m 5 ej#ln~>w<hdhmlà<gn ug
gafjqw<y hdwnh§lbdbd~CQRVydeagq_fgaf!|xwnchdgif P hzg^bzcgaff>w<hlniw<ydfÀj[]RVydg_Ð pi
fydc·~Il mà<ydcg O$P »<ÓRgij3agjj[gqj[gihdf>w<h5 eij[ln~Cwnhdhmlnà<gqdg¡¹i¹¡gÛÜce4RVydgahdigfwnhVjºlnbdbd~> iRVydeigaf¡ln~>jgachdlj[<ga¹qgahkj P Ñ6ljgahdfCw<h 5 eij[ln~Cwnhdhmlnà<g gifj¡lH¾ydfÀj[eig^zg Ûël w<h qagRVy5 ga~C~>g bdcw dyd>fg ~Cl¹ a¹qgºdeicgahzagºdgbQwnjgahkj[>ga~9lnyBt|Qw<chdgaftzg~ C¹qbxeimlnhdigdg#Ædlncà<gR yzg~IlbdcgaffCwnhlaw<ydfÀj[QR yzglnbzbd~CQR yzeag 1~IljgahdfCw<h gih^iCcaydjåw<y1<gacÀjÅgafÀjln~Cw<cfåeiàkln~Cggah ¹qw1dyz~Cg qº~IljgahdfCw<h 5 eij[ln~Cwnhdhmlnà<gj»n»/l PÌ h"g gj~>gafÅeRVymljCw<hzfzgiw<ydbd~Clnà<g¡Ð O?PVO ÓågjÐ O?P »nÓåbxgiyz<gihVjijcg@ÛÜw<c¹qga~>~Cga¹qgahkjjc[lnzyd>jgafÅbmlncÅ~Cg
f#Ædea¹¡l¡e4R yz>ln~CgihkjÅdg@~Il mà<yzcg O?P » P
!"#$
−pi = F/Sm−p Ck = d0S/U0 Ze u
Smv ZL Zm/S2
m−Ck −Ck C0 i
WAÂ O$P » É1#Ædei¹l!cgibdceifgahkj[lj[Û/dgife4RVymlj[>w<hdfÅdg@awnydbd~Ilà<g P
Ú w<ydc~Il!bdcga¹qCâacg@bmlncjCg@dg~Il!¹¡gifydcgÐÜgahZbzceafgahdigºdg~IlbdcgiffCw<hlaw<ydfÀj[QR yzg −pi = F/SmÓÊ
~CgÅj[clnhdfzydijgaydcÛÜw<hzij[>w<hdhmlhVjgah ¹qCicw<bzÆdw<hdgbxgiyzj j[cg¹¡w dea~>Cfebdlncåydh àneahdeic[ljgaydczgtáÆdeingahd>hdw<hkj~IlRj[gahzfCwnh¡>hVjgachdg uTh
ÐNj[gahzfCwnh q·aCcaydjÏw<yz<gicj)R yzg~Ò w<hq#Ædgic#ÆdgUq@¹¡gifydcgac[ÓÏgij~ C¹qbQeamlnhzagChkj[gichdg ZTh
fw<hkjdw<hdhdeigafbmlnc
uTh =−piSmU0
jω d0(S2mZL + Zm)
gij ZTh =1
jω C0
+
(
U0
ωd0Sm
)21
ZL + Zm/S2m
, Ð O?PVO=Z Ó
agURVyd6aw<hzdyd>jbQw<ydcÅ~Cg@¹qw<hkj#lnàngRgt1bxeic>¹¡gihVj[ln~6lny^f#Ædei¹lqea~>gaijcQR yzg·e4RVyd>ln~>gahkjdw<hdhde mà<yzcg O?P égah^deaiw<hdhdgiij[lnhkj~Clºfw<ydcag·à<eahzeac[lhVj ug
P
uThZe u
ug
Z
ZTh i
WA O?P é É1#Ædei¹l!ea~>gaj[cQRVydgReRVydnl~Cgahkjlny^¹qw<hkj#lnà<g PÚ w<ydc~ClZfgiaw<hzdg bmlncÀj[Cgdg~Il"¹¡gifydcgÐëgah bdceafgahdig^ ydhzgj[gahzfCwnh²5 ej#ln~>w<hdhmlà<g ug 6= 0 ÓÖÊD~Cgj[clnhdfzydijgaydcÛÜwnhdijCw<hdhdlnhkjgihMea¹qgijj[gayzcaÊÏ~IlÛÜwncag"gt j[eic>gaydcgZgafÀj hVyd~C~>g(Ð pi = 0 Ógij~Ilbzcgaff>w<h
p Ðë~Il²bdcgaffCwnh c[l)¿nw<hdhdeigçÓ^gifj"agi~C~Cg RVyd·lnbdbdlnc[l >j^lnyBt¶|Qw<chzgaf^dg~Ò >¹¡bQeadlnhdag ZLÐmà<yzcg O?P »nÓ P
Ñ-giw<¹qbxw<cÀj[gi¹¡gihVj zy j[clnhdfdydijgaydc^gafÀjZzeaac>j^bmlnc fw<h >¹¡bQeadlnhdagei~Cgiij[c]RVydg ZThP Ñ-ga>ciyd>j
eRVydnl~Cgahkjdy_¹qw<hkj#lnà<ggafÀjRagi~CydGdgº~Cl mà<yzcg O?P ébQw<ydc~CgRVydgi~D~Cl¡deicgahzagºdgbxwj[gahkjCga~ uTh = 0gijR~CldeicgahzagdgºbQwnjgahkj[>ga~ uggafÀj·lH¾ydfÀj[eagzg¹lnhzCâacgDqig RVy5 gi~C~>gºbdcw dyd>fgº~Cl¹ a¹qg3#ÆVyzj[g!dg
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ydhj[clnhdfdydijgaydc·ea¹qgijj[giydcaÊ-bdcgahd~Il ¹ a¹qg3ln~Cgiydc RVydgq~Cg¹qCacw<bdÆdwnhdgºfwn>j@gaha>ciyd>j@w<y1<gacÀj·w<y#Æmlcà<etbdlnc~Ò >¹¡bQeadlnhdag Ze
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u
Cref
ie
ur Ze
ir: 2b0* !$C
; 5 GHKV
. *Z>F#%5L?/5-D#%C&>F&L=!$&G# _>2>F* !$C
Mpr
. *Z>F#%5L?/5-]CHKF!$!$&G# _>2>F* !$C
Mpe
WÏA O?P ½ CfbQw<f>j[ÛGbxw<yzc~ eij#l~Cw<hdhdlnà<gtgih"bdcgaffCwnh pdf#Ædea¹¡l¡zg·bzcChzaCbQg P
haw<ydbz~Cgayzclaw<ydfÀj[QR yzg_ÐÜ)l) >j[e!dg!<wn~Cyd¹qg VcÓtgifj@ÛÜgac¹¡e q fgaf·gt jcea¹q>jeaf@bmlnc@dgiyBt¹qCacw i
bdÆdw<hzgafaÊ~ ydh_ÛÜw<hdj[CwnhdhmlnhkjRgihei¹¡gjjgaydciÊ5~ lnyzjcg3gihceaigabzjgaydc¡Ð mà<ydcg O?P ½nÓ P Ñ6lalç j[eºÛÜw<c¹¡eig3bmlnc~ gahdfga¹º|z~Cg m¹¡>acw<bdÆdw<hzgaf i iw<ydbd~>gaydconZl¡yzh"<wn~Cyd¹qgàneaw<¹qeijc]RVydg V ÐÜdxeacgahkjdg Vc
)lc>~-Chdi~Cyzj~Cgif/<w<~>yd¹¡gif ÛÜcw<hkj#lyBt!dgif¹qCicw<bdÆzw<hdgaf ¹qgahkjCw<hdhzeafÏlnyqbmlnc[là<c[lnbzÆdg O?P » PQO Ó P Çågaw<ydbz~CgayzcÏbQgay1jål)nw<Ccdxeacgahkj[gifÛÜwnc¹qgaffga~>w<hZ~>gRjß¿1bQg3zg¹qCicw<bdÆzw<hdg@yzj[>~C>fe@gij~Il!àkln¹q¹¡gRzg·ÛÜceRVydgahzagafRg t1bd~Cw<j[eigÐÜiÛ P» PVO Êmlhdhdgt1g·ÝgijÅcgÛ P j»n»/lNÓ PËß~gifjqln~>w<cf·bxwnff>|d~Cgqdg¹qgafydcgicº~ C¹qbQeamlnhzagqea~Cgiijc]RVydg¡dgjc[lnhdfÀÛÜgacÀj ZE = ur0/ie
gihkj[cg~Cgafdgay.t ¹¡>acw<bdÆdw<hzgafaÊ9c[lnbzbxw<cÀjdg ~Il^j[gahzfCwnhqiCciyd>j3w<yzngacj ur0
lnyBt(|xwnchdgifdy§¹qCicw<bzÆdw<hdgce iagibzj[giydcfydc~>giw<ydc[lhVj ie jc[l)<gicf[lhVjq~>g¹¡>acw<bdÆdw<hzgea¹qgijj[giydc P ÑGlZjgahdfCw<h1q_iCcaydjºw<yzngacÀj ur0q^¹qgafydcgicgifj3deadyz>j[gqdgq~Il j[gihdf>w<h ur
¹¡gifydceag q jc[l)<gicfº~>gqbzce)l¹¡bd~>malj[giydcR5 C¹qbxeimlnhdig Zebmlnc~ClZj[ga#Æzhd]RVydg^5 ChdfgacjCw<h²zg jgahdfCw<h P Ñ-g^iw<ydclnhkj ie gafÀjqdeadyz>j!dg ~Il_¹qgafydcg dg ~ClZj[gahzfCwnhie/(jωCref)
Êmiw<hdhmln>fflnhkj~Cl3ln~Cgiydcdg@~Il)lbmlnaj[e CrefP
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Ñ gihdfgi¹º|d~Cgxm¹¡>acw<bdÆdw<hzgaf i aw<ydbz~Cgayzc nÐdà<ydcg O$P ½<Ó@bxgiyzjj[cg cgabdceafgahkj[e dgÛël w<hÛÜw<c¹¡gi~C~>gnÊ5 lnbdcâafq~Cg iCciyd>jºdg ~Il mà<yzcgÐ O?P »<Ó3gj¡~>gafe4R ydlj[>w<hdf^Ð O$P Y Ó3gjZÐ O?PVO ékÓÖÊ/bmlnc~>g f#Ædei¹lea~>gaijcQR yzge4R yz>ln~CgihkjºÐ mà<ydcg O?P Þ<ÓåwIH Zar
gafj~Ò >¹¡bQeamlhdag@lniw<ydfÀj[]RVydg@dy^¹qCacw<bdÆdwnhdgtceiagibzj[giydcdwnhdhdeag@bmlc~Ò g tzbzcgaff>w<h(Ð O?P Y Ó
Zar =1
S2r
[
(
U0r
ωd0r
)21
Ze + 1/(jωC0r)+ Zmr
]
, Ð O?PVO ½<Ó
Segj Sr
ej#lnhkjÏ~CgifÏfydcÛëlagafϹqw<|dC~>gaf/zgaf±¹¡gi¹º|dclnhdgaf±dgÅ~Ò ei¹¡gjj[giydc±gijÏdyqceiagab1j[gayzcaÊVgijÏwIHq~CgiCciyd>jei¹¡gjj[giydcÅdg@agjj[g mà<yzcgqÐ O$P ÞkÓåceabQw<hdqq~Ò e4RVymlj[>w<hÐ O$PQO é<Ó P
Mpeie Zme/S2
e ZLe pe pi Zar pr
qe Seve Srvr ZLr
%'&#f#()&#$
WAÂ O?P Þ É1#Ædei¹l!ea~>gaj[cQRVydg@eRVyd>ln~>gahkjÅdy^dCfbxwnf>j>Û6bQw<ydcÅ~ eij[ln~Cwnhdhmlnà<gbmlncÅceia>bdcw aj[e P3 lnbdcâifÅ~ eRVymljCw<hÐ O$P ÞkÓÖÊd~Ilºbzcgaff>w<h pr
lny1j
pr = − 1
Ze
[
(
U0r
ωd0r
)2
+ Zmr
(
Ze +1
jω C0r
)
]
jω d0r
U0rSrur .
Ð O?PVO ÞkÓ
Ú lncln>~C~>gaydcfaÊ ~ g4uq)lni>je Mprzy^¹qCicw<bzÆdw<hdgRceaagibzj[giydcfa eaac>j3Ðëe4R PO?P \ Ó
Mpr =
(
ur
pr
)
ir=0
≈ ur0
pr
,
ÐÜ~ lnbdbdcwt1>¹ljCw<h (pr)ir=0 ≈ prgafÀj-¾yzfj[meaglnyqbmlnclnà<c[lbdÆdg O?P » P é RVydEfyz>jÖÓÖÊkgijÏagi~C~>g Mpe
dgÅ~ ea¹qgij ijgaydcÐdà<ydcg O?P ÞkÓ
Mpe =qeie
=Seve + pe(Zme/S
2e )
−1
ie,
Seveeij[lnhkj/~>gdei|d>j9dg~IlŹ¡gi¹º|dc[lhdgzgå~ ea¹qgijj[giydc9gj pr
~IlÅbdcgiffCw<hºfydcD~Cl¹qga¹3|dc[lnhzgådyceaigabzjgaydc PÑ-gafgt1bdcgiffCw<hdfbdceiaeazgahkj[giftdgafg4uq)la>jeaf Mpe
gij Mprzw<hdhdgihVj~ gt1bdcgaffCw<h"dyZbzcw dyd>jdgif
gu¡alnaj[eif
MpeMpr ≈ur0
ie
Seve + pe(Zme/S2e )
−1
pr
= ZEYT ,Ð O?PVOY Ó
wIH YTgifj!lnbzbxgi~Ce~Òld¹¡jj[lnhdagqlniw<ydfÀj[]RVydgdgj[clnhdfÀÛÜgacj3dgaf3dgayBt(¹¡>acw<bdÆdw<hzgaf P Çågijj[gkRVymlhVj>je
cgabdceafgihkj[g_~>g"clnbdbQw<cjqdy§dea|dj qe dg"~Cl¹qga¹3|dc[lnhzg^zg"~Ò ei¹¡gjj[giydcqgah iw<ydcÀj i iCcaydjlaw<ydfÀj[QR yzgÐÜf>jymljCw<hwIH pefgic[lnjhVyd~±fydc~Cl mà<ydcg O?P ÞkÓtfyzc~Cl bzcgaff>w<h pr
fydc~Cl^¹qga¹3|dc[lnhzgqzyceiagibzj[giydc
"! " " ' v
Ð qe/prÊGde dhd>jCw<hdgq~Il"hzw<c¹qg j»<»lNÓ 6gi~C~Cg!bQgayzj ijcgzeij[gic¹qChdeig¡bdlnc~>g¡aln~Ciyd~±gah#Ædw<CfCff[lnhkjºyzh
¹¡w dâi~CgRlniw<ydfÀj[]RVydg·lnbzbdcw<bzcCeRbQw<ydccgabdceafgahkj[gic~CgRfÀ¿ fj[âi¹¡g@dg@¹qCicw<bdÆzw<hdgafÅiw<ydbd~>eaf PÝChzf ÊQ~CgifdgiyBtxRVymlhVj>jeaf ZE
gij YTbxgiyz<gihkj ij[cgºawnhdhVydgaf¡Ðëcgafbxgiij[<gi¹¡gihVj@bdlnc~Cl¹¡gifydcgºgij
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bdÆdw<hzgaf3ej#ln~>w<hdf·dgq~Iln|Qw<c[l j[w<>cg 5igabQgahddlnhkjga~>~Cgqlnd¹qgij·ydh(igacÀj#ln>hhdw<¹3|dcgq5 ÆV¿ bQwnj[Æzâafgifdw<hkjC~gafÀjhzeaagifflnCcgdgRneacmgacÅ~Cl3nl~CCz>j[e P
$&% %% 38F5 ,7*, A&< 1(,.*, *D5E<?19,.-D>6,7,.198;:ѱ yzjC~C>f[ljCw<h¡dg@~Il3jga#ÆdhdQRVydgzg·~Clºj[gahzfCwnh^>hdfeiceig·bQw<ydcÅ~Clº¹¡gifydcg·zg·~Ò >¹¡bQeadlnhdagea~Cgiijc]RVydg
dgj[clnhdfÀÛÜgacjºfydbdbQw<fg RVydg ~ClZbdcgiffCw<h piÊ/aceaeag bmlcº~ ea¹qgijj[gayzcaÊDfydc~Il_¹qga¹3|dc[lnhzgdy ceiagab1j[gayzc
gafÀjChdzeabQgahdmlhVjg dg~Il"#Æmlncàng ea~Cgiijc]RVydgdg agi~Cyd i aÐ O$PQO?P ½<ÓÖÊ9ig R yzaw<hzdyd>j q_aw<hzÛÜwnhddcg~CgafC¹qbxeimlnhdigafÅlniw<ydfÀj[]RVydgifzy"¹qCicw<bdÆzw<hdgRceiagibzj[giydcgih"iCcaydjÅw<yz<gicjgij#Æmlcà<e@bmlncÅydh^bdce)ln¹qbd~C im)l j[gayzc P ̱hZg xgijR~Ò >¹¡bQeamlhdag·lniw<ydfj]RVydg Zar
h5 gafÀjRbmlnf~ C¹qbxeimlnhdigTqa>cayz>jw<yz<gicjdy_¹qCacw ibdÆdw<hzgceiagab1j[gayzcÐÜdw<hdhzeagbmlncD~Ilcga~Il j[Cwnh O?P ¸kÓ9¹¡lnCfGbd~Cy1j nj ~Ò >¹¡bQeadlnhdagÏdeibxgihdmlnhkjÏdg~ C¹qbQeamlnhzag5 gahkj[ceag!dybdcealn¹qbd~Cm)ljgaydcºÐëdwnhdhdeagbdlncR~Ilcga~Clj[>w<h O?P Y ÓdyÛëlnjsRVydg~>gº¹¡>acw<bdÆdw<hzgºceiagab1j[gayzcgafÀjg giij><gi¹¡gihkj#Ædlncà<e PËß~xgifjbQw<ffC|d~>gzg<eicmgac RVydgagifdgay.t C¹qbQeamlnhzagafÏfwnhVjåj[câifnw<CfChdgifåawn¹¡bzjgjgahVy dgafå)lncln i
j[eicCfÀj[QR yzgafÅdgaf¹¡>acw<bdÆdw<hzgafÐ åcdga~ ¾c ZFO Þn¼qgj Z5O\ ¼kÓgijdy"bdcealn¹qbd~Cm)ljgaydcy1j[C~>CfeqÐ åcdga~ ¾c5Ì»nÞ Y é j Þ/lNÓ@ÐÜj[ln|d~>g)lny O?PVO Ó P
ÑGÉ O Ú ÑGÉd» ÚÐ O=Z Þ<¼kÓ Ð ZFO\ ¼kÓ
áGgahdfCw<h^dg@bQw<~IlncCflj[>w<h U0Ð@Ó »¼<¼ »n¼<¼
Çlnbdlnaj[e mnlny^cgibxwnf n C0ÐÜb+X Ó ½<½ OY ʽ
Äl)¿<w<h^Chkj[eic>gaydc a Ðë¹q¹Ó ¸zÊé Z Ê Þk½±w<~>yd¹qgteRVydnl~Cgahkj Veq
Ðë¹q¹ 3 Ó OZ<\ ¸zÊ é>fj[lnhdag@¹qga¹º|zc[lnhdg i ei~Cgiij[cw1zg d0 = εSm/C0
Ðë¹q¹Ó ¼zÊ O\ ¼zÊ OZ
è8D O?PVO Ïln~>gaydcfTm jß¿ bxgif n^zgaf)lnclnijeac>fj]RVydgafÅdg@¹qCicw<bdÆzw<hdgaf ZFO Þn¼!gj ZFO=\ ¼ PÇåw<hdhmln>fflnhkjÅ~IlÛÜceRVydgahzagnÊEigafÅdw<hdhzeagafÅbQgac¹¡gjj[gihkjÅdgR)ln~>ayd~>gac~Ò >¹¡bQeadlnhdaglnawnydfj]RVydgUqºa>c i
aydjÅw<yzngacjdy^¹qCacw<bdÆdwnhdg (Za)i=0 = γRPR/(jω Veq)Ð » P »nÞ<Ó P
Ú w<ydcD~CgÏbdcealn¹¡bz~Cd)ljgaydcGyzj[>~C>fenÊç~Ò >¹¡bQeadlnhdagÏ5 gahkj[ceag ZegifjDdw<hdhdeigbmlnc 1/Ze = jωCe+1/Rel)<ga Ce ≈ ¼1ʼk½ºb+XÊ Re ≈
O ΩP
Çågafdw<hzhdeagifqgj~Cgifºcgi~IljCw<hzf Ð O$P Y Ó@gij"Ð O?P ¸<Ó·bQgac¹qgijj[gihVj!zg)ln~>ayd~>gac ÐÜiÛ P j#ln|d~>g)lny.t O?P »"gj O?P ébxwnydcdgay.t nl~CgayzcfÅgt j[c a¹qgafdg@~IlºÛÜce4RVydgahdigçÓÅ~ w<cdcgRdg@à<c[lhddgayzczg~Ò >¹¡bQeamlhdagRlaw<ydfÀj[QR yzggiha>cayz>j@w<yzngacÀjºdy¹¡>acwnbdÆdw<hdg (Za)i=0 = Zm/S
2mgij~ eaalncjSq agjj[gqln~Cgiydc@f±~Cg!¹qCacw<bdÆdwnhdg!gafj
#Æmlncà<e@bmlnc~CgRbzce)l¹¡bd~>malj[giydc@ÐÜagafdgiyBtRVymlnhkj[j[eiffw<hkjiw<hkj[>h z¹¡gihVjÅzeaacw<Cff[lnhkjgaf#Ó PÇågij@eaalncj@gifj@hdeià<~C>à<g)l|d~CglnChzfhRVydgÊ5bmlncfydj[g^Ðëzy_Ûëln>j R yzgagijj[g!C¹qbQeamlnhzag Zar
h5 gifj·lny1j[cgRVydgº~ C¹qbQeamlnhzag@j[gac¹¡>hmln~>gSRVydg3nw<>j~Ilal)1j[edgºawnydbd~Ilà<gçÓÖÊx~Ò >hda>dgahdig3dg3agijj[g!lnbzbdcw=t1C¹¡lj[>w<hfydc!~Òlnz¹¡jj#lhdag law<ydfÀj[QR yzg^zg"#Æmlcà<g dy ¹¡>acw<bdÆdw<hzg ea¹qgijjgaydc YLe = 1/ZLe
Ê ag RVyd9¾ydfjmg
'<' ()*+#,=# !"$-. 8/, 0.2&-B"$=/,4=*5%36 #-.,D0.:;-<=-. 8/,D-A 0."?-A@#-. 8/,$C 0<*+2&!8*FEG-L@#*Fk2&*
:;0<J88-L8-K0."<%3@#"<%M0
"/0.6 8-.!"?- _ _
Zm/S2mÐÒÈ P f P ¹ −5 Ó i ¾ O Ê » PQO ¼ 11 i ¾ O Ê» PQO ¼ 8
(
U0
ωd0Sm
)21
Ze + 1/(jωC0)ÐëÈ P f P ¹ −5 Ó Ð »1Ê éM¾ O Ê O Ó PVO ¼ 7 Ðë¼zÊ» \ ¾ O Ê \ Ó PVO ¼ 2
èÏ8 9 O?P » Ëß¹qbQeamlnhzag (Za)i=0 = Zm/S2mgijeaalncj Za − (Za)i=0
bQw<ydcydh^¹qCacw<bdÆdwnhdg Z5O\ ¼ P"/0.6 8-.!"?- _ _
Zm/S2mÐÒÈ P f P ¹ −5 Ó i ¾ Y Ê Þ PVO ¼ 9 i ¾ Y Ê Þ PQO ¼ 6
(
U0
ωd0Sm
)21
Ze + 1/(jωC0)ÐëÈ P f P ¹ −5 Ó Ð O Ê ¼M¾¼zÊ O ÞkÓ PVO ¼ 5 O Ê O M¾ÅÞzÊ Þ
èÏ8 9 O?P é Ëß¹qbQeamlnhzag (Za)i=0 = Zm/S2mgijeaalncj Za − (Za)i=0
bQw<ydcydh^¹qCacw<bdÆdwnhdg Z5O Þ<¼ P
bd~>ga>hdga¹qgahkjt~Ò y1j[C~>Cflj[>w<h dgº~Il!j[gi#Ædhd]RVydgzg~IlqjgahdfCw<h>hdfeiceig P Ëß~9gafÀjTqhdwnjgacteiàkln~Cgi¹¡gihkjsRVydgnÊdyÛëlnj RVydgq~Cgaftln~>gaydcf@dg YLe
gij Zarfw<hkjbzc[lj]RVydga¹qgahkj·Czgahkj[QR yzgaf@bxwnydcR~Cg¹qCicw<bdÆzw<hdgceiagibzj[giydc
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Mpeie pe 1/Ye 1/Yr pr
qr = 0SvrSve
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k0
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[
pe
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=
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Zisin k0` cos k0`
[
1 0Yr 1
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pr
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]
, Ð » P ékÓ
gjzw<hdhdg
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(
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j sin k0`+ Yr cos k0`
)
pr .
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YT0 =Sve + Yepe
pr
= j sin k0`
(
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Zi
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)
+ cos k0` (Ye + Yr) ,
fw<>jYT0 = j sin k0`
[
S
ρ0c0+ρ0c0S
YeYr
]
+ cos k0` (Ye + Yr) .Ð » P[Z Ó
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p(z) = A(e−j k0z +Bej k0z) ,
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ln¹º|dClnhkj[gif3ÐÜbdcgiffCw<h5ÊEj[gi¹¡bQeaclj[yzcgçÓgij RVydgºigijjg3deibxgihdmlnhzagºdwn>j ijcg3bzcCfggih_awn¹¡bzjgdlnhdf~Cgafceifyd~j#ljfG ej#ln~>w<hdhmlnàngϹ¡ln>f-hdg±bdcw<bQw<fgbmlf9dgϹqwH¿ngah·bxwnydc9iga~Cl P Ñ-gÏbdlnc[lnànc[lnbdÆzgå» P » P ½bdceifgahkjgådgifj[gic¹qgafDiw<ccgaj[>ÛÜfiÊnbQw<ydc9bdcgihddcgågahawn¹¡bzjgå~Ò >h5mydgihdagdgagif bdcgiffCw<hºgj/j[gi¹¡bQeaclj[yzcgnÊ dej[gac¹¡>hdeafbmlnc9Älf¹ºydffgihj » Y ÊI»n¸`l gijDyzj[>~C>feafGbmlnc9ydhºà<clnhdºhzw<¹º|dcgådg~Cln|Qw<c[ljw<Ccgaf6dg¹¡ej[cwn~Cw<à<>gÏlnaw<yzfj[QRVydg P
% % 8&/7/B*- 52198&: 5 6*/ 1 >?* :68&/ A (19, * Ñ6gifRea#Æmlhdà<gaf@j[Æzgac¹q]RVydgifRmlnhdfR~>gmyd>dg^Ðëzw<hkj@~Il¡jga¹qbxeic[ljydcge<w<~>ydgmlnhdfR~>g3j[gi¹¡bzf@lçnga!~Il
bdcgiffCw<hEÓGgijÏl)<gi~Ilbmlcw<1dg~Ilal)1j[e@Ð&RVyddcgifjg R ydlnf i Cfwnj[Æzgac¹qgçÓ6¹¡w dmgahkj/~>g#Æmln¹qbdgbdcgaffCwnhgahcgiàklnczgag RVy5 C~/fgac[l>j@fDagaf@g gj[f@h5 gt1CfÀj#ln>gahkj@bmlnf P Ñ6l iw<ccgaj[Cwnhbdcw<bQw<feagbmlncR~Clhdw<c¹¡ggafÀjlffw a>eagTqydh^¹¡w dâa~>gtdg@)l) >je@eij#l|d~C5bdlnc@gic|QgacTj O »/l Ptlnhdf@ag!bmlc[lnà<clnbdÆdgÊ5~ gt1bdcgaffCwnhdg!~Ò >¹¡bQeamlhdag5 yzhdg¡al)1j[e^ÐÜc[lnbdbQw<cÀj@dg!~IlbdcgaffCw<h q ~Il
fw<cÀj[>gdg~Il!)l) >je3fyzc~Cg·dei|d>j q¡~Ò gihVjceigçÓg t1a>jeagºbdlncyzh_à<eahzeac[l j[gayzc m>de)ln~ n fga~>w<hLRVydg3 gifjRyzhà<eahzeac[l j[gayzcdg@dei|d>jw<y"dg·bdcgiff>w<h^gafjeij[ln|d~>CgR5 lnbdcâaf Rgac|xgicÐÒl P ÓÊm~Ò g tzbzcgaff>w<hZdg·~ C¹qbQeamlnhzag5 ydhdgåal)1j[egt1aj[eagbmlncGydhdgfw<ydcagåceagi~C~CgÏgah3gafjDgahdfyd>jgådeidydj[gRÐÜ| P ÓÊçj[w<yH¾w<ydcf95 lnbdcâifWRgac|xgic P Ñ6g#Ædw<gt3 ydhzggt1bdcgiffCw<h!dg~Ò >¹¡bQeadlnhdag5 ydhdgalç j[eej#ln|d~>CgbmlncWRgac|xgic lnbdbz~C]RVyml|d~CgÏmlnhdf ~>gåalndcgdg·~ eij#l~Cw<hdhdlnà<gRdg·¹¡>acw<bdÆdw<hzgafÅbmlncÅceia>bdcw aj[e@gafÀjgihdfydj[g·gt1bd~>]RVydeÐë P Ó P Ì h mhÊx~Ò g tzbzcgaff>w<hZdg~Il^aw<ccgaj[>w<hj[Æzgac¹q]RVydghdw<c¹¡ln~>CfeigKj »<»/låln>hdf R yzg~Il"Ûël w<h(zw<hkjga~>~Cggifjy1j[C~>Cfeaggifjg t1bd~C>a>jeagÐë P Ó P* 4:;@W0.8*+!"?-*+"? !,%76 #- eV ##-k"?* %&0|-8"<%M0A-@#*F !8-k, 8/"?- &F%780A*+2&-)(x#- #0 !%&U
8-L@#-B,,=%3@gic|QgacaÊ dlnhdfq~ lncjCa~>gChkj>j[yz~Ce "! "! " " "!
# "!" "! j O »/lÒÊ1¹qw<hkj[cg RVydgt~Ò >¹¡bQeadlnhdaglniw<ydfj]RVydgdgj[clnhdfÀÛÜgacj5 ydhdgal)1j[eqdg¡nw<~Cyz¹¡g V gjºdgqfydcÀÛëlnag A ÐÜc[lnbzbxw<cÀj3zg~ eaalncj·>hdfj[lnhkj#lnhde¡dgbzcgaff>w<h(mlnhzf~Cl)l) >je fydc~>g dea|dj!dg ~Il_fw<ydcagçÓÖÊ wIH C~h5 ¿²lbdlnfdg bdcwnbmlnàkljCw<hÊ9gifjqdxeacgahkj[gÊ fgi~Cw<h1RVydg ~Cg#Æmln¹qb_lniw<ydfÀj[]RVydg3gafjà<eahzeace3bmlncyzhdg3fw<ydcigm>de)ln~>gn dg3dei|d>jÐëà<eihdeaclj[giydczg3dei|d>jtlç¿<lnhkjRydhdgC¹qbxeimlnhdigtCh mhdCg)Óåw<y dg@bdcgiffCw<hÐëà<eihdeaclj[giydcdg@bdcgiff>w<h"lç¿<lnhkjydhdg@C¹qbxeimlnhdigRhVyd~C~>gçÓ Ptlnhdf@~Cgalnf@5 ydhdg!fw<yzcag!dg!dei|d>jaÊ5~ C¹qbQeamlnhzaglniw<ydfÀj[]RVydg ZV
zgq~Cl)l) >jeqfi eiac>j)Ê5lbdcâif@gic|QgaciÊ
ZV =γP0
jω V
1 + (γ − 1)EV
γ, Ð » P ½<Ó
gijÏbQw<ydc ydhdgfwnydcigdgÅbdcgaffCwnh5ÊV~ C¹qbxeimlnhdigdg~IlR)l) >jenÊkhdwnjeag ZPgifj)Ê<jw<y ¾w<yzcf±5 lnbdcâif Rgac|xgicaÊ
ZP =γP0
jω V
1
γ − (γ − 1)EP, Ð » P ÞkÓ
wIH ~CgtÛëlnj[giydc γP0/(jω V ) cgabzceafgahkj[g~ lnd¹q>jj#lnhdig5 ydhdg·)l) >jeTq!bdlncw<>fc>à<Cdgif PÑ6giftaw g4uqiCgahkjf EVgj EP
fw<hkjRej[cw<>jga¹qgahkj~>Ceif q~Cl¡à<eiw<¹qeij[cCg·dg3~Cl¡alç j[e P Ëß~Cffw<hkjtzw<hdhdeifmlnhdf~Òlcj[>a~>g¡dgRgac|xgicj O »lgij!aln~Ciyd~Ceif5 lnbdcâif~Il_ceiÛÜeacgahdig j O ¼/l·Ðë~>gdeij[lnC~dg agif)ln~>ayd~>fgafjdw<hdhze·dlnhdfÅ~CgRbmlc[lnà<clnbdÆdg@é P »<Ó P
_G__ W()-."/J!#%36 #-., -<:;0<J##-., -.a !,*5E-K@W 8 2feg0A=*+23##*5E-K#-., :9%3"?/G@!J#8-.,|@#*F
0."<%3@#"<%M0
Ì hfgtceiÛÜeic[lnhkjrq3igafaln~Ciyd~CfiÊ1>~Qlnbdbmlncl Cj RVydgtigafaw g4uqiCgahkjfbxgiyz<gihkj ij[cgteaicj[fåfw<ydf~Cl·ÛÜw<c¹¡gdy^de<ga~>w<bdbQga¹qgahkj¹¡w mln~xfydnlhVjUp
EV,P =
∞∑
n=n0
DNn
1 +MnX2V,P
, Ð » P Y Ó
l)<giXV =
A
V
1 − j√2
√
γαT
ωgj XP =
A
V
1 − j√2
√
αT
ω, Ð » P \ Ó
wIH αT = λ/(ρ0CP ) = c0`hÐëbdlncRde dhd>jCw<h_zg `h Ógifj@~Cg3aw g4uqa>gahkj@dgdydf> >jejÆdgac¹¡QR yzgdymyzCdg P ÇågRdeinga~>w<bdbQga¹qgahkjgafjÅzeadydjdgR~Ò g tzbzcgaff>w<hÐÒé P » Z Ê é P »nÓÊzzgR~ ea)lcj¹qwç¿<gah dgtjga¹qbxeic[l i
jydcg@mlnhzf~Cl)l) >j[e PÑ-gaftj[gic¹qgaf Nn
Ê MnÊ D gj n0
deibxgihddgahkj·dg~Clà<eaw<¹qeijc>g3dg~Cl)l) >je P Ëß~>fRfw<hkj@dw<hzhdeafRmlhdf~>g·j[ln|d~Cgalny" PQO dg3~Òlhdhdgt1g¤bQw<ydcdxeacgahkj[gifÛÜw<c¹¡gifÅdg3alç j[e P Ñ Chdd>ag n bxgiyzjtcgabdceafgihkj[gacydhfgayd~6ChzdCignÊdydh^dw<yz|d~CgjÅw<y ydh^jc>bd~Cgj5 Chdd>agif PÇågde<ga~>w<bdbQga¹qgahkj!awnhVngacà<g^lnffg ~Cgihkj[ga¹qgahkj^ÐÜfydcÀj[w<y1j!gah²¹qwH¿ngahdhzgaf!gijÆdlnyzj[gifºÛÜce4R yzgahdigaf#ÓÖÊ
igDRVyd9cgahdZfw<h_y1j[C~>Cflj[>w<h bxgiy_bdclj[QR yzg P h_de<gi~Cw<bdbQga¹qgahkjsqaw<yzcjj[gac¹¡gbmlncÅj[clnhdfÛÜwnc¹qeag3zgÑ6lnbd~Clnagdeazyd>jzgt~ gt1bdcgaffCw<hÐÜeR P é P é<¸1Ê Åé P »<Ó±dgt~Ò ei)lncÀj¹qwH¿ngah dgjga¹qbxeic[ljydcgmlnhdfå~Il3alç j[enÊgifjbdcw<bQw<feRbmlnc @gic|QgacbQw<ydcÅgt1bdcC¹qgac EV
gj EPp
EV,P = 1 −XV,P +G1X2VP
+3
4
√πG2X
3VP, Ð » P ¸kÓ
wIH^~Cgifaw g4uqa>gahkj[f G1gj G2
fw<hkjdeijgac¹¡>hdeafbmlc~Ilà<eawn¹¡ej[c>gtdg@~Ilalç j[eqÐÜj[ln|d~>g)lny^ PVO Ó P
4:;@W0.#*+#"?-L*+"? #,4%36 8-LzeQ #8-x"?*8%&=0K-8"<%M0A-@#*F ##-L, #/"?-K0A-.272&-Ñ-gaf±C¹qbxeimlnhdigaf ZV
gij ZPaln~Ciyd~Ceigaf±hdgaw<ccgafbxwnhddgahkj RVy5 q·dgafÏ)lfÏ>de)lyBtxÊ ~ClRfw<ydcagaw<¹qbQw<c i
j[lnhkjjw<y ¾w<ydcfÅgahZbzc[lj]RVydg·ydhdg·C¹qbxeimlnhdig·>hkj[gachdg mhd>g P Ý>hdfÒÊdaw<hzfCzeac[lhVj RVydg3~>g@#Æmln¹qbZlniw<ydf ij]RVydg·mlnhdfÅ~Il!)l) >je@gafjaceae·bmlncÅydh"à<eahdeic[ljgaydcdg@jß¿ bxg·áÆdeingahd>h_5 C¹qbxeimlnhdig Zd
ÊRgac|xgicSj O »/lgih"deidydj~Ò >¹¡bQeadlnhdagtlnaw<yzfj[QRVydg Zc
dg@~Ilºalç j[e@mlnhdfÅig·alnfÅhdw<h^Cdealn~ p
Zc =ZP + Zd
1 + Zd/ZV
. Ð » PVO ¼kÓ
Ý hzwnj[gic RVydgfE~ C¹qbxeimlnhdig Zdgifjà<c[lhddgnÊ Zc ≈ ZV
Ðëàneahdeic[ljgaydcdgzea|djÖÓ gijf Zdgifjbxgj[j[gnÊ
Zc ≈ ZPÐëà<eihdeaclj[giydcÅdg@bdcgiff>w<hEÓ P
" @!@!27%3"?*5%& 2feV0<*F23##*FEG-K#-x:;%7"?G@hJ8#-.,D@#*Fk0."<%3@#"<%M0tlnhdf~>g)lfqzg ~ eij#l~Cw<hdhdlnà<ggih bdcgiff>w<h dg ¹qCacw<bdÆdwnhdgafbmlcºceiaCbzcw a>jenÊ9~CgfÀ¿1fÀj[âi¹¡g dg ~Cl
mànydcg O?P ½@gifjydhdg)l) >j[egt1i>j[eigbdlnc±ydh¹qCicw<bdÆzw<hdgyzjC~C>fegihea¹qgijj[giydc q·~Ò ydhzgzgfgafåg t j[cei¹¡j[eifgjÛÜgic¹qeagTq~ lnyzjcgRgt jcea¹q>jeRbmlncydh^¹qCacw<bdÆdwnhdgRceiagab1j[gayzc P
lnc >fgah^deazyd>jln~>w<cf)R yzgnÊmf~CgR#Æmln¹qb dgRbdcgaffCw<h^gafÀjÅydhdÛÜw<c¹qgtmlnhzf~Cl3)l) >jenÊm~Ò g tzbzcgaff>w<hdg¡~ C¹qbxeimlnhdigqlnaw<yzfj[QRVydg¡zg~Il )l) >je¡fgayd~>g_ÐÜgah(fydbdbQw<f[lhVj3bmlncÀÛëln>jga¹qhdgij·c>à<Czg~Il fydcÀÛëlnag¡dyceaagibzj[giydc#Ógifjdw<hdhdeig_bmlnc¡~ gt1bdcgaffCw<hÐ » PVO ¼kÓÖÊÏw H Zd = 1/Ye
gifj ~Ò >¹¡bQeadlnhdag^lniw<ydfÀj[]RVydgZdy¹qCicw<bdÆzw<hdgRea¹qgijj[giydcTj O ½l P3 lnyzj[cgbdlncjaÊ~ C¹qbxeimlnhdiglnawnydfj]RVydgdgaf^¹¡>acw<bdÆdw<hzgaf eij[ln~Cw<hzf^zg~Cln|Qw<c[ljw<Ccg_eij[lnhkj j[câif
à<clnhddgÊ lc Cfq<eacmgRVydg_~ gaccgiydcfydc~>gafceafyd~>j[lj[f¡gahdà<gihddceag_bmlc~ lnbdbzcw=t1C¹¡lj[>w<h Zc ≈ ZV
" ! "# "#$ _Gv
gafÀj>hzÛÜeacCgayzcgTq¡¼1ʼ<¼ O ¶gah"|mlnffgafÛÜceRVydgihdagif P Çågjjgºlnbdbzcw=t1C¹¡lj[>w<h bxgiyzjdw<hd ijcg·aw<hdfCdeiceigaw<¹q¹qgtfy5uqflnhkj[g·aw<¹qbzj[gRjgahVy"zg·~ClbdceiaCfCw<h cga#Ædgic#Ædeig3mlnhzf~>gafej#ln~>w<hdhmlnàngaf PÚ lncGln>~C~CgiydcfiÊi~Ò g t1bdcgiff>w<h·dg EV
Êçlnbzbmlnc[lCfflnhkjGmlnhdf-iga~C~>g dg ZVÐëe4R P » P ½<ÓgifjG#Ædw<>f>g gah@ÛÜw<hdijCw<h
dgÅ~Iltàkl¹¡¹qgdgÅÛÜce4R yzgahdigafåg t1bd~Cw<j[eiggijÏdgafÏd>¹¡gihdf>w<hdf/dgifiw<ydbd~>gaydcf±yzj[>~CCfeaf P ±y¡~>gaf d>¹¡gihdf>w<hdfdgafaw<ydbz~Cgayzcfy1j[C~>CfeafgijR~>gafà<ln¹¡¹qgafdg·ÛÜceRVydgihdagif@gt1bd~Cwn>j[eigafiÊx~>g3de<ga~>w<bdbQga¹qgahkjR¹qw1dln~Ð » P Y Ólydhdg·aw<hkngacàngahdig!lnffg º~>gahkj[g Egih_g xgijaÊq O ¼<¼<¼ ê Êx>~5Ûëlnyddcln>jàklcdgic O ¼n¼ O ¼<¼ºjgac¹qgafbxwnydcl)nw<Ccydhdg@bdceaa>f>w<h qqÞ#ÆdcgafÅf>à<hdm)lj>ÛÜffydcÅ~Il3ln~>gaydcdg EV
j » Z l PÑGl ln~Cgiydc@w<|zjgahVydgl)nga~>g!deinga~Cwnbdbxgi¹¡gihkj q aw<ydcÀjjgac¹¡g!bmlncRjc[lnhdfÀÛÜw<c¹qeagqdg¡Ñ6lnbd~Ilag"Ð » P ¸kÓ
gafÀjj[câifÅbdcw #Ædg@dgRagi~C~>gt)ln~>ayd~>eagUqbmlncÀj[>cådy^deinga~>w<bdbQga¹qgahkj¹¡w mln~ P Ì h^àklncmlnhkj O ¼<¼ O ¼<¼jgac¹qgafmlnhdfå~Cgzei<gi~Cw<bzbxgi¹¡gihVjŹqw mln~9Ð » P Y ÓÖÊ1iga~Cyz i ixgifjjcâifbdcw #Ædgtdy zei<gi~Cw<bzbxgi¹¡gihVjÐÒ» P ¸kÓ q½3#Æzxcgaff>à<hdd)lj>ÛÜfaÊzlny^dgaffydftzg O ¼ ê @bQw<ydcdgaf¹qCacw<bdÆdwnhdgafjß¿ bxgÑGÉ O Ú gjén¼ ê bQw<ydcÅdgifÑGÉd» Ú P Ñ6gdeinga~>w<bdbQga¹qgahkj ÐÒ» P ¸<ÓRbQgay1jºdw<hzj[cg¡yzj[>~C>femlnhzfj[wnyzj[gq~Il àkln¹q¹¡gzg¡ÛÜce4RVydgahdigafºg t1bd~Cw<j[eig¡~>w<cfdgafÅej#ln~>w<hdhmlnàngaf P Çågijjgeiac>j[yzcgRdg EV
fgi¹º|d~Cg@~Clº¹¡>gayBt lnmlnbzjeag·lny bdcw<|d~Câi¹¡g|j » Z l P̱h mh5ÊÅ~Il²)l) >jeÛÜw<c¹qeagbmlnc ~>gaf ¹¡>acw<bdÆdw<hzgaf gij"~Cgiw<ydbd~>gaydciÊaw<yzbd~Cgiydc d>j^dgmà<clnhd´<w i
~Cyd¹qgn(w<y§aw<yzbd~Cgiydc¡z>jKmgah w<hddgifbd~Clnhdgafon<ÊÏbQgay1j j[cg^awnhdf>deaceagZiw<¹q¹¡g ydhdg")l) >je"dg ÛÜw<c¹¡gm`RVymlnf i ¿ ~CChzdcQR yzgn_ÛÜgac¹qeag q^fgafgt j[cea¹q>jeaf P Ñ-gafaw g4uqa>gahkj[f G1
gij G2#Ædw<>fCf@fw<hkjºdw<hz¡igayBt
aw<ccgifbQw<hdmlnhkj qagjj[g·à<eiw<¹¡ej[cCgtmlhdfÅ~Cgtj#l|d~Cgalny^ PQO p
G1 =8R + πR2
π(2R + 1)2gij G2 =
R3 − 6R2
3√π(2R + 1)3 ,
Ð » PVO?O Ó
wIH R gifj~CgRclnbdbQw<cjÅ~>w<hdà<ydgiydcfyzczIln¹qâijcgtdy^i¿ ~>Chdzcg P S-."$%&1J8-./:9%36 8-@#0."?#%3,0A-K@#*F 2&* 8G/:;- w 8' FF _3lbdcâifå~Cgc[lCfwnhdhdga¹qgahkjdg lc CfÏgt1bQw<fetdlnhdfå~>gbdlnc[lnànc[lnbdÆzgbdceaieadgihVjaÊz~Ò >¹¡bQeamlhdaglniw<ydf i
j[QR yzgdgj[clnhdfÛÜgicj zg~Cl)l) >jegt1aj[eigbdlnc/ydh!¹qCacw<bdÆdwnhdgea¹qgijj[giydc 5 lnd¹q>jj[lnhdig YegafÀjfgahdfC|d~>g i
¹¡gihkjåeiàkln~Cg q ZVP ÝChdf Êk~ lnd¹q>jj#lnhzaglnawnydfj]RVydgtdgj[clnhdfÛÜgicjådyf¿ fÀj[âa¹qgt¹qCicw<bdÆzw<hdgaf i iw<ydbd~>gaydc
bdcgihmlnhkjºgihiw<¹¡b1j[gq~Il"zCffCbml j[Cwnhzydglny.tg gj[f·j[Ædgic¹q]RVydgaf·mlnhzf3~Il )l) >jenÊGhdwnjeag YT1fi eiacj)Ê
mlnhdfÅ~>gR)lnfÅwIH ~CgR#Ædln¹¡b^dg@bdcgaffCw<h^gafÀjydhd>ÛÜwnc¹qgnÊYT1 =
1
ZV+ Ye + Yr ,
fw<j)Ê15lbdcâif~Ò g tzbzcgaff>w<h"zg ZVÐÒ» P ½<Ó
YT1 = jωV
γP0CTH + Ye + Yr ,
Ð » PVO »<Ól)<ga
CTH =γ
1 + (γ − 1)EV, Ð » PVO ékÓ
wIH EVgafjdwnhdhde@bmlnc·ÐÒ» P ¸<Ó P
ÇågijjgÛël w<h5 gt1bdcC¹qgac~Cgafg xgijfåj[Ædgic¹q]RVydgifåfwnydfå~Cl·ÛÜw<c¹¡g5 ydhdgtiw<ccgaijCw<h¡gafÀjagi~C~Cgcgj[gih yzgbmlnc~Clºhdw<c¹qg·ÇÌ ËÞ O ¼n¸ Z`i »j»<»l6bxw<yzcydh"#Ædln¹¡b dg@bdcgaffCwnh"ydhz>ÛÜw<c¹¡g PËß~mbxgiyzjÏbmlnc[l Cj[cgawnhVjc[lnzCijw<Ccg RVydgÅ~ lnd¹q>jj[lnhdig Ye
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gafjqyzjC~>Cfeigmlnhzfq~>g)lfqw H§~>gaw<yzbd~Cgiydc!gafjydh²awnydbd~Cgiydcdj mdg à<clnhd(nw<~Cyd¹qgn<Ê9~Cgjgac¹qg dw<¹qChdlnhkjºmlnhzfagijj[g g tzbzcgaff>w<h§gifj!dwnhd agi~Cyd
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iw<hkj[gahdlnhkjt~Cg@<wn~Cyd¹qg V ÊE~Cgiflnz¹¡jj#lhdagifzgaf¹qCicw<bzÆdw<hdgifl)¿<lnhkjRln~>w<cfÅydhdg·ChBmydgahzag3ceadyz>j[g·fyzc~>gceafyd~>j[lj P Ñ-gaflnd¹q>jj#lnhdigaf±dgifå¹qCacw<bdÆdwnhdgaf±fwnhVjdw<hzyzjC~>Cfeigaf±mlnhzf~Ò g tzbzcgaff>w<hÐÒ» PQO »<Ó/dg YT1mlhdfÅ~Cgt|dy1jÅ5l)nw<Ccydhzg@lnhmln~>w<à<Cgl)<ga@~Ò g tzbzcgaff>w<h(ÐÒ» P »<ÓdgR~Òlnz¹¡jj#lhdag YT0Êd~ClºdxeacgahdigRgahkj[cg
~>gafÏdgayBtgt1bdcgaffCwnhdfeij[lnhkjå~>gjgac¹¡gaw<ccgiij[Û CTH1>f[lnhkj)qbdcgahdzcggihawn¹¡bzjg~Cl·zCffCbml j[Cwnhqzydg
lny.t g xgijfjÆdgac¹¡QR yzgaf PÇågåbdcChdiCbQgzgaw<ccgiij[>w<hjÆdgac¹¡QRVydglnbzbd~CQR yzeagÏdgåÛël w<h3¹ºyz~>j[>bd~C>)lj>ng±lnynw<~Cyd¹qgåzgå~Il)l) >je
gifj9lnmlnb1j[e±mlnhdf-igijj[g±¹i¹¡g hzw<c¹qg lny@)lfGw H·>~¿@lbdcw<bmlnà<lj[>w<ht5 w<hddgif9bz~IlnhdgifÏÐë~>Cà<hdg qaw<hdfÀj#lnhkjgafceabmlncÀj[>gaf#Ó P ѱ lnd¹q>jj#lnhzaglnawnydfj]RVydgzgjc[lnhdfÀÛÜgacÀj YT1
dyfÀ¿ fj[âi¹¡g¹¡>acw<bdÆdw<hzgaf i aw<yzbd~Cgiydc±bdcgihmlnhkjgih"iw<¹¡b1j[g@~Cgifg xgij[fÅj[Æzgac¹q]RVydgifgafÀjtln~>w<cfåeaac>jg·dlnhdfÅ~Ilºhdwnc¹qg
YT1 = j sin k0`
[
S
ρ0c0CTH +
ρ0c0S
YeYr
]
+ cos k0` (Ye + Yr) .Ð » PVO=Z Ó
Çågijj[ggt1bdcgiffCw<h_Ð » PVO=Z ÓDgifjaw<hdfÀj[cyd>jgzgÅÛël w<hq·iw<hk<gicà<gic<gicfÏ~ gt1bdcgaffCw<h_Ð » PVO »<ÓDgih¡|dlnffgafÛÜceRVydgahzagaf P Ì hg xgijaÊ5RVymlnhd k0`
dg CgahkjbQgij>jaÊ gihÛëln>f[lnhkjyzf[lnà<gdgafåcga~Clj[>w<hdf ρ0c20 = γ/χT ≈ γP0gj V ≈ S` Êm~ gt1bdcgaffCwnh(ÐÒ» PQOZ Óådw<hdhzg
YT1 ≈ j k0`S
ρ0c0CTH + j k0`
ρ0c0S
YeYr + Ye + Yr ≈ jωV
P0γCTH + Ye + Yr ,
RVyd6awnccgifbQw<hd^|dCgih q"ÐÒ» PVO »<Ó PÝ hdwnjgac RVydg~Clhdw<c¹qgÇÌ ËÞ O ¼<¸ Z/i »qbzceaiw<hd>fg!5 lnbdbd~>]RVydgac~Iliw<ccgaijCw<h CTH
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eij#lhVjtydhdgºiw<ccgaijCw<hZdg3<wn~Cyd¹qg PÚ w<ydcÏlnyzj[lnhkj)Êk~Cgjgac¹¡gÅgah YeYr
eij[lnhkjÏbxgj[>j±bdlnc±c[lnbdbQw<cÀjÏlnyBtqlny1j[cgifaÊkhdgÅbmlflbdbd~CQRVydgac ~Cltaw<ccgaj[>w<hCTH
qºigtj[gic¹qgtCh5dydgahdigRbxgiy ~Cgceafyd~>j[lj P ѱ gt1bdcgaffCwnh"dg~ lnd¹q>jj[lnhdig YT1cgijgahVydg@CigifjÅdw<hz
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j[ceag Zte ydh(jyd|Qgi¿ ~C>hddc]RVydgwnyz<gicj qZfw<h lnyzjcgg t1jcei¹¡j[eÐÜC¹qbxeimlnhdig¡jgac¹qChdln~Cg!hVyd~C~>gçÓÖÊ9dg
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fyzc·~>gdea|z>j qte gahkj[clnhkjdlnhdf@~Cgj[yd|QgnÊzfa eaac>jSj \ ÊÀ»né`lZte =
pte
qte= Zt0 tanh j kt`t ,
Ð » PVO ½<ÓwIH ~ C¹qbxeimlnhdigt)lc[lnj[eacCfÀj[]RVydg Zt0
gifjdw<hzhdeag@bmlnc
Zt0 =ρ0c0
πa2t
√
(1 −Kv)[1 + (γ − 1)Kh]
Ð » PVO ÞkÓ
gijÅ~>gRhdw<¹º|dcg@5 wnhddg ktbmlc
k2t = k2
0
1 + (γ − 1)Kh
1 −Kv, Ð » PVOY Ó
l)<gaKh,v =
2
kh,vat
J1(kh,vat)
J0(kh,vat). Ð » PVO\ Ó
Ñ-gafÅhdw<¹º|zcgafÅ5 w<hdzg@jÆdgac¹¡QRVydgRgijÅ CfR yzgayBtxÊEcgifbQgaij>nga¹qgahkj khgij kv
Êdfw<hkjdw<hdhzeafÅbmlnc
kh =1 − j√
2
√
k0
`h
gj kv =1 − j√
2
√
k0
` ′v, Ð » PVO ¸kÓ
`h = λ/(ρ0c0CP ) gj ` ′v = µ/(ρ0c0)
ej#lnhkjcgafbQgaj[<ga¹qgahkj!~>gaf3~>w<hdà<yzgaydcfº)lnclnijeacCfj]RVydgifj[Æzgac i¹¡QRVydgtgijÅ CfRVydgaydfg PÑGlbdcgaffCwnhq~ gahkjceag 5 ydh ceifg)ly dg n j[yz|xgif¡alnbdC~>~Iln>cgif5 C¹qbQeamlnhzag 5 gahkj[ceag Zte
gifjq~Il¹i¹¡gbdcgiff>w<h pte
RVy5[qq~Ò gihVjceigR5 yzh"fgayd~-j[yz|xg P Ñ-g·zea|djj[wnj[ln~q~ gahkjceag@dgRig@ceafg)lny^gafÀjeaàkln~8q~Il!fw<¹q¹qgdgiftdei|d>j qte qq~ gahkj[ceagºzg3#Æml$RVydgºjyd|QgnÊQfw<j n qte P ѱ C¹qbQeamlnhzag·5 gihkj[ceig5 ydh_ceifgalnydg n j[yd|Qgaf)lbdC~>~Iln>cgafÏ5 C¹qbxeimlnhdig@5 gihkj[ceig Zte
gifjdw<hd@eaà<ln~Cg q Zte/nP
_ _ W()-."/J!#%36 #-., -<:;0<J##-., -.a !,*5E-K@W 8 2feg0A=*+23##*5E-K#-., :9%3"?/G@!J#8-.,|@#*F
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SG/-."$%&>27%&0A- 23*L@#0.,-.!"?- 8- W-., "?*F@!%72323*F%3-.,$C@#@W,0A-#*F!, 2&*x8/:;- w ' FF _hhzw<¹º|dcg n zgj[yz|xgif)lnbzC~C~ClnCcgaf/wnyz<gicj[fdlnhdfå~Clbmlcw<Edyiw<ydbd~>gaydch5 lnbdbQw<cjgahkjydhdgaw<hkjc i|dy1j[Cwnh^fCà<hzmalj[<gRmlnhzf~Cl3ln~Cgiydcdg@~ lnd¹q>jj#lnhdigtlniw<ydfÀj[]RVydg@dgRjc[lnhdfÀÛÜgacÀjgjmlnhdfÅ~Cgifceifyd~j#ljf
5 eij[ln~Cwnhdhmlnà<gRVydg"mlnhdfq~Cgifq|dlnffgaf¡ÛÜceRVydgihdagifaÊåà<ln¹¡¹qg dg ÛÜce4R yzgahdigafmlhdf¡~Cl$RVydga~>~Cg~Clbzcgaff>w<hlniw<ydfÀj[]RVydggafÀjåfydbdbQw<feigydhz>ÛÜw<c¹¡gmlhdf~ClR)l) >j[e P ÝChzf ÊkdlnhdfÏagijj[gàkln¹q¹¡gzg|mlffgifÛÜceRVydgahzagafiÊ~Òlnz¹¡jj#lhdag lniw<ydfÀj[]RVydg^dg j[clnhdfÛÜgicj YT2
dy f¿ fjâa¹qgZ¹ºydhzdg n j[yz|xgif¡alnbdC~>~Iln>cgifºfa eaic>jZÐÜ~Cw<5 lndz>j[>w<h dgifdei|d>jfaÊm>a5fw<cj[lnhkjÅdg@~Il)l) >jeçÓ
YT2 = jωV
γP0
CTH + Ye + Yr +n
Zte
.
tlnhdfydhdgà<ln¹¡¹qg!dg¡ÛÜceRVydgihdagifbd~Cydfea~>gi<eignÊ9~ lnd¹q>jj[lnhdig YT2dg c[lnj j[cggt1bdc>¹¡eig¡fga~>w<h
ydh ¹qw dâa~>gbdcgahmlhVjÅgih aw<¹qbzjg~Ilbdcw<bdlnàkljCw<h¡5 wnhddgafåbd~Ilnhzgafgj~Il·dCfÀj#lnhdigsRVydfeabmlcgR~Cgifjyd|xgifalnbdC~>~Iln>cgifdgifŹ¡>acwnbdÆdw<hdgif P ÇågabQgahdmlhVjaÊmmlnhdfigijj[g@àkln¹q¹¡gdgtÛÜce4RVydgahdigafiÊQ~Cl3bdceafgihdag·dgtjyd|xgifalnbdC~>~Iln>cgifCh5dydgahdig3bQgay_~Cl!ln~>gaydcÅdg·~ lnd¹q>jj#lnhzag@lniw<ydfj]RVydgdg·jc[lnhzfÛÜgacÀj YT2
dy"f¿ fjâa¹qgnÊR yzbQgayzjln~>w<cf j[cg·iw<hdfCdeiceag@e4R yz>ln~Cgihkj[gTq~ gt1bdcgiffCw<hÐ » PVO=Z Ó PÝ>hdfÒÊ9~Ò giccgiydcºiw<¹¡¹qCfggih²aw<hdfCdeic[lnhkj!~Òlnz¹¡jj#lhdag¡lnawnydfj]RVydg dgjc[lnhdfÀÛÜgacÀj YT2
>hddeabQgah imlhVjgRdgt~Cld>fj#lhdagsRVydxfeibmlncgR~>gafÏj[yd|QgafalnbdC~>~Iln>cgif dgaf¹qCicw<bdÆzw<hdgafgafÀjfydbzbxw<feagRbQgij>j[ggij YT2gifjln~Cw<cfeaicj[gnÊmdlnhdfÅ~Ilºhdwnc¹qg|j»<»l Ê
YT2 = YT1 +n
Zte
, Ð » P »n¼kÓwIH~ lnd¹q>jj[lnhdig YT1
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YT =
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j sin k0`
(
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ρ0c0+ρ0c0S
YeYr
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+ cos k0`(Ye + Yr) +n
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∆rad ,Ð » P » O Ó
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Mp1 = Mp0.Cor(P0, P1).Cor(T0, T1) .Ð » P »<»<Ó
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Mp1dB = Mp0dB + (P1 − P0)CP + (T1 − T0)CT ,ÐÒ» P »né<lkÓ
ϕ1 = ϕ0 + (P1 − P0)ϕP + (T1 − T0)ϕT .ÐÒ» P »nén|EÓ
wIHMp0,1 = 10Mp0,1dB/20ej ϕ0,1 ,
gijwIH CPÊ CT
gij ϕPÊ ϕT
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PP (Ω) = a0 + a1Ω + ... + a9Ω9 , PT (Ω) = b0 + b1Ω + ...+ b9Ω
9 ,
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gij CTdw<hdhdgihkjdwnhd
CP (Ω) =CP250
PP (Ω250)PP (Ω) , CT (Ω) =
CT250
PT (Ω250)PT (Ω) , Ð » P » Z Ó
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2mÐëe4R P O?P ¸<ÓÊ wIH~Il
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bxwnydc@~ eij#l~Cw<hdhdlnà<gºgihbdcgaffCw<hbmlnc·ceiaCbzcw a>je P Ì ~>~CgcgibdceifgihVjg~ C¹qbQeamlnhzag!lniw<ydfj]RVydgqdgq~ gah ifgi¹º|d~CgŹqga¹º|zc[lnhdg i #Æmlncà<glnccCâacg P Çågjjg#Æmlncà<glnccCâacggafÀjawnhdfj>j[yzeagÅmlnhdfÏ~ ln>c aw<h dhdegihkj[cg~Il¹¡gi¹º|dclnhdggij~Ò ei~Cgaj[cw1dglnccCâicgnÊVmlhdf~Cgifåj[cw<ydfmlnhdfigijj[gtei~Cgaj[cw1dggij~Cl3)l) >jetlncc>âacg P Ñ-gj[cwny5 eaàkln~>Cflj[>w<hdgbzcgaff>w<hÐNj[yd|QgalnbdC~>~Iln>cg)Óh5 l"ydhc<~>gRVydg¡mlnhdf~Cgaf@j[câaf·|mlnffgafÛÜceRVydgihdagif^Ðëgihdgaffw<yzfdg3»n¼ ê çÓÖÊEÛÜceRVydgihdagiftgah"dgaÆdwncfzg~Cl!à<ln¹¡¹qg@5 yzj[>~CCflj[>w<hÆml|d>jydga~>~CgRdgif¹qCacw<bdÆdwnhdgafeijyddCeif¡>aÐÒÑGÉ O Ú ÊÏÑGÉd» Ú Ó ÏlnChzf Ê ~Cliw<hkj[cC|dyzjCw<h dg^ag^j[cwny mlnhzf~Il#Æmlcà<gZlnccCâacg bxgiyzj j[cghdeaàn~CCàneag P tlnhdf~Cg@)ldcg·5 C¹qbxw<cÀj#lnhkjfÅjc[l)lnyBt^dg·¹qw1dei~C>f[ljCw<hxj » Y ÊÀ» \ l ÊEÄlnf¹ºydffgah_bzcw<bQw<fg·ydhdg¹¡w dei~CCflj[>w<hlnffgawn¹¡bd~>âijgzgagjågihdfgi¹º|d~>g¹qga¹º|zc[lnhdg i #Æmlncà<glcc>âacgfw<ydfÏÛÜw<c¹qgÅ5 yzha>ciyd>jea~>gaijcQR yzgReRVyd>ln~>gahkj PÚ w<ydcDydhdgÅlnbdbz~CCalj[>w<h@j[ga~>~Cg R yzg~ eij[ln~Cw<hzhmlnà<gbmlncDceaiCbdcw1i>j[eÊ~Ò >¹¡bQeadlnhdag (Za)i=0
bQgayzj/fC¹qbd~>g i¹¡gihkj ijcgÅlnff>¹¡>~Ceag q~Ò >¹¡bQeamlhdag5 yzhqiCcaydjDceafwnhmlnhkj±feicCgRÐëa>ciyd>j9jß¿1bQgTmnÄÑGÇ ndÓ9gij/aw<hdfCdeiceigaw<¹q¹qg>hddeabQgahzmlnhkj[gRdgifaw<hzd>jCw<hdfln¹º|zIlnhkj[gifRÐÜmlnhdf~Cl¹qgafyzcgtwIH igafiw<hddj[CwnhdflncCgahkjåbxgiyEÓ PÌ ~C~Cgbxgiyzjln~Cw<cf ijcg@eaicj[g@fw<yzf~ClºÛÜw<c¹qgDp
(Za)i=0 =1
jωCa+ jωMa +Ra ,
Ð » P »<½<Ófw<j
(Za)i=0 =1
jωCa(1 − ω2MaCa + jωRaCa) .
ѱ lnd¹q>jj#lnhzaglnaw<yzfj[QRVydg qa>ciyd>jw<yzngacj Ya = 1/(Za)i=0 ydh¹qCicw<bdÆzw<hdg·bxgiyzj ijcgºeiac>j[g
fw<yzf~Cl3ÛÜw<c¹qgRfyz>lnhkj[g¡Ðëhzw<c¹¡ln~C>feigDj »<»/lIÓ
Ya = jωVeq
γRPR, Ð » P »nÞkÓ
wIH PRgafÀj3~Cl bdcgiff>w<hfÀj#lj]RVydg!mlnhzf~>gaf@aw<hzd>jCw<hdfRfÀj#lnhzmlncdgqceÛÜeacgihdagqgij γR
~Cg!c[lbdbxwncj@dgif#Æmln~>gaydcf¹¡lnff]RVydgif qbzcgaff>w<hgj q·nw<~Cyz¹¡gÅawnhdfj[lnhkjmlnhzfåagifå¹ a¹qgafÏaw<hzd>jCw<hdfiÊkgijwIH~>gnw<~Cyz¹¡g
Veq =CaγRPR
1 − ω2MaCa + jωRaCa
, Ð » P » Y ÓgafÀj lnbdbQga~>e_~Cg^<w<~>yd¹¡gZeRVyd>ln~>gahkj dy¹qCicw<bzÆdw<hdg P Ëß~tcgabdceafgahkj[g~Cg^<w<~>yd¹qg_aw<¹qbd~>gt1g_5 ydhàkl gahzÛÜgic¹qe mlnhdfqydhdg^)l) >je"cCà<>dgnÊ/fyd|d>ff[lhVj¡dgifawn¹¡bdcgaffCwnhdf i zeij[gihkj[gaflndIl|mlj[QRVydgafiÊ dg^jga~>~Cg
v _ W()-."/J!#%36 #-., -<:;0<J##-., -.a !,*5E-K@W 8 2feg0A=*+23##*5E-K#-., :9%3"?/G@!J#8-.,|@#*F
0."<%3@#"<%M0
fw<cjgkRVydg~ C¹qbQeamlnhzag!dgig<w<~>yd¹qg¡iw<¹¡bz~Cgt1g¡dgàkl fwn>jeaàkln~>gkqZ~Ò >¹¡bQeadlnhdagqlniw<ydfÀj[]RVydgdy¹qCicw<bdÆzw<hdg P Ú lc3awnhVngahkj[>w<hdj»n»/l ÊGag¡<w<~>yd¹¡gqgafÀjgt1bdcC¹qegih(ÛÜw<hdijCw<hdgfl"ln~Cgiydcºlny.t|dlnffgafÛÜceRVydgahzagaf Vbf = CaγRPR
gijdg_~Ilbdyd~>f[ljCw<h§dgZceafwnhmlnhdig ω0 = 1/√
MaCadgZ~Ò >¹¡bQeadlnhdag
(Za)i=0bQw<ydcÅ~Il`R yzga~C~>g (Za)i=0
gafÀjceagi~C~>g p
Veq =Vbf
1 − ω2
ω20
+ jωRaCa
. Ð » P » \ Ó
Ñ-g@nw<~Cyz¹¡gRe4R yz>ln~Cgihkj Veq ydh^¹qCacw<bdÆdwnhdgRgafÀjdwnhd@gahkj[>âacgi¹¡gihkjzeij[gic¹qChde@bmlcfw<h^<w<~>yd¹¡g
lny.t|mlnffgaf3ÛÜceRVydgihdagif VbfÊ-~Cl^bdyz~Cf[l j[Cwnhzgceifw<hdlnhdag ω0
gij~ ln¹qw<cjCffga¹qgahkj Ra P Ñ g t1bdcgiff>w<h
ÐÒ» P »ÞkÓfyzbdbxwnfg RVydg~ lnd¹q>jj#lnhzag YagafÀj@ChddeibxgihdmlnhkjgdgafRiw<hddj[Cwnhdftln¹3|dIlnhkjgafaÊxga~>~Cg3gifj@g tzbzc i
¹qeag bmlcaw<hk<gihkj[CwnhHj »<»/liw<¹q¹¡g ~Òlnz¹¡jj#lhdaglnaw<yzfj[QRVydg dy <w<~>yd¹¡gaw<¹qbd~Cg t1g VeqÐë~Cyz i ¹ a¹qg
>hddeabQgahzmlnhkjdgafaw<hzd>jCw<hdfln¹º|dClnhkj[gaf[Óåmlnhdf~Cgifaw<hzd>jCw<hdfåfj#lhdmlnc"dg@ceiÛÜeacgahdig PÇågijj[g!gt1bdcgaffCw<hdg!~ C¹qbQeamlnhzaglniw<ydfÀj[]RVydg ydh¹qCacw<bdÆdwnhdgºh5 gafÀjRVy5 ydhdgqlnbdbzcw=t1C¹¡lj[>w<h5Ê
jw<yzj[gÛÜw<Cf6fy5uqf[lnhkj[gåbxw<yzc9~>gÏaln~Ciyd~VdgÏ~ g4uq)lni>je5 ydh¹¡>acwnbdÆdw<hdg±bmlc9ceaa>bdcw aj[e P Ëß~ gifjDagabQgahzmlnhkjq hdwnjgacRVydgigijj[glnbdbdcw=tz>¹l j[Cwnhgafj Czgzgfgahdf"mlnhdf igacj[lnChzf^alnf P Ì h g xgij)Êej#lnhkj <eicmeRVydg·~ g4uq)la>jeRgah"bdcgiff>w<h^5 yzh"¹qCicw<bdÆzw<hdgRgafÀjzeabQgahdmlhVjgzgafaw<hdz>j[>w<hdfÅln¹3|dIlnhkjgafdgRÛël w<hfCà<hdm)l j[>ngj » Y Ê »n¸`lÒÊ C~Qfgacln>jåln|dfydczgt5 ydh bxwnChkjåzg ydgj[Æzeaw<c]RVydgzgÛëlnCcgyzf[lnà<gdgt~Ò g tzbzcgaff>w<hÐÒ» P »n½<ÓDdg (Za)i=0
Ðë>hddeabQgahzmlnhkj[gÅdgifiw<hddj[Cwnhdf ln¹º|zIlnhkj[gif#ÓDmlhdf±~ gt1bdcgiffCw<hZÐ O?P[\ Ó9fga~>w<h¡~Cl$RVydga~>~Cg~Ò gu¡alnaj[e Mp
gifjt>hk<gacfgi¹¡gihVjtbdcw<bQw<cÀj[Cwnhdhdga~>~Cg qq~ C¹qbxeimlnhdig (Za)i=0P hdg·j[gi~C~Cg·lnbzbdcw #Ædg3bQgay1j
igabQgahdmlhVjijcg·gahk Cf[là<eag3l mh^5 wn|dfgic<gicsRVymln~>>j#l j[>nga¹qgahkjÅ~ ChBmydgahzag@dCcgaj[g·dgafbmlncln¹¡âj[cgafdy¹qCicw<bdÆzw<hdgRfyzc~Cl3ln~Cgiydcdg@fw<h^g4uqalnaj[e PÑ-gRcgabQw<cÀjÅdgafgt1bdcgaffCw<hzfºÐ O?P ¸<ÓÊ6ÐÒ» P »ÞkÓågijÐÒ» P » \ ÓåmlnhdfÅ~Ò g tzbzcgaff>w<h(Ð O?P[\ ÓådgR~ g4uqalnaj[eRdw<hzhdg
Mp = − U0
Sd0
Veq
γRPR=
Mpbf
1 − ω2MaCa + jωRaCa
, Ð » P »n¸kÓRVyd6h5 gafÀj RVy5 ydhdglnbdbzcw=t1C¹¡lj[>w<hgijwIH Mpbf
gafj~Il3ln~>gaydczg MplnyBt|mlnffgafÛÜce4R yzgahdigaf P
Ý>hdfÒÊzgah^w<|dfgacÀnlhVj~Il3ln~>gaydczg·~Ò guq)lni>j[e MpÊmC~5lbdbmlncln>j RVydg p
~Cl3ln~Cgiydcdy <wn~Cyd¹qg@lnyBt|mlnffgafÅÛÜce4R yzgahdigaf Vbf = CaγRPRgifjdw<hzhdeagRbmlc |Mpbf | ~Il3ln~>gaydclyBt |mlnffgifÛÜceRVydgahzagafdy^¹¡w dyd~>gtdg Mp
Ð mà<ydcg» P[Z ÓÊ ~Cl_ÛÜceRVydgihdag^dg ceifw<hdlnhdag F0 =
1
2π√
CaMa
gafÀjl jj[giChkj[gRVymlnhz§~Cl_bdÆmlnfg"dg MpgafÀjzg
¸n¼ EÐ mà<ydcg3» P ½<ÓÖÊ ~Òl¹¡w<cÀj[>ffgi¹¡gihkj Ra
Êziw<hdhmln>fflnhkjÅ~Cgafnl~Cgayzcfdg Vbfgij F0
Ê1gafÀjÅdeadyz>jdy clnbdbQw<cj |Mpbf ||MpF0
| =
ω0RaCaÊ |MpF0
| eij[lnhkj~>gR¹¡w dyd~>gtdg@~Ilºln~>gaydcdg MpbQw<ydc f = F0
Ðmànydcg·» P Z Ó P 0<J##-x8-x:;-.,= 8-L#%3-."$-Çågijj[g ¹¡ej[Ædw dgÊWqbdceifgahkj¡w<|dfw<~Câj[gnÊDgafjqfw<¹¡¹¡ln>cga¹qgahkj3deaic>jg mlnhdf!~Cl_hdw<c¹qg^ÇÌ ËºÞ O ¼<¸ Z`i
» j»<»l±gijeij#l>j·yzj[>~C>feag!bmlc·zghdw<¹3|dcgiyBt~Iln|Qw<clj[w<>cgiftdg¹qeijcw<~>w<à<>gnÊiw<¹q¹¡g!~ ËÍÈtÎkj é<¼`lÒÊ-bxw<yzc,2465#%27!$5*,#%&:=HKCF!%#$5 5+*, !1 d 274#%*Z>27-0!@L#$Cd #%-0!&;:HK2-*#%+C-/C#@27,;=>2#%2> !$C&#$*Z$#c b5,GHKC&I0G*
ba27,-0!]L627#D$2)ba27,G#D27G=` 4/2$$&de#%C&I0G-6>F&Vbf
;TZ2de#%C&I0G&-/>FE:=^#%C&$5-627-/>F0 = ω0/2π
F!D,Yd 2> !$&G#D:=EI0G/2 * !$CQ = ω0/RaCa = 1/Ra
√
Ma/Ca
M 5G Dd 2> !$&G#:=YL6&#"!%&1/Q
X n : !"!%i-57!%* 5- :=Dd 2>F!$G/#:=YI0G/2 * !$C;/>@?5*Z$* LB5G#:=&>5HKHK5=:=*S!%C&:=9-/57!%27!$*,5-;<- 3 2JL/2 b #@27*,HK-0!i:=9"&-/DL? 8="*ZI0GKL65G#i>2#%2> !$C&#$*Z$#G/-9* HKLBC&:2-/>F92>F5G6f!%*,I0G;>$3 &"!1L65G#@IG/5*<,,]- 3 f!1L/2#$!$&-0G/D:/27-/,L#%C&$-0!1:=5=>FGHK-0!n / -/"!$* !$G=! 2a!$*,5-627:= . C!$#%5,5+*
"# # " " " v '
102
103
104
−42
−41.5
−41
−40.5
−40
−39.5
−39
−38.5
−38
−37.5
−37
h/#$CI0G-/> M X
|Mpbf |
|MpF0|
|Mpbf |/|MpF0|
F0
 » P Z ^ÎqÐ Û O Ú Ó!"$#%4u'& &)(*+,-#.0/"¹'(1&)32dÆ03/4*ß¿253ÑGÉd» Ú P
102
103
104
60
80
90
100
120
140
160
180
h#%C&I0G-6>F M6 X
789: ;
F0
 » P ½ Ú Æ =< Ð0aàVÓ>"?#.u@& &A(*B+,-#%/"¹@(&A=2dÆ3/C*ß¿D25Ñ9Éz» Ú P
vG__ W()-."/J!#%36 #-., -<:;0<J##-., -.a !,*5E-K@W 8 2feg0A=*+23##*5E-K#-., :9%3"?/G@!J#8-.,|@#*F
0."<%3@#"<%M0
+ *B)¹'(1/) < 2 ¹¡â *B < CaÊ Ma
A Ra>?#.(>¹@25+A /&A q &A(1 &A(*63 ) * (Za)i=0
0 < ¹'(1&)32dÆ03/ </ 253&A" << )/ *B(1)1a¹'A/ * P( P #" #
Vbf " "!
/M¹'(1&)32zÆ3/ < * 0*B(1( < + A/ +i¹@ **B) / < / &A3201A0 2 + Iq &A * )* 3/ * ?# D*Bt*B +a¹'(*+ < *Û )¹'+A 2 1@¹'(1&A 32dÆ=/ 03/ *4(5Û 0*¹@ < A4 31z¹@"+R ( 1)/ *aÊ &AA( i &A(/-#%+)* / * 2 <C 1(C¹'A/ *+ P / *B=( < (>âa¹'t¹'(1&A 32dÆ=/!Ð)/"à=+A/+) /^¹'(1&A 32dÆ=/ @¹' < 0"21 <2 *B(* R "1 < A.t 0* < ÓÊ ) / *q 2 3( *B+) C < * 0*B(1( < + &A<¹q¹@4+A&)A20*AÐ mà3 » P5P Ó P Ñ #.(>¹@25+A /&A+A1)&)*(]R >* / < Û A * ZE = ur0/ie
¹' < 0+A>A/ *B !1 < )Bt¹'(1&)32dÆ03/ <2 )¹')* "+ *B)¹'(1/) 1,2 D0(* < u'& &)(*B+ < < A.t^¹@(&A=2dÆ3/ < Ð+4R PO$PQOY Ó p
MpeMpr = ZEYT .
Ñ à ¹q¹' 0Û +4R A/&) < + * / * &#Æ3( < (1 *B)11 R > 20 < < (3/ &)3 < *(]R < * < 202 < +A /0(>Û 3¹@)* 1 < )* < *[Æ0A¹'(]R < 2 *B(* < Ê ?# z¹@(** /&A &A3 < *B(QR0 * / < Û A * YT
25AD* 13 < *B 3//0+AnÊ # 20â < ?#. tD2 << (3/"Ð » P »<ÓÖÊ3&A<¹'20*>*B)/' < +R *(13/ < ÐÒ» P » Ó *Ð » P » \ ÓÖÊ < = < Û 3¹@
γP0YT
jω≈ Vc + Vf + Vfe + Vfr + (Vbf + Vbfe + Vbfr)
γP0
γRPR, Ð » P é<¼kÓ
A& < /0=* *(13/ < + /(1 < / < &) 2 à3 2dÆ * +0* 2 à3 2dÆ » P » Ê )* IH 1 < 3d¹' < Vf
)* Vbf
< 3/ *C A *B(Û < ¹@(&A 32dÆ3/0 m¹@ < +n P
. *,>#$5L?5- aHE"G/#$C
Vf
. *,>#$5L?5-/#%C&>L=!%G#
. *Z>F#%5L?5-CHKF!$!$&G#
ur0
ie
; *ZHK*,>#$5HKCF!$#%*ZIG/
Veff
[ *Zf!%5-
. *Z>F#%5L?/5-#%C&>F&L=!$&G#
. *Z>F#%5L/?5-C&HE!"!%G#
u′
r0
i′eY*B` Y 08`
 » P Î < 3d¹', A&)*(>Û Bt =<< < Û +R )/&A < P / < / < )&A3/0 *Bi¹@2 < Ê 1¹'(1&)32zÆ3/ m¹' < +=n < * i¹@2 &A+ 2 / 2( < *B3/ ¹i¹@( ¹¡â *B )*·¹ /( #./0 ( < ¹'(1&)<¹'+)*(]R Ê &)ABt i &A( +)* / * + * 3//+ < -#.0/ 253(1/ *, ( i¹@)/ < (3//) P Ñ #.(>¹@25+A /&A!+)1)&)*B (]R *B / < Û ) * Z ′
E < * 3 < ¹@ < +A )* 1!2( < *3/ < * ¾ < *+
¹ /0(CâA rq + ( < A $#%+aà 1(*B+! < 0ABt@(C¹'25+A /0&A < ZE * Z ′
EÐ mà3 » P EÓ P Ñ =1d¹'Åà3+A i
¹@+ *B (]R Dq@$# / *4 2( < *B3/KR0( 25A¹')* &A)* *B +ià (*B+ < (C¹'2 +) /&) < +)1)&)*B (]R < < *,/=*+Veff
P Ñ ,2(*! < 4u'& &A(*+ < MpeMpr+)* / * (1/0+A25A/ / * < &)3/(*B(1=/ < / < 1 < R A1 < (1
< *C0+)*B)¹'(1/+Ê(1 (A/ *MpeMpr = Z ′
EY′T = ZEY
′T = ZEYT ,
"# # " " " vGv
&AUR (3//
γP0Y ′
T
jω= γP0
YT
jω≈ Vc + Vfe + Vfr + (Vbfe + Vbfr)
γP0
γRPR+ Veff .
Ð » P é O ÓÌ / +aà / * < tD2 < < (1=/ < ÐÒ» P é<¼kÓ>)*3ÐÒ» P é O ÓÊ( (1)/ *
Veff = Vf + VbfγP0
γRPR. Ð » P ék»<Ó
Ñ 31d¹' à3+)<¹@+ *B (]R Veff¹' < 0+ q ?# / * 2( < *B3/ )2+ < )/ *>=/& < <¹q¹' 31z¹@
Û 3/ * Vf0 ¹@(&A=2dÆ3/m¹' < +=n * < 3/ =1d¹' +R ( 1)/ * Bt =< < < Û +R A/0&A < Vbf&A3 (Cà=+ Û &)*A γP0/(γRPR)
P Ç> 3d¹'ºà3+An¹@+ *B(QR Veff < * 225A+ 1 31d¹' A& *B(Û
¹@(&A 32dÆ3/0 / < 1 < &)3/(*B(1=/ <4 ¹ ( / * < P Ñ 31d¹'Û 3/ * Vf+)* / * &A3// 202=tD( i
¹ *( a¹'A/ *^Ð !» P é PQO ÓÖÊ 3d¹' Bt =< < < Û +4R A/&) < Vbf ¹'(1&)32dÆ03/xm ¹@ < +n < *
13 < +A0(* " ¹@ < 0 31d¹'Rà3+)<¹@+ *B (]R Veffq $# / * 20( < *3/ P
(1( P #$ F0ѹ'(1&A 32dÆ=/ m¹' < 0+n < *2 &A+ q $#%t*B +a¹'(*B+ #./ *B 5 ¹ a¹' ( ¹¡â *BÊ!$# 0*
t*B +a¹'(*B+C * C+)* / *åÛ A¹@+)42 >/ * / < & *B)!+a¹')* *B)RÐ 4u'& &)(*+4&An¹@2tD4&)3// )Ó5tD+ < / 20( < *3/¶¹ /(,-#%/ ( < ¹'(1&)<¹'+)*(]R +)* 13//0+A P Ñ #%4u'& &A(*+ A/ 20 < < (3/ 0¹@(&A=2dÆ3/, +A&)A20*A3Ðë¹'(1&)32dÆ03/|m¹' < +=ndÓ < #%+A&)(* (&A(±Ð +R P8O?P \ Ó
Mp =ur0
pr=ur0
ue
ue
pe
pe
pr,
IH ue < * *B)/ < (3/ 22(]R +A .t =/ < *B / < &)*A+a¹')* *BA0 * pe
2 << (13/ &)3 < i*B(]R UR -#%(1-(>¹@25 < q'?#.)/ *B+)" * 5 PÑ * / < & *B) +a¹')* *BA0à3+A/dâ) /0 3/ &)3 < *B(]R q 0/Û +4R0A/&)2 D&#Æ0 Û + iR0A/&) 0 + < 3/ /&A < 225 < +) +)&AA2D*BA0 P Ñ ( < < * ¾ < *+A ¹ /(>âA Kq &)LR 2 << (3/ < ¹@i¹ / @ +A&)A20*AiÊ0)* =/&C *A/ < (13/q"&A(1 &A(* 3 ) *)q < < 3 / < ur0
Ê **A(Cà=/!/!¹ tD(C¹ d¹æÐ + < 3/ /&) *B 5çÓ P Ñ < 2 << (13/ < Bt tD*+i¹@(*B+ < 0 * 5 < 3/ * 3 <A/ R *B0 )* 2zÆ =< 0 pr/pe
< * Ÿn¼ P Ñ 2dÆ =< 22 = * ue/pe+)* / * &A3/0/0·ÐA1
< * +A0(*B>! 2dÆ =< !!?#.u@& &A(*B+!&)<¹'21 t0 *B / < &)*A +i¹@ **A < 225 < +) &A3/0/0çÓÊ 2dÆ =< C0 ?#.u'& &)(*B+ )/ 2 < < (1=/ Mp
< * 3 < +A0(*B ¹' < ur0/ueP Ì / +A25+)* / *
$#%32 +) *(13/ 253"( +)A/ * < Û +R )/&A < 3( < (/ < $#%/ @$# 0*B nÊ A ' Û +4R A/&) + < 3/ /&A ¹'(1&A 32dÆ=/nÊ62 = R0A1 Mp
/ 2zÆ =< ¸n¼ dÊ < * 0+)*B)¹'(1/+) 2 (1/ *B)253 *(13/ P
(1(( P " 9 RaÑ 1)C0 $# ¹'3 *( < < a¹'A/ * Ra
< * +)(*B ?#. t020 < < (3/ Ð » P »n¸<Ó $#%4u'& &A(*B+ MpP Ý
Û +R )/&A "+ < 3/ /&A F0Ê |Mp| = |MpF0
| *
Ra =1
ω0Ca
|Mpbf ||MpF0
| ,Ð » P é<ékÓ
1 < ¹@ < |Mpbf | * |MpF0| +)* / * +)*A¹@(/+ < à= 2dÆ0(]R a¹'A/ * Ð mà P » P[Z Ó)* A& Ca =
Vbf/(γRPR) Ð P Ó P( P " "! Ñ #%+A25.R0 &)*B0A1 < *km &#Æ /0(CâA n A/ *B)¹' -#.+ 30*B(3/ < *A&#Æ/(QR < !¹@ < < &)3 < i*B(]R < P Ý4(1/ < ( Ê < *A&#Æ/0(]R < 0¹@ < >&)(*B+) < 20+A&)+Ai¹¡¹'A/ * < 3/ * & *B)11i¹@)/ * /3/0/+A <& A1 <!< 30Câ A/ *C/&)A * (/ /0<¹ #.(/ *A <à *B(1=/ < P
v _ W()-."/J!#%36 #-., -<:;0<J##-., -.a !,*5E-K@W 8 2feg0A=*+23##*5E-K#-., :9%3"?/G@!J#8-.,|@#*F
0."<%3@#"<%M0
Ç>3/&)A/ / * + *BA¹@(/ *(13/" 3d¹@ .t =<< < Û +R )/&A < Ê==0*B ÏÛ (*8R > *A&#Æ/(QR < < *(*0*B(3/ < * / 2 D&)+A+,&) 0*B)Bt )/ *Bi¹@2 < Ê2 < (A < 2 Câi¹@ < -#%304*Æ+A3 (]R < 2 < A/ * P á630* -# 53Ê < 0ABt 3d¹' < &)<¹'2 + < / < 3/ * 2 < ¹ a¹'/ *B @Ðëà=+A<¹'+)*(]R )* &)3 < *B(]R )Ó P # D*B 2 *aÊ31¹'(1&)32zÆ3/Um ¹@ < +n /-#%+)* / * 2 < 1(>¹@)/ *+nÊ (1) <¹@i¹ / /-#% < *>2 =< t & *Ba¹'A/ *> ¹i¹@rR -#%A/ 0*(1( <B *B(1=/ /3¹ P Ì / /5ÊD?# 22=tD(C¹ *(13/< )13/ R )1 $#%(1/5 0A/&)'0 < * < *[Æ)¹'(]R < < * /+aà3(Cà3 nÊ-&)<¹@2D*B *B)/ ' à ¹q¹@ºÛ +4R A/&) < &#Æ3( < (1 )*" < 0(C¹'A/ < (13/ < <¹ (1/0nÊ5/-#. < * 2 =< *3 ¾ 30 < (A/ +)( +A / < 2 *B(QR nÊ 3d¹@,0 &)32A + * / * ( 1 PÚ 30 &ADR ( < *, ¹@ < Û +R A/0&A + < 3/ /&)nÊ- 2 D&)+A+ < * << ( *Bâ < 3 Ðë¹ *B+)(A Ê *a¹'2 < Ó q¹')* *B A/ " Ê--# 0* / *,21 < R -#.( < * =< < (Vu'&A(1 )/ 2 *(]R 0*B 3 )!/ * / < & *B) +a¹')* *BA0>3/ * < & &)*+A( < *(]R < 2 ) )/ * )*B < Bu < ¹q¹@)/ * (1)/&A3/0/0 < P Ì / /5Ê Bt A/ (1 3/ < Û +R A/0&A + < =/ /&) ÐNÛ +4R0A/&) << +A +)çÓÖÊ!1 <3/ < ( 1 < 25A A/ * 3(14/ (1/5A/&) /3/ /+ià31(>à3 01 < ,1 + < * *,)*")1 < / < 3/ *2 = * / * 2 =< 2 ( < < )/'&)<¹'20*B! / < ¹@ < P á630* &)A&A(0&A3/ *B ( 0rq + 1( < A < ¹' < < 0Û +4R0A/&) 4 + < 3/ /&) )*>-# ¹@= *B( << a¹'A/ *RÐ&R ( < *>+A0(*!0 F0
)* VbfÓ R (5( < R )/ *>-# )*B
=<< (C¹'2 +A&A( < < )* 3/0& 2 )m A/ * 1 < n &A<¹'20* *B)/ < ¹'ç¿3)/ < *BA&#Æ0/(]R < (>¹@253 * / * <¹@( < A/ " PÝ4(/ < (ÒÊ 5 &A302 53 *3(1 < ¹@+ *B=1<à3( 3/ * 20+iÛ +)+·¾ < R -#[q 2+ < A/ * < (C¹'(*B)Uq ¹@ < " 31d¹', A& *B(Û Bt =<< < Û +R )/&A < 2 ¹'+)*Æ" < < *B(*B0*(13/5Ê < A253 i* / * Bt 3//+) < &)3/ < *& *B),3 Bt 1) < *ß¿2(]R < 2 304 Û +R )/&A 0 + < 3/ /&A *$# ¹'3 *( << a¹'A/ * PÑ < *B)&#Æ/(]R < @¹' < -#%(C¹'2 +) /&),¹'(1&)32zÆ3/ < 2+ < A/ *+A < / < 1,2 à= 2dÆ0 < ( i / * < 3/ * & *B)11i¹@)/ * < *B)&#Æ/(]R < ¹'( < < A/ " ºÑ6ÈÌ )* A/ < à3 / < /<¹ ABt 53 *B3( < ¹'+)*3<à3(1 P Ç> < *B)&#Æ/(]R < ÊD21 < < (>¹@201 < q·¹@ **B C)/ " Ê < 3/ * < < (521 <2+)&A( < < P
" c0<J88-x%M0A/*B%-Ë < # à=(* 0 & &AAC2 202=tD(C¹ *B(3/ < < &)&A << ( < 1)4 < 2 ¹¡â *B < Vbf
Ê F0)* Ra-#%/¹'(1&)32zÆ3/ -# 2â < A < 3/@4u'& &A(*+C0+)*B)¹'(1/+)C253 < 1) < 220=tD(C¹ *B( <
VbfÊ F0
)* RaÐ 3//+) < &A3/ < *B &)*A 3 1) < 0*B)/0 < 3 < "2+)&A+A0A/ * < +)* 13// à3 < Ó P
Ñ < ¹'+)*Æ < 04+ *BA¹@(/ *(13/'+A&)(*B < / < &A 2 à3 2dÆCtD23(*A/ *>Û (* R0nÊ03 < -#%/+ * 3// à= 2 ! +A&)(12 D&)(*B+C *B 3( < ¹@(&A=2dÆ3/ < ÊD < ( *(13/ < < A < (1/ *(* < 2 30!1 <& & *B+)( < *B(QR0 < Ð Vbf
Ê F0Ê RaÓ -#%/¡¹@(&A=2dÆ3/ /¹'D( A/ * A/@(A/'$#%4u'& &)(*+ 0*B)/04253 1 <
)Bt D*B < ¹'(1&A 32dÆ=/ < D*B(1( < + < / < 2 *B&A3 -#%+)* 13/0/ à3 2 +A&A(2&A(*B+Ð&iÛ P // t0 Ê P » P »<Ó P( P # " #$
Vbf ""!
Ì / =< < < Û +R )/&A < Ê-?#.u'& &)(*B+ Mp < * R =< ( i (1/0+A25A/ / *B 0 F0
)* RaP Ñ 31d¹'"+R ( i
1)/ * Bt =< < < Û +R A/0&A < < *C03/&"+)*A¹@(/+,(1/0+A25A/ ¹¡¹'A/ * "&A < 2 ¹qâ)* < PË < # à3(* "+)*A¹@(/A ?#.u@& &A(*B+,,*3( < ¹@(&A=2dÆ3/ < 2 ¹@+ *[Æ,"+)&A(2&A(*B+ / < < & (*+ < 31z¹@ < ( +))/ * < P Ñ &)<¹'2 ( < =/@ < + < * * < 0*A/ < 253 &#Æ R C& (*+2 )¹')*" +)( 1 < 3d¹@ <" Bt =<< < Û +R )/&A < < *B=( < ¹'(1&A 32dÆ=/ < )/ < ( / * 12 *B&A3 +A&)(*C / < 1 < &#Æ+a¹ 0 mà3 » P Y.PÑ < & &A < /+A&) < <B ( < 253 &A **¹'+)*ÆD0qÐ& 1&A0- < YT
Ê MpdB)* DiffdB
Ó < =/ * )&)*B0+ <2 / 1<à3(&A(A=0& 1&)<R ( )*3 /)q $#%+A&) / / )2+ < )/ * *B(3/3à= 2dÆ0(]R < DiffdB
P Ñ < 1) < (/(*( < 0 < Vbf
< 3/ *6 < 1) < 22=tD(C¹ *( < Ð=//+A < &)3/ < *B0&)*B) = A <
"# # " " " v N
Vbf1AVbf2
AVbf3
ZE12aAZE13a
AZE23a ZE12b
AZE13b
AZE23b
M12a = ZE12a YT12a
M13a = ZE13a YT13a
M23a = ZE23a YT23a
M12b = ZE12b YT12b
M13b = ZE13b YT13b
M23b = ZE23b YT23b
Mp1dBaAMp2dBa
AMp3dBa Mp1dBb
AMp2dBb
AMp3dBb
Diff1dBADiff2dB
ADiff3dB
V ′bf1
AV ′
bf2
AV ′
bf3
; 2 &G#@* -* !$*Z27,&g2LL#%5`=* H92a!%* b&
. "G/#$:2-/ >5GL,G/#12 . &$G#%D:/27-/,i>F5GL,G#g4
>&27Z>FG >27Z>FG/
>&27Z>FG >27Z>FG/
* ABC&#$&-/>Fg:=&MpdBa
F!MpdBb
4/"&#$ba27!$*,5-+#@27L?*ZI0G
; 2 &G#%HK5=:=* ./Cg:=&Vbf
;>@?5*,$*,&27LL/#$5`=*,HK27!$*,b&HE&-0!LB5G#gHK*,-*,HE*Z$#1,&
DiffdB
:
9::
Vbf
: 0
f
 » P Y Î+)*Æ < ( (,2 30 +)*A¹'(1/0A 1 < 3d¹@ < +R ( 1A/ * <C Bt =< < < Û +R A/0&A < P
v _ W()-."/J!#%36 #-., -<:;0<J##-., -.a !,*5E-K@W 8 2feg0A=*+23##*5E-K#-., :9%3"?/G@!J#8-.,|@#*F
0."<%3@#"<%M0
2+)&A+)A/ * + * 3// à=çÓ P Ñ < )Bt')/(>âA < + * 25 < < &#Æ+a¹ Ð < ) *(13/')*Ϲ'D( & *B(1=/ < Ó< 3/ * )&)*+A < ¹ / )11i¹@)/ * 2 / 325+A *B) P Ç>A10( i &)(Ź@( 220=tD(C¹ *B( a¹'A/ *@1 < 1) < < Vbf
q +)(1/ ¾ )&)*B) / < 1@2<à= ¹q¹@' & 1&) A/(Û 3/& *B(3/ < 3/ < A *B(1=/ < A2 + < )/ * *(13/ < à3 2dÆ(QR < < DiffdB
P Ë !&A << -#% A& *B) < ¹'D( & *B(1=/ < < 1 <Vbf
R / (1 < *(C¹'DR 1 < DiffdB
< =/ *·¹'(1/(>¹ 1 < < 0 0/à ¹q¹@ Û +R )/&A < 21 <+)*A/2 << ( 1 PË + 1a¹'A/ *aÊ0&A **R¹@+ *[Æ ,0+)*B)¹'(1/ *B(1=/ /+)&A < < (*B"$#%0*B(1( <B *(13/ -# ¹'3(1/ < *B 3( < & (*B+ <( +A A/ *B < / -#.0*B)/(1 ¹'3(1/ < )Bt A < Vbf
253>&#Æ R t¹'(1&A 32dÆ=/C)/ < ( / *1 2 =*B&A=1 +)&A (*"2 , < &#Æ+i¹ mà=q» P Y Ê-&)R ( 25A¹@ *"-#% A&)*A /!¹'ç¿3A/0/< < 1) < < Vbf
0*B)/ < 2 = &#Æ R ·¹@(&A=2dÆ3/ P / < 2 *(]R Ê à ¹¡¹' "Û +R A/0&A < &#Æ03( < (1 2 30 2201(]R ) &A)* *B_¹'+)*Æ < * 0$#%30 O ¼<¼ »n¼<¼<¼ '2 = 0 < ¹'(1&)32zÆ3/ < ÑGÉ O Ú *!»n¼n¼ ½¼<¼<¼ '2 30 <ÑGÉd» Ú P Ú =4 < Û +4R A/&) < (/zÛ +A (1A0 < Ê $#%(1/5 0A/&) < Û (* <C &A= < *(]R < < ¿ < *[âa¹' < 1+ < * *2 )0* 2 A/0 < 2 32 = *B(3/ < /3/ /0+aà3(Cà3 1 < )*!253 < Û +R )/&A < 21 < +A +A < Ê$#%4u'& &A(*+,/-#% < * 2 < (1/0+A25A/ / *" < 1) < F0
* RaP
(1( P #$ F0
" 9 RaÑ < A < Û +R )/&A 0 + < 3/ /&A F0
)* ?# ¹'3 *B( < < i¹@)/ * Ra
< 3/ *|q + *BA¹@(/A&A3/H¾ 3(1/ *Bi¹@)/ * & &) < 2 ¹qâ)* < à3( << )/ * qÛ 3/ ( < ( 1' / < ¹i¹@qà ¹q¹@ qÛ + iR A/&) < Ð P Ó P / < &A **à ¹¡¹' @Û +4R0A/&) < Ê $#%4u'& &A(*+,+A25A/0 3/&RÛ 3 *Bi¹@)/ * &A <A.t 2 ¹¡â *B < q Û 3( < PË < # à3(* (1&)(@+)*A¹@(/A $#%4u'& &)(*+@@*B=( < ¹@(&A=2dÆ3/ < 2 ¹'+)*[Æ0D@ +A2(&A D&)(*B+ / < /à ¹q¹@ 0Û +R A/0&A < R ( (/&A0* 1 < Û +R A/0&A < 0C + < =/ /&)4 < ¹'(1&A 32dÆ=/ < P Ñ <4u'& &A(*+ < 0*A/ < 25A¹@ **B)/ * 13 < ÊA # 20â < $#% < A *B(1=/, < mà30 < » P ½!)*D» P Z )*$#%+R *B(1=/Ð » P »n¸kÓÖÊD0,+A0(14 A> Û +R )/&A"4+ < 3/ /0&A F0
* ?# ¹'3 *B( < < i¹@)/ * Ra2 30
&#Æ R ¹'(1&A 32dÆ=/ A/ < ( / * 20&A+A0 +)&A(*B / < 1 < &#Æ+a¹ mà=^» P[\5P Ñ <& &A <!< =/ * )&)*+ < A/ 2 A/ / * )/ &)<¹'20*B, < A < < Vbf
20+ a¹'A/ * + *B)¹'(1/+) < Pá =0*B &))* *B 2 D&)+A0 < * A& *B+) 2 0/ 1<à3(&A(A & 1&)Ð +)*A¹@(/ *B(3/ *B=* 1i¹@)/ * 0*<¹ *B( < +A)Ó P Ñ < A < (1/(*B( 1 < < F0
)* Ra
< =/ * < A < 220=tD(C¹ *B( < Ð=//+A <&A3/ < *B &)*A 3 1) < 2 +A&)+A)/ * +)* 13// à3çÓ P / < 2 *B(QR nÊ(1 < 5u * @*â < 2 )-#%(*B+) *(13/ < 2 = **A(/" < 1A0 < < * < 2 = 1 < F0
* RaP
Ç>)* *B¹@+ *[ÆnÊ &)*A1a¹'A/ * )/ < à3 tÚ Ñ6ÝHÊ 1¹'A(11) &)<¹@20<¹'( < )/ *B 0(11(*B+ * < (C¹'21(&A(*B+40·¹'( < A/ " 2 = 0*A/(!*B3 < < 2 ¹qâ)*B < =1d¹',+R ( 1A/ * PÑ #.(/&A3/ +A/(A/ *¹ ¾ ) ,&) 20&A+A0+ < *R " + *B)¹'(1/ *(13/ 0 < Vbf
< * A&)*+A 2 0/¹'+)*ÆD0xm ( +))/ *(1)11=n P Ñ < A ) < &)<¹q¹@( < < < 1 < ¹' < 0 < / < < ( +A A/ *B < & (*+ < < &)d¹ A/ *)Ê &)TR ( 25AD*C A/0"1 < + < * * < 25A )2& *B( 1 < )* (C¹'2+)&A( < P Ë < * =/&"(1/( < 25A/ i< 1 253 0*B(1( < A &))**¹'+)*ÆD0 R C1 < *B)&#Æ/(QR0 < ¹' < 0425A¹@ **A/ *> #. D*BA/0(1 /53//0 A2 D0&)*B( (1(*+ P
" F !#" %$
& &(' *) / ,+@ < .- +A&)(* < /10 243657-6 < * &A3/ < *B(*B+) -#%/ 202 )(1 +A&)(12 D&)(*B+8:9 5;5;< =>A ?>A@!B--#%/ / DC < ) )&)*3(A 8FE 2 9.G IHD&('J+ 5A 0 A(B> * -#.0/LKNMO3P 2-*Z"?J[ #%* H927#%8N 274B5#@2a!$5#$8K5dJQg>5G/"!$*Z>
! "#$%'&( *)+, - ,/.103254 687
ZE12 9 ZE13 9 ZE23
F01 9 F02 9 F03 Ra1 9 Ra2 9 Ra3
M12 = ZE12 YT12
M13 = ZE13 YT13
M23 = ZE23 YT23
Mp1 9 Mp2 9 Mp3
ϕ1 9 ϕ2 9 ϕ3 Mp1dB 9 Mp2dB 9 Mp3dB
F ′01 9 F ′
02 9 F ′03 R′
a1 9 R′a2 9 R′
a3
:-;"<=?>@;
ACBDE;"=?>@<GFH?FEIFJB%D;"<KBL?L?>@MN$FOPBI@FEQR;"< ACB%D;S=?>T<GFEHUFVI@FBDE;<KB%L?L?>@MN,FEOPB%IFQ;"<W B%D W =?D
W B%D W =?D
W BD W =?D W B%D W =UD
ACB%D;S=?>T<GOXM$Y?FVZU[";"<KY,;<F0 \ Y,[SI;">OXFH?[S;<Y] BL?>@^"<ZU_a`Ub,`dc \ LUB>FHeI;">LMDJBI@FEMRHI;SDD;"<1fe=?;
ϕ(F0) = 90 g
ACBDE;"=?>@<OXM,Y,FEZU[S;<1Y,;"<Ra \Y,[SI;">OXFH?[S;<KY] BL?>@^"<["f`ab?` hRhi;SIKZU_U`Ub?` j
k l m m:9m : m:
F0
k l m m:
9m : m:
Ra
onPp 2436<qsr+)* 'D0 < ( (1,253 0+)*B) +'(1/) 1 <Xt +eu )/&A < " + < =/ /&) * +@3 *B( << +@)/ * < 3
6 ! #"$&%')(+*-,./0 1*$ #"$&234(%$ )5% 6(
(%(789
onPp 243653q MA' *) />0 +@ < !2 = $#%+)* 1=// .0 < +@(&A=2J'3/ < 2 +)&A(120&A(*+>0*(1( < +) &):<; 3
& < +@(&A=2J'3/ < &)32+ < (1/ < (Pu A < 2 + +'2( / & *B) <'< 3/ *@2 &A+ < / < /A/&)A(/ *&)1 < .->&) u (C2 ) +')* #.( < 3A'1 < C < *>= +@ 0(* &A3 < *B( u .3 &6 *BA/ < (1=/ 253 ( <B *(13/ < *2 D0(*,2 / 0 +A/0+A *BA0C@?D*A / * &A3/ *BA3+ADC $# (14-#.0/ 3* += *B*3
& 0 +A/+) *A '$# / C < A A&)*3(A 2 D0(* / *B)/ < (3/ ugC ?#.)/ *+)FE G ) $# 22 H
A(1 0 +A&)(12 D&)(*B+ 8 /10 3N24365 B3 M> < ( 0 / !-#%A/ *B +A < *'&)3/(*B(1=//+ 2 ?# 202 )(1 8 +'21( / & *B(1=/ -/ *B .0 .3 3 3 B * *B / < +'( < C < < < 3 *B( <I J )*2K 8 *A/ < (13/ uBB )1(+A < < 2 )&)*B( +@)/ * +'(1&) H
2J'03/ + +@ **B)- 2 >$#%(1/ *A +'+A0( (! < 3/ 20+ +@2( / & *B) 8 * C25ML ; GN 5 B - )*OC =(1QP 0?# / DC < )3 & < ( 0 / *B3 /+ 2 "1 +'(1&)327'3/ +A&A)20*B) < * +A&A02 +)+ < "$#%A/ *+A5R 6 0?# 202 )(1 +)&A(120&A(*+ u ( *B / <"t 3 +@ )*"1 *B / < +')*")2( <,<B < 3 *B(S 8 *B)/ < (3/ uA
BTC 3(1MPVU@0 ?# / DC < A A&)*3(A:3
W =0* 1 2&A << < +' < < * D*B;+ *B( < +*3 & # / DC < A A& *B3 (1A (1/ < ( u $# 22 )(1>0 +A&A(2&A(*B+ < 3/ * 2(1=*+ < 2 3(/ *A 0 >X=&)+C / 0 (&A(1) + 1( < +'A/3Y ( < A= < (1&*3 M> 1 0 (&A(A25A +@ * t (1 $# &eu ( < (*(13/ < + < 0* * < +' < ')* @ < A/ 0 ( < *A / < / t A(1'0& 1&) ; ?D&A) 3 & < &Ru ( < (*B(3/ << =/ * A/ < (** (*B+A < 2 / 2 0 + +@ )/ZY ( < 4= =< (1& )* r *B 3& 2 (1/&)(125 &A 2 0 +,+' A25 < < "1 +'D#=A1 2=2 < + 2 /3 +' M ;[ E G 5]\ H 2&^62;2_ *+)&A (* / < ,2 *0 27'243 243
& # 202 )(1 @+A&)(12 D&)(*+@25A +@ * -#%a` A& *B) 1'1(A/ )/ *B@1 < (b` +)A/ * < < ( 0 / G? /0+A&A << (1 <C +@ < .3 H3/ 2(/&A(2 t 3/& *B(3// +@)/ * (1/ < ( u < < +A++'A/ * < &)3/ < *B(*0*B( t < < 3/ * 3//+ < / <!<B/ &(' < CD/0320*B( u A/ //@?0 =O3Gc / / C&A3/0(*(13//0A +'(1A?'1 < (b` +)A/ * <!< ( 0 / G? - &A * 202 )(1 25A +')*ZC?#.D*B(1( < *B),'&('3( < (1 A/ *B (b` +))/ * <I/ *B < 2 =<< H ' 0* 8FE - 2 G - E G;G <d B *( < 25 < "-# +'21( / & *BA0 < 0 (1/ + 0 01 < 8:G C 9*G J= B 3
& < 2 + +'21( / & *BA0 < < +@(&A 32J'3/0 < &A=/ *(1)//A/ * &(' &A/ / + < ( < * /&A 253 ( <B *(13/-#%A/ (1 3/ Efe
Ω )/ 2 1g=)1 < 0 / +'21( / & *BA0 -#%( +'25+A /0&A> #.)/ *+)Pu =< ( H (1/ / /(1*3*M>A25A/ / *- < +'21( / & *A < /-#%+)* / * 2 =< 2 t (* < --(1 h?D( < *B' < & 2 &)(*B+ < 2 < (* < /=/ /+ 0 1( 0 1 < A/ *B
) ! "#$%'&( *)+, - ,/.103254 6
1) A/ *+A * + =<< .- &) <' +'21( / & *A < /-#%3/ * 3/& 2 =< / ( +'25+A /0&A -#%A/ *+A (1/ / /(1*3 [ < *4=/& /+)&A <<B (1 -#%+'21 C3)4 *A&('/( u *B)/ < (3/ (/ < +)+A 2 =C0+)*B) +'(1/) 1A0C$#%( +'2 +) /&)!+AA& *B( u *B / <"t ) *- 2253 * *B)/ < (3/ < 3 *B( @ +A&)A20*A C,&A(1 &A(* 3 A *ur0
< 1,&)3 / * ie A/ A/ *+A "$#%++')* *B)3& "2 + +'2( / & *B) L+'(1&)327'3/,+ +@ **B) &A3/ *B(A/ *CA/ 21 < & 2 &)(*B+" + t +A A/&) CrefC *B A < u A1 < *+' < 0+ 1 &)3 / * #.)/ *+) ie 8 /10 E 3 9 B3 & & 2 &)(*+ + t +A)/&A Cref0*B(1( < +A &:<; < * +)* 13//0+A"2 t (& / * *C + 0 01(g=) +@)/ *!+ H +)* 13/0/+A & M [-; 3& < +'(1&A 32J'=/ < + +@ **A *,+)&A)20*B) < 3/ *, A2 + < A/ *B+ < < /10 243 E G 2 C1) < &('0++ +eu ( 1A/ * W '+ A/(/ -5( +'2 +) /&) +)1)&)*B (u ZThe
253 $#%++')* *B) )* *B)/ < (3/ C &A(&A0(* 3 A *ur0
)*"(D+@25+A /&A ZThr253C1 +A&A)20*B) 8 & t 3 ) *(13/ E 3 E \ )*"2 .0 2J'0 E 3 E 3 9 B 3 & < & 2 &A(*B+ <
2 =< (*B < < 3/ * /=*B+) < CpA)* CpB
3 / < / 2 +'(1A *B +@2 < - *BA/ < (1=/ +AA& *B( u ug
< * 221( u +A < @$#%+AA&)* (g=A 0+@(&A 32J'3/0C+ +@ **B)3 & *B)/ < (3/ ur0
< * 0 +A/+)+),2 > 2 < < (1=/ 22(u +) < 0> +@ + /40+@(&A 32J'3/0 +A&)A20*A -D2 < < (1=/ / < & (*+C&)+A+)42 +'(1&)32J'03/ ++')* *BA03 A? *A/ < (13/ <uA
)* uB< 3/ * A +A <C G? < 3 *B(1 < S * K "$# 22 )(1 0 +A&)(12 D&)(*B+ 8 /10 0I243 E G B 3
ug
Yie ZThe
Cref CpB u
i ≈ 0gB FB hB
rB uB
ur0
ZThr
ur CpA
i ≈ 0gA
FA hA
rAuA
onPp 243 E G q/r < 0 "$#%( +'2 +) /&) +AA&)*( u0 0"* / <"t A * ZE 2 +'(g=A ,2 *B(1*3 # 2B= < 1 < 0AG? &A(&)(* < A2 + < A/ *B+ < /10 243 E G - < *B)/ < (3/ < uA
* uB25A A/ * *Bh?D2 ( +'+A <
A/ t 3/& *B(1=/ ie )* ur02
uB = hBFBgBu = hBFBgB−ie
jω (Cref + CpB)
8 243]\B)*
uA = hAFAgAur = hAFAgA1/(jωCpA)
ZThr + 1/(jωCpA)ur0 .
8 243 9 B& < u / *(*B+ < uA
- uB- ug
* Cref+)* / * <'< A01 <,0 /A0 < +' < 1 < - (1 < * /0+A&A << (1
-#%+A1(D+@(/A < @?020 < < (3/ < 8 243]\J-243 9 BC1 < 0*B < *A +' < 8 0 (1/ < - / *B .0 < --& 2 &)(*B+ < 2 =< (*B < B 3 "! B#aMR>@B%IMRFE>@;$8;"HRI@>@BDE; Y,;<&%oHUY,=U<!I>@F;"<&'CD; W I>@Ffe=?;<
! #"$&%')(+*-,./0 1*$ #"$&234(%$ )5% 6((%(789
K63"&A t (.- / < )&A3/ &A3021 +' < < < * h` )&)*B0+ < "1 +'3/ * .0 /10 243 E;E / <eu A + +@ *B)/ < (3/ +AA& *B( u 4-#%A/ *+A uref
< *!0*(1( < +)C253h?D&A(*B)>+AA& *B( u +'A/ *! < AG?+'(1&)32J'03/ < 3
u′
g uref u′ u′
BCpB rB
r Cref
YgB FB hB
ZThe
i ≈ 0
uref u′
r u′
ACpA rA
gA FA hAZThr i ≈ 0
onPp 243 E;E qsr < "$#%( +'2 +) /&) +AA& *B( u ",*B / <"t ) * ZE )G?D(g= +@"2 *B(.3& 0 (1/ gB
< *C0 G 1=O-?#.(D+@25+A /&A ZThe/-# 2253 *B,3/0& &A0/"&A3/ *B ( D*B(1=/ &A(&A0(* *
*A/ < (13/ u′B< #.+)&A (* 1= <
u′B = hBFBgBu′ = hBFBgB
Cref
Cref + CpBuref .
8 243 B& *B)/ < (3/ u′A < * h?D2 ( +'+A )/ t 3/& *B(1=/ *B)/ < (3/ uref
-# 2 = < 4&A(&)(*>+A1)&)*(u /10 243 E;E - < 3(*
u′A = hAFAgAu′r = hAFAgA
1/(jωCpA)
ZThr + 1/(jωCpA)uref .
8 243 N B; / / / - )/ t ( <B / * , 2253 * < ) *(13/ < 8 243 -1243 \B>)* 8 2 3 9 - 243 N B -0(1 (A/ *
u′BuB
=jωCref
−ieuref
)* uA
u′A=
ur0
uref,
&)u ( =//"2 = $#%h?D2 << (13/ "$#%( +'25+A /0&A +)1A& *B (u ",*B / <"t ) * ZE
ZE =ur0
ie=
−1
jωCref
uA
uB
u′Bu′A
.8 243;< B
& 1) ?#.(D+@25+A /&A +)1)&)*B (u * / <"t A * ZE < *'3/& )/ *B(g=) +@)/ *'+ *B) +'(1/+) 2
+' < " < 2253 * < uA/uB)* u′A/u′B - &)3// ( << / * 1) " & 2 &)(*B+ Cref
3
!"#%$& '!"' %()** + (),
&.-0/ '21436587:9 - 369:; / ;=< -?> 7 + - 7 -=> ;A@B58< -?>C/?- -CD7E9:1F@ - 9HGJI >6- ;A7EI6< - 65 6KL58M 0 9E143J'65uNI - -OKQPR5 > G:ICS'G:1 >6/?-< -T/?- 9H7U145 >6- G 61VG - GW7 '6;?MV9:5uNI - GXGHI69WK - GuNI - K8K - GY9 - 3MZG -?> 7[K - G\7 -?/ ' > 5 uNI - G]< P^;=7U14KLM >6> 1 0Z-_-?> 369 - G:GH5LM > 321V99:; / 5L3C9:M / 587:;.- +`;A7 '6Ma< - G-uaIC5\3 - IC@ -=> 7b1VK8MZ9:G =7E9 - 9 - +`58G - G -?> uNI - Gc7E5LM > 3MdP - GH7eK -J/ 1VG > M47U1.+ + -=> 7f< - G+`Ma<6==K - G[3J' CBGH5uNI - G > MZ9 +g14KL5LGH;?Gh< - KQP -=> G - + 6K - +`5 / 9:MV3J'6M >6- G Hi/ MZI63CK - IC93j 1 > G /?-k/ '21V365l7E9 - -B<C5 `;?9 -=> 7EGN+`Ma<6=?K - GW3J' CaG:5 uaI - G_<6; / 9:5l@41 > 7[K -k/ '61.+`3g1 / MVI6GH7:5uNI - <61 > G]Km1 / 1n@a587E;t MZ9 +'; - 361V9_K - G +`5 / 9:MV3J'6M >6- G - 7kK -o/ MZI636K - I69 -2G:M > 7k<6;A@ - K8MZ363;?G - 7 / M;+`361V9:;=G3
cd5 > G:5 -p1V369 =?G'1n@ZMZ589b3MZG:;#K - Gb;RuaI6147E58M > Gq< - KQPr1 / MZI6Gc7E5uNI -g- 70<6; /2> 5hK - G ' Ca3MV7 '6=?G - G`1V<J+'5LGHG:5 CK - G -I >6- ;A7EI6< - <6;=7E1V5LK8KL; - <6I / '21*+`3s1 / MZI6Gc7E5uNI - <21 > GfK -b/ 14GfMZtu5LK - Gc7oG:IC363MVG:;qI > 5 t MZ9 + - - / M.+`3C7 - 7 -?> I< - G - ` - 7:Gv7 ' - 9 +'5uNI - Ge7 - K8GuNI - 369:MZ3MZGH;g321V9 e - 9 - 9^ E 2_:- - GH7 - ?B3MVG:; -g- 7v<65LG / IC7:; - 3 &%- Gv9:;?GHI6K87E147EGM C7 -?> I6G CKwP^5LGHG:Ix< -`/=- 7:7 - ;A7EI6< - G:M > 7 -?> GHI6587 -g/ M.+`32149:;?Gv1VIG? +'MB<#=?K - Gy< -g/ '21*+`3zI > 5 t MV9 + -'- 7b14I+`Ma<6==K - < - 369HMZ321 0 147E58M > < P^M > < - Gk3CKm1 >6- GT369:; / M > 58G:;=G32149=h1VKLKL1 0 ' ^ _:3
|g=~ ' A I e [ ` = oC, 'Tg
& PR1.+'36KL5l7EI6< - <CI +`MVIC@ - + -=> 7b1 / MVI6GH7:5uNI -g- GH7qGHICS'GE1.+ + -?> 7 t 1V5 6K - 3MZI69v3MZIB@ZMZ589 -=> KL5 > ;n1V9H5LG - 9K - G_;euNI2147:5LM > G 3 &%- G_@41V9:5L16K - Gh<6; / 9:5l@41 > 7KQPR;A7U147h<7C > 1.+'5uNI -y- 7T7 ' - 9 +'Ma<7C > 1.+'5uNI - <6I2I658< - G:M > 7 qKQPR; / 1V9c7T5 > Gc7U1 > 7U1 > ;y< - 369 - G:G:58M > p 321V9T9E1V3C3MZ9c7fCqKm1b369 - G:GH5LM > GH7E147E5 uNI - P0
-qKL1q@a5l7 - GHG - 321V9c7E5 / I6KL1V5L9 - ~v -qKQPR; / 1V9c7T5 > Gc7U1 > 7U1 > ;y< - +g1VGHG - @ZMVKLIJ+'5uNI - ρ′ 321V9_9:1V363MZ9H7fCqKL1 +`1VGHG - @ZMVKLIJ+'5uNI - Gc7U147:5uNI - ρ0
-qKQPR; / 1V9c7T5 > Gc7U1 > 7U1 > ;y< - 7 - +'3;?9E17EI69 - τ 361V9T9E1V363MZ9c7fC0KL1J7 - +`3;?9:147EI69 - GH7U17E5uNI - T0
-qKQPR; / 1V9c7T5 > Gc7U1 > 7U1 > ;y< P -=> 7E9:MV365 - +`1VG:GH5uNI - s 32149_9E1V363MZ9c7fC0KQP -=> 7E9:MV365 - +`1VG:GH5uNI - GH7E147E5 uNI - S0
3&%- GT321V9:1.+=A7E9 - GPuNI65 -?> 7:9 -?> 7<21 > GkKm1b<C; /2> 5l7E58M > < - Gk369HMZ369:58;=7:;?GT<6I#2I658< - G:M > 7kK - G_G:I65l@V1 > 7:G qK - G / M - S / 5 -?> 7EGT< - @a5LG / MZGH587:;e< -e/ 5LG:1V5LK8K - + -?> 7 µ - 7k< - @ZMZK8IJ+ - η -qK -o/ M - S / 5 -?> 7k< -v/ M > <6I / 7E58M > 7 ' - 9 +'5uNI - λ -qK -o/ M - S / 5 -?> 7k< -v/ M;+'369 - G:G:5 65LK8587:;d5LGHMV7 ' - 9 + - χT
-qK - G / '61VK - I69:G +`1VGHG:5uNI - Gf321V9I > 5l7E;J< - +g14G:G - Cg369 - G:GH5LM >u/ M > Gc7U1 > 7 -J- 7MC'@ZMZK8IJ+ -v/ M > GH7U1 > 7 Cp- 7 Cv
- - 7kK - IC9_9E1V363MZ9c7 γ -qK - 7U1VIG?u< Pr1 /?/ 9HMZ5LGHG - + -=> 7e< - 3C9 - GHG:58M > 361V9oI > 5l7E;q< PR1 /?/ 9:MZ58G:G - + -?> 7e< - 7 - +'3;?9E17EI69 - C#@VMZKLI7+ -/ M > GH7E1 > 7 β = (∂P/∂T )V3
\ E
' 6 I & /( '92 7* % 70 27a 4(% )$ D4$)% * (
&%- Go;euNI217E5LM > GoKL5 > ;n1V589 - GO'6M;+'M 0 = >6- G 0 MZIC@ - 9 > 1 > 7eK - +`MVIC@ - + -=> 7y<CIs6I65L< - GHM > 7e1VI > M;+ 69 - < -7:9:MZ58G 8 ^6< _O243 9 3 E B
q *-,. G$ ) )/( 9 8 ;euNI2147:5LM >st M > <21.+ -?> 7U1VK - < - Km1q<7C > 1*+`5 uaI - B1
c0
∂ ~v
∂t+
1
ρ0c0~grad p = `v ~grad div ~v − ` ′
v~rot ~rot ~v +
~F
c0,
8 43 E B
MVt `v - 7 ` ′vGHM > 7k< - GTKLM >J0 I - I69:G / 149E1 / 7E;?9H5LGc7E5uNI - GT<6; /2> 5 - GT32149
`v =1
ρ0c0
(
4
3µ+ η
) - 7 ` ′v =
µ
ρ0c0,
- 7kMZt c0 =√
γ/ρ0χT
- GH7kKm1 / ;?KL;=9:5l7E;o1V<C5m1217E5uNI - <CIGHM > 3q *-,. G$ ) 4$ !/(# # $ V)2* 10! 8 uNI65B7E9E14<6I6587 Km1 / M;+'369 - G:G:5 65LK8587:;]<CI06I65L< - B<21 > GKL1uNI - K8K - KwP^; / 1V9H7J5 > Gc7U1 > 7E1 > ;'< - +`1VG:G - @ZMVKLIJ+'5uNI - ρ′ - GH7 - ?B369:5D+`; -?> t M >C/ 7E58M > < - Gv@41V9H5m1CK - G5 > <6;=3 -=> <21 > 7 - G p - 7 τ 361V9 ρ′ = (p− βτ)γ/c20
-
ρ0c0 div ~v +γ
c0
∂
∂t(p− βτ) = ρ0c0q ,
8 43 2;Bq *-,. G$ )7$)( /( %$( * 34$ ] ' )V* ) 1* G)( <21 > GKm1uNI - KLK - KQPR; / 1V9c7d5 > G H7E1 > 7U1 > ;x< P -?> 7E9HMZ365 - s - GH7 - ?B369:5D+`; -?> t M >6/ 7:5LM > < - G@41V9:5L16K - G5 > <6;=3 -=> <21 > 7 - G p - 7 τ 361V9s =
CP
Tτ +
βχT
ρ0p -
(
1
c0
∂
∂t− `h ∆
)
τ =γ − 1
βγ
1
c0
∂ p
∂t+
h
c0CP,
8 43 B
MVt `h = λ/(ρ0c0CP ) <6;?GH5 0Z>C- I >6- KLM >J0 I - I69 / 1V9E1 / 7E;=9:58GH7E5 uNI - < - <65 `I6GH5LM > 7 ' - 9 +'5uNI - 3&%- GY7 - 9 + - G ~F - q - 7 h 9 - 369H;?G -?> 7 -=> 7[KwP^5 > 6I -?>6/=- < - G:MZIC9 /?- G[1 / MZI6Gc7E5 uaI - G]9 - GH3 -=/ 7:58@ - + -=> 7[< -Pt MZ9 /?- -< - <6;6587 - 7T< -y/ '21VK - I69T<21 > GTK - <6M;+`1V5 >6-d/ M > GH5L<6;=9:;*3 ;W> < - 'CMZ9:GT< - GkGHMZI69 /?- G - /?- GT7 - 9 + - GTG:M > 7 > I6K8G3
|` . = A"! #%$ "& '( )*) ! + ,- . #/$ # 0. )1#2$435)6$ & )6$7 ~98;:=< $
e - 9 - 9^ E 2_:-O<CM > 7yK - Gd9:;?GHI6K87E147EGdG:M > 7DCDKm1 21VG - <CI +'Ma<6=?K - 9 - 7 -=> I <21 > GoKm1 > MV9 + - M ;[?> E G 5]\ H24-T369HMZ3MZG - < P^;=7:I6<65 - 9K -s/ '21.+'3 1 / MZI6Gc7E5uNI - <21 > GI >C-x/ 1n@a587:; -?> 3C9 -?> 1 > 7 -?>/ M.+`3C7 - K - G - ` - 7:G7 ' - 9 +`5 uaI - GD14G:G:M / 58;?Gg1VI +'MZIC@ - + -=> 7#1 / MZI6GH7:5uNI - 3AKMZI69 / M > Gc7E9:IC5L9 - G:M > 9E1V58G:M >6>C- + -?> 7 -]5LK >6- t 1V587I6G:1 0Z- uaI - < - KQPR;RuNI2147E58M > < -g/ M > <6I / 7E58M > < - Km1 / '21VK - I69 - 7b< - Km1KLMV5W7 ' - 9 +`Ma<7C > 1.+'5uNI - K85 > ;?1V9:58G:; -- ?C3C9:5 +`1 > 7TKm1 65l@41V9:5L1 >6/?- <6I+'5LK85 - I 8 ;RuNI2147E58M > G 73 B 3
& PRM @ - 7b< -g/?- 321V9E1 0 9E1V37' -`- GH7q< - 369HMZ3MVG - 9vI >6-`- ?B36KL5 / 147:5LM > <6;A7U1V58KLKL; - <6I 9E145LG:M >6>6- + -?> 7JG:I65l@B532149 e - 9 - 9 - 7k< - +'M > 7E9 - 9_K - GkK85 +'587 - G]< -y/?- 9E1V58G:M >6>C- + -?> 7 3
,+U +U )+e *P)+)+,, )#' & 4 6
! #"$ &%'()*+,.-$0/213%5467()*+89-7;:'+,7-$67:<*+,=-$>/(?:&@7(,/56$ &.- <CM;+g145 >6- < PR;A7EI6< - D - GH7 / M > GH7:587EIC;]361V9 K - 2I65L< -Y/ M > 7 -=> IJ<21 > G%I >6-W/ 1n@a587:;Y< - t MZ9 + - uNI - K / M > uNI -
t - 9 +`; - 321V9XKm1eG:I69 t 1 /?- S 3 &%-f/ '21*+`3#1 / MZI6Gc7E5 uaI -f- Gc7 0 ; > ;?9:;TCJKwP^5 > Gc7U1 > 7[5 > 5l7E5m14K t = 0 321V9XI >6- GHMZI69 /=-< - @V149:5m17E5LM > < - 369 - GHG:5LM > MZI< - <C; 65l7k@VMZKLI7+`5 uaI - 3 &%- G<65 + -?> GH5LM > Gk<CI <6M;+`1V5 >6- D GHM > 7dGHI6363MZG:; - G3 - 7E5l7 - G_< - @41 > 7kKm1qKLM >70 I - IC9_< PRM > < - 3e - 9 - 9; / 9H587pKwP^;euNI2147:5LM > < -X/ M > <CI / 7:5LM > < - KL1 / '214K - IC9 -?> KwPR16G -?>6/=- < - G:MZI69 /?- G\< -[/ '61VK - I69^ E 24- ;Ru 3E _%GHMZI6GTKm1 t MV9 + - <6M >6> ; - 361V9 8 73 B
(
1
c0
∂
∂t− `h ∆
)
τ =γ − 1
βγ
1
c0
∂ pg
∂t.
8 73 \B& 1 > M47U147:5LM > pg
<C;?G:5 0Z>6- -3MZI69 e - 9 - 9 - KwP^; / 149H7k5 > Gc7U1 > 7E1 > ;e< - 3C9 - GHG:58M > 369HMB<CI6587k321V9TI >L0 ; > ;=9E147 - I691V<21V3B7E;.- /=- 7X; / 149H7X< - 369 - GHG:5LM > ;=7E1 > 7 CyKwP^MZ9:5 0 5 >C- < - KQPR; / 1V9c7]5 > Gc7U1 > 7U1 > ;< - 7 - +`3;?9:147EIC9 - τ 1VI`G - 5 > <CI2I658< - 3
K 1V9k1V58KLK - I69:G - -=> t 145LGE1 > 7TICGE1 0Z- < - KL109 - KL147E58M > KL5 > ;n149:5LGH; -ρ′g = ρ0χT (p− βτ)
<21 > GTKQPR;RuaI6147E58M > < -y/ M > <6I / 7:5LM > < - Km1 / '61VK - I69 8 73 B -C5LK@a5 -?> 7(
1
c0
∂
∂t− γ`h ∆
)
τ =γ − 1
βχT ρ0
1
c0
∂ ρ′g∂t
,8 73 9 B
MZt ρ′g 9 - 3C9:;?G -?> 7 - KQPR; / 1V9c7d5 > GH7E1 > 7U1 > ;v< - +g14G:G - @ZMVKLIJ+'5uNI - 369HMB<6IC587321V9kI >L0 ; > ;?9:147 - I69f14<21V3C7:;.- /?- 7; / 1V9H7'< - +`1VG:G - @VMZKLIJ+'5uNI - ;A7U1 > 7- /?- 7H7 -/t MZ5LG H / 5 - 707EMVI @HMZI69HG'G - K8M > e - 9 - 9- CsKwP^MZ9:5 0 5 >C- < - KwP^; / 1V9H75 > Gc7U1 > 7E1 > ;o< - 7 - +`3;?9:147EIC9 - τ ^ E 2 -6;eu 3 E 2_:3e - 9 - 9k369:MZ3MZG - <6M >6/ < - 9:;=G:MZIC<69 - KwP^;euNI2147:5LM > < -J/ M > <6I / 7E58M > < - KL1 / '214K - IC93MZI69 τ <21 > G< - IG?/ 1VGT<65b`O;=9 -=> 7:G qKL1e@41V9:5L147E58M > < - 7 - +'3;=9E147:I69 - τ 1e3MZIC9[MZ9H5 0 5 >6- I >6- @41V9H5m147:5LM > < - 369 - G:G:58M > pg
369:Ma<6I65l7 - 321V9[I >0 ; > ;?9:147 - I69k< - @V149:5m17E5LM > < - 369 - G:GH5LM > 8 5D+`3;?<21 >6/?-o> I6K8K - B-qKL1x@V149:5m17E5LM > < - 7 - +`3;?9:147EI69 - τ 1)3MZIC9#MZ9:5 0 5 >6- I >6- @41V9:5L147E58M > < - @VMZKLI7+ - +`1VGHG:5uNI - 1/ρ′g369HMB<CI6587 - 321V9TI >0 ; > ;?9E17 - IC9< - @41V9H5m147:5LM > < - <C; 65l7 8 5 +'3;?<21 >C/?- 5 >4/2> 5 B3& P^;euNI217E5LM > < - / M > <6I / 7E58M > < - Km1 / '21VK - I69 - GH7#<CM >6/ 5 > 7 - 9H369:;A7E; - 5 / 5 / M;+,+ - I >6- ;RuNI2147E58M > < -
<65 `I6GH5LM > <21 > G.Km1uNI - KLK - K - G -=/ M > <I+ - + 69 -Y-?> pg
8 ;eu 373 \B MZI -?> ρ′g 8 ;Ru 343 9 B - ?C3C9:5 + - Km1 / M > 7E9H5 6IC7:5LM >< P^I >6- G:MZIC9 /?- 8 < - @41V9H5m147:5LM > < - 369 - G:G:58M > MZI< - +g1VGHG - @ZMVKLIJ+'5uNI - B[<21 > GTK - 3C9:M /?- GHG:I6Gk< - <65b`OI6GH5LM > 3
A 00 6%%(7 a5) 1 4 ! 4$((]9-?>- ` - 7-_KwP^; / 149H7#< - 369 - GHG:5LM > MZI < -
@ZMZK8IJ+ - +g14G:G:5 uNI -q-?> 7E9:1 > 7e<21 > GoKQPR;RuaI6147E58M > < -0/ M > <6I / 7:5LM > < - KL1 / '21VK - I69 >6- 9 - 369H;?G -=> 7 -'-?> 1VI / I >/ 1VGqKQPR5 > 2I -=>6/?- < P^I >6- GHMZI69 /=- - +`1V5LGJ<6; / 9:5870KQPR;A7U147q7 ' - 9 +`Ma<7C > 1.+'5uNI - <6I)2I65L< --=> 7EMZIC7'3MZ5 > 7q<CI<6M;+`1V5 >C- < P^;=7EIC< - 3
;W> 9:;?1VKL5l7E;.-CG:5%< - GdG:MVI69 /=- Gk< - <C5 `;?9 -=> 7 - G > 147:I69 - GGHM > 7d9H;?321V9c7E5 - GT<21 > GkK - <CM;+g145 >6- -6K - I69 / M > 7E9H5 H6IC7:5LM >- GH7#3C9:5LG - -?>/ M;+'3C7 - 361V9DK - G`7 - 9 + - G ~F - q - h <61 > GDK - GD;RuaI6147E58M > GD< - : 1n@a5 - 9 H Ha7:MB - G-h< -/ M > G - 9c@4147E58M > < - Km1,+`1VGHG -b- 7f< -b/ M > <6I / 7E58M > < - Km1 / '61VK - I69 8 73 E - 73 24- 73 B 3 HB5 /?- GdG:MZI69 /?- Gy1 0 5LGHG -=> 7G:I69K - G t 9:M > 7:5g=?9 - G<6Iu<6M;+`1V5 >C- - - K8K - GkG:M > 7MCg369 -=> <69 -q-?> / M;+`3B7 - <61 > GfKQP - ?C3C9 - GHG:58M > < - G / M > <C587E58M > G1VIG? t 9:M > 7E5 =?9 - G3
cd5 > G:5 -[KwP^5 > 7 - 9:369H;=7U17E5LM > uNI65 - GH7 t 1V5l7 - < - G;RuNI2147E58M > G< - <65b`OI6GH5LM > uNI - e - 9 - 9 / ' - 9 / ' - C 9H; HG:MZIC<69 -o- GH7T5 >C- ?C1 / 7 - 37K 1V9T1V58KLK - I69HG-N5LK > P - GH7T3214G t 1V587[I6GE1 0V- <21 > G_GHM > 9E1V58G:M >C>6- + -?> 7_< - KQPR;RuNI2147E58M > < -: 1n@a5 - 9 H HN7EMB - G > 5< - KQPR;RuaI6147E58M > < -f/ M > G - 9H@4147:5LM > < - Km1I+g1VGHG - -C;euNI217E5LM > G[3MZI69c7U1 > 7 > ; /?- G:GE145L9 - GQCJKm1< - G / 9H5L3C7:5LM > <6I+'MZIC@ - + -=> 7k1 / MZICGH7E5 uNI - <21 > GTG:M >-=> G - + 6K - 3 & 1 +`;A7 '6Ma< - < - 9H;?G:MVKLIC7:5LM > IC7:5LK85LG:; -f- 7KwP^5 > 7 - 9:369H;=7E147E58M > < - G9H;?G:ICK87U17EG_M C7 -?> I6GkG:M > 7k<6M >C/ CqM CG - 9c@ - 9d< --t 1C=M >/ 9:5l7E5 uaI - 3
6 I & /( '92 7* % 70 27a 4(% )$ D4$)% * (
" +)@> &'567 ' "$ &%5( *+, -$6 "$ &+ / 5/+8 ! !
cd369 =?Gp1n@ZMZ589%3MZGH;XK - G.;euNI217E5LM > G.< - <C5 `I6G:58M > 8 73 \ - 7 73 9 B - e - 9 - 99 - +`1V9uNI - uNI - K - I69 t MZ9 +bICKm147:5LM >- GH7G:5 +'5LKL1V5L9 -T- 73uNI P - KLK - GT3 - IC@ -=> 7d<6M >6/ =7E9 - ; / 9H587 - GTG:MZICGkKL1 t MZ9 + -(
1
σ
∂
∂t− ∆
)
τ(~r, t) =1
σ
∂ θg(t)
∂t3MZI69 ~r ∈ D , t ≥ 0 ,
8 73 > 1 BMZtKm1J@41V9H5m1CK - τ - Gc7kG:I6363MZGH; -e/ 1VI6G:1VK - - - 71n@ -=/
θg(t) =γ − 1
βγpg
- 7 σ = c0`h ,
<21 > GTK -o/ 14Gk<C; / 9H587k321V9_KQPR;RuNI2147E58M > 8 73 \B- - 7
θg(t) =γ − 1
βχT ρ0
ρ′g- 7 σ = γc0`h ,
<21 > GTK -o/ 14Gk<C; / 9H587k321V9_KQPR;RuNI2147E58M > 8 73 9 B3c /=- 7H7 - ;RuaI6147E58M > < - <65 `I6GH5LM > - e - 9 - 9[1VGHG:M / 5 - Km1 8 K - G B / M > <6587:5LM > 8 G(BY1VI? t 9HM > 7E5 =?9 - G - 7hKm1 / M > <65 H
7:5LM > 5 > 587E5L1VK - G:IC58@41 > 7 - G-29 - G:3 -?/ 7E58@ - + -?> 7
τ(~r, t) = 0 , 3MZIC9 ~r ∈ S , t ≥ 0 ,8 73 > B
τ(~r, t) = 0 , 3MZIC9 ~r ∈ D , t = 0 .8 43 >V/ B
& 1v@41V9:5L16K - θg(t)- GH7hG:I6363MZGH; - I > 5 t MZ9 + - <61 > GhK - <6M;+`1V5 >C- D 3JM - 7:7 - ' Ca3MV7 '6=?G - 5 +'36KL5 uaI - <6M >6/
uNI - -CG - KLM > K -k/ 1VG / M > GH5L<6;=9:;*-CKwP^; / 1V9H7X< - 369 - G:G:58M > pgMVIDKwP^; / 1V9H7X< - +`1VGHG - @VMZKLI7+`5 uaI - ρ′g - GH7_I > 5 t MZ9 + -<21 > GvK - <6M;+`1V5 >C- - /=- Gy@41V9:5L16K - Gf;=7E1 > 7eG:I6363MZGH; - G / 1V9:1 / 7:;?9H5LG - 9y5 / 5 Km1#G:MZIC9 /?- C#KwP^MZ9:5 0 5 >C- <6I / '21.+'3
1 / MZI6Gc7E5uNI - 8 'Ca3MV7 '#=?G -v- 9H9:M > ; - B3j 1 > G K -[/ 1VG\369H;?G -?> 7 -Z<CI t 1V587 uaI - K - G <65D+ -=> G:58M > G <CIq<6M;+`1V5 >6- D GHM > 7Y3 - 7E5l7 - G < - @41 > 7YKL1dKLM >70 I - IC9< P^M > < - - KL1D369 - G:G:58M > 1 / MZI6GH7:5uNI - p - GH7yI > 5 t MZ9 + - <21 > GoK - <6M;+`1V5 >C- D 8 C / M;+'369:58Gd<21 > GoK - G / MZI / ' - GK85 +'587 - Gp7 ' - 9 +`5 uaI - G(B3 j 1 > G /=-T/ 1VG -ZG:52KL1 +g1VGHG - @VMZKLIJ+'5uNI - ρ′g - Gc7]I > 5 t MV9 + - <21 > G D 8 C / M.+`369H5LG\<61 > GK - G / MZI / ' - G[K85 +'587 - G(Bp1VKLMZ9HG-Z<OPr1V3C9B=?G]Km1y9 - Km147:5LM > ρ′g = ρ0χT (p− βτ) - τ <6MZ5l7]1VI6G:GH5 A7E9 - I > 5 t MZ9 + - <61 > GD
8 C / M;+'369H5LG\<21 > GYK - G / MZI / ' - GWKL5D+`5l7 - G B - /?- uNI65 > P - GH7]321VGYK -h/ 1VG 3cd5 > G:5:-4KQPR; / 1V9c7W< - +g14G:G - @VMZKLI7+`5 uaI -ρ′g
>6- 3 - IC7_321VG =7:9 - I > 5 t MZ9 + - <21 > G[7EMZIB7[K - <6M;+`1V5 >6- D 58K - GH7 t M >6/ 7:5LM > <6I#3MZ5 > 7X<21 > GXK - G / MZI / ' - GK85 +'587 - G3jd- +`1 > 5 =?9 - C - ?C3C9:5 + - 9Km1'G:MZK8IC7E58M >L0 ; > ;?9:1VK - <6I 369HM 6K =+ - 7 - +`3MZ9 - K 8 73 > 17- - / BGHMZI6GfKL1 t MZ9 + -< P^I > 369:Ma<6I65l7q< -#/ M > @ZMZK8IC7E58M > < - Km1 @41V9H5m1CK - t 1n@ -=/ I >6-st M >6/ 7:5LM > < - e 9 -?-?> - e - 9 - 901VGHG:M / 5 - I >369HM 6K =+ - < - e 9 -?-=> 8 <6M > 70KL1 G:MZK8IC7E58M > - GH70Km1 9H;?3M > G - 5 +'36I6KLGH5LM >C>6- K8K - <6I G CaGH7B=+ - B uNI65 - GH70; / 9:5l7GHMZI6GkKm1 t MZ9 + - GHI658@41 > 7 -
(
1
σ
∂
∂t− ∆
)
g =1
σδ(t) 3MVI69 ~r ∈ D , t ≥ 0 ,
8 73 N 1 B
g = 0 3MZI69 ~r ∈ S , t ≥ 0 .8 73 N B
H - 9H; t ;?9:1 > 7k1VKLMV9:GOCbKQPRMVIC@a9E1 0Z- < - Mh1V9HG:Km1 - 7Z1 -0Z- 9M^ E G -C363 315 N - E 5;57-72 ]\J- G _:- e - 9 - 9h9H;?G:MVIC7K - 369:M CKg=+ - 8 43 < B[;RuNI658@41VK -?> 7<C 8 73 N B -6<61 > GkK - uNI - K%Km1bGHMZI69 /=- 5D+`3CI6KLGH5LM >6>C- KLK - δ(t) - GH7T7E9:1V<6I65l7 - GHMZI6GKL1 t MV9 + - < -y/ M > <6587:5LM > 5 > 587:5m1VK -
,+U +U )+e *P)+)+,, )#' & 4
(
1
σ
∂
∂t− ∆
)
g = 0 3MZI69 ~r ∈ D , t > 0 ,8 736<Z1 B
g = 0 3MZI69 ~r ∈ S , t ≥ 0 ,8 736< B
g = 1 3MZI69 ~r ∈ D , t = 0 .8 736< / B
& P^;euNI65l@41VK -=>6/?- < - G3C9:M 6K =+ - G 8 43 N B - 7 8 736< B - Gc7369:; / 5LGH; - -?> 1 >6>6- ? - M -?> +'M > 7E9:1 > 7/uNI PR58KLG3MVG:GB==< -?> 7Km1 + + - G:MZK8IC7E58M >0 ; > ;?9:1VK - 3
K 1V9 1V5LK8K - IC9:G-FKQP6' Ca3MV7 '6==G - G - K8M > Km1auaI - KLK - Km1 t M >6/ 7:5LM > θg(t)- GH7 I > 5 t MZ9 + - <61 > G K - <6M.+g1V5 >6- < P^;=7EIC< -
5 +'36KL5 uNI - uNI - KQPR; / 1V9c7o5 > GH7U1 > 7E1 > ;b< - +g1VGHG - @VMZKLI7+`5 uaI - MZIuKwP^; / 1V9H7f5 > Gc7U1 > 7U1 > ;b< - 369 - GHG:5LM >u- GH7yI > 5 Ht MZ9 + - <21 > G /?- <6M;+`1V5 >6- 3 [ K - Gc7\<6; +`M > 7:9:;]36KLICG'61VIC7 - 7p<21 > GpK - 321V9:1 0 9E143J' - > 343 243 2_< - KL1k9H; t ;=9 -?>C/?- ^ < _uNI - G - ICKK - < - I?C5 =+ -T/ 1VG - pg
I > 5 t MZ9 + - - - Gc7[GHI6G /=- 3C7:5 6K - < -T/ M > <CI65L9 - CoI > 9:;=G:I6Kl7U147 / MZ9:9 -?/ 7[3MZI69Km143 - 7:587 - / 1n@B5l7E; / M > G:58<6;?9H; - 5 / 5:314363 - KLM > G uaI - K - Gq@V149:5m1 6K - G pg
- 7 ρ′g >C- 9 -?> < -?> 7 > I6K8K - + -?> 7/ M;+'3C7 - <6Iu7 CB3 - < - G:MZIC9 /?-'- ? / 587U1 > 7yK - 2I658< - <21 > GyKL1 / 1n@a587E; +`1V5LG uaIOP - KLK - Go<6; / 9:5l@ -?> 7vKwP^;=7E147f7 ' - 9 H+`Ma<7C > 1.+'5uNI - <6I 6I65L< -v-?> 7:MZIC7d3MZ5 > 7d<6I <6M;+`1V5 >6- 3cd5 > G:5 - K -v/ 1VGfMZt pg
- GH7fI > 5 t MV9 + -e- GH7oK - G - I6K7E9:1V587:;f<21 > GTKL10GHI6587 - < -y/?- <6M / IJ+ -?> 7-2G:MZ5l7
θg(t) =γ − 1
βγpg
- 7 σ = c0`h ,8 7365 B
;=7E1 > 7 -?> 7 -=> <6I uNI - K - G / 1VK / I6K8GhG:M > 7kG:5D+`58Km1V589 - G[3MZI69_KwPR1VIC7:9 -o/ 1VG3j 1 > GqK - I69vMZIC@a9E1 0V- +U C +U + G+U U ^ E G _ - Mh1V9HG:Km1 - 7 Z1 -0Z- 9b<6;A@ - K8MZ363 -?> 70< - GG:MZK8IC7E58M > G_14I3C9:M 6K =+ - 8 736< B[3MZI69_<65b`O;=9 -=> 7 - G t MZ9 + - G_<CI<6M.+g1V5 >6- D 3&.- 321V9E1 0 9E1V37' - G:IC58@41 > 7y1D3MZI69 6IB7y< - 369H;?G -=> 7 - 9yKL1 +`;A7 '6Ma< - < - 9:;?GHMZKLIB7E5LM > IC7E58KL58G:; - 1V5 > G:55uaI -K - Gf9:;=G:I6Kl7U147:GoM C7 -=> I6Gy<21 > GoK -b/ 1VGyMZtuK - <6M;+`1V5 >6- D - Gc7yI >s/ CaKL5 > <69 - t - 9 +';C#G - G - ?a7E9H;+'587E;=G3 &%- G1VIC7:9 - G t MZ9 + - GW< -/ 1n@B5l7E;=GX9H;?3 - 9H7:MZ9:58; - GX<21 > GWKwPR1V9H7:5 / K - < - e - 9 - 9f^ E 2_ G:M > 7]7E9:1V587:; - G]<61 > G]KwPR1 >6>6- ? -j 3
+,/6 *+8 " +,6$ /( " %! #( 6 &&.- <6M;+`1V5 >6- < PR;A7EI6< - D - GH7h5 / 5 I >D/ CaKL5 > <69 - /2> 5< - KLM >J0 I - I69 ` -N< - 9:1 CVM > a - 7[1VIG?`32149:MZ58G]9H5 0 58< - G8 5D+`3;?<21 >6/?- < - 361V9:MZ5%5 >7/2> 5 - B31M - <6M;+`1V5 >6- 10I > @VMZKLIJ+ - V - 7kI >6- GHI69 t 1 /=- 7EMV7E1VK - A 3M[M;+`3B7 - 7 -=> Iz< - Km1#G C +';=7:9:5 -0/ CBK85 > <C9:5uNI - <CIz<6M.+g1V5 >6- < P^;=7:I6< - -.K - 369:M CKg=+ - 8 43 < BfG=PR; / 9:5l7 - -=>/ MaMZ9:<CM >6> ; - G / CaKL5 > <69H5uNI - G -
(
∂ 2
∂r2 +1
r
∂
∂r+∂ 2
∂z2 − 1
σ
∂
∂t
)
g = 0 3MVI69 r ∈ (0, a) , z ∈ (0, `) , t > 0 ,8 73 E G 1;B
g = 0 3MVI69 r = a , z = 0 MZI ` , t > 0 ,8 73 E G B
g = 1 3MVI69 r ∈ (0, a) , z ∈ (0, `) , t = 0 .8 73 E G / B
$* 7 $ 6 6( *5!/% #(/M - 3C9:M 6K =+ - 1V<J+ - 7T< - GTG:MZK8IC7E58M > GQCb@41V9:5L16K - GhG:;=321V9H; - G< - Km1 t MZ9 + -
g(r, z, t) = fr(r)fz(z)h(t) .8 43 E;E B
6 I & /( '92 7* % 70 27a 4(% )$ D4$)% * (
&%- 9 - 3MZ9c7T< -y/?- 7 Ca3 - < - G:MVKLIC7:5LM > <21 > GTKwP^;euNI2147:5LM > < - <65b`OI6GH5LM > 8 73 E G 1 B[<6M >6>6- 1VKLMV9:G1
fr
∂ 2fr
∂r2 +1
rfr
∂ fr
∂r+
1
fz
∂ 2fz
∂z2 =1
σh
∂ h
∂t.
8 73 E 2;BKMZI69 uNI P^I >6- t M >6/ 7E5LM > < - Km10G - I6K - @V149:5m1 6K - t 3CI65LGHG - A7E9 - 5L< -?> 7E5 uaI - C'I >6-t M >6/ 7:5LM > uNI65 >6- <6;=3 -=> <uNI - < - GT@41V9:5L16K - G_< P - GH321 /?- -8uNI - KLK - G uNI - GHMZ5 -=> 7K - G_@41VK - I69:GT< - r ∈ (0, a) - z ∈ (0, `)
- 7 t ≥ 0 -25LK - GH7> ; /?- G:GE145L9 - uNI -e/ '21uNI - + - + 69 - < - KQPR;RuNI2147E58M > 8 73 E 2;B[GHMZ587h; 0 1VK)CqI >C- + + -f/ M > GH7E1 > 7 -e> MV7E; - α 5LK@a5 -=> 7d14KLMZ9HG1
h
∂ h
∂t= −σα ,
uNI65.<6M >C>6- 3MZIC9 h(t)h(t) = e−σαt .
8 73 E B;Y> G:I65l@V1 > 7kK - + + - 9:1V5LGHM >6>6- + -=> 7T3MZI69_K - G t M >C/ 7E58M > G fr(r)
- 7 fz(z)-2KwP^;euNI2147:5LM > 8 73 E 2;B[<6M >6>C-
(
∂ 2
∂r2 +1
r
∂
∂r− αr
)
fr(r) = 08 43 E \Z1 B
- 7(
∂ 2
∂z2 − αz
)
fz(z) = 0 ,8 43 E \ B
1n@ -=/ αr + αz = α 3& 10GHMZKLIB7E5LM > Cq@41V9H5m1CK - G[G:;?361V9:; - G 8 73 E;E BX3 - IC71VK8MZ9:G A7E9 - ; / 9H587 - GHMZI6GkKm1 t MZ9 + -g(r, z, t) = ϕ(r, t)ψ(z, t) ,
MZtK - G t M >6/ 7E5LM > G ϕ(r, t)- 7 ψ(z, t) GHM > 7<C; /2> 5 - GT9 - G:3 -?/ 7E58@ - + -?> 7321V9ϕ(r, t) = fr(r)e
−σαrt - 7 ψ(z, t) = fz(z)e−σαz t .
; KLK - G_GHM > 7GHMZKLIB7E5LM > G_9 - G:3 -?/ 7:58@ - + -?> 7d< - GT369:M6Kg= + - G_G:I65l@V1 > 7:G (
∂ 2
∂r2 +1
r
∂
∂r− 1
σ
∂
∂t
)
ϕ = 0 3MZI69 r ∈ (0, a) , t > 0 ,8 73 E 9 1 B
ϕ = 0 3MVI69 r = a , t > 0 ,8 73 E 9 B
ϕ = 1 3MVI69 r ∈ (0, a) , t = 0 ,8 73 E 9 / B
- 7(
∂ 2
∂z2 − 1
σ
∂
∂t
)
ψ = 0 3MZIC9 z ∈ (0, `) , t > 0 ,8 73 E > 1 B
ψ = 0 3MZIC9 z = 0 MZI ` , t > 0 ,8 73 E > B
ψ = 1 3MZIC9 z ∈ (0, `) , t = 0 .8 73 E >V/ B
&%- GhGHMZKLIC7:5LM > G ϕ(r, t)- 7 ψ(z, t) G:M > 7_9 -?/ ' - 9 / '6; - GkG:MZICGhKm1 t MZ9 + - < - <6;=@ - KLMV363 - + -=> 7EG_G:IC9hKm1 21VG -< - G t M >6/ 7:5LM > G_3C9:MZ369 - Gk1VGHG:M / 5L; - Gk1VIG?#369:M CKg=+ - G 8 73 E 9 B - 7 8 73 E > B 58KO@a5 -?> 7
,+U +U )+e *P)+)+,, )#' & 4 7
ϕ(r, t) =+∞∑
n=1
AnJ0
(
λnr
a
)
exp−σλ2
nt
a2 ,8 73 EN B
ψ(z, t) =
+∞∑
n=0
an sinnπz
`exp
−σn2π2t
`2,
8 73 E < B
MZt'K - GY@41VK - I69HG]3C9:MZ369 - G λnGHM > 7[K - GW9E1 / 5 >C- GW< - 9E1 >J0 n < - J0(λn) = 0 3 &%- G / M - S / 5 -?> 7:G An
- 7 anG:M > 7
< - G / M > GH7E1 > 7 - G< P^5 > 7:; 0 9:147E58M > uNI65.G:M > 7k<6;=7 - 9 +`5 > ; - Gk361V9_K - G / M > <C587E58M > G_5 > 587:5m1VK - G3& 1 t M >C/ 7E58M > ϕ(r, t) <6MZ587TG:147E58G t 1V589 - CqKL1 / M > <C587E58M > 5 > 587E5L1VK -
ϕ(r, 0) = 1 =
+∞∑
n=1
AnJ0
(
λnr
a
)
,
uNI65X<CM >6>6- - 1V3C9B=?Gv369:M @ -?/ 7:5LM > G:I69JKm121VG - < - G J0(λmr/a)- 7 / M;+`3B7 - 7 -?> I < - KwP^MZ9H7 '6M 0 M > 1VKL5l7E;b< - G
;?K8;+ -?> 7EG_< -y/=- 7:7 - 61VG - 8 1 >6>6- ? - ; -2;eu 3 ; 3 \ - 7 ; 3 9 B -
An =2
λnJ1(λn).
8 73 E 5 B& 1 t M >6/ 7:5LM > ϕ(r, t) G?P^; / 9:587T<CM >6/y-?> <6; /2> 587E5l@ -
ϕ(r, t) =+∞∑
n=1
2
λn
J0(λnr/a)
J1(λn)exp
−σλ2nt
a2 .8 73 2 G B
&.- G / M - S / 5 -?> 7:G anG:M > 7b<6;=7 - 9 +'5 > ;?GJ< - Km1+ + - t 1C?M > -?> 9 - 3MZ9H7E1 > 7bKwP - ?B369 - GHG:5LM > < - ψ(z, t)<21 > GvKm1 / M > <C587E58M > 5 > 5l7E5m14K - <6I 3C9:M 6K =+ - 8 73 E > B - /?- uNI65[<6M >6>6- - 1V369 =?GJ3C9:M @ -=/ 7E58M > GHI69bKm1 21VG - < - G
sinnπz
`
- 7 / M;+'3C7 - 7 -?> I< - KwP^MZ9H7 '6M 0 M > 1VKL5l7E;d< - GT;?K8;+ -?> 7EGT< -y/?- 7:7 - 61VG - -
an =
4/(nπ) G:5 n - GH7k5 +'321V589 -0 G:5 n - Gc7k321V5L9 - .
& 1 t M >6/ 7:5LM > ψ(z, t) G?P^; / 9H587k<6M >C/
ψ(z, t) =4
π
+∞∑
n=0
1
2n+ 1sin
(2n+ 1)πz
`exp
−σ(2n+ 1)2π2t
`2.
8 73 2 E B
&.- 369HMB<CI6587_< - GkG:MZK8IC7E58M > G ϕ(r, t)- 7 ψ(z, t) <6M >6>C- KL1b9:;?3M > G - 5 +'36I6KLGH5LM >C>6- K8K - g(r, z, t)
g(r, z, t) =4
π
+∞∑
m=0
+∞∑
n=1
2
λn
1
2m+ 1
J0(λnr/a)
J1(λn)sin
(2m+ 1)πz
`
exp
[
−σt[
λ2n
a2 +(2m + 1)2π2
`2
]]
.8 73 2;2;B
6 I & /( '92 7* % 70 27a 4(% )$ D4$)% * (
& 1kGHMZKLIC7:5LM > τ(r, z, t) <CIJ369HM 6Kg= + - 8 73 > B - GH7 1VK8MZ9:G%<6M >6> ; - 321V9.K - 369HMB<CI6587.< -]/ M > @VMZKLIB7E5LM > 3MZ9c7U1 > 7GHI69TKm1b@41V9:5L16K - t
τ(r, z, t) = g ∗t∂ θg
∂t.
8 73 2 Bj 1 > GeK -0/ 1VGJ< - Gv3 - 7:587 - G / 1n@a587E;=G 8 / 1n@a587:;?GJ<CM > 7JK - Gy<65 + -?> G:5LM > GyG:M > 7e7E9B==GJ5 >#t ;?9H5 - IC9 - GDCKm1#36KLICG
3 - 7:587 - K8M >J0 I - I69q< P^M > < -#/ M > GH5L<6;=9:; - -Y7EMZIB7 -?> 9 - GH7U1 > 707:9B=?G0GHI63;=9:5 - I69 - G2CuKwP^;?321V58G:G - I69q< - G / MZI / ' - GK85 +'587 - G]7 ' - 9 +`5 uNI - G B -2K -d/ '21.+'3< - 369 - G:G:58M >- Gc7I > 5 t MZ9 + -- 7 >6- <6;=3 -=> < uaI - < - Km1J@41VK - I69 +'M C -?>C>6-<21 > GpK - @ZMZK8IJ+ - V < - KQPR; / 1V9c7.5 > Gc7U1 > 7E1 > ;W< - 7 - +`3;?9:147EI69 - -nK8I65 H + + -5t M >6/ 7E5LM > < - GHM > 369:MZ3C9 - 0 9E14<65 -=> 7<21 > GWK - G / MZI / ' - GXK85 +'587 - Gp7 ' - 9 +`5 uaI - G 8 @VMZ5L9Y5 >#t 9E1f;Ru 3 43 9.G MZI032149 - ? - +'36K - ^ <7-Z;Ru 3 43 > 9 _ B34cd5 > G:52G - I6K -KL1q@41VK - I69 +'M C -=>6>6- < - Km1b7 - +`3;?9:147EIC9 - <21 > GkK - <6M;+`1V5 >C- D < - @ZMZK8IJ+ - V
< τ >=1
V
∫∫∫
V
τ dV
- GH7]IC7E58K - 3 ; KLK - G?P^; / 9:587W<OPr1V3C9B=?G]KwP - ?B369 - GHG:5LM > 8 73 2 Bp< - τ - 7 / M;+'3C7 - 7 -?> I`< -k/=- uaI - θg(t)- Gc7XI > 5 t MZ9 + -
<21 > G D -< τ >=< g > ∗t
∂ θg
∂t,
GHMZ587-6<21 > GTK - <6M;+`1V5 >6- < - MVI69:5 - 9-CKwP - ?B369 - G:GH5LM > < - θg;A7U1 > 7 /?- KLK - <6M >6> ; - 361V9_KwP^;euNI2147:5LM > 8 73 5;B -
< τ >= jω < g(ω) > θg(ω) = jωγ − 1
βγpg(ω) < g(ω) > .
& 1#@41VK - I69 +'M C -=>6>6- G:3217E5m14K - -G:I69yK - @ZMZK8IJ+ - V - < - Km1#9:;=3M > G - 5 +'36I6K8G:58M >6>6- KLK - < g > <21 > GeK -<6M.+g1V5 >6- < - MZI69H5 - 9o<CM >6>6- -%< Pr14369B==G 8 73 2.2;B - KL1#369:MZ3C9:5L;A7E; 8 ; 3 B - 7eKL1#7:9E1 > G t MZ9 +`; - < -,& 1V3CKm1 /=- ^ E -2.543736< _ 8 9E1V3C3 - KL; - <61 > GkKwPR1 >6>6- ? - B-
< g >=8
π2
+∞∑
m=0
+∞∑
n=1
1
(m+ 1/2)2λ2n
1
jω +λ2
nR2 + (m+ 1/2)2π2
(1 + 2R)2
A2
V 2 σ
,
MZt R = `/(2a)- 7hMZt A
V=
2
a+
2
`<6;=G:5 0Z>6- K - 9E14363MV9H7hGHI69 t 1 /?- G:IC9[@ZMZK8IJ+ - 3 & 1J@41VK - I69+`M C -?>6>6- < -
KL1q@41V9H5m147:5LM > < - 7 - +'3;?9E17EI69 - < τ > G=PR; / 9:5l71VKLMZ9HG-6<61 > GTK - <6M.+g1V5 >6- < - MZI69H5 - 9-
< τ >=γ − 1
βγpg
+∞∑
m=0
+∞∑
n=1
8
π2
1
(m+ 1/2)2λ2n
1
1 +λ2
nR2 + (m+ 1/2)2π2
(1 + 2R)2 X2
,8 73 2 \B
MZt X - Gc7<CM >6> ;e321V9 8σ = c0`h
BX2 =
A2
V 2
c0`hjω
.
M - 7H7 - G:MVKLIC7:5LM > <6I#369:M6Kg= + - 8 736< B >6-d/ M > @ - 9 0Z- 9:1V365L< - + -?> 7XuNI - 3MZI69h< - G_3 - 7E5l7 - G[@41VK - I69:GX< - ω8 243 243 2 3r1 B /=- 7:7 -t MZ9 + - < - GHMZKLIC7:5LM > G - 9H;=@]=?K - <CM >6/ 3 - Iu3C9E147:5uNI - C`KwP^I6GE1 0V- 3 >6- 1VIB7E9 - t MZ9 + - < -GHMZKLIC7:5LM > - @V14Km16K - G:I69eI >6- 36K8I6GvKL1V9 0V- 0 1.+ + - < - t 9:;RuaI -?>6/=- -p1;=7:;`3C9:MZ3MZG:; - 361V9 e - 9 - 9 - K8K -q- Gc7369H;?G -?> 7E; - 1VI321V9:1 0 9E143J' - / 5 H < - GHG:MZICG3
,+U +U )+e *P)+)+,, )#' & 4
$*)9$ 4$(959/( % 6(9( 1 ( & 6%* 14M[M;+,+ - <21 > G'K - 32149E1 0 9:1V3J' - 369H; / ;?< -=> 7-[K - G'G:MZK8IC7E58M > Gq<6I 369:M CKg=+ - 8 73 E G BbGHM > 7g9 -?/ ' - 9 / '6; - G
G:MZICG t MV9 + - < - G:MZK8IC7E58M > G Cb@41V9:5L16K - G_G:;=321V9H; - Gg(r, z, t) = fr(r)fz(z)h(t) = ϕ(r, t)ψ(z, t) ,
MZt ϕ(r, t)- 7 ψ(z, t) GHM > 7kG:MZK8IC7E58M > G_9 - G:3 -?/ 7E58@ - + -?> 7< - GT369HM 6Kg= + - G 8 73 E 9 B - 7 8 73 E > B3& 19:;?GHMZKLIB7E5LM > < - Gq3C9:M 6K =+ - G 8 73 E 9 - 73 E > B - GH705 / 5 - ` -=/ 7:I6; - <21 > GbK - <6M;+`1V5 >6- < -& 1V36KL1 /?- - -=>
9E1.+ -?> 1 > 7 /?- GT< - IG?369HM 6Kg= + - GQCq< - Gk369:M6Kg= + - GT1VIG? / M > <65l7E5LM > Gh5 > 587E5L1VK - G > I6K8K - G 3;W> 3MZGE1 > 7 ϕL(r, t) = 1 − ϕ(r, t) -6K - 369:M CKg=+ - 8 43 E 9 BX< - @a5 -?> 7 -=>- ` - 7
(
∂ 2
∂r2 +1
r
∂
∂r− 1
σ
∂
∂t
)
ϕL = 0 3MZI69 r ∈ (0, a) , t > 0 ,8 73 2 9 1 B
ϕL = 1 3MZI69 r = a , t > 0 ,8 73 2 9 B
ϕL = 0 3MZI69 r ∈ (0, a) , t = 0 ,8 43 2 9 / B
G:MZ5l7 -B<21 > GkK - <CM;+g145 >6- < -& 1V36KL1 /?-(
∂ 2
∂r2 +1
r
∂
∂r− q2
)
ΦL = 0 3MVI69 r ∈ (0, a) ,8 43 2 > 1 B
ΦL =1
s3MVI69 r = a ,
8 73 2 > BMZt ΦL(r, s)
- GH7kKL1J7E9E1 > G t MZ9 +`; - < -& 1436Km1 /=- < - ϕL(r, t) - s - Gc7kKm1b@41V9:5L16K - < -& 1V36Km1 /?-o- 7 q2 = s/σ 3& 1qGHMZKLIC7:5LM > < -e/=- 369:M CKg=+ - G?P^; / 9:5l7ΦL =
1
s
I0(qr)
I0(qa).
8 73 2 N B& 1q@41VK - I69 +'M C -=>6>6- < ΦL >
< - ΦLGHI69TK - @ZMVKLIJ+ - V G?P^; / 9H587
< ΦL >=1
s
2
a2
∫ a
0
I0(qr)
I0(qa)r dr ,
G:MZ5l7 - -?>/t 1V5LG:1 > 7TI6GE1 0V- < - KL1q9 - KL147E58M > 8 ; 3 Bh< - KwPR1 >6>C- ? -D; -
< ΦL >=2
sqa
I1(qa)
I0(qa).
8 73 2.< B&%- 9 - 3MV9H7h< - G_<6;A@ - K8MZ363 - + -?> 7EGh1VG C +'3C7EMV7:5uNI - Gh< - G t M >6/ 7:5LM > G[< - = - G:G - K +'Ma<65 / ; - G I1(qa) - 7 I0(qa)8 ;Ru 3 ; 3 N B -2K85 +'587:;d1VI?#7E9HMZ5LG_369 - +'5 - 9:Gh7 - 9 + - G -2<21 > GTKwP - ?B369 - GHG:5LM > 369H; / ;?< -=> 7 - -<6M >6>C- 3MZI69 < ΦL >
< ΦL >≈2
sqa
[
1 − 1
2qa− 1
8q2a2 + O
(
1
(qa)3
)]
,
uNI65%G?P^; / 9:587 -=>6/ MZ9 - -
< ΦL >≈2σ1/2
a
1
s3/2− σ
a2
1
s2 − σ3/2
4a3
1
s5/2+ O
(
σ3
s3
)
.8 73 2.5 B
6 I & /( '92 7* % 70 27a 4(% )$ D4$)% * (
j PR1V369B==G_K - G[7E9:1 > G t MV9 +'; - G_< - & 1436Km1 /=- 5 > @ - 9HG - G 8 2*57343 B- 8 2.57373 9 B - 7 8 2.57373 > B]< - Km1J9H; t ;?9 -?>6/=- ^ E _9:1V363 - KL; - GT<21 > GkKQPr1 >6>6- ? - -2KL1J@V14K - IC9 +`M C -=>6>6- < ϕL >
< - ϕLG:IC9kK - @ZMZK8IJ+ - V <CM >6>6-
< ϕL >≈2√π
2
a(σt)1/2 − 1
4
(
2
a
)2
σt− 1
24√π
(
2
a
)3
(σt)3/2 + O(
(σt)2)
,8 73 G B
GHMZKLIC7:5LM > @41VKm1 6K - 3MZI69]< - G]7 - +'36GXG:ICS'GE1*+,+ -?> 7X3 - 7E587:G 8 -?> 9 -0 149:<g<CID321V9:1.+=A7E9 - 1/σ B 3 & 1vGHMZKLIB7E5LM >< ϕ > <6I3C9:M 6K =+ - 8 43 E 9 B - GH7k<6;=<6I65l7 - < - Km1bGHMZKLIC7:5LM > < ϕL >
32149_Km1q9 - Km147:5LM > ϕ = 1 − ϕL
< ϕ >≈ 1 − 2√π
2
a(σt)1/2 +
1
4
(
2
a
)2
σt+1
24√π
(
2
a
)3
(σt)3/2 + O(
(σt)2)
.8 73 E B
& 1 t M >6/ 7:5LM > ψ(z, t)- Gc7<C;=7 - 9 +'5 > ; - < - Km1 + + - t 1C=M > uaI - KL1 G:MZK8IC7E58M > ϕ(r, t) 3 ;W> 3MZGE1 > 7
ψL(z, t) = 1 − ψ(z, t) -2K - 369:M6Kg= + - 8 73 E > B[<6M >6>C- -2<21 > GkK - <6M;+`1V5 >C- < -& 1V36Km1 /?-
(
∂ 2
∂z2 − q2
)
ΨL = 0 3MZI69 z ∈ (0, `) ,8 73;2V1 B
ΨL =1
s3MZI69 z = 0
- 7 ` , 8 43 2 B
MZt ΨL(z, s)- Gc7kKm1J7E9:1 > G t MZ9 +'; - < -& 1V36KL1 /?- < - ψL(z, t) 3& 10GHMZKLIB7E5LM > < -y/?- 369:M6Kg= + - G=PR; / 9:5l7
ΨL =1
s
sinh qz + sinh q(`− z)
sinh q`.
8 73 B& 1q@V14K - IC9 +`M C -?>6>6- < ΨL >
< - ΨLGHI69TK - @VMZKLIJ+ - V G?P^; / 9:587
< ΨL >=2
sq`
sinh q`/2
cosh q`/2,
8 73]\BGHMZ587
< ΨL >=2
sq`
[
1 + 2
+∞∑
n=1
(−1)ne−nq`
]
.8 73 9 B
K 1V9k7E9:1 > G t MV9 +`147E58M > < - & 1V36KL1 /?- 5 > @ - 9:G - <6M >6> ; - <21 > GfK - Gd9 - Km17E5LM > G 8 2.573736< 9 B - 7 8 2.57373 9 B_< - KL19H; t ;?9 -?>6/=- ^ E _ 8 1 >6>6- ? - B-6Km1J@41VK - I69 +'M C -?>C>6- < ψL >
< - ψLG:I69TK - @ZMZK8IJ+ - V <6M >6>6-
< ψL >=4
`(σt)1/2
[
1√π
+ 2
+∞∑
n=1
(−1)nierfcn`
2√σt
]
,8 73 > B
MZt5 - 9 t / <6;=G:5 0Z>6- KwP^5 > 7E; 0 9E14K - < - KL1 t M >6/ 7:5LM > <OP - 9:9 - I69k<6; /2> 5 - 321V9
ierfc x =
∫ ∞
x
erfc t dt , 1F@ -?/ erfc x =2√π
∫ ∞
x
e−t2 dt .
& 10G:MZK8IC7E58M > < ψ > <6I369:M CKg=+ - 8 43 E > B - Gc7f1VK8MZ9:G[<6M >6> ; - 361V9
,+U +U )+e *P)+)+,, )#' & 4 U
< ψ >= 1 − 2
`(σt)1/2
[
2√π
+ 4+∞∑
n=1
(−1)nierfc n`
2
1
(σt)1/2
]
.8 73 N B
&.- 369HMB<6IC587k< - GG:MZK8IC7E58M > G < ϕ >- 7 < ψ > <6M >6>C- 1VK8MZ9:GTKL10@41VK - I69T< - Km1 t M >6/ 7:5LM > < - e 9 -?-=> -?>
+`M C -?>6>6- GHI69TK - @ZMVKLIJ+ - < g > <6I369:M6Kg= + -
< g >≈ 1 − 2√π
A
V(σt)1/2 +
8R + πR2
π(2R + 1)2
(
A
V
)2
σt+R3 − 6R2
3√π(2R + 1)3
(
A
V
)3
(σt)3/2
− 4
2R + 1
A
V(σt)1/2
+∞∑
n=1
(−1)nierfc n2R + 1
(σt)1/2A/V+ O
(
(σt)2)
.8 73;< B
& P - ?B369 - GHG:5LM > < - Km1J@41VK - I69 +'M C -=>6>6- < τ > < - τ GHI69TKm1J@ZMVKLIJ+ - V 8 ;eu 373 2 B
< τ >=< g > ∗t∂ θg
∂t,
<6M >6>C- -C<61 > G_K - <6M;+`1V5 >C- < - MZIC9:5 - 9-NKm1 t M >6/ 7:5LM > θg;A7U1 > 7 /=- KLK - 9 - 7 -=> I - <21 > G_K - 32149E1 0 9:1V3J' - 369:; / ; H
< -?> 7 8 ;Ru 373 5;B -< τ >= jω < g(ω) > θg(ω) = jω
γ − 1
βγpg(ω) < g(ω) > .
cd5 > G:5 - -?> t 145LGE1 > 7dI6GE1 0Z- -<21 > GoKQP - ?C3C9 - GHG:58M > 8 73;< B-< - Km1`7:9E1 > G t MZ9 +`; - < - & 1V3CKm1 /=- ^ E -2.543736< 9 _:-MZtKm1J@41V9H5m16K - < -& 1V36KL1 /?- s - GH7k9 - +`3CKm1 / ;o321V9 jω
8 7E9E1 > G t MZ9 +`; - < - MZI69:5 - 9 B - - 7k< - Km1b369HMZ369:58;=7:;+∞∑
n=1
(−x)n =1
1 + x,
Km1J@41VK - I69 +'M C -?>C>6- < τ > G?P^; / 9:5l7T<21 > GTK - <CM;+g145 >6- < - MZIC9:5 - 9-
< τ > ≈ γ − 1
βγpg
[
1 −X +8R + πR2
π(2R + 1)2 X2 +
3
4
√π
R3 − 6R2
3√π(2R + 1)3 X
3
+2
2R + 1
X
1 + e2(2R+1)/X+ O
(
X4)
]
,8 73;5 B
MZt X - GH7<6M >C> ;y321V9X2 =
A2
V 2
c0`hjω
.
& 1G:MVKLIC7:5LM > 8 73 E Bf3MZIC9 < ϕ > ;A7U1 > 7v5LG:GHI - < PRI > <6;A@ - KLMZ363 - + -?> 7J1VG C +'3C7EM47E5uNI -0-=> s = jω -Km1G:MZK8IC7E58M > 8 73.5 Be3MZI69 < τ >
> P - GH7q@41VKL16K - uNI - <21 > GbKm1+ - G:I69 - MZt ω > P - Gc70321VGb3 - 7:587- /=- uaIC51V3632149E1V5l7_5 / 5 <6I t 1V587_< - KL1b<6; /6> 587:5LM > <6I321V9:1.+2==7E9 - X 3
>6- 143636KL5 / 147:5LM >> IJ+';?9H5uNI - +'M > 7E9 - uNI - K - 7 - 9 + - 2
2R + 1
X
1 + e2(2R+1)/X14363MV9H7 - I >6- 3C9:; / 5 H
G:58M > 7:MZI @HMZI69HGh5 >t ;?9:5 - I69 - C 10−6 <1=GHI69hKL1v@41VK - I69[< - < g > / P - GH7h3MZIC9 /=- Km1uaIOPR58K > P - Gc7_321VGh9 - 3MZ9H7:;
<21 > GTKL1 > MZ9 + - M ;[ > E G 5]\ H 2^ 2;2_ 3
6 I & /( '92 7* % 70 27a 4(% )$ D4$)% * (
6 R 4 6% 8* G O2!$* 7 ' %$( * 9%/( #)(;Y> 9:;?GHIJ+';.- KL1@41VK - I69I+`M C -=>6>6- GH32147E5L1VK - < - KQPR; / 1V9c7b5 > GH7U1 > 7E1 > ;`< - 7 - +`3;?9:147EI69 - < τ > 3 - IC7
<6M >6/ =7:9 - ; / 9:587 - - -?> t M >6/ 7E5LM > < - KwP^; / 1V9H7Y5 > GH7E1 > 7U1 > ;_< - 369 - G:GH5LM > pg-ZGHMZI6GYKL1 t MZ9 + -A0 ; > ;?9:1VK - GHI658@41 > 7 -
< τ(ω) >=γ − 1
βγpg(ω)EP ,
8 73 \ G BMZt Ep
3 - IC7 A7E9 -y- ?C3C9:5 +';o< - < - IG?+g1 > 5g==9 - G
EP =+∞∑
m=0
+∞∑
n=1
8
π2
1
(m+ 1/2)2λ2n
1
1 +λ2
nR2 + (m + 1/2)2π2
1 + 2R2 X
,8 73 \ E B
8 - ?B369 - G:GH5LM >- ?C1 / 7 - 43 2 \ B -6MZI
EP ≈ 1 −X +8R + πR2
π(2R+ 1)2 X2 +
3
4
√π
R3 − 6R2
3√π(2R + 1)3 X
3
− 2
2R + 1
X
1 + e−2(2R+1)/X,
8 73 \2;B
8 - ?B369 - G:GH5LM > 73;5g@41VKm1CK - 3MZI69 X GHICS'GE1.+ + -?> 7y3 - 7E5l7B -1n@ -=/ R = `/(2a) - X2 =A2
V 2
c0`hjω
- AV
=
2
a+
2
`
- 7kMZt λn<6;?GH5 0Z>C- K - d?;=9:M'< - 9E1 >70 n < - J0(λn)
3
- * (8$:' (,:'+,6 *4670-7; &(, ! -7>/(?:'( *%&%- 9 - 3MZ9c7< - KL1J@41VK - I69 +`M C -?>6>6- < τ >
8 73 \ G BX<61 > GkKm1bKLMV57 ' - 9 +'Ma<7C > 1.+'5uNI - KL5 > ;n149:5LGH; -< ρ′ >= ρ0χT (pg − β < τ >) ,
MZt < ρ′ >- GH7kKL1 +'M C -=>6>6- G:I69TK - @ZMZK8IJ+ - V < - KQPR; / 1V9c7T5 > Gc7U1 > 7U1 > ;y< - +g1VGHG - @ZMVKLIJ+'5uNI - -6<6M >6>C-
< ρ′ >=1
c20[γ − (γ − 1)EP ] pg .
8 73 \ B& 1 +`M C -=>6>6- < - KQPR; / 1V9c7]< - +`1VGHG - @ZMZK8IJ+`5 uNI - < ρ′ >
- Gc7[9 - K85L;QCyKm1o@V149:5m17E5LM > < - @VMZKLI7+ - < - Km1 / 1n@a587:;δV < - Km1 t 1C?M > GHI658@41 > 7 - 8
Vm<6;=G:5 0Z>6- K - @ZMZK8IJ+ - +g14G:G:5 uNI - <6I 2IC5L< -b- 7 Vm0
K - @ZMZK8IJ+ - +g1VGHG:5 uaI -Gc7U147:5uNI - B
< ρ′ >
ρ0=< ρ >
ρ0− 1 =
Vm0
< Vm >− 1 =
−δ < Vm >
< Vm >= −δV
V.
& Pr1V<7+`5l7:7U1 >6/?- 1 / MZI6Gc7E5uNI - YP = 1/ZP< - KL1 / 1n@a587E;*-F9E1V3C3MZ9c7<6IJ<6;658732149:5L;A7U1VK -?> 7E9:1 > 7 8
−jω δV BGHI69TKwP^; / 149H7T< - 369 - G:GH5LM > uNI65 -?> 9:;=G:I6Kl7 - 8
pgB - - Gc7k<6M >6/ <6M >C> ; - 321V9
YP =−jω δV
pg= jω
V
ρ0c20
[γ − (γ − 1)EP ] ,8 73 \4\B
uNI65 > P - Gc7v1VIC7E9 - uNI - KwP - ?B369 - G:GH5LM > 8 243 > BdG:5 +'36K - + -?> 7 + -?> 7E58M >6> ;J<21 > GoK - 321V9E1 0 9E1V37' - 243 243 2g<6;?<658;5CKQP - ?C3MZGH;eGHI /?/ 5 > 7< - G / MV9:9 -=/ 7:5LM > G_7 ' - 9 +`5 uaI - G_369H; / M > 5LGH; - Gk321V9TKL1 > MZ9 + - M ;[ > E G 5]\ H 2 3
)+U ! C ) ( (X +U ! 6
j 1 > G /=- 7:7 -y- ?B369 - G:GH5LM > 8 43 \4\ B - EP
- Gc7T<6M >6> ;e361V9hKwP - ?B369 - GHG:5LM > 8 73 \ E B]MVI 8 73 \2;B 3 j 1 > GkK - Gh< - IG?/ 1VG -O58K - GH7o3MZG:GH5 6K - < - @V;?9H5 /2- 9 uNI - G:5pK - G - ` - 7EGf7 ' - 9 +`5 uNI - GfGHM > 7 > ; 0 KL5 0 ;?G 8`h → 0 -OGHMZ587 X → 0 B -KwPR1V<J+'587H7U1 >6/=- YP
/ MZ9H9 - GH3M > <&CqKQPr1V<7+`5l7:7U1 >6/?- 1 / MZI6Gc7E5 uaI - 14<65m16147E5 uaI - < - Km1 / 1n@a587:;.3;W> - ` - 7 -K - 7 - 9 + - EP
- Gc7o; / 9:587dG:MZICGdKm1 t MZ9 + - < PRI > <6;=@ - KLMV363 - + -=> 7 C / MVI69H77 - 9 + - 321V9k7E9E1 > G Ht MZ9 +'; - < -I& 1V3CKm1 /=- 8 73 \2;B-6K - 9H;?GHI6K87E147 EP = 1 3MZI69 `h → 0
- GH7k5 + +';?<65L147 3HB5 /=- 7 - 9 + - EP
- GH7o; / 9H587G:MZI6GdKm1 t MZ9 + - <OPRI > <C;=@ - KLMZ3C3 - + -=> 7O+`Ma<21VK 8 73 \ E B - -?> > ; 0 K85 0Z- 1 > 7kK - G- ` - 7:GT7 ' - 9 +`5 uNI - G -25LKO@a5 -=> 7
EP =8
π2
+∞∑
m=0
1
(2m+ 1)2
+∞∑
n=1
4
λ2n
= 1 ,
-?> @ - 9H7EI< - G369HMZ369H5L;=7:;?G 0 ; > ;?9:1VK - GTG:I65l@V1 > 7 - G
4
π
+∞∑
m=0
1
2m+ 1sin
(2m+ 1)π
`z = 1 ∀z ∈ (0, `)
8 73 \ 9 1 B+∞∑
n=1
2
λn
J0(λnr/a)
J1(λnr/a)= 1 ∀r ∈ (0, a) ,
8 43 \ 9 B
uNI65+bI6Kl7E5L3CKL5L; - G]321V9 - KLK - G H + + - G - 7k5 > 7:; 0 9H; - GT9 - G:3 -?/ 7E5l@ - + -?> 7G:I69 z ∈ (0, `)- 7 r ∈ (0, a) 9 - G:3 -?/ 7:5 H@ - + -=> 7k<6M >6>6-?> 7
8
π2
+∞∑
m=0
1
(2m+ 1)2 = 1- 7 +∞
∑
n=1
4
λ2n
= 1 .
;W>4/2> -2G:5 EP = 1 -6KQPr14<J+`5l7:7E1 >6/?- 1 / MVI6GH7:5uNI - < - Km1 / 1n@a587:; YP
8 43 \4\ BX@V14IC7
YP = jωV
ρ0c20
,
/?- uNI65 / MZ9H9 - G:3M > < C5 -?> CyKwPR1V<J+'587:7E1 >6/=- 1 / MVI6GH7:5uNI - < - Km1 / 1n@a587:;kGH5 K - +'MZIC@ - + -=> 7[< -k/ M.+`369 - GHG:5LM > G H<6;=7 -?> 7 - G - GH7kG:IC363MVG:;v1V<65m1 2147E5 uNI - 3
R 6() & 1g9 - Km147:5LM > < ρ′ >
ρ0= −δV
VIC7:5LK85LG:; - 361V9 e - 9 - 9 8 - 7o9 - 369:58G -v/ 5 H < - GHG:I6G Bf1V<J+ - 7
uNI - K - GY361V9:MZ58G\< - Km1 / 1n@B5l7E;_GHM > 7X361V9 t 1V5l7 - + -=> 7\9:5 0 5L< - G- /=- uaIC5 > P - GH7X321VGYK -_/ 1VG -?> 369:147E5 uNI -_-?> 9E1V58G:M >< - KL1 369:;=G -=>6/?- <6I +'5 / 9HMZ3J'6M >6- 9H; /?- 3C7 - I69 8 K -g/ '21.+'3)< - <6;=36Km1 /?- + -?> 7q<CI +'5 / 9:MZ37'6M >6- ;+ - 7:7 - I69- Gc7 - ?B369:5D+`;d321V9hK - t 1 / 7 - I69 δV B3 & P - ` - 7T< - KQPr1V<7+`5l7:7U1 >6/?- <6I+`5 / 9:MV3J'6M >6- 9:; /?- 3C7 - I69k1VI69E14587h3CI A7E9 -5 > 7E9HMB<CI6587032149 e - 9 - 9 -=> t 1V58GE1 > 7'I6G:1 0Z- < - KQPR;RuNI2147E58M > < -/ M > G - 9H@4147:5LM > < - Km1 +g1VGHG - -]GHI658@41 > 7'Km1369:;=G -=> 7E147E58M > <6M >6> ; - I > 3 - I36K8I6G_KLMZ5 > 8 7373 E -6;euNI217E5LM > G 73 \ > C 73 \ 5 B3
|gc| "( # +[ - '%) ) ! # ) - - ) + ,- . ) ! $ .; ( ) =$ ( "!% '%)
P^M - 7k< -y/?- 321V9:1Z9E143 -o- Gc7d<OP - ?C3C9:5 - 9_K -o/ 21'3< - 3C9 - GHG:58M > I > 5 t MV9 -o/ 9:;=;y<21 > GTI >6- 43 -hH7E5l7 - / 1F@a5l7E;0< - @ZMZK8I -
V- 7v< - GHI69 t 1 /?- A 32149e< - Gy@B5 69E17E5LM > Go< - 321V9HMZ5LG5^ 363"!$# &% H # > _! M - 7:7 -
6 I & /( '92 7* % 70 27a 4(% )$ D4$)% * (
`
r
a
z~ξ \ ~ve
onPp ! # qMBK85 > <C9 - t - 9 `; C0G - G - ?a7E9H; '587E;=GT321V9_< - G '5 / 9:MZ3CM >6- G !
/ 1F@a5l7E; - GH7[< - t MZ9 -T/ aKL5 > <69:5 uNI -T- 7 t - 9'; - CyG:M >`- ?a7E9H; '587E; z = 0 321V9]I >6- - 69E1 >6- 1 > 5'; - < P^I >'MZIC@ - -?> 7@a5 69E147:MZ5L9 - < -e/ 61`3< - <6;=36Km1 /=- -?> 7 ξ - 7k< - @a587 - G:G - ve = jω ξ - 7 CqG:M >- ?a7:9:; '587:;z = ` 361V9_I >6- - 69:1 >6- < PR1V<'587H7U1 >C/?- 1 / MZI6GH7:5uNI - Yr
ZI69 - ! # !%- 7 - 9 - 43 - 7:587 -/ 1n@a587:; <6;?GH5 >C- I >6-f/ 1n@B5l7E;f<CM > 7_K - G[<65 -=> G:58M > G[9 - Gc7 -?> 7_7:9B=?G[5 >#t ;=9:5 - I69 - G CJKL136K8I6Gd3 - 7:587 - K8M > ZI - I69k< PRM > < - λ0
/ M > G:58<6;?9H; - g145LGT<6M > 7fK - G<65 -?> GH5LM > G9 - GH7 -?> 7f7:9B=?GG:I63;?9H5 - I69 - GMCKQPR;=321V5LGHG - I69T< - G / MVI / - GKL5 `5l7 - G]@a5LG / MV7 - 9'5uNI - G δv,h
λ0 >>3√V
- 7 δv,h <<3√V ,
MZtδv ≈
√
2` ′v
k0
,- 7 δh ≈
√
2`hk0
.
%546$( *+ -7;/( '*#0:'( *#%
q I G$ ) 7/('9 G 6 Uj I t 14587'uaI - K - Gd<65 -?> GH5LM > Gd< - KL1 / 1n@a587:;bG:M > 7o7:9B=?Gd5 >#t ;=9:5 - I69 - GMCDKm1`KLM > VI - I69< P^M > < -q- 7-uNI -K -q/ 21'3s< - 369 - G:G:58M > >6- 369:;=G -?> 7 - 321VGo< - @41V9H5m147:5LM > GfG:3217E5m14K - Gf1VI @VMZ5LGH5 > 1 - < - Gv32149:MZ58G /=-/ 61`3 < - 369 - GHG:5LM > - Gc7oG:I63C3MZGH;bI > 5 t MZ9 - <21 > Gk7EMZIB7 - KL1 / 1F@a5l7E;! 149dG:IC587 - ~grad p = ~0
- 7fKL1@a5l7 - GHG - 321V9c7E5 / I6Km145L9 -d- GH7'uNI21VGH5 > I6K8K -o-=> 7EMVIC7k3MZ5 > 7T< - KL1 / 1n@a587E; /?- uaIC5 > PR5 `36K85uNI - 361VGT3MZIC914IC7U1 > 7XI > <C; 65l7[< - 321V9HMZ5 > ; VKL5 - 16K - - 7XK - G]3 6; > M= >6- GX< - @a5LG / MZGH587:;?G > Pr1436321V9:1V5LGHG -?> 7[321VG !
q I G$ ) 4$ !/(# # $ &* V 60!; K8K - 3 - IC7 A7E9 -y- ?C3C9:5'; - < PR1V369 =?GTKwP^;euNI217E5LM > ! G:MZICGkKL1 t MZ9 - 5 > 7:; Z9:1VK - GHI658@41 > 7 -
∫∫∫
V
[
∂ ρ′
∂t+ div ρ0~v
]
dV = 0 ,
GHMZ587∫∫∫
V
∂ ρ′
∂tdV + ρ0
∫∫
A
~v. d ~A = 0 . ! \ >
- 9H7E1V5 >6- Gv361V9:MZ58Go< - Km1 / 1n@a587:;0;=7U1 > 7v1 > 5'; - Go<OPRI > `MVIC@ - -=> 7y@a5 C9E147:MZ5L9 - < - <6;?36KL1 /?- -=> 7~ξ 5 / 5 -?> z = 0 '5 / 9HMZ36M >6- ; - 7H7 - IC9 - 7k< PR1VIC7:9 - Gk;=7E1 > 7 / 1V9:1 / 7:;?9:58G:; - Gk321V9_K - I69T5'3;=<21 >6/=- 369 - G:GH5LM > @a587 - G:G - Z 5 / 5 -?> z = ` '5 / 9HMZ36M >6- 9H; /=- 3C7 - I69 /=- 7H7 - ;RuNI2147E58M > 3C9 -?> <#Km1 t MZ9 -
GHI658@41 > 7 - <61 > GTK - <6Mg1V5 >6- t 9H;euNI -?> 7:5 - K ∫∫∫
V
jω ρ′ dV + jω ρ0
∫∫
A
~ξ. d ~A+ ρ0
∫∫
A
p
ZdA = 0 .
! \ N
)+U ! C ) ( (X +U !
j I t 1V587h< - KwP^I > 5 t MV9'587E;k<6I / 21'3< - 369 - G:G:58M > 58KO@a5 -?> 7∫∫
A
p
ZdA = p
∫∫
A
1
ZdA =
pA
Z,
! \
MZt 1/Z <6;=G:5 >6- KL1q@41VK - I69 'M -?>6>C- < - KwPR1V<'587H7U1 >6/=- < - 321V9:MV5 - 7Km1q9 - Km17E5LM > ! \ N < - @a5 -?> 71VK8MZ9:G 1
ρ0
∫∫∫
V
ρ′ dV + δV +pA
jω Z= 0 ,
! \
MZt δV =
∫∫
A
~ξ. d ~A <C;?G:5 >6- KL1`@41V9:5L147E58M > < - @ZMZK8I - < - Km1 / 1n@a587E;JKL5L; - 1VI? @a5 69:147E58M > G< - Km132149:MZ5 `5 / 9:MV36M >6- ; - 7:7 - I69 -=> z = 0 !
q I #9$ J7%( 6 '2* # 6( 6 4&3* 109 - KL147E58M > K85 > ;?1V9:58G:; -
ρ′ = ρ0χT (p− βτ)
9 - 3MZ9H7:; - <21 > GkKm1b9 - KL147E58M > < - Km1q9 - Km17E5LM > ! \ 3 - 9 - 7T< P^;?KL5 `5 >6- 9[Km1J@41V9:5L16K - ρ′ - 7k<6M >6>C-[
1 +A
jωZχTV
]
p− β
V
∫∫∫
V
τ dV =δV
χTV,
!% ;RuaI6147E58M > uaIC5 `M > 7:9 -_-?> 32149H7E5 / I6K85 - 9 uNI - K -h/ 61`30< - 369 - GHG:5LM >'>C- <C;?3 -?> <'321VGY< - τ(~r, t) `1V5LG< - G:1J@41VK - I69 'M -?>6>C- G:I69[7:MZIC7_K - @VMZKLI -
V < - Km1 / 1F@a5l7E; 1VKLMZ9HG - uNI - τ @41V9H5 - < - t 1C?M >5 `3MZ9H7E1 > 7 - <61 > GTK - G / MZI / - GK85'587 - GX7 - 9 `5 uNI - G !q I #9$ ) 4$ ]$ ) * V ) 1* (
MVI6GG:1 t MZ9 - ! - K8K - G=PR; / 9:5l7k<21 > GTK - <CM&g145 >6- t 9H;euNI -=> 7:5 - K (
jω
c0+ `h ∆
)
τ =γ − 1
βγ
jω
c0p ,
! # MZI -?>C/ MZ9 -
τ =c0`hjω
∆τ +γ − 1
βγp .
! MZt `h <6;=G:5 >6- I >6- KLM > ZI - I69 / 1V9E1 / 7E;=9:58GH7E5 uNI - < - <65 `I6GH5LM > 7 - 9'5uNI - ! M&'3C7 - 7 -=> I#< - KQPRI > 5 Ht MZ9 `5l7E;f<CI / 21 `3< - 369 - G:GH5LM > p /=- 7H7 - < - 9 > 5g==9 - 9 - Km147:5LM > <CM >6>6-
∫∫∫
V
τ dV =c0`hjω
∫∫
A
~grad τ. d ~A +γ − 1
βγV p .
!
( +,/6 *+ 1G:MVKLIC7:5LM > < - KwP^;euNI217E5LM > ! # uNI65WG:147E58G t 1V5l7 CKL1 / M > <C587E58M > τ = 0 G:I69eK - G t 9:M > 7E5 =?9 - G 3 - IB7
=7:9 - ; / 9:587 - K - t 1 / 7 - I69 ejω t - GH7kM&'5LG
τ =γ − 1
βγp[
1 − e−j khu]
, ! \
6 I & /( '92 7* % 70 27a 4(% )$ D4$)% * (
MZt u - Gc7dKL1'3MZGH587E58M > <6I 3MZ5 > 7 / M > GH5L<6;=9:;J< - KL1 / 1n@B5l7E; 9 - 3;?9H; - 361V99:1V363MZ9H7<C0Km1 > MZ9 g1VK - G:MV9H7U1 > 7 -C0Km1bGHI69 t 1 /=- < - Km1q321V9HMZ5 u = 0 G:IC9kKL1b321V9HMZ5 ! - 7H7 - t MV9 - < - G:MVKLIC7:5LM > - Gc7 / MZ9:9 -?/ 7 - GHMZI6Gg9H;?G - 9H@ - < P^I >6- 321V9H7 uNI - KQPR;=321V58G:G - I69'< - G / MZI / - GK85'587 - Gh7 - 9'5uNI - GdG:MV587T7E9 =?Gd3 - 7:587 - 32149d9E14363MV9H7d1VIG?<65 -=> G:58M > Gk< - KL1 / 1F@a5l7E; / P - Gc7 H C H <6589 - uNI - KL1t M >C/ 7E58M > e−jkhu <6; / 9HMZ5LGHG - 7E9 =?GT@B5l7 - 3MZI69TI > 3MZ5 > 7'uNI65.G=PR; / 1V9c7 - < - KL1'361V9:MZ5 - 7k< Pr14IC7E9 - 32149H73uNI -GHMZ587 t 1V5l7_K -o/ CMZ5b?
kh =−1 + j√
2
√
k0/`h , !
3MZI69 uNI -g/=- 7:7 - t M >C/ 7E58M > 9 - 3C9:;?G -?> 7 - C5 -?> I >6- M > < - < - <65 `I6G:58M > 7 - 9'5uNI - uaIC5X3; > =A7E9 - <21 > GvKL1/ 1F@a5l7E; Cq32149H7E589_< - GT32149:MZ58G ! 1V9TGHI6587 - GHI69TKm1qG:I69 t 1 /?- u = 0
∂ τ
∂u= −γ − 1
βγp
1 + j√2
√
ω
c0`h- 7∫∫
A
~grad τ. d ~A =
∫∫
A
∂ τ
∂udA = −1 + j√
2
√
ω
c0`hAγ − 1
βγp ,
/=- uNI65%<6M >6>6- 2> 1VK - -?> 7T3MZI69_KwP - ?B369 - GHG:5LM > ! ∫∫∫
V
τ dV =
[
V − 1 − j√2A
√
c0`hω
]
γ − 1
βγp .
! > %- 9 - 3MZ9H7[< -d/=- 9:;=G:I6Kl7U147[<21 > GhKL1v9 - Km17E5LM > ! % <CM >6>6- KQP - ?B369 - G:G:58M > G:IC58@41 > 7 - 3MZIC9[KwPR1'36KL5l7EI6< - < -KL103C9 - GHG:58M > <61 > GTKm1 / 1F@a5l7E;
p = − γδV
χTV
1
1 +γA/V
jω χTZ+
1 − j√2
(γ − 1)A
V
√
c0`hω
. ! N
MZ9HG / MZI / - G_KL5'587 - G KwPR1V<'587H7U1 >6/=- < - Gh361V9:MZ58G - GH7 > I6K8K - <21 > GXK -d/ 1VGh369H;?G -?> 7_GE14I t 3MZIC9XKQP - ?B7:9:; H'587:; z = ` uaIC5X1#3MZI69y1V< `5l7:7U1 >6/?- 1 / MZI6Gc7E5uNI - Yr
! 1D@41VK - I69 `M -?>6>6- 1/Z < - KQPr14<`5l7:7E1 >6/?- < -32149:MZ5 - GH71VKLMV9:Gh<6M >6> ; - 321V9
A
Z=
∫∫
A
1
ZdA =
∫ 2π
0
∫ a
0
Yr
πa2 r dr dθ = Yr ,
/=- uNI65%<6M >6>6- 3MVI69_KwP - ?B369 - G:GH5LM > ! N < - Km1b369 - G:GH5LM > 1 / MZI6GH7:5uNI -
p = − γδV
χTV
1
1 +γ
jω χTVYr +
1 − j√2
(γ − 1)A
V
√
c0`hω
. !
%- 6I65L< -y-=>#t - 9 `;y<21 > GKL1 / 1n@B5l7E; - Gc7d< - KQPr1V589 uaIC5%3 - IC7 A7E9 -e/ M > GH5L<6;=9:; / M& - I > 1]dy32149 t 1V5l7 /=- uNI65p5 `3CKL5uNI - ρ0c20 = γ/χT ≈ γP0
! PR1V<'587H7U1 >6/=- 1 / MZICGH7E5 uNI - < - 7E9:1 > G t - 9H7 YT =Sve + Yep(0)
p(`)<6IG aGc7>= - `5 / 9:MV36M >6- G Hi/ MZI63CK - IC9 ;eu ! !$# G=PR; / 9:5l7k5 / 5
YT = − jω δV
p+ Ye = jω
V
γP0
[
1 + (γ − 1)A
V
1 − j√2
√
c0`hω
]
+ Ye + Yr . !
+)# +U d+U ) 7
- 7:7 -D- ?B369 - GHG:5LM > ! < - KwPR1V<'587H7U1 >C/?- 1 / MVI6GH7:5uNI - < - 7E9:1 > G t - 9H7 YT
- GH7 /=- KLK - 369HMZ3MZG:; - 321V9k58;=7 - 7 %-?/ MZKLK85 >6- 7^ &% _B1V5 > G:5 uNI - 361V9[1VKLKL1&^ _a3MZI69pK - G;=7E1VKLM >6> 1 - G -=> 21VGHG - G t 9:;euNI -=>6/?- G ! - 7:7 -- ?B369 - G:GH5LM >)- GH7b1VI6G:GH5[; 1VK - C KQPr1V< `5l7:7U1 >6/?- YP
7E9HMZIC@V; - 32149 e - 9 - 9 ;Ru ! ! \4\ ;eu ! ]\ < - KwPR1V9H7:5 / K -< - e - 9 - 9T^$# _ EP
;A7U1 > 7T<6M >6> ;y321V9hKL1bG:MZK8IC7E58M > C / MZI69H7_7 - 9 - 361V9h7E9:1 > G t MV9'; - < - 1V3CKm1 /=- ;Ru ! ! \ ;eu ! !#< - e - 9 - 9 CbKwP^MZ9:<C9 - # -?> X - ?B369 - G:GH5LM > uNI65 > P - GH7_3214G /?- KLK - 9 - 7 -?> I - <61 > GhKL1 > MZ9 - ;[6> # % \ H ^ _ !
1DGHMZKLIB7E5LM > ! 3MZI69fK -t 1 / 7 - IC9 −jω δV/p - GH7o58< -?> 7E5 uNI - CgKm1`G:MZK8IC7E58M > < - e - 9 - 9 ! \4\ 3MVI69 −jω δV/pg 6MV9'5LGbKQPr1V< `5l7:7U1 >6/?- YruNI65 > P - GH7'3214G`369H5LG -#-?> / M&'3C7 - <21 > G0K - 'MB<6==K - < -
e - 9 - 9 -?> ; / 9:58@41 > 7 1/√
j = (1 − j )/√
2 !
|g < $ # - A! "! ) #? ! ) j 1 > GoK -J/ 14GoMZtuK -v/ 21'3u< - 369 - G:GH5LM >u> P - Gc7y36KLICGfI > 5 t MZ9 - <21 > GoKm1 / 1n@B5l7E; K - Gf<65 -?> G:5LM > Gd< -
Km1 / 1F@a5l7E; >6- G:M > 7v36KLI6Gy3 - 7:587 - Gy< - @41 > 7vKm1DKLM > ZI - I69y< PRM > < - I > 14IC7E9 - 'Ma<6=?K - 3 - IC7 =7E9 - 3C9:MZ3MZG:;3MVI69o9 - 369:;=G -?> 7 - 9JK -q/ 21'3x1 / MZI6Gc7E5 uaI - <61 > GeKm1 / 1n@a587:; Km1#369:MV321 147:5LM >u- GH7 /?- KLK - < PRM > < - Ge36KL1 >6- GG:I65l@41 > 7JKQPr1? - < - KL1 / 1n@a587:; / BK85 > <C9:5uNI -'- 7JKwPR1'MZ9H7:5LG:G - -?> 7v< - GvM > < - Gb<)1VI? - ` - 7EGv@a58GuNI - I? - 77 - 9'5uNI - G - Gc7q369H5LG -=>)/ M&'3C7 - <21 > GJK -g> M& 69 - < P^M > < -D- 7q321V9vKwP^5 > 7E9HMB<CI / 7:5LM > < P^5'3;=<21 >6/=- Gb< -321V9HMZ5%1VIG? - ?a7E9H; '587E;=G ^ H #!_! - 'Ma<6=?K -f- Gc7 -?> 321V9c7E5 / I6KL5 - 9 /=- KLIC5%369:MZ3MZGH;e361V9h1VK8Km1& ^ _ !j 1 > G]I > GHMZI / 5< -/ Km1V9c7E;T<6I'<65LG / MZI69HG /?- 9H7E1V5 >6- GX< - GX1V36369HM9?B5 g147:5LM > G t 1V587 - GW<21 > G /=- 32149E1Z9:1V3 -3 - IC@ -=> 7h321V9:1 L7:9 - I > 3 - Ig9:1V365L< - G ! ;Y>`/ M&'36KL; -=> 7W< - KL1o9:;?GHMZKLIB7E5LM > 3C9:MZ3MZG:; - <61 > G /?- 321V9E1 Z9E1V3 - I >6- 1V363C9:M / - 36KLI6G_<C;=7U145LKLK8; - <CI - 369:M6Kg= -f- GH7<6M >6> ; - <21 > GTKQPr1 >C>6- ? - e !
4 67()*+ * /* %5 >)/?- 9H7E1V5 >)> M 69 - < P a3MV7 6==G - G`GH5'36KL5 2/ 147:9:5 /?- Ge3 - IC@ -?> 7 =7E9 - 9 - 7 -=> I - G'3MZIC9bG:5 `36K85 2- 9yK - G
;euNI2147:5LM > G[< - KQPr1 / MZI6Gc7E5 uaI - ! # ! - 7 ! ^ ! N _! ; KLK - GXGHM > 7_369H;?G -=> 7E; - Gk1VI t I69 - 7QC - GHI69 -< - K - I69_I6GE1 - !
.- G BGc7>= -k/ M > G:5L<C;?9:; - GH7 /?- K8I652<6; / 9:5l7W321V9WKm1 VI69 - !$# <6;6IC7X<6I`361V9E1V9E1V3 - ! I >6-T/ 1n@a587E;/ aKL5 > <69H5uNI - < - 9E1 ZM > a - 7y< - K8M > ZI - I69 ` - Gc7 - ? / 5l7E; - CGHM >s- ?a7E9H; '587:; z = 0 32149fI > `5 / 9HMZ36M >C-; - 7H7 - IC9d< PR1V<'587H7U1 >6/=- C / 589 / IC587MZIC@ - 9H7 Ye
I > '5 / 9:MZ3 6M >6- 9:; /?- 3C7 - I69o< PR1V<'587:7E1 >6/=- 1 / MZI6Gc7E5uNI -C / 5L9 / I65l7_MZIC@ - 9c7 Yr
- Gc73CKm1 / ; C0KQP - ?a7E9:; `5l7E; z = ` ! 1 - C9E1 >6- < - KQPR; - 7:7 - I69 - Gc7GHI6363MZG:; - 1n@ZMV5L9#I > 9E1 ZM > ; Z1VK C /=- K8I65o< - Km1 / 1n@a587:;! M >
`MVIC@ - -=> 7 - GH7'14G:G:5 `58KL;C /?- KLI65[< PRI > 365LGc7EM > 36KL1 > - KLK -g/ 9H; - I > <C; 65l7 Sve
uaI61VG:5hI > 5 t MZ9 - G:I69GE1 GHI69 t 1 /=- ! P^; / 149H7q5 > GH7E1 > 7U1 > ;#< - 369 - G:G:58M > p ;=7E1 > 7 uNI21VGH5 I > 5 t MZ9 - G:I69qI >6- G -?/ 7E58M > <6I)7EI - 5LK - GH7 / M > G:58<6;?9H; / M& - 5 > <6;=3 -=> <21 > 7'< - Km1 / MBMZ9H<6M >6> ; - r - 7'3 - IC7 A7E9 -> MV7E; p(z) ! 1V90G:IC587 - KL1/ M&'3MZGE1 > 7 - 9E14<65m1VK - <CI V9E1V<65 -?> 7h< - 369 - G:GH5LM >- Gc7 > ; ZKL5 - 1CK - 321V9h9:1V363MZ9H7OCbGE1 / M`3MZGE1 > 7 - 1?B5m14K -- 7 321V9TG:IC587 - KL1 / M&'3MVGE1 > 7 - 9:1V<65L1VK - < - KL1J@a587 - G:G - 32149H7E5 / I6KL1V5L9 - 3 - IC7 A7E9 -y> ; ZKL5 Z; - vr << vz
! 1V9p1V5LK8K - I69:G -?> 9E1V58G:M > < - KQPR5 `3MZ9c7U1 >6/=- < - G - ` - 7:G K85L;?GCkKm1h@a5LG / MZG:5l7E;]< -[/ 5LG:1V5LK8K - -?> 7 Km1_@41V9:5L147E58M >< - Km1q@B5l7 - G:G - 321V9c7E5 / I6Km145L9 - v = vz
G:I65l@41 > 7dKL10@41V9H5m16K - r - GH7k7E9 =?GG:I63;?9H5 - IC9 - C'GE10@V149:5m17E5LM > G:I65l@V1 > 7Km1b<6589 -=/ 7E58M > z ∣
∣
∣
∣
∂
∂zv(r, z)
∣
∣
∣
∣
<<
∣
∣
∣
∣
∂
∂rv(r, z)
∣
∣
∣
∣
. ! > %
5 > 14K - -?> 7G - I6K - KL1 / M&'3MZGE1 > 7 - 1?B5m1VK - < - KQPR;RuNI2147E58M > < -5: 1F@a5 - 9 H 7EMB - G - GH7DC - ?C3C9:5 - 9 - K8K -
6 I & /( '92 7* % 70 27a 4(% )$ D4$)% * (
369 -?> <Km1 t MZ9 - 1436369:M / 6; - GHI658@41 > 7 - (
1
c0
∂
∂t− ` ′
v1
r
∂
∂rr∂
∂r
)
vz(r, z) = − 1
ρ0c0
∂
∂zp(z) , r ∈ (0, a) , z ∈ (0, `) .
! > #F1 c /=- 7H7 - ;euNI217E5LM > ! > #F1 G:M > 7T14G:G:M / 58; - G_K - G[< - IG? / M > <6587:5LM > G]G:IC58@41 > 7 - G_G:I69_KL1 / M`3MZGE1 > 7 - z < - KL1@a587 - GHG -
vz(0, z)9 - GH7 - 2> 5 -y- 7 vz(a, z) = 0 ∀z . ! > #
1'GHMZKLIC7:5LM > 3MZI69 vz< -e/?- 369:M6Kg= - ! > #F1 G?P^; / 9H587 -?> 9:; Z5 - 21V9 `M > 5uNI - -=> M& - 7H7U1 > 7kK -
t 1 / 7 - I69 ej ω t vz(r, z) =
j
ωρ0
∂
∂zp(z)
[
1 − J0(kvr)
J0(kva)
]
, ! >
MZtK -o> M& 69 - < PRM > < - < - <65 `I6GH5LM > kv
- GH7k<6; 6> 5.321V9kv =
1 − j√2
√
k0/` ′v .
1q@41VK - I69 'M -?>C>6- < -v/=- 7:7 - @a5l7 - GHG - G:I69TI >C- G -?/ 7:5LM > <6I7EI - 'MB<6==K - M > < - 36Km1 >C- G?P - ?B369H5 -1VK8MZ9:G / M& - GHI6587
< vz >a=2
a2
∫ a
0
vz(r, z)r dr =j
k0ρ0c0
∂ p(z)
∂z
[
1 − 2
kva
J1(kva)
J0(kva)
]
. ! >
PR;RuaI6147E58M > < -y/ M > G - 9c@V17E5LM > < - KL1 g1VGHG - ! 369 -=> < KL1 t MZ9 - G:5 `36K85 ; -
jωγ
c20(p(z) − βτ(r, z)) + ρ0
∂
∂zvz(r, z) = 0 ,
! > \ - 7G:10@41VK - I69'M -=>6>6- G:I69TI >C- G -?/ 7:5LM > <6I7EI - G?P^; / 9:5l7
∂
∂z< vz >a +
jω
ρ0
γ
c20(p(z) − β < τ >a) = 0 .
! > ;Y> 2> <6I t 1V587 uNI - Km1u@41V9:5L147E58M > < - 7 - `3;?9:147EI69 - τ G=Pr1 >C> ICK --?> r = a C KwP^5 > Gc7U1V9q< - KL1u@a5l7 - GHG -32149H7E5 / I6KL1V5L9 - ! > % G:1g@V149:5m17E5LM > GHI658@41 > 7oKL1'@41V9:5L16K - r - Gc7f7E9 =?GoG:IC3;=9:5 - I69 - CgGE1g@41V9H5m147:5LM > G:I65l@41 > 7KL1J@V149:5m1 6K - z !1cd5 > G:5 KQPR;RuaI6147E58M > ! 3MZI69_KL1 / M > <6I / 7:5LM > < - KL1 / 21VK - I69_369 -?> <Km1 t MZ9 - 1V36369HM / 6; -GHI658@41 > 7 -
(
1
c0
∂
∂t− `h
1
r
∂
∂rr∂
∂r
)
τ(r, z) =γ − 1
βγ
jω
c0p(z) , r ∈ (0, a) , z ∈ (0, `) .
! > > 10G:MZK8IC7E58M > < -y/?- 7:7 - ;RuaI6147E58M > uNI65.GE17E5LG t 1V587h1VIG? / M > <65l7E58M > G
τ(0, z) 9 - Gc7 - 2> 5 -e- 7 τ(a, z) = 0 ∀z , ! > N G=PR; / 9:5l7
τ(r, z) =γ − 1
βγp(z)
[
1 − J0(khr)
J0(kha)
]
, ! >
MZtK -o> M& 69 - < PRM > < - kh
- Gc7k<6; 2> 5%321V9kh =
1 − j√2
√
k0/`h ;
+)# +U d+U )
GE1b@41VK - I69'M -?>6>C- GHI69TI >6- G -=/ 7E58M > G?P^; / 9H587
< τ >a=γ − 1
βγp(z)
[
1 − 2
kha
J1(kha)
J0(kha)
]
, ! >
%- 9 - 3MZ9H7< -J/=- 7H7 - ;RuNI2147E58M > ! > <21 > GKwP^;euNI2147:5LM > ! > / M > <6I65l7TC'I >6- G -?/ M > < - 9 - KL147E58M >-?> 7E9 -369 - G:GH5LM >- 7T@a587 - GHG - ∂
∂z< vz >a= − jω
ρ0c20
[
1 + (γ − 1)2
kha
J1(kha)
J0(kha)
]
p(z) , ! N %
uNI65 1n@ -=/ Km1s369 - `5 =?9 - ;eu ! ! > 'M > 7E9 - uNI - K - 3C9:M 6K = - 3 - IB7 A7E9 - 7E9:1V587:;32149g1 > 1VKLM&V5 - 1n@ -=/Km1g76;?MZ9H5 - < - GyKL5 >6- GMC / M > GH7E1 > 7 - Gv9H;?321V9c7E5 - G ! ;W> - ` - 7 K - Go;euNI2147:5LM > G ! > - 7 ! N % -?> > MV7U1 > 7w(z) = S < vz >a
K - <6;6587f1 / MVI6GH7:5uNI - MZt S = πa2 <C;?G:5 >6- KL1`G -=/ 7:5LM > <69HMZ587 - <6I / aKL5 > <69 - G:M > 7< - Km1 t MZ9 -
∂ p
∂z+ Zvw = 0
- 7 ∂ w
∂z+ Yhp = 0 ,
! N # MZtKwP^5'3;=<21 >6/=-f-?> G:;=9:5 - Zv
- 7kKwPR1V<'587:7E1 >6/=-d-?> 321V9:1VKLK =?K - Yh32149kI > 587:;o< - KLM > VI - I69_G=PR; / 9H58@ -=> 7
Zv =1
S
j k0ρ0c01 −Kv
- 7 Yh = Sj k0
ρ0c0[1 + (γ − 1)Kh] ,
! N 1n@ -?/
Kh,v =2
kh,v
J1(kh,va)
J0(kh,va).
! N
@7( (8: +,6#*46$ -7( / : /* $-$ # 1h3C9:MZ321 147E58M > < - GM > < - G G:I63C3MZGH; - Gp36KL1 >6- G <21 > GK -\/ aK85 > <69 -\- Gc79:; V5 - 32149%KwP^;euNI2147:5LM > GHI658@41 > 7 - M B7 -?> I -y-=> ;=KL5'5 > 1 > 7hKm1J@41V9H5m1CK - w <21 > GTK - GT;RuaI6147E58M > ! N #
(
∂ 2
∂z2 + k2z
)
p(z) = 0 , z ∈ (0, `) ! N \
MZtk2
z = −ZvYh = k20
1 + (γ − 1)Kh
1 −Kv
, 1F@ -?/ Re kz > 0- 7 Im kz < 0 .
! N pMZIC9g7EMZICG#K - G'7 a3 - G< - / MZIC36K - I69 - 7D7:MZIC7 - Km1 Z1 - < - t 9:;RuaI -?>6/=- G / M > G:5L<C;?9:; - |kh,va| > 8 1V5 > G:5OK -f/ aK85 > <69 - 3 - IC7 A7E9 -o/ M > GH5L<6;=9:;oG:M - I > 7EI - 4KL1V9 - |kh,va| >> 1 ! > <6;A@ - K8MZ363 - -?> 71VG '3C7EM47E5uNI - C KQPRMV9:<69 - # < - GbK8M > ZI - I69:G / 1V9:1 / 7:;?9:58GH7:5uNI - G ` ′
v
- 7 `h < - G t M >6/ 7E58M > GJ< - - GHG - K/ M > 7 -?> I - Gk<21 > G Kh,v<6M >C>6- 1VKLMZ9HG_3MVI69 kz
k2z ≈ k2
0
[
1 +1 − j√
2
2
a√
k0
(√
` ′v + (γ − 1)
√
`h )
]
, ! N >
G:MZ5l7kz ≈ k0
[
1 +1 − j√
2
1
a√
k0
(√
` ′v + (γ − 1)
√
`h )
]
. ! N4N
6 I & /( '92 7* % 70 27a 4(% )$ D4$)% * (
1 / M > <65l7E58M > 1VIG?#KL5'587 - GXG:I69_KL1bG:I69 t 1 /?- <6I `5 / 9HMZ36M >C- ; - 7H7 - IC9kG=PR; / 9H587 -?>/t 1V5LG:1 > 7TI6GE1 - < -KL103C9 - '5g==9 - 9 - Km17E5LM > ! N #
− 1
Zv
∂ p
∂z= Sve − Yp p
3MVI69 z = 0 , ! N
MZt Yp
- GH7kI >6- 1V<'587H7U1 >C/?- 9 - 369:;=G -=> 7E1 > 7dK - G - ` - 7:GT< -y/ MZI / - GdK85'587 - GX7 - 9 `5 uNI - GT1VI@ZMV5LG:5 > 1 - < -KL10361V9:MZ5 @ZMV5L9 ^ ! !$#!_ - 7d1 >6>6- ? -
Yp =S
ρ0c0
1 + j√2
√
k0(γ − 1)√
`h . ! N
;Y>z = ` KQPRM > < -[- GH7\9H;=2; / 65 - G:I69\I > 36Km1 > < P^5'3;=<21 >6/=- Yr+Yp /?- uNI65a<6M >6>C- 3MVI69 KL1 / M > <6587:5LM >1VI?KL5 `5l7 - GXG:I69TK - 9:; /=- 3C7 - I69 < Pr14369B==GKL1b369 - `5 =?9 - ;RuNI2147E58M > ! N #
1
Zv
∂ p
∂z= −(Yr + Yp) p
3MZI69 z = ` . ! %
10GHMZKLIB7E5LM > < - KwP^;euNI2147:5LM > < - 369HMZ321Z147E58M > ! N \ 3MZI69_KL1q369 - G:G:58M >- Gc7d< - KL1 t MZ9 -p(z) = B(e−j kzz +R` ej kzz) ,
! # MZtK - G / M > GH7U1 > 7 - G B - 7 R`
G:M > 7k<6;=7 - 9 `5 > ; - Gk361V9_K - G / M > <C587E58M > GT1VI? t 9:M > 7:5g=?9 - G ! 1 / M > <65l7E58M > 1VIG? t 9HM > 7E5 =?9 - G ! &% -=> z = ` <6M >6>6- 1VK8MZ9:G
R` = e−2j kz`Yi − (Yr + Yp)
Yi + (Yr + Yp),
! MZt Yi 1V3C3 - KL; - KQPr1V< `5l7:7U1 >6/?- 587E;=9E147:58@ - <6I / aKL5 > <69 - G?P^; / 9:587
Yi =
√
Yh
Zv=
j kz
Zv=
S
ρ0c0
√
[1 + (γ − 1)Kh](1 −Kv) , !
GHMZ587 -?> 369 -=> 1 > 7K - <6;A@ - KLMZ363 - -?> 7d1VG '3C7EM47E5uNI - < - G t M >6/ 7:5LM > G_< - - GHG - KCqKwP^MZ9H<69 - K - 36KLI6G 214G
Yi ≈S
ρ0c0
[
1 +1 − j√
2
2
a√
k0
(−√
` ′v + (γ − 1)
√
`h )
]
, ! ]\
;Y> 2> KL1 / M > <65l7E58M > 1VIG? t 9:M > 7E5 =?9 - G ! N -=> z = 0 <6M >6>6-
B =Sve
Yi(1 − R`) + Yp(1 +R`).
! - * ( $:' (,:'+,6 *4670-7; &( !
1 uNI21 > 7:587:;eIB7E5LK - <21 > GkKL1 `;A7 6Ma< - < P^;=7E1VKLM >C> 1 - 32149_9:; / 5L369HM / 587:; - GH7kKQPr1V< `5l7:7U1 >6/?- 1 / MZI6Gc7E5 uaI -< - 7E9:1 > G t - 9H7 YT uNI6MV7E5 -?> 7<6Iv6IG?e< - @a587 - G:G -[-?>e/ MVI69H7 H / 589 / I65871 / MZI6Gc7E5uNI - 369HMB<CI6587321V9.K - `5 / 9:MV36M >6-IC7:5LK85LG:; -=> ; - 7:7 - I69y321V9yKm1#369 - GHG:5LM > 1 / MZI6GH7:5uNI - 1Z58G:GE1 > 7eGHI69vKL1 - 69E1 >6- IC7E58KL5LGH; -q-=> 9:; /?- 3C7 - I69 #&! !
YT =Sve + Yep(0)
p(`).
! >
U , o +e @o U
; KLK - G?P^; / 9H587 -?>/t 1V5LG:1 > 7TI6GE1 - < - GT;euNI2147:5LM > G ! # ! - 7 !
YT = j sin kz`
[
Yi +(Ye + Yp)(Yr + Yp)
Yi
]
+ cos kz` (2Yp + Ye + Yr) . ! N
;W> 214G:G - G st 9:;euNI -=>6/?- G KwP - ?B369 - G:GH5LM > ! N < - @a5 -=> 7YT ≈ j kz` Yi + 2Yp + Ye + Yr ,
! & G:MZ5l7 -?>/t 1V5LG:1 > 7TI6GE1 - < - G - ?B369 - G:GH5LM > G ! ]\ ! N - 7 ! N]N 1n@ -?/ S = πa2
YT ≈ jωV
ρ0c20
[
1 + (γ − 1)
(
2
a+
2
`
)
√
c0`hjω
]
+ Ye + Yr ,
G:MZ5l7 YT ≈ jω
V
γP0[1 + (γ − 1)X] + Ye + Yr ,
! 1n@ -?/
X =A
V
1 − j√2
√
`hk0
,
K - 9:1V363MZ9H7TGHI69 t 1 /?- @VMZKLI - < - KL1 / 1n@a587E;o@41VIC7 AV
=2
a+
2
`!
- 7:7 -g- ?C3C9 - GHG:58M > < - KwPR1V<'587H7U1 >6/=- YT
- Gc7 65 -?> Km1 - CKQPRMZ9H<69 - # -?>X uaI - KwP - ?B369 - GHG:5LM > ! \4\ ! \ 7E9HMZIC@V; - <21 > GTI > < - Gk< - IG? `Ma<6;=K - G[< - e - 9 - 9 - 7T7E9HMZIC@V; - ; 1VK - -?> 7_<21 > GK - `Ma<6==K -
43 - 7E5l7 -o/ 1F@a5l7E; 9 - KL147E58M > ! !
|g ! - ' ) ) % . $ ) ! - ( ` '_) )[ - ! -W MZIB7 - GXK - G ';=76MB< - G[< -/ 1VK / I6K < - KQPr1V< `5l7:7U1 >6/?- 1 / MZI6Gc7E5uNI - < - 7:9E1 > G t - 9c7[IB7E5LK85LGH; - G C /?- HMVI69XM > 7
;=7:;k3C9:;?G -?> 7E; - Gh<61 > G /?-k/ 21V365l7E9 - ! pMZI69Y7 - 9'5 >C- 9 I >6- @V;?9H5 2/ 147E58M >0> I `;=9:5uNI - < - GX369HMZ3MZGY7 -?> I6GXG:I69/?- Gk<C5 `;?9 -=> 7EGT9:;=G:I6Kl7U147:G - GH7k369H;?G -=> 7E; - 5 / 5!
+ -5/?:&@7( 9-7 7 & *+ 6 $* + .- 'MB<#=?K - 3C9:MZ3MZG:;]321V9 Z1V9c@a5LG ^$# _ - 79 - 369:58G.321V9Km1 > MZ9 - ;[0> # % \ H ;eu ! !$# !$# G=Pr1V3C36I65 -G:I69vK - 9E1V58G:M >C>6- -?> 7v< - e - 9 - 9v369H;?G -?> 7E;#<21 > GJK - 32149E1Z9:1V3 - ! ! %-g/ 14GJ9 - 7 -=> I)361V9 Z149H@a5LG - GH7/?- KLI65pMZtuKwP^; / 1V9H7o5 > GH7E1 > 7U1 > ;b< - `1VG:G - @ZMZK8I'5uNI - ρ′g 1VGHG:5'5LK8;ZCDI > 7 - 9 - < - G:MZI69 /?-q- GH7yI > 5 t MZ9 -
<21 > GTKL1 / 1n@a587E; /Rt ! ! !$# ! - 3 -?> <21 > 7 / M& - 5 > <65uNI6;D<61 > GqK - 321V9E1 Z9E1V3 - ! !$# K - G -=/ M > < - C9 - < - KQPR;RuNI2147E58M > < -/ M > <6I / 7:5LM > < - Km1 / 61VK - I69 7 - KLK - uNI P - KLK -k- GH7 t MZ9 bI6KL; - 321V9 e - 9 - 9 ;eu ! ! \ - 7 ! >6- 9 - 369:;=G -?> 7 - 321VGKwPR1 / 7:5LM > < P^I >6- G:MVI69 /=- <21 > GoK - <6M&`1V5 >6- `1V5LGKwP^;=7E147 - <6Iu2IC5L< - <61 > GfKL1 / 1n@a587:;! p1V9o1V58KLK - I69:G <21 > GvI >6-D/ 1n@B5l7E;`<6M > 7bK - GJ<C5 -?> GH5LM > GeG:M > 7J3 - 7E5l7 - Gv321V9v9E1V363MZ9c7 C KL1K8M > ZI - I69e< PRM > < - 43 - 7E5l7 -/ 1n@a587:; /?- uNI65 - Gc7kK -f/ 1VGT5 / 5 K -o/ 21'3< - 369 - G:GH5LM > 1 / MVI6GH7:5uNI -o- Gc7kI > 5 t MV9 - ^ > ! ! !_ /?- uNI65- Gc7h5 >C/ M&'32147:5 6K - 1n@ -?/ KwP a3MV7 #=?G - < PRI >D/ 21'3g< P^; / 1V9H7[5 > Gc7U1 > 7E1 > ;d< - `1VG:G - @VMZKLI'5uNI - I > 5 t MZ9 - / 1V9_K -o/ 21 `3< - 7 - `3;?9:147EI69 -o> P - GH7kK8I65 H - 321VGTI > 5 t MZ9 - !cd5 > G:5 K - 9H;?G:ICK87U17T3MZIC9_KwPR1V<'587:7E1 >6/=- 1 / MZI6GH7:5uNI - < - 7E9:1 > G t - 9H73uNI65%< - @a9E1V5l7 =7E9 - IC7E58KL5LGH; - GH7 /?- KLI65<6;=7 - 9 `5 > ;3MZI69]KwP a3MV76=?G - pg
I > 5 t MZ9 - ;eu ! ! \4\ - 7 > M > 321VG /=- KLIC53C9:MZ3MZG:;f1 / 7:I - K8K - -?> 7]321V9Km1 > MV9 - ;eu ! !$# ! # !
6 I & /( '92 7* % 70 27a 4(% )$ D4$)% * (
: MV7:M > G -?> 2> uNI - KwP - ?B369 - G:GH5LM > < - KwPR1V<'587H7U1 >C/?- < - 7E9:1 > G t - 9H7 YT;A7U16K85 - 361V9 e - 9 - 9 ;Ru ! ! \4\ - GH7k;euNI65l@V14K -?> 7 - C /=- KLK - ;=7E16KL5 - <21 > GkKL1 / 1V<C9 - < - KwP^;=7:I6< - <6I / 21'31 / MZI6Gc7E5 uaI - <61 > GTI >6- 43 - 7:587 -/ 1F@a5l7E; ;Ru ! ! GHMZI6Gk9:;?G - 9c@ - < - 369 -=> <69 -e-?>/ M`3C7 - KwPR1V<'587H7U1 >6/=- < - G '5 / 9:MZ3 6M >6- G !
+ ( &(,*+ -7 #%6$/ () + 6 (,5:0-7* %! & + -5/%- GuNI217E9 - `Ma<6==K - Ge369H;?G -=> 7E;?Gq3C9:; / ;?< - -?> 7q3 - IB@ -?> 7 A7E9 - 69H5g==@ - -?> 7 / 1V9E1 / 7E;=9:5LGH;?G / M& -
GHI6587 qK - <6;=@ - K8MZ363 - -?> 7 'Ma<21VK ;eu ! ! \4\ ! \# ;A7U1CKL5 32149 e - 9 - 9 ^$# _ - 7k369H; / M > 5LGH;e361V9_K -y> MZ9 - ;[ N 1 > 7:;?9:5 - I69 - C'KL1 > MZ9 - ;[6> # % \ H ^ # _ qK - G -=/ M > < 'Ma<6=?K - <6587TC / MZI69c7o7 - 9 - ;A7U1CKL5321V9 e - 9 - 9 ;Ru !& ! \]\ ! \ 3MVI69fKwP a3MV76=?G - pg
I > 5 t MZ9 - qK - 'MB<6==K - 58G:GHI <CI / 1VK / I6K <CI / 61`3 1 / MZI6Gc7E5uNI -q-=> 43 - 7E5l7 - / 1F@a5l7E; ;Ru ! ! 369HMZ3MZG:;q32149T5L;A7 - 7 %-?/ MZKLK85 >6- 7k1V5 > GH5KuNI - 361V9 h1VKLKL1& - 7k;RuaIC58@41VK -=> 7 C'KQPRMV9:<69 - K - 36KLICG 214Gk< - Gd<C;=@ - KLMZ3 H3 - -?> 7:G14G `3B7EMV7:5uNI - G C /=- K8I65 < - e - 9 - 9T3MZI69 pg
I > 5 t MZ9 - qK - `Ma<6==K - ;A7U1CKL5632149 e - 9 - 9 ;eu ! !$# ! # 3MZIC9\KwP B3MV76=?G - ρ′g I > 5 t MZ9 - 369:; / M > 58G:;_32149Z149H@a5LGD^ # _ - 7d1 / 7EI - KLK - -?> 7 > MV9`1VKL58G:;^ _ !%- G_9:;=G:I6Kl7U147:G_M C7 -?> I6Gk3MZIC9 / 21 / I > < -o/=- G `Ma<6==K - G[G:M > 7 / M`32149:;?G > I';?9H5uNI - -=> 7 ! ;Y> 76;?M H
9H5 - K - Gv7E9HMZ5LGv369 - `5 - 9HG `Ma<6==K - G / 5l7E;?GJG:M > 7q;RuaIC58@41VK -=> 7EG / 1V9b9 - 3MVGE1 > 7qGHI69bK - G - G a3M47 6==G - G !j 1 > GbKm1369:147E5 uaI - 5LKY3 - IC7 =7E9 - 5 > 7:;?9 - G:G:1 > 7q< -`/ M&'321V9 - 9 > I `;=9:5uNI - -=> 7 /?- GJ7E9HMZ5LG 'MB<#=?K - GJ1 2>< - <6;A7 - 9 `5 >6- 9TK - uNI - K%< - GT7E9:MV5LGT3 - IC7 =7:9 - K - 36K8I6Gk1V<21V3C7:;y3MZIC9kK - GT;=7E1VKLM >6> 1 - G ! %- Gk9:;=G:I6Kl7U147:GT369:; HG -?> 7E;=G'3MZ9c7 -?> 7JG:I69bK - @VMZKLI - 1 / MZI6Gc7E5 uaI - < - 7:9E1 > G t - 9c7 VT
<6I)G aGH7B= - '5 / 9HMZ36M >6- G Ho/ MVI636K - I69 369HMZ3MZ9H7:5LM >6>C- K7CqKwPR1V<'587H7U1 >C/?- < - 7:9E1 > G t - 9c7 YT
VT =γRPR
jωYT ,
MZt γR- 7 PR
GHM > 7d9 - GH3 -=/ 7E5l@ - -=> 7dK - 9E14363MV9H7< - G / 1V361 / 5l7E;?G / 14KLMZ9H5 uNI - G Cp- 7 Cv
`1VG:GH5uNI - G < -KQPr1V589 - 7kKL1b369 - G:G:58M > GH7U17E5uNI - <21 > GTK - G / M > <65l7E58M > G_GH7E1 > <2149:<< - 9:; t ;=9 -=>6/?- !pMZIC9f< - G / M& `Ma<6587:;?Gk< P - ?B36KLMV587U17E5LM > < - Go9:;?GHI6K87E147EG K - @ZMVKLI - < - 7:9E1 > G t - 9c7 VT
- GH7 > MZ9`1VKL58G:;32149DGE1x@41VK - I69g1VIG? 21VG:G - G t 9H;euNI -?>C/?- G VTbf
-=> KwPR16G -?>6/=- < P - ` - 7EGD7 - 9 `5 uaI - GD<6; 2> 5 - / M& - KL1GHM& - <6Is@ZMZK8I - Z;?M&';=7:9:5 uaI - < - Km1 / 1n@a587:; - 7e< - Ge@VMZKLI - Go;RuNI658@41VK -?> 7EGe1VIG? 21VG:G - G t 9:;RuaI -?>6/=- G< - G `5 / 9:MV36M >6- G VTbf = V +Vbfe +Vbfr MVt'K - G\@VMZKLI - G ;euNI658@41VK -?> 7EGY< - G `5 / 9HMZ36M >C- GYG:M > 7WK85L;=GCK - I69HG14<`5l7:7E1 >6/?- GT1 / MVI6GH7:5uNI - Gk9 - G:3 -?/ 7E58@ - G Ye,r
-?> 21VGHG - G t 9:;RuaI -?>6/=- Gd361V9 Vbfe,r = γRPR/jω Ybfe,r ! ! !R1 ! .- G Z9:1V365 uNI - G< - G ZIC9 - G ! ! ! \9 - 369H;?G -=> 7 -?> 7Y<CM >6/ K - 9E1V3C3MZ9c7 VT
VTbf
-?> t M >6/ 7E5LM >< - KL1 t 9H;euNI -=>6/?- !
%- GY<65 -=> G:58M > G < - G / MZIC36K - I69:G G / 6; g1VG -=> 1 >6>6- ? - c - 7WK - G / 149E1 / 7E;?9H5LGc7E5uNI - GY< - G '5 / 9:MZ3CM >6- GIC7:5LK85LG:; - G<21 > G /=- 7H7 - 1V3C36KL5 / 147:5LM >> I';?9:5 uNI - G:M > 7f<6M >6> ; - Go<21 > GfK - G7E16K - 1VIG? !$# - 7 ! ! %- <C;=@ -@HK8MZ363 - -?> 7 `Ma<21VK ;Ru ! ! \# - GH7 / 14K / ICKL;y1n@ -=/ # %%? # %&%J7 - 9 - G !
[ Ko1V36361V9E1 87 -?> 369 - '5 - 9gKL5 - I uNI - KL1 <65 `;?9 -?>6/=-s-?> 7E9 - K - G#9H;?G:ICK87U17EGM C7 -?> I6G3MZIC9#K - 'MB<#=?K -> MV9`1VKL58G:; - 7 3MZIC9pK - Gp1VIC7E9 - G`Ma<6=?K - G / 21 `3b< - 369 - G:GH5LM > I > 5 t MV9 - > P - GH7 321VG > ; ZK85 - 16K - / 149 - KLK -3 - IC7 147H7 - 5 > <69 - % %&T< -?> 9 - 1V9:<e< - Km1T369:; / 58G:5LM > 9 -?/ - 9 / 6; - 3MVI69pK - G.;=7E1VKLM >6> 1 - G.< - '5 / 9:MZ3CM >6- G 1 / 7:I - K8K - -?> 7JKm19 - 369HMB<CI / 7:5 658KL587:;0G:I69bKL1 <6;A7 - 9 `5 > 147:5LM > < P - S / 1 / 587:; -?> 369 - G:GH5LM > <61 > GbK - G 61VG:G - Gt 9H;euNI -?>C/?- G - GH7 < - KQPRMV9:<69 - < - % % #q % % _< ^ \ _ - 7 KQPR5 >6/?- 9H7:587EIC< -W- ?B3;=9:5 -=> 7E1VK -W- GH7\;A@41VKLI6; - C -?> @a5L9HM >% % q< !
+ 6
P - ?B369 - GHG:5LM > C / MZI69c7v7 - 9 - ;=7U1 6KL5 - 361V9 e - 9 - 9e3MZIC9eKQP a3MV7 6==G - pgI > 5 t MZ9 - ! ;Ru !
! \ ! \4\ - GH7f; 1VK - C0KwP^MZ9H<69 - # -?> X C'KQP - ?B369 - G:G:58M > MC7 -=> I -J-?>/ 1VK / I6KL1 > 7TK -y/ 21'3 < - 369 - GHG:5LM >-?> 43 - 7:587 - / 1n@a587:; ! ;eu ! ! C /?-?/ 5Y369B==G-uNI - e - 9 - 9 > PR5 > 7:9:Ma<6I65l7d3214GoK - Gf1V<'587:7E1 >6/=- G Ye- 7 Yr! 1 / M > 7E9H5 6IC7:5LM > < - G]7 - 9 - Gh< P^MZ9H<69 - G:I63;?9H5 - IC9OC # - GH7_1VG:G - df9H;?<6I65l7 -/ 1V9 <21 > GhKm1 1 - < -
t 9:;RuaI -?>6/=- GTIC7E58KL58G:; - Z; > ;?9E14K - -?> 7_3MZI69[K - G[;=7U14KLM >6> 1 - G G:IC3;=9:5 - I69 - C% d - KLK - 9 - GH7 - 5 >#t ;?9H5 - I69 -C % % #< @41VK - I69 `1?B5`1VK - M C7 -?> I - 3MZI69'< - G '5 / 9HMZ36M >6- G ! P - ?B369 - G:G:58M > C KwP^MZ9:<69 - #-?>X
- GH7y<6M >6/ G:IBS`G:1 > 7 - 3MZI69 - ?B369:5 - 9fK - G - ` - 7:Gd7 - 9'5uNI - Gf<21 > G /=- 7H7 - 1 - < -t 9:;euNI -=>6/?- G9 -=/ M& g1 > <6; - 321V9fKL1 > MZ9 - ! W MZIC7 -t MZ5LG 3MZI69 t 1V589 - < - G - G:IC9 - G CD< - G t 9:;RuaI -?>6/=- Ge36KLI6G21VG:G - G 5LKG - 9E1V5l7v3C9:; t ;=9E1CK - 3MZI69eM C7 -?> 589vI >C- 36K8I6G V9E1 > < - 369H; / 5LG:58M > < P^IC7E58KL58G - 9yK - <C;=@ - KLMZ3C3 - -=> 7 C / MZI69H77 - 9 -f/ M&'36K - 7hMVI - K - <6;=@ - K8MZ363 - -?> 7 `Ma<21VK uNI65 / M > @ - 9 - 36K8I6Gh9E14365L< - -=> 7 -?> 7E9 =?G 21VGHG - Gt 9:;RuaI -?>6/=- G -?> 7 -?> 1 > 7 / M&`3B7 - < - Km1b9 - `1V9uNI - <6M >6> ; -e-?>2> <6I321V9:1Z9E143 - ! ! \ !
MZI63CK - IC9 MZI636K - I69 MZIC36K - I69_< -/ aK85 > <69H5uNI - # / aK85 > <69H5uNI - # V9E1 > <#@VMZKLI - #j 5m12==7:9 -o- ?B7:9:; `5l7E;?G # > # > M > ZI - I69 - ?a7E9H; '587E;=G % % # j 5m12==7:9 -o/=-?> 7E9 - # > # \ M > ZI - I69 /?-?> 7:9 - \ N #
"! p !$# q j 5 -=> G:58M > G_< - G / MZI636K - I69HG !
# # # Y9:M t M > < - IC9 t 9HM > 7U1VK - % \ % \ # N > # NYWMZK8I - 61VG:G - G t 9:;eu ! 3 N #0\ N N # # > 9:;RuaI -?>6/=- < - 9:;=G:M > 1 >6/?- d &%&%&% %%% %&% %&% 1 / 7 - I69T< - uNI21VK8587:; < H # N N H # N N H % \ H % \
"! p !q 1V9:1 / 7:;?9:58GH7:5uNI - GT< - G `5 / 9:MV36M >6- G !
6 I & /( '92 7* % 70 27a 4(% )$ D4$)% * (
102
103
104
−0.2
0
0.2
0.4
0.6
0.8Gerber modal, p
g unif.
Gerber CT, pg unif.
Petite cavitéGerber CT, ρ
g’ unif. (norme)
!
onPp ! qYWMZK8I - 1 / MZI6Gc7E5uNI - < - 7:9E1 > G t - 9c7 > MV9`1VKL58G:; VT/VTbf-=> t M >6/ 7E58M > < - Km1 t 9:;RuaI -?>6/=- G - K8M ><65b`O;=9 -=> 7:G `Ma<6=?K - G_3 aG:5 uNI - G3MZIC9_< - G `5 / 9HMZ36M >C- G # - 7TI >/ MZI636K - I69 / aK85 > <69H5uNI - !
102
103
−0.1
0
0.1
0.2
0.3
0.4Gerber modal, p
g unif.
Gerber CT, pg unif.
Petite cavitéGerber CT, ρ
g’ unif. (norme)
!
onPp ! qYWMZK8I - 1 / MZI6Gc7E5uNI - < - 7:9E1 > G t - 9c7 > MV9`1VKL58G:; VT/VTbf-=> t M >6/ 7E58M > < - Km1 t 9:;RuaI -?>6/=- G - K8M ><65b`O;=9 -=> 7:G `Ma<6=?K - G_3 aG:5 uNI - G3MZIC9_< - G `5 / 9HMZ36M >C- G # # - 7kI >/ MZI63CK - IC9 / aKL5 > <69:5 uNI - !
+
102
103
−0.05
0
0.05
0.1
0.15
0.2
0.25
0.3Gerber modal, p
g unif.
Gerber CT, pg unif.
Petite cavitéGerber CT, ρ
g’ unif. (norme)
!
onPp ! \q YWMZKLI - 1 / MZI6GH7:5uNI - < - 7:9E1 > G t - 9c7 > MZ9 g14KL5LGH; VT/VTbf
-=> t M >6/ 7E5LM > < - Km1 t 9:;euNI -=>6/?- G - KLM ><65 `;?9 -?> 7EG `Ma<6==K - Gk3 aG:5 uaI - Gy3MVI69< - G'5 / 9:MZ3CM >6- G # # 1n@ -?/ I > / MZI636K - I69<6587 4< - Z9E1 > <@ZMZK8I - !
* (),'%$& ( " (z!"%%%$&
*D # % W* % * % * %%
1 > MZ9 - ;[ > # % \ H ^ _CG:I69\KQPR;A7U1VK8M >6> 1 - 32149\9:; / 5L3C9:M / 587:;[< - G `5 / 9HMZ36M >C- G\;A7U1VK8M > G\< - Km1M H9E147:MZ5L9 - 3C9:MZ3MZG - I > 'MB<6==K - 1 / MVI6GH7:5uNI - <6I#G aGH7>= - < - < - IG? '5 / 9:MZ3CM >6- G ; - 7:7 - I69 - 7_9H; /?- 3C7 - I69 / MZIC36KL;=GX32149[I >6-/ 1n@a587:; / aK85 > <69H5uNI - /t ! ! ! - 'MB<#=?K - uNI659 - 3MZG - G:I69XKm1y369:MZ361 147:5LM > < P^M > < - G36Km1 >6- Gy<21 > GeKL1 / 1n@a587E; > ; ZK85 - K - G 'MB< - GoG:I63;?9H5 - I69:G 'MB< - Gy9E1V<C5m1VIG? uNI65Y3MZI69:9:1V5 -=> 7J1n@VMZ5L9yI >6-5 > 6I -?>6/=-g> M > > ; VKL5 - 16K - G:I69JK - Ge9:;=G:I6Kl7U147:GJ< P^;=7E1VKLM >6> 1 - ! P - ?B58GH7 -=>6/?- < -g/=- G 'MB< - G - GH7J<6I - 14It 1V587 uNI - K -/ 21'3#< - <6;=36Km1 /=- -?> 7_< - KL1 - C9E1 >6- <6I `5 / 9HMZ36M >C- ; - 7H7 - I69 > P - Gc7T321VG_I > 5 t MZ9 - /t ! 5 >#t 9:1 - 7iuNI - Km1o321V9c7E5 - 'M 658K - < - KL1 - 69E1 >6- < - G `5 / 9HMZ36M >C- G > PR1e3214G - ?C1 / 7 - -?> 7hK - -<65m1 =A7E9 - uaI - Km1 / 1n@B5l7E; !$c5 > GH5 KL1q369 - G:G:58M > G:I69kKm1 - 69:1 >6- <6I9:; /?- 3B7 - IC9 > P - Gc7d3214GkI > 5 t MZ9 -o- 7 < - 36K8I6G /?-o> P - Gc7361VG - ?C1 / 7 - -?> 7TK -f/ 21'3< - 369 - G:GH5LM > 'M -?> uNI65 - GH7 - G:I69H;o321V9 /=- K8I65 Hi/ 5 / 1V9hG:1 - 69:1 >6-o> PR1q321VG > M > 36KLICG_I > 'MZIC@ - -?> 7k< - 365LGc7EM > 36KL1 > !
1VG bI6GHG -?> ^ __369HMZ3MVG - < - 369 -?> <69 --=> / M&'3C7 - K - G 'MB< - G09E1V<65L1VIG?x321V90I > 7 - 9 -#/ MZ9H9 -?/ 7E5 tCD1V363CKL5uNI - 9< -t 1C?M > bI6Kl7E5L3CKL5 / 147E5l@ - CgKQPr14<`5l7:7E1 >6/?- 1 / MZI6GH7:5uNI - < - 7E9:1 > G t - 9H7d<6;=7 - 9'5 > ; - G - KLM > K -`Ma<6==K - 3 aG:5uNI -J> MZ9`1VK85LG:; ! %-J/ 14K / ICKp< -v/?- 7:7 -J/ MZ9H9 -?/ 7E58M > 9 - 3MZG - GHI69dI > `Ma<6==K - g1476; `147:5uNI -9:5 ZMZI69 - IG?z1VIz<6;?361V9H7 `1V5LGe1V< - 7 2> 1VK - -=> 7 /?- 9H7E1V5 >6- G a3MV7 6==G - G uaIC5X9 - Kg=A@ -=> 7b36KLIC7BAV7e< - KQPR5 >GH7EI65l7E58M > Km1 > 17EI69 - <65LGHG:5832147E5l@ - <6Iu2I65L< -b> P - GH7y361VGf369H5LG -J-?>s/ M&'3C7 -b/ 1V9 - KLK - GHMZ9H7y< - GoM -=/ 7:5 t Gf< -KwPR1VIC7 - I69 C /=- 7 -=> <69HMZ587 ! ;Y> - ` - 7 58K]3C9 -?> < 65 -=>x-=> / M&'3C7 - < - t 1C=M > 1 > 1VK N7:5uNI - K - t 1V5l7 uaI - Km1369 - G:GH5LM > G:I69hK - 9:; /=- 3C7 - I69 > P - GH7_361VG[I > 5 t MZ9 - C / 1VI6G - <6ID369HM K > M > I > 5 t MV9 - <CI / 21'3g< - <6;?3CKm1 H/?- -=> 7y< - Km1 - C9E1 >6- < - KwP^; - 7H7 - I69`1V5LG >6- 7E5 -?> 7 / M`3C7 - uNI - < - t 1C=M >s- '365L9H5uNI - < - KwP - ` - 7<6I / 21'3< - <C;?36KL1 /?- -=> 7 > M > I > 5 t MZ9 - <6I<C5m1V3C9E1& - 9H; /?- 3C7 - I69kG:I69TK -o/ 61`3< - 369 - G:GH5LM > ! P - GH7T1V5 > G:5uNI - <21 > G /?-f/ 61V36587:9 - K - 'MB<#=?K -k- Gc7_9 - 369:58G[< -3t 1C?M > 1 > 1VK N7E5uNI - G:I69hKL1 214G - <OPRI >C-G:MZK8IC7E58M > `Ma<21VK - <6Iu3C9:M 6K = - `1476; `147E5 uNI - uNI65 9 - 3MZG - G:I69fKL1 t MZ9 bI6KL147E58M > 5 > 7E; V9E1VK -v- 7-uNI65 < -G:I69 / 9HM L7d369 -=> < -=>u/ M&'3C7 - Km1 > 17EI69 - <65LGHG:5832147E5l@ - <CIs6I65L< - /?- uNI65 ;=@a587 - < PR1F@VMZ5L9TCDK - 7:9E1V5l7 - 9G:MZICGKm1 t MZ9 - < - 7 - 9 - G / MZ9H9 -?/ 7E5 t G1q3MZGH7 - 9H5LMZ9H5 !<=~ ' - A!/ ! ;( ) ! - y) #%$ ;& '( )
.- G aGH7B= - ;=7:I6<65L; 9 - 369H;?G -=> 7E;uG:IC9DKm1 ZI69 - \!$# - GH7 / M > Gc7E587:I6; < - < - I?'5 / 9HMZ36M >6- G KQPRI >; - 7H7 - IC9X< - 9E1 ZM > ae KwPR1VIC7:9 - 9:; /?- 3B7 - IC9h< - 9:1 ZM > ar / MZI636K8;?G[361V9XI >6-d/ MZKLM >C>6- < - 2I658< -k/ M > 7 -=> I -<21 > GXI >D/ BK85 > <C9 - CvG -=/ 7:5LM >D/ 5L9 / I6KL1V5L9 - < - 9:1 VM > a G:I63;?9H5 - I69h1VI?g9:1 ZM > G ae
- 7 ar
- 7]7E9 =?Gh3C9:M / - < -> N
) A ) 1% 14$a)& 4 ) 0 )* )( ) 7 * 6! 0)(+* $( * #9$
'9/"( 1* & %( * 1 * 9
/=- IG? H / 5! %- G - 69E1 >C- G[< - KQPR; - 7:7 - I69 - 7h<CID9:; /?- 3B7 - IC9hG:M > 7[1 > 5 `; - G]9 - GH3 -=/ 7:58@ - -=> 7_< - G]@a587 - G:G - Gve 5'3MVG:; - - 7 vR
<6I - 1VI / 21'3D< - 3C9 - GHG:58M > 9 - C?I#GHI69hKm1 - 69:1 >6- ! .- 369:M6Kg= -k- Gc7_G:I63C3MZGH;CoG ';=7:9:5 - 1?B5m1VK - > ;n1 > 'MZ5 > G\3MZI69\< - GW9E1V58G:M > G < P - ?B3MZGH; KQPr1 > ZK - θ 1V36321V9:1 L7\1VI0<6; CIC7W<6I / 21V365l7E9 -`1V5LG[Km1qG ';=7E9H5 - 9 - GH7 - GHI6363MZG:; - <61 > G /?- 9H7U145 >6- GkGH5'36KL5 2/ 17E5LM > G !
aae ar
ve vR
`
z
V
onPp \ ! # q j 5LGH3MZGH587:5 t 3MZI69TKwP^;=7E1VKLM >C> 1 -f-=> 3C9 - GHG:58M > G / 6; g10< - 369:5 >6/ 583 - !
8 467()*+ +$-7( (,/ ' : +7-7**+ (86 &+*! & 6$ 9/ ( &+,* /( %! #(8/
%- Gy369:5 >6/ 58321VK - Gd@41V9H5m16K - Gf<C; / 9H58@41 > 7yKwP^;=7E147o7 - 9'Ma< > 1'5uNI - <6Is2I658< - 145L9 G:M > 7eK - G - GuNI - 369:; / ;=< - -=> 7 ! # !j 1 > GWKL1fG:I65l7 - K - 369HM 6Kg= -_- GH7 - ?B369H5';QCo321V9H7:5L9\< - GY7E9HMZ5LGY;RuaI6147E58M > G t M > <61 -=> 7U1VK - GY< - KwPR1 / MZI6G H7:5uNI -J-=> 2I658< - 7 - 9 `MF@a5LG"uNI - IG? 14KL5 > ;n1VGk5 5L5 5L585 C`GE1n@VMZ5L9KwP^;euNI2147:5LM > < - : 1F@a5 - 9 H 7EMB - G /=- KLK - < -/ M > G - 9H@4147:5LM > < - Km1 g1VGHG -0- 7 /=- K8K - < -0/ M > <6I / 7E58M > < - Km1 / 214K - IC9 ! j 1 > G /?- Gy;RuaI6147E58M > G I >s/?- 9c7U1V5 >> M 69 - < P a3MV7 #=?G - G0G:5'36K85 2/ 147E9H5 /?- Gy3 - IC@ -=> 7 A7E9 - 9 - 7 -?> I - G ! ; K8K - GvG:M > 7q3C9:;?G -?> 7E; - G'1VI t IC9 - 7 C - GHI69 - < - K - I69TI6G:1 - ! ;Y> 369 - '5 - 9TK85 - I KwP^; / 149H75 > Gc7U1 > 7U1 > ;v< - 369 - G:GH5LM > ;=7E1 > 7 uNI21VGH5pI > 5 t MZ9 - G:IC9I >C- G -=/ 7E58M > <6I / BK85 > <C9 - 5LK - Gc7 / M > G:58<6;?9H; / M& - <6;=3 -=> <21 > 7y3 - Iu< - G / MaMZ9H<6M >6> ; - G r - 7 θ !'cd5 > G:5 K - G / M&`3MZG:1 > 7 - Gq< - Km1@a5l7 - GHG - ~v G - KLM > r - 7 θ G:M > 7q3 - 7:587 - Gb< - @41 > 70GE1 / M`3MZGE1 > 7 - 1?B5m14K - vz
KL1> M47U147:5LM > vz
> P - Gc7_IC7:5LKL58G:; - 5 / 5uaI - 3MZIC9[< - GPuaI - GH7:5LM > Gh< - GH5'36KL5 2/ 147:5LM > G]ICK87E;=9:5 - I69 - G ! p1V9_1V58KLK - I69HG -=> 9E1V58G:M > < - KwP^5'3MV9H7U1 >6/?- < - G - ` - 7:GqKL5L;=GZCKm1@a5LG / MZGH587:;g< -D/ 5LG:1V5LK8K - -?> 7 Km1@41V9H5m147:5LM > G:IC58@41 > 7bKL1@41V9H5m16K - r < - KL10@a587 - GHG - 32149H7E5 / I6KL1V5L9 - vz
- Gc7d7:9B=?GG:I63;?9H5 - I69 - C`GE10@41V9:5L147E58M > G:I65l@V1 > 7Km10<6589 -?/ 7E58M > z /=- uNI65%5 `36K85uNI --?> 321V9H7:5 / I6KL5 - 9 uNI - K - t 1 / 7 - I69TKL58;MCqKm1J@a5LG / MZGH587E;y< - @VMZKLI - 3 - IC7 =7:9 -y> ; ZKL5 Z;!5 ! 1 / M&'3MZGE1 > 7 - 1?B5m14K - < - KwP^;euNI217E5LM > < -+: 1F@a5 - 9 H 7EMB - GJ369 -=> <)Km1 t MV9 - 1V3C369:M / 6; - GHI658@41 > 7 - ;RuaI6147E58M > < - 1 -=>GH MZ5LG - I658KLK - /Rt ;Ru ! ! > #y1
(
1
c0
∂
∂t− ` ′
v1
r
∂
∂rr∂
∂r
)
vz(r, θ, z) = − 1
ρ0c0
∂
∂zp(r, θ, z) .
\!$# c /?- 7:7 - ;euNI217E5LM > \!$# G:M > 71VGHG:M / 5L; - GTK - GT< - IG? / M > <65l7E58M > GhGHI658@41 > 7 - GTG:I69TKm1 / M&`3MZG:1 > 7 - z < -Km1J@a587 - GHG -
vz(0, θ, z)9 - Gc7 - 2> 5 -o- 7 vz(a, θ, z) = 0 ∀z, θ . \!
5L5 ! 1`G -=/ M > < - ;euNI2147:5LM > 369:58G -v-?> / M&`3B7 -e- GH7 /?- K8K - < -J/ M > <6I / 7E58M > < - Km1 / 21VK - I69 ! j I t 1V587'uNI - KL1@41V9:5L147E58M > < - 7 - `3;?9:147EI69 - τ G=Pr1 >C> ICK -f-=> r = a
- 7h< - Km1eG ';=7E9H5 - <CI#G aGH7B= - KL1e7 - `3;?9:147EIC9 -- Gc7k7:9B==G_3 - I<6;=3 -=> <21 > 7 - < - KQPr1 > ZK - θ - 7kGE1q<6;=9:5l@Z; - G - KLM > z - GH7 > ; ZKL5 - 1CK - < - @41 > 7kGE1q<6;?9H58@V; -
12 @+ $+ ) +
G - K8M > r <21 > G'K - G / MVI / - G`KL5 `5l7 - Ge7 - 9'5uNI - G !p1V9'G:I65l7 - KQPR;RuNI2147E58M > < -/ M > <6I / 7:5LM > < - Km1/ 21VK - I69D^ ;eu ! ! > _%369 -?> <Km1 t MV9 - 1V36369HM / 6; - G:I65l@V1 > 7 -
(
1
c0
∂
∂t− `h
1
r
∂
∂rr∂
∂r
)
τ(r, θ, z) =γ − 1
βγ
jω
c0p(r, θ, z) ,
\! C0KL1uNI - K8K - GHM > 71VG:GHM / 5L; - GK - G / M > <65l7E5LM > G
τ (0, θ, z) 9 - GH7 - 2> 5 -o- 7 τ(a, θ, z) = 0 ∀z, θ . \ ! \ 5L585! - GH7 - C - ?B369H5 - 9oKL1g7E9:MV5LG:5 = - ;RuaI6147E58M > t M > <21 -?> 7E1VK -b/ M > GH5L<6;=9:; - /?- K8K - < -'/ M > G - 9H@4147:5LM > < -
Km1 `1VG:G - jω ρ′ + ρ0div ~v = 0 .
;W>/t 1V58GE1 > 7TI6G:1 - < - Km1b9 - Km17E5LM > < PR;A7U147ρ′ =
γ
c20(p− βτ) ,
\ ! 3MZIC9_;?KL5 `5 >6- 9XKm1b@41V9:5L16K - ρ′ KQPR;RuNI2147E58M > < -y/ M > G - 9c@4147E58M > < - Km1 g14G:G - 3 - IB7d14KLMZ9HG =7E9 - ; / 9H587 -
γ
ρ0c20
∂
∂t(p− βτ ) +
1
r
∂
∂r(rvr) +
1
r
∂
∂θvθ +
∂
∂zvz = 0 .
\ ! > .- GyG:MZK8IC7E58M > Gd< - Gy;euNI2147:5LM > G \!$# C \! > 3 - 9 - 7:7 -?> 7 -?> GHI6587 - < P - ?B369:5 - 9oKwP^;euNI2147:5LM > < - 3C9:M H
321 17E5LM > <6; / 9H58@41 > 7#Km1 369:MV321 147:5LM > < P^I >6- M > < - uNI21VGH5d36KL1 >6- <61 > GDKL1 / 1n@a587:; -?> 9 - '36Km1 Cn1 > 7`K - GuNI21 > 7E5l7E;?G vz τ - 7 p 32149XK - I69HGX@41VK - I69:G 'M -?>6>C- GhGHI69[I >C- G -=/ 7:5LM > <CI / aKL5 > <69 - /?- uaIC5 / M > <6I65l7OCeI >369:M6Kg= - 14IG?KL5 `5l7 - G[<6;=3 -=> <21 > 7I > 5uNI - -=> 7< - KL1 / MaMZ9:<CM >6> ; - 1?C5L1VK - !
, 4 67()*+9-7 $ &+ (,()#*+ P^; / 1V9H7W5 > GH7E1 > 7U1 > ;_< - 369 - G:GH5LM > p ;=7E1 > 7 / M > G:5L<C;?9:; / M& - 7:9B==GY3 - I`<6;=3 -=> <21 > 7]< - G / MaMZ9:<6M >6> ; - G
r, θ <21 > GdK - G / MZI / - GoKL5 `5l7 - Gh@a58G / M47 - 9'5uNI - GdG:I69K - G321V9HMZ5LGkKm147:;?9:1VK - G M > < - uNI214G:5 3CKm1 >6- - 7fKL1/ M&'3MZGE1 > 7 - vz< - Km1s@a587 - G:G - 321V9H7:5 / I6Km1V589 - ;=7E1 > 7 t MZ9H7 - -?> 7D<C;?3 -?> <21 > 7 - < - r <21 > G#K - G / MZI / - G
KL5 `5l7 - Gh@a58GuNI - ICG - G K - G -=/ M > < - 69 - < - KQPR;RuNI2147E58M > \!$# - GH7dG:I6363MZGH;b5 > <C;?3 -?> <21 > 7d< - r /?- uNI65/ M > <6I6587MC; / 9H5L9 - KL1gG:MVKLIC7:5LM > vz<6Is369:M6Kg= - \!$# \! -?> 9:; V5 - 21V9 `M > 5uNI - G:MZICGoKm1 t MZ9 -
1V36369HM / C; - K - t 1 / 7 - I69 ejω t - GH7kM&'5LG /t ! ! \! #
vz(r, θ, z) ≈j
k0ρ0c0
∂
∂zp(r, θ, z)
[
1 − J0(kvr)
J0(kva)
]
, \! N
MZtK -o> M 69 - <OPRM > < - < - <C5 `I6G:58M > kv
- Gc7k<6; 2> 5361V9
kv =1 − j√
2
√
k0/` ′v .
j 1 > GvKm1#G:I65l7 - KQP - ` - 7J<C5LG:GH5L3217E5 t <x1VIG? / MVI / - GJKL5'587 - Gd@a58GuNI - ICG - Gb1VI > 5l@ - 1VIx< - GJ32149:MZ58GeKL1 H7E;=9E1VK - Gb< - KL1 / 1n@a587E; - GH7'369H5LG -=> / M&`3B7 -D-=> 369 -?> 1 > 7'KL1 @V14K - IC9 `M -?>6>6- GHI658@41 > 7 r <6I 369HM Kh<CI/ 21'3< - @a587 - G:G - 5 `3MZG:;o361V9K - G / MZI / - GKL5'587 - G[@a58GuNI - ICG - G 1V< - 7H7U1 > 7k< - Km10G:MZ9c7 - KwP B3MV76=?G -uNI -J/=- 7 - ` - 7f<658G:GH5L32147:5 t KL58;v14IG? / MZI / - GfK85'587 - Gh@a5LGuNI - I6G - G -?> r = a 1 ` -?/ 7 - < - Km1 - `1 > 5 =?9 -
7 ) A ) 1% 14$a)& 4 ) 0 )* )( ) 7 * 6! 0)(+* $( * #9$
'9/"( 1* & %( * 1 * 9
KQPRM > < - 36Km1 >C- ! \ - 7'KwP^M > < - uNI21VGH5T36Km1 >C- !cd5 > G:5 Km1 @41VK - I69 'M -?>C>6- 9 - 7 -?> I - 5 / 5_3MZ9c7 - G:I690KL1G - I6K - t M >C/ 7E58M > [1 − J0(kvr)/J0(kva)]
uNI65%@41V9H5 - 7E9 =?Gk9E1V3658< - -?> 7<21 > GkKm1 / MZI / - KL5'587 - @a5LG"uaI - I6G - K - t 1 / 7 - I69 ∂
∂zp(r, θ, z) t M >6/ 7:5LM > < - r - 7 θ ;A7U1 > 7kG:MZ9c7E5 < - KQPR5 > 7E; V9E1VK - G:I69 r - 7 θ !
- 7H7 - @41VK - I69`M -?>6>6- 321V9c7E5 / I6K85g=?9 - > MV7:; - vz G=P - ?B369:5 - 1VK8MZ9:G / M& - GHI6587
vz ≈ j
k0ρ0c0
∂
∂zp(r, θ, z)
1
πa2
∫ 2π
0
∫ a
0
[
1 − J0(kvr)
J0(kva)
]
r dr dθ
=j
k0ρ0c0
∂
∂zp(r, θ, z)
[
1 − 2
kva
J1(kva)
J0(kva)
]
. \ !
: MV7EM > GT1VI321VG:G:1 - 3MZIC9_K - G - GHMZ5 > G_< - Km1bGHI6587 - uNI - K - G / M&'3MZGE1 > 7 - G vr
- 7 vθG - KLM > r - 7 θ < - KL1@a587 - GHG - 321V9c7E5 / I6Km145L9 - uNI65 G:M > 7 / M >#t M > <6I - Gy1n@ -=/ K - I69@V14K - IC9`M -?>6>6- 1VIuG -?> Gy5 > <65uNI6;J369:; / ;?< - H
-?> 7 / 1V9 > M > 1` -=/ 7E; - Gk5 / 5 321V9_KL1J@B58G / MZG:5l7E; 3 - IC@ -?> 7 A7E9 -y- ?C3C9:5'; - G -?> 369 - `5 =?9 - 1V36369HM9?B5 g147:5LM >32149
vr(r, θ, z) =j
k0ρ0c0
∂
∂rp(r, θ, z) ,
\!
vθ(r, θ, z) =j
k0ρ0c0
1
r
∂
∂θp(r, θ, z) .
\! # % jd- Km1 - t 1C=M > uaI - 3MZI69YMC7 -=> 5L9\KQP - ?B369 - G:G:58M > < - vz
\! N KL1fG:MZK8IC7E58M > τ < - KwP^;euNI2147:5LM > \! / M&`3B7 - 7 -=> I< - G / M > <65l7E5LM > G \ ! \ - GH7k<6M >C> ; - 321V9 ! \!$#
τ(r, θ, z) =γ − 1
βγp(r, θ, z)
[
1 − J0(khr)
J0(kha)
]
, \!$#&#
MZtK -o> M& 69 - < PRM > < - kh
- Gc7k<6; 2> 5%321V9
kh =1 − j√
2
√
k0/`h ,
- 7XG:1y@41VK - I69 'M -=>6>6- G:I69WI >6- G -?/ 7:5LM > G?P^; / 9:5l7 -?> 1V< - 7H7U1 > 7WK - G - GW1V36369HM9?B5 g147:5LM > G uNI -T/?- K8K - G9 - 7 -=> I - G369:; / ;?< - -=> 7kGHI69 vz ;Ru !1\!
τ(r, θ, z) =γ − 1
βγp(r, θ, z)
[
1 − 2
kha
J1(kha)
J0(kha)
]
. \!$#
cd5 > G:5 KL10@41VK - I69 `M -?>6>6- < - KQPR;RuaI6147E58M > \! > 1VI G -?> Go< - Gd;RuaI6147E58M > G \! - 7 \ ! # / M&'3C7 -7 -?> I0< - G\;RuNI2147E58M > G \! \! \! # % - 7 \ ! # / M > <CI6587 <65L9 -?/ 7 - -?> 7CfKQPR;RuNI2147E58M > < - 369:MV321 147:5LM >GHI658@41 > 7 -
[
k20
1 + (γ − 1)Kh
1 −Kv+
1
1 −Kv
(
1
r
∂
∂rr∂
∂r+
1
r2
∂ 2
∂θ2
)
+∂ 2
∂z2
]
p(r, θ, z) = 0 , \!$#
MZtK - G t 1 / 7 - I69HGTKL5L;=GQCqKm1b@a5LG / MZG:5l7E; - 7 CqKL1 / M > <6I / 7E58M > 7 - 9'5uNI - Kv- 7 Kh
G:M > 7k<6M >6> ;?Gk321V9
Kh,v =2
kh,va
J1(kh,va)
J0(kh,va).
\!$#0\
) + + 7 U
PR; / 1V9c7`< - 369 - GHG:5LM > p <6;=3 -=> <21 > 7`3 - I< - G / MBMZ9H<6M >6> ; - G r, θ G - G'<6;?9H58@Z; - G`321V9H7:5 - KLK - GbG - KLM > /?- G/ MaMZ9:<CM >6> ; - Gf<21 > GfKwP^;euNI217E5LM > \ ! # G:M > 7f< - Gd7 - 9 - Gd3 - 7E5l7EG'uaIC5\3 - 9 - 7H7 -?> 7d< PR; / 9H5L9 - 1 −Kv ≈ 1-?>/t 1 / 7 - I69 !6cd5 > G:5%KQPR;RuNI2147E58M > \ ! # 3 - IC7 =7E9 - ; / 9H587 - G:MZI6GkKm1 t MV9 - G:5'36K85 ; -[
1
r
∂
∂rr∂
∂r+
1
r2
∂ 2
∂θ2 +∂ 2
∂z2 + k2
]
p(r, θ, z) = 0 , \! #
1n@ -?/k2 = k2
0
1 + (γ − 1)Kh
1 −Kv
≈ k20
[
1 +1 − j√
2
2
a√
k0
(√
` ′v + (γ − 1)
√
`h )
]
. \!$# >
cd5 > G:5 K - 'MZIC@ - -?> 7_<6Ig6I65L< - <21 > GXKm1 / 1n@a587E; - Gc7_<6; / 9:5l7h321V9]KwP^;euNI217E5LM > < - 3C9:MZ321 147E58M > \!$# <6M > 7qK - GqG:MVKLIC7:5LM > Gv<6MZ5l@ -?> 70G:147E58G t 1V589 - K - G / M > <65l7E58M > Gq1VI? t 9HM > 7E5g==9 - G -?> z = 0- 7 z = ` 369 -=> 1 > 7-?> / M&'3C7 - KL1 t MV9 - <CI / 21'3 < - @a5l7 - GHG - < - G - 69E1 >6- G0< - `5 / 9HMZ36M >C- G 9:1V5LGHM > <OPA7E9 - < - G
MZ3;?9E17 - IC9:G -?> r - 7 θ < - KwP^;euNI2147:5LM > \!$# 1V5 > GH5uNI - K - G - ` - 7:G_< - G / MZI / - GkK85'587 - GW@B58G / MV7 - 9'5uNI - GG:I69 /=- GT321V9HMZ5LG !
< "( # +[ - `%) !/ ! ) + - . )6$ ( . ) -%._ $ '( ) ) 35$ ) ) !
1 / MZK8M >6>6- < - 2I658< -/ M > 7 -=> I - <21 > G]Km1 / 1n@a587:; - GH7XK - G:5g= - < PRI >g/ 21'3#1 / MZI6Gc7E5uNI - 21V9 `M > 5uNI -/ 9H;?;0321V9oKL1g@B5 69E17E5LM > < - Km1 - 69:1 >6- <6I '5 / 9:MZ3CM >6- ; - 7H7 - I69oG:5l7EI6; -'-=> z = 0 !7\!$# ! %- G
321V9HMZ5LGXG:587:I6; - G -?> z = 0- 7 z = ` 369:;=G -=> 7 -?> 7k9 - G:3 -?/ 7:58@ - -?> 7TK - G_1V<'587H7U1 >6/=- G Ye
- 7 YruNI65 G:M > 7hK - G
1V<'587H7U1 >6/=- GX1 / MVI6GH7:5uNI - G[< - G '5 / 9:MZ3 6M >6- GC / 5L9 / I65l7[MZIB@ - 9c7 ! .- G - ` - 7:Gh< - <65LGHG:5L36147E58M > @a58GuNI - I? - 77 - 9'5uNI - GXGHM > 7[369H5LG -?>D/ M&'3C7 - 321V9]K -T/ 6MZ5.?'< P^I >6-'t M >6/ 7:5LM > < - e 9 -=-?> uaIC5G:147E58G t 1V5l7WG:I69 /?- GX321V9HMZ5LGz = 0
- 7 z = ` CqI >6-o/ M > <6587:5LM > 1V<21V3C7:; - / M > <65l7E58M > '5b?a7 - <61 > G_Km1auaI - KLK - Km1 t M >6/ 7:5LM >#- 7TGE1q<6;=9:58@V; -G:M > 7kKL58; - G_G:I69TK - G_321V9HMZ5LG_32149_KwP^5'3;=<21 >6/=- @a5LG / MV7 - 9'5uNI - < - 32149:MZ5 ! 1q369HMZ321Z147E58M > < - GTM > < - Gk<21 > GTK -o/ aKL5 > <69 -o- GH7k9:; Z5 - 361V9TKwP^;euNI2147:5LM > \! #
(∆ + k2)p(~r) = 0 , ∀~r ∈ D . \!$# N
c /?- 7:7 - ;euNI217E5LM > < - 3C9:MZ321 147E58M > G:M > 71VGHG:M / 58; - GTK - G / M > <6587:5LM > GT1VIG? t 9HM > 7E5g==9 - GkGHI658@41 > 7 - G
ρ0c0vz = ρ0c0ve − ζz0p ,3MZI69 r ∈ (0, ae) , θ ∈ (0, 2π) , z = 0 ,
\! # Z1 ρ0c0vz = −ρ0c0vR + ζz`p ,
3MVI69 r ∈ (0, ar) , θ ∈ (0, 2π) , z = ` , \! #
ρ0c0vz = −ζz0p ,3MVI69 r ∈ (ae, a) , θ ∈ (0, 2π) , z = 0 ,
\! # / ρ0c0vz = ζz`p ,
3MZI69 r ∈ (ar, a) , θ ∈ (0, 2π) , z = ` , \! # V<
MZt ζz0
- 7 ζz`G:M > 7]K - GW1V<'587:7E1 >6/=- GY1 / MZI6Gc7E5uNI - GWG:3; / 5 uNI - G]< - G]361V9:MZ58G -?> z = 0
- 7 z = ` 7E9E14<6I65LG:1 > 7K - G - ` - 7EGk< - G / MZI / - GK85'587 - GX7 - 9 `MF@a5LG"uNI - I6G - GG:I69TK -o/ 61`31 / MZI6GH7:5uNI - <21 > GkKL1 / 1n@a587:; 1 >6>6- ? - - 7
vr = 0 3MZI69 r = a , θ ∈ (0, 2π) , z ∈ (0, `) , \!$# - / 1V9%K - G - ` - 7:G.< - G / MZI / - GKL5 `5l7 - G7 - 9'MF@B58GuNI - I6G - G Km17E;?9:1VK - G G:M > 7p369H5LG - G -?>e/ M&'3C7 - <61 > G.K -Y> M 69 -
< P^M > < -y/ M`36K - ? - k !.- GT1V<'587H7U1 >6/=- G ζz0
- 7 ζz`9 - GH7E1 > 7T3 - 7E5l7 - G / M`3C7 - 7 -=> I< - GT;euNI2147:5LM > G \! \!$#0\ /=- G / M > <65 H7E58M > G_1VI? t 9:M > 7:5g=?9 - G \ ! # G?P^; / 9H58@ -?> 7 1 >6>6- ? -
∂ p
∂n= −∂ p
∂z=
j k0ρ0c01 −Kv
vz ≈j k0ρ0c01 −Kv
ve − j k0ζz0 p \!$# 1
7 ) A ) 1% 14$a)& 4 ) 0 )* )( ) 7 * 6! 0)(+* $( * #9$
'9/"( 1* & %( * 1 * 9
3MZI69 r ∈ (0, ae) θ ∈ (0, 2π) z = 0 ∂ p
∂n=∂ p
∂z= − j k0ρ0c0
1 −Kv
vz ≈ j k0ρ0c01 −Kv
vR − j k0ζz` p \!$#
3MZI69 r ∈ (0, ar) θ ∈ (0, 2π) z = ` ∂ p
∂n= −∂ p
∂z= −j k0ζz0 p
\!$# / 3MZI69 r ∈ (ae, a) θ ∈ (0, 2π) z = 0
∂ p
∂n=∂ p
∂z= −j k0ζz` p ,
\!$# < 3MZI69 r ∈ (ar, a) θ ∈ (0, 2π) z = `
∂ p
∂n=∂ p
∂r= 0 ,
\!$# - 3MZI69 r = a θ ∈ (0, 2π) z ∈ (0, `) !
pMZIC9]I >6- M > < - 21V9 `M > 5uNI - K -T/ 21'3`< - 3C9 - GHG:58M > 1 / MZI6Gc7E5 uaI - p(~r) <21 > G]I > <6M&`1V5 >C- D K85'587E;32149_I >6- G:I69 t 1 /?-t - 9 `; - S 3 - IC7 =7:9 - ; / 9:5l7TG:MZI6GTKL1 t MV9 - 5 > 7E; Z9E14K - GHI658@41 > 7 - ^ > ! !_ K - t 1 / 7 - IC9ej ωt - GH7M&`58G - 7kK - @ -=/ 7 - I69 ~r - Gc79 - 369H;?G -?> 7E;v5 / 5O321V9_K - G_7E9:MV5LG / MaMZ9H<6M >6> ; - G r θ - 7 z
p(~r) =
∫∫∫
DG(~r, ~r0)f(~r0) dD0 +
∫∫
S
[
G(~r, ~r0)∂
∂n0
p(~r0) − p(~r0)∂
∂n0
G(~r, ~r0)
]
dS0 . \!%
1 t M >6/ 7E58M > f(~r0)9 - 369:;=G -=> 7 - KwPR1 / 7:5LM > < - G[G:MVI69 /=- Gh9H;?321V9c7E5 - G CvKwP^5 > 7:;?9:5 - I69W<6Ig<6M&`1V5 >6- D <61 > GK -h/ 1VGY3C9:;?G -?> 7 f(~r0) = 0 ! 1 t M >6/ 7:5LM > G(~r, ~r0)
- GH7WI >6-Xt M >6/ 7E5LM > < - e 9 -?-=> GE147:5LG t 1V5LG:1 > 7CfKwP^;euNI217E5LM >(∆ + k2)G(~r, ~r0) = −δ(~r, ~r0) <61 > G D \ ! #
MZt δ(~r, ~r0) - Gc7bKL1<658GH7:9:5 CIC7E58M > < -Dj 589E1 / - 7 C< - G / M > <C587E58M > Ge1VIG?zK85'587 - Go13C9:5LMV9:5 uNI - K / M > uaI - GGHI69KL1qG:I69 t 1 /?- S ! 10<6;?9H58@Z; - ∂
∂n0
9 - 369H;?G -?> 7 - Km1q<6;?9H58@V; - G:I65l@V1 > 7kKL1 > MZ9`1VK - K8M / 1VK - C'KL10GHI69 t 1 /?- SMZ9H5 -=> 7:; - @ - 9:GkKwP - ?a7E;=9:5 - I69_<6I<6M&`1V5 >6- ! - 7H7 - 9 - KL147E58M > Z; > ;=9E1VK - 58G:GHI - <6I 7 6;=MZ9B= - < - e 9 -?-?> ^ > ! ! _k143636KL5 uNI6; - CzKL1 / 1F@a5l7E; < -/ MZI636KL1 - 3 - 9 - 7W<OP - ?C3C9:5 - 9 1VG:G - dTG:5 `3CK - -?> 7 V9>X /=- 14I / CMZ5b?0<OPRI >C- t M >6/ 7:5LM > < - e 9 -?-=> 1V<21V3C7:; - KL103C9 - GHG:58M > 1 / MZI6Gc7E5 uaI -e-=> 7:MZIC7T3MZ5 > 7T< - KL1 / 1n@a587E; !;Y> t 1V5LG:1 > 7[ICGE1 - < - G / M > <C587E58M > G]1VIG? t 9:M > 7:5g==9 - G \ ! # 1?q - 3MZI69[K -/ 61`3`< - 369 - GHG:5LM > K -/ CMZ5b?< P^I >6-t M >6/ 7E58M > < - e 9 -?-=> G:147E58G t 1V58GE1 > 7TG:IC9kKQP -=> G - 6K - < - Gk321V9HMZ5LGh< - KL1 / 1n@a587E;y14IG? / M > <6587:5LM > G
∂
∂nG(~r, ~r0) = −j k0ζz0G(~r, ~r0) ,
3MZI69 r ∈ (0, a) , θ ∈ (0, 2π) , z = 0 , \! V1
∂
∂nG(~r, ~r0) = −j k0ζz`G(~r, ~r0) ,
3MZIC9 r ∈ (0, a) , θ ∈ (0, 2π) , z = ` , \!
∂
∂nG(~r, ~r0) = 0 , 3MVI69 r = a , θ ∈ (0, 2π) , z ∈ (0, `) ,
\ ! / 3 - 9 - 7<OP - ?C3C9:5 - 9TKQPR5 > 7:; Z9:1VK - < - G:I69 t 1 /?- 1V3C321V9E145LG:G:1 > 7k<21 > GkKm1q9 - Km147:5LM > \ ! % uNI65.9 - 369:;=G -=> 7 - KL1369 - GHG:5LM > 1 / MZI6Gc7E5uNI -y-=> 7:MZIC7k3MV5 > 7_< - Km1 / 1F@a5l7E; < - KL1 g1 > 5g==9 - GHI658@41 > 7 -
p(~r) =j k0ρ0c01 −Kv
[∫∫
Se
ve(~r0)G(~r, ~r0) dS0 +
∫∫
Sr
vR(~r0)G(~r, ~r0) dS0
]
, \!
12 R) @+ $+ @+ K) + X) +) 7 6
Se
- 7 Sr;=7E1 > 7dK - GTG:I69 t 1 /?- G9 - GH3 -=/ 7E5l@ - Gf< - Km1 - 69:1 >6- < - KwP^; - 7H7 - IC9 - 7<6I9H; /=- 3C7 - I69 K - G - ` - 7:G< - G / MZI / - G7 - 9'M @a58GuNI - ICG - GY;A7U1 > 7p3MV9H7E;=Gp361V9pK -it 1 / 7 - I69 Kv
uNI65a5 > 7 - 9H@a5 -?> 7 <61 > GKwP^5'3;=<21 >6/=- <CI`58KL5 - I - 7[361V9XKL1 t M >6/ 7:5LM > < - e 9 -=-?>D- KLK -hH - <61 > G]K -k> M& 69 - <OPRM > < - k 3MZIC9XK - G - ` - 7EGXKm147:;?9:1VIG?- 7[<21 > GXK - G[1V<'587H7U1 >6/=- G]G:3; / 5 uNI - G ζz0
- 7 ζz`1VI > 5l@ - 1VID< - G - 69:1 >6- GX< - G `5 / 9HMZ36M >C- G - 7hK - I69
3;=9:5836;?9H5 - !4c > MV7 - 9WKm1oG ';=7:9:5 - ; - 7:7 - I69 H 9:; /?- 3C7 - I69 <21 > G]KwP - ?B369 - G:GH5LM > <6I / 21'3`< - 369 - GHG:5LM >1 / MVI6GH7:5uNI - /=-e/ 61`3<6;=3 -=> < < -e/=- GT@a587 - G:G - G3uaI - KuNI P -?> G:MZ5l7_K - I69 / 14I6G - !<c| #/$ ) 0="! ) ? "!/+ - A! ) 35$ ) ) ! #%$ "& '( ) eD
)T $ #/$ #/$ ) 1 t M >6/ 7E58M > < - e 9 -?-=>g- GH7XG:MZK8IC7E58M > <6ID369HM 6Kg= - <C; / 9H587X321V9]KwP^;euNI2147:5LM > \! # - 7h< - G / M > <C587E58M > G1VIG? t 9:M > 7E5 =?9 - G \! V1 /
(∆~w +∂ 2
∂z2 + k2)G(~w, z; ~w0, z0) = −δ(~w − ~w0)δ(z − z0)<61 > G D ,
\! \ 1 ∂
∂n0G(~w, z; ~w0, z0) = − ∂
∂z0G(~w, z; ~w0, z0) = −j k0ζz0G(~w, z; ~w0, z0)
3MZIC9 z = 0 , \! \
∂
∂n0G(~w, z; ~w0, z0) =
∂
∂z0G(~w, z; ~w0, z0) = −j k0ζz`G(~w, z; ~w0, z0)
3MZI69 z = ` , \! \ /
∂
∂n0G(~w, z; ~w0, z0) =
∂
∂r0G(~w, z; ~w0, z0) = 0 3MVI69 r = a ,
\ ! \V< 1n@ -?/ ∆~w =
1
r
∂
∂r
(
r∂
∂r
)
+1
r2
∂ 2
∂θ2!
1DGHMZKLIB7E5LM > < -q/?- 369HM 6Kg= -J- GH7y9 -?/ - 9 / 6; - GHMZI6GyKm1 t MV9 - < P^I > <C;=@ - KLMZ3C3 - -=> 7^ > !$#&! \ / _G:I69TKL1 21VG - < - G t M >6/ 7:5LM > G_3C9:MZ369 - GTMZ9c7 6M > MZ9'; - G ψt
µν(r, θ)<6I / aKL5 > <69 -
G(~r, ~r0) =∑
µ,ν=0,1...
t=c,s
gµν(z, z0)ψtµν(r0, θ0)ψ
tµν(r, θ) .
\!
.- G t M >6/ 7:5LM > G_369:MZ3C9 - GTGHM > 7dGHMZKLIC7:5LM > Gh<6I369:M CKg= - 1VIG?g@41VK - I69:GT369HMZ369 - G(∆~w + k2
wµν)ψtµν(r, θ) = 0 ∀r ∈ (0, a)
- 7 ∀θ ∈ (0, 2π) , \! > 1
∂
∂nψt
µν(r, θ) = 0-?>
r = a- 7 ∀θ ∈ (0, 2π) ;
\! > - K8K - G[G?P^; / 9H58@ -?> 7 t = c, s 9 - GH3 -=/ 7:58@ - -=> 7d3MZI69 cosµθ
- 7 sinµθ ψc
µν(r, θ) =1
aµbµν
cos µθJµ(kwµνr) , \! N 1
ψsµν(r, θ) =
1
aµbµν
sin µθJµ(kwµνr) , \! N
- 7_K - G[@41VK - I69HGh369HMZ369 - GT1VG:GHM / 5L; - G kwµνG:M > 7_5 `3MZGH; - G[321V9_Km1 / M > <C587E58M > 1VI? t 9HM > 7E5g==9 - G_<6I#369:M6Kg= -
\! > / M > <6587:5LM > G_< -D:- I g1 >C> kwµν = γµν/a ,
\!
7 ) A ) 1% 14$a)& 4 ) 0 )* )( ) 7 * 6! 0)(+* $( * #9$
'9/"( 1* & %( * 1 * 9
MZt γµν
ν = 0, 1, 2, 3... 9 - 3C9:;?G -?> 7 - K - ν + 1 5g= - d?;=9:M`< - KL1'<6;=9:5l@Z; - 369 - `5 =?9 - < - Km1 t M >C/ 7E58M > < -
- G:G - K Jµ GHMZ587 J ′
µ(γµν) = 0 ! - 3C9:M 6K = - 369H;?G -=> 7 - <CM >6/ I >6- <6; Z; > ;=9 - G /?-=>6/?- < PRMV9:<69 - !%- G / M - S / 5 -?> 7EGk< -y> MZ9 g14KL5LG:147E58M > aµ
- 7 bµνG:M > 7k<6M >C> ;?Gk321V9
a2µ =
∫ 2π
0
cos2 µθ dθ =
∫ 2π
0
sin2 µθ dθ = (1 + δµ0)π , \!
- 7b2µν =
∫ a
0
J2µ(kwµνr)r dr ,
GHMZ587 -?> IC7E58KL58GE1 > 7_KL1b9 - KL147E58M > ^ ! \! _%36I658G_K - G_9 - KL147E58M > G ^ ! \ N !$# - 7 _%< - KL109H; t ;=9 -?>C/?- ^$# _
b200 =a2
2
b2µν =a2
2
(
1 − µ2
γ2µν
)
J2µ(γµν)
3MZI69 (µ, ν) 6= (0, 0) .
\!&%
%- <6;=@ - KLMV363 - -=> 7 \! < - KL1 t M >6/ 7:5LM > < - e 9 -=-?> GE147:5LG t 1V587 65 -=> 145 > GH57CqKm1 / M > <6587:5LM >g-?>/t 9:M >GH7:5g=?9 - G \! \Z< ! - Gc7 - C - ?C3C9:5 - 9JK - G / M - S / 5 -?> 7EGq<CI)<6;=@ - KLMV363 - -=> 7 gµν(z, z0)
< - g1 > 5g=?9 - C /=-uNI - KwP - ?B369 - GHG:5LM > < - G G:147E58G t 1VGHG - CDKQPR;RuaI6147E58M > < - GoM > < - G \! \ 1 - 7e1VI? / M > <65l7E5LM > G -=> t 9HM > 7E5 =?9 - G \! \ - 7 / ! [ K @B5 -?> 7T7EMZIB7 / 1VK / I6K t 14587 1 >6>6- ? -D[ [ ! #
gµν(z, z0) = −cos(kzµνz + ϕ0µν) cos(kzµν(z0 − `) − ϕ`µν)
kzµν sin(kzµν`+ ϕ0µν + ϕ`µν),
\! #F1 GH5 z < z0 - 7k321V9
gµν(z, z0) = −cos(kzµνz0 + ϕ0µν) cos(kzµν(z − `) − ϕ`µν)
kzµν sin(kzµν`+ ϕ0µν + ϕ`µν),
\! # GH5 z > z0 MVt ϕ0m ϕ`m
G:M > 7k<6M >6> ;?Gk321V9 ;eu ! [ ! N
tanϕ0,`µν = − j k0ζz0,`µν
kzµν.
\!
<# < ;( # +[ - `%) !/ = + , - .%- / 21'3< - 369 - G:GH5LM > 1 / MZI6Gc7E5 uaI - <21 > GgKm1 / 1n@a587:; - GH7#M B7 -?> I -=> 9 - 3MZ9c7U1 > 7`Km1 t M >6/ 7:5LM > < -e 9 -=-?>s- ?B369H5'; - <21 > GoK - 32149E1Z9:1V3 - 369:; / ;?< -?> 7 \! <21 > GoKwP^;euNI2147:5LM > \! ! M&`3B7 - 7 -?> Iz< -
KL1 t MZ9 - < -/?- 7:7 - t M >6/ 7E58M > < - e 9 -=-?> Km1s369 - G:GH5LM > p(r, θ, z) 1V3C321V9E1 L7'1VK8MZ9:G0G:MVI6GgKL1 t MZ9 - <OPRI ><6;A@ - KLMZ363 - -?> 7G:I69TK - G t M >6/ 7E58M > G_369HMZ369 - G ψt
µν(r, θ)
p(r, θ, z) =j k0ρ0c01 −Kv
∑
µ,ν=0,1...
t=c,s
[
gµν(z, 0) < ve|ψtµν >e +gµν(z, `) < vR|ψt
µν >r
]
ψtµν(r, θ) ,
\ !
MZt Km1 > MV7E147E58M > < ve|ψtµν >e
<6;?GH5 >6- KwP^5 > 7:; Z9:1VK - GHI69 Se<6I 369HMB<CI6587 ve(r, θ)ψ
tµν(r, θ)
- 7MZt <vR|ψt
µν >r<6;=G:5 >6- KwP^5 > 7E; Z9E1VK - GHI69 Sr
<CI369HMB<CI6587 vR(r, θ)ψtµν(r, θ)
!
) + 7
1qGHMZKLIC7:5LM > 3C9 -?> <<6M >6/ Km1 t MZ9 -
p(r, θ, z) =∑
µ,ν,t
ptµν(z)ψ
tµν(r, θ) ,
\!]\ 1n@ -?/
ptµν(z) =
j k0ρ0c01 −Kv
[
gµν(z, 0) < ve|ψtµν >e +gµν(z, `) < vR|ψt
µν >r
]
. \!
j 1 > G`Km1uG:I65l7 - < - KQP - ?B3MZGH; K -/ 21'3< - @a587 - GHG - ve
/ 21 `3 @B5 69E17EMZ589 - G:MVI69 /=- - GH7gG:I6363MZGH;5 > <C;?3 -?> <21 > 7e< - KwPR1 > VK - θ - 7J361V9yG:I65l7 - / M&`3B7 - 7 -=> Iz< - Km1#G ';=7E9H5 - < - 9:;A@ZMZK8IC7E58M > < - KL1 / 1n@a587E;- 7g< - /?- KLK - <6I `5 / 9:MV36M >6- 9:; /?- 3C7 - I69 K - / 21 `3< - 369 - G:GH5LM >- GH7#K8I65 H - 5 > <6;=3 -=> <21 > 7D< - θ - 7 / M > G:; / IC7:58@ - -=> 7#Km1 @a587 - G:G - vR; 1VK - -?> 7 ! -=/ 5 - Gc7 / MZ9:9HM MV9:; -?> / 1VK / I6KL1 > 7`KQPR5 > 7E; V9E1VK - ve,R<6;?GH5 > 1 > 7T5 > <65b`O;=9 - -?> 7 ve
MZI vR
< ve,R|ψtµν >e,r=
∫ ae,r
0
r0 dr0ve,R(r0)
∫ 2π
0
dθ0ψtµν(r0, θ0) ;
\! > < PR1V369B==GoKwP - ?B369 - G:GH5LM > \! N < - G ψt
µν KwP^5 > 7E; Z9E14K - G:I69 θ0 - Gc7 > I6K8K - 3MZI69 µ 6= 0 - 7y< - G:IC9 / 9HM L7 - KLK -- Gc7; 1VK - -?> 7 > I6KLK - 3MVI69 t = s µ = 0 1 >6>6- ? - [ [ ! !cd5 > G:5 KL10GHMZKLIB7E5LM > \!]\ < - @a5 -=> 7
p(r, z) =∑
ν
pc0ν(z)ψ
c0ν(r) .
\! N jd- `1 > 5g==9 - C / 1VK / I6K - 9vK - G pc
0ν(z)<6M > 7JKwP - ?B369 - G:GH5LM >)- GH70<6M >6> ; - 321V9bKQPR;RuaI6147E58M > \! 5LK - Gc7> ; /=- GHGE1V589 - < P - ?B369H5 - 9vKm1@a5l7 - GHG - vRuNI65X<6;?3 -?> < - KLK - - <6I / 61`3 1 / MZICGH7E5 uNI - p(r, z) ! pMVI69/?- t 1V5L9 - 321V9 - ?a7 -=> G:58M > < - KL19 - Km17E5LM > SrvR = −Yrp(`)
9 - KL5L1 > 7JKQPr1V< `5l7:7U1 >6/?- Yr<6I '5 / 9:MZ3 6M >6-
9:; /?- 3B7 - IC9_1VIG?`369 - G:GH5LM > G - 7X@B5l7 - G:G - G_1 / MZI6GH7:5uNI - GhKLMZ9HGuNI - K -d/ 61`3g1 / MVI6GH7:5uNI -f- GH7h5 > <6;=3 -=> <21 > 7_< -r Km1b369 - G:GH5LM >- Gc7k; / 9H587 -
SrvR(r) = −Yr(r)p(r, `) , \!
MZtKwPR1V<'587H7U1 >6/=- Yr(r) 9 - 369:;=G -=> 7E1 > 7dKQPr14<`5l7:7E1 >6/?- 1 / MZICGH7E5 uNI - KLM / 1VK85LGH; -ot M >6/ 7:5LM > < - r <6I9:; /?- 3 H7 - I69 - Gc7J; / 9H587 - <21 > GvKL1GHI6587 - GHMZI6GvKm1 t MZ9 - G:I65l@41 > 7 - -=> t M >6/ 7E58M > < - G:1@41VK - I69 `M -?>6>6- G:I69eKm1G:I69 t 1 /?- <6I9:; /?- 3C7 - I69 < Yr >r
Yr(r) =< Yr >r +δYr(r) , \!
δYr(r);=7E1 > 7k7 - KKuNI - < δYr >r= 0 !
cd5 > G:5 K - t 1 / 7 - I69T<6;?3 -?> <61 > 7d< - vR< - Gk9 - Km147:5LM > G \! - 7 \! N G=PR; / 9H587
−Sr < vR|ψc0ν >r=< Yrp(`)|ψc
0ν >r
\! \ % - 7 321V9T9 - 3MV9H7T< - KwP - ?B369 - GHG:5LM > \! N 3MVI69 z = `
−Sr < vR|ψc0ν >r=
∑
ν′
pc0ν′(`) < (< Yr >r +δYr)ψ
c0ν′ |ψc
0ν >r , \! \#
G:MZ5l7 < Yr >r;=7E1 > 7T5 > <C;?3 -?> <21 > 7< - r
−Sr < vR|ψc0ν >r=
∑
ν′
pc0ν′(`)
[
< Yr >r< ψc0ν′ |ψc
0ν >r + < δYrψc0ν′ |ψc
0ν >r
]
. \! \
7 ) A ) 1% 14$a)& 4 ) 0 )* )( ) 7 * 6! 0)(+* $( * #9$
'9/"( 1* & %( * 1 * 9
j I t 1V587 uNI - < δYrψc0ν′ |ψc
0ν >r3 - IC7 =7:9 - ; / 9:587T5 / 5%G -?> GH5 6K - -?> 7k; 1VK)C < δYrψ
c00|ψc
00 >r
K - 7 - 9 -M&'5LGW<6M >6> 1 > 7XI >6-d/ M > 7E9H5 6IC7:5LM > G:IC363MVG:; -> ; VKL5 - 16K - C p - 7 ψc
00;=7U1 > 7[5 > <C;?3 -?> <21 > 7X< - r 5LK23 - IB7
A7E9 - 9 - '36Km1 / ;o321V9 < δYr >r< ψc00|ψc
00 >r= 0 < P^MZt−Sr < vR|ψc
0ν >r=∑
ν′
pc0ν′(`) < Yr >r< ψc
0ν′ |ψc0ν >r .
\! \ %- 9 - 3MZ9H7]< -/?- 7:7 -- ?B369 - G:GH5LM > <21 > GXKwP - ?B369 - G:GH5LM > \! N < - Km1o369 - G:GH5LM > / M`3C7 - 7 -?> ID< - Km1y9 - Km147:5LM > \! / M > <6IC587fC -?> ; / 9:5l@V1 > 7 < Yr >r= Yr
p(r, z) =j k0ρ0c01 −Kv
∑
ν
[
g0ν(z, 0) < ve|ψc0ν >e
− Yr
Srg0ν(z, `)
∑
ν′
pc0ν′(`) < ψc
0ν′ |ψc0ν >r
]
ψc0ν(r) ,
\ ! \4\
MZt K - G'7 - 9 - G < ψc0ν′ |ψc
0ν >r<6;=G:5 >6-?> 7DK - G05 > 7:; Z9:1VK - G'GHI69'Km1sG:I69 t 1 /?- Sr
< - Gg3C9:Ma<6I6587:G ψc0νψ
c0ν′ <6M >6> ; - G -?> 1 >6>6- ? - [ 361V9JK - G - ?B369 - G:GH5LM > G [ ! ##F1 ! %- Gv7 - 9 - G - ?B369H5`1 > 7b< - G / MZI636KL1 - G -?> 7E9 -
<65b`O;=9 -=> 7:G `Ma< - G ν 6= ν ′ GHM > 7o7E9B==Gy3 - 7E587:Gf< - @V1 > 7eK - Gy1VIC7:9 - G - 7e3 - IC@ -=> 7v<CM >6/ =7:9 -q> ; ZKL5 Z;?G K - GMZ9H<69 - GT< - Z9E1 > < - I69TGHM > 7<CM >6> ;=G<21 > GTK - 321V9:1Z9:1V3 -M[ ! ! 1'369 - GHG:5LM > 1 / MVI6GH7:5uNI - p(r, z) - GH7f<CM >6/ 9 - 7 -?> I - GHMZI6GdKL1 t MV9 - -?> t 1 / 7:MZ9:58GE1 > 7KwP^5 > @ - 9:G - < - Km1GHI69 t 1 /=- S = πa2
p(r, z) =j k0
1 −Kv
ρ0c0S
∑
ν
[g0ν(z, 0)S < ve|ψc0ν >e −Yr0νg0ν(z, `)p
c0ν(`)]ψ
c0ν(r) ,
\ ! \
MZt Yr0ν = Yra2
a2r
< ψc0ν |ψc
0ν >r!
1 / M&'321V9:1V5LGHM > < - G - ?B369 - G:G:58M > G \ ! N / M&'3C7 - 7 -?> I< - \! - 7 \! \ 3 - 9 - 7_< - <6;A7 - 9 `5 >6- 9K - G / M - S / 5 -?> 7EG pc
0ν(z)
pc0ν(z) =
j k0
1 −Kv
ρ0c0S
[g0ν(z, 0)S < ve|ψc0ν >e −Yr0νg0ν(z, `)p
c0ν(`)] ,
\ ! \ > GHMZ587 3MVI69 z = `
pc0ν(`) =
j k0
1 −Kv
ρ0c0S
g0ν(`, 0)S < ve|ψc0ν >e
1 +j k0
1 −Kv
ρ0c0S
Yr0νg0ν(`, `)
, \! \ N
/=- uNI65%<6M >6>6- 3MVI69_KwP - ?B369 - G:GH5LM > < - pc0ν(z)
pc0ν(z) =
j k0
1 −Kv
ρ0c0S
[
g0ν(z, 0)S < ve|ψc0ν >e
− Yr0νg0ν(z, `)
j k0ρ0c01 −Kv
g0ν(`, 0) < ve|ψc0ν >e
1 +j k0
1 −Kv
ρ0c0S
Yr0νg0ν(`, `)
]
; \! \
7C7
K - 3C9:Ma<6I6587TG / 1VKL1V5L9 - < ve|ψc0ν >
- GH7 / 14K / ICKL;f<21 > GKQPr1 >6>6- ? -M[ [ ! ! 1z369 - G:G:58M > 1 / MZI6Gc7E5uNI --?> 7:MZIC7D3MZ5 > 7`<6I / aKL5 > <69 - ;=7U1 > 7D1V5 > G:5 -?> 7E5 =?9 - -=> 7D<6; 2> 5 - 321V9'K - G
;euNI2147:5LM > G \! N - 7 \! \ 5LK29 - GH7 - 1VK8MZ9:G Ce<6;=7 - 9'5 >C- 9WKwP - ?B369 - G:GH5LM > < - KQPr14<`5l7:7E1 >6/?- 1 / MZICGH7E5 uNI - < -7E9:1 > G t - 9H7 YT
<CIG a7B= - !
< ( -0- !/+ ) +[ 0- '/) ) - $ !/ )6$ -.-y/ 21 `3< - 369 - G:G:58M > - 7kK -y/ 21'3< - @a5l7 - GHG -v> PR;A7U1 > 7321VGTI > 5 t MZ9 - G_<61 > GkKm1 / 1n@a587:; KwPR1V<'587 H
7U1 >C/?- 1 / MZI6Gc7E5 uaI - < - 7:9E1 > G t - 9c7 9:1V363MZ9H7[<6ID<6;6587[< - KQPR; - 7:7 - I69 -=>#/ MZIC9H7 Hi/ 589 / IC587[1 / MZICGH7E5 uNI - G:IC9[Km1369 - G:GH5LM >-?> z = ` - Gc7I >6- t M >6/ 7:5LM > < - r !
PR1V<'587H7U1 >C/?- 1 / MZICGH7E5 uNI - < - 7:9E1 > G t - 9c7 > MZ9 - ;["> # % > \ H ^ _ - GH7q<C; 2> 5 - 5 / 5 -=> 9 - `36KL1 HCn1 > 70K - G uNI21 > 7:587E;=G uaIC5h<6;?3 -?> < -?> 7`< - Km1 @41V9H5m16K - r 321V9bK - I69b@41VK - I69 'M -=>6>6- G:I69qKL1 G:IC9 t 1 /=- <CI7E9:1 > G:<CI / 7 - I69T5'36K85uNI6; 321V9 - ?a7 -?> GH5LM > < - Km1q9 - Km147:5LM > ! # C0GE1n@ZMV5L9
YT =Se < ve(r) >e + < Ye(r)p(r, 0) >e
< p(r, `) >r
, \! \
Ye;=7U1 > 7TKQPr14<`5l7:7E1 >6/?- 1 / MZICGH7E5 uNI - KLM / 1VK85LG:; - t M >6/ 7:5LM > < - r < - KwP^; - 7H7 - IC9 ! ;Y> ; / 9H58@41 > 7
Ye(r) =< Ye >e +δYe(r) , \!%
- 7p(r, 0) =< p(0) >e +δp(r, 0) ,
\! # K - G -?/ M > <7 - 9 - <6I > I';?9E17 - IC9k< - @a5 -=> 7
< Ye(r)p(r, 0) >e = < (< Ye >e +δYe(r))(< p(0) >e +δp(r, 0)) >e
≈ < Ye >e< p(0) >e + < Ye >e< δp(0) >e + < p(0) >e< δYe >e ,
G:MZ5l7< Ye(r)p(r, 0) >e≈< Ye >e< p(0) >e .
\! 1V9TG:IC587 - -?> 3MZGE1 > 7
Ve =< ve >e=2
a2e
∫ ae
0
ve(r0)r0 dr0 , \!
- 7Ye =< Ye >e=
2
a2e
∫ ae
0
Ye(r0)r0 dr0 , \! \
5LK@B5 -?> 7YT =
SeVe + Ye < p(0) >e
< p(`) >r,
\! G:MZ5l7 -?> 9 - 3MZ9c7U1 > 7kKwP - ?B369 - G:GH5LM > < - Km1b3C9 - GHG:58M > p(r, z) GHMZI6GkGE1 t MZ9 - \! N
YT =
SeVe + Ye
∑
ν
pc0ν(0) < ψc
0ν >e
∑
ν
pc0ν(`) < ψc
0ν >r
. \! >
76) A ) 1% 14$a)& 4 ) 0 )* )( ) 7 * 6! 0)(+* $( * #9$
'9/"( 1* & %( * 1 * 9
[ K%9 - GH7 - 14KLMZ9HG C - ?B36KL5 / 5l7 - 9_K - G / M - S / 5 -=> 7:G pc0ν(`)
- 7 pc0ν(0) !
%-o/ M - S / 5 -?> 7 pc0ν(`)
- GH7<6M >6> ;e361V9_KwP - ?B369 - G:GH5LM > \ ! \ N
pc0ν(`) =
S < ve|ψc0ν >e
1 −Kv
j k0
S
ρ0c0
1
g0ν(0, `)+ Yr0ν
g0ν(`, `)
g0ν(0, `)
, \! N
- 7K -o/ M - S / 5 -?> 7 pc0ν(0)
- Gc7k<6M >6> ; -=> t M >C/ 7E58M > < - pc0ν(`)
361V9_Km1b9 - Km17E5LM > \! \ >
pc0ν(0) =
j k0
1 −Kv
ρ0c0S
g0ν(0, `)
[
S < ve|ψc0ν >e
g0ν(0, 0)
g0ν(0, `)− Yr0νp
c0ν(`)
]
. \!
%- <6;=7E1V5LKO<6I / 1VK / I6K < - G / M - S / 5 -=> 7EG pc0ν(`)
- 7 pc0ν(0)
- Gc7<CM >6> ;y<21 > GkKQPr1 >6>6- ? - [ [ ! > ! .- 9H;?GHI6K87E147M B7 -?> I3MZIC9_K -o/ M - S / 5 -?> 7 pc
0ν(`)369 -=> <Km1 t MZ9 - 1 >6>6- ? -D[ 9 - Km147:5LM > G [ ! - 7 [ ! %
pc0ν(`) =
SeVeS
a0b0νV0ν
YT00ν + Yr0ν
[
Yp0ν
Yi0νj sin kz0ν`+ cos kz0ν`
] , \!
MZt V0ν- GH7fI > / M - S / 5 -?> 7d<6;?3 -?> <61 > 7o<6I 369HM K< - @a587 - GHG - < - KwP^; - 7H7 - IC9 / 1VK / I6K8;v<61 > GfKQPr1 >C>6- ? -Z[ [ ! 9 - Km147:5LM > G [ ! #F1 / MVtKQPr14<`5l7:7E1 >6/?- YT00ν
9 - KL147E58M >+[ !% G?P^; / 9:587
YT00ν =
[
Yi0ν +Y 2
p0ν
Yi0ν
]
j sin kz0ν`+ 2Yp0ν cos kz0ν` , \! > %
- 7MVtK - t 1 / 7 - I69 Yp0ν- GH7<6M >C> ;y321V9 [ !
Yp0ν =S
ρ0c0ζz0ν
- 7 Yi0ν =S
ρ0c0
kz0ν
k0(1 −Kv) .
\! > # p1V9k1V5LK8K - IC9:G K - G / M - S / 5 -?> 7EG pc
0ν(0) G=PR; / 9:5l@ -=> 7 1 >C>6- ? - [ 9 - Km147:5LM >+[ ! #
pc0ν(0) = pc
0ν(`)
[
j sin kz0ν`Yr0ν + Yp0ν
Yi0ν+ cos kz0ν`
]
. \! >
PR1V<'587:7E1 >6/=- 1 / MZICGH7E5 uNI - < - 7E9E1 > G t - 9H7kG?P^; / 9:587T<CM >6/ 2> 1VK - -?> 7TG:MVI6GKL1 t MZ9 -
YT =
1 + Ye
∑
ν
SV0ν
a0b0ν< ψc
0ν >e
cos kz0ν`+Yp0ν + Yr0ν
Yi0νj sin kz0ν`
YT00ν + Yr0ν
(
Yp0ν
Yi0ν
j sin kz0ν`+ cos kz0ν`
)
∑
ν
SV0ν
a0b0ν
< ψc0ν >r
1
YT00ν + Yr0ν
(
Yp0ν
Yi0νj sin kz0ν`+ cos kz0ν`
)
. \! >
j 1 > GJK -'/ 1VGb361V9H7:5 / ICKL5 - 9yMZt K - GJ9:1 VM > G ae ar
- 7 a G:M > 7J; 14IG? KwPR1V<'587H7U1 >C/?- / MV9:9:5 Z; - Yr0ν \! \ - Gc7; 1VK - C Yr!
7
5C< - 3CKLI6G 5LK > P q1f369HMZ321Z147E58M > uNI - < P^M > < - GY36KL1 >6- G <21 > GYKL1 / 1n@a587:; ν = 0 - 7WGH5CK - G - C9E1 >6- G< - G '5 / 9:MZ3CM >6- GYM > 7]I > 'MZIC@ - -=> 7XI > 5 t MZ9 - 1VK8MZ9:G < ψc
00 >e,r=1
a0b00
- 7 V00 = 1 /?- uNI65 <6M >6>C-3MVI69 YT
YT = j sin kz00`
[
Yi00 +(Ye + Yp00)(Yr + Yp00)
Yi00
]
+ cos kz00`(Ye + Yr + 2Yp00) , \! > \
MZtsK -0> M& 69 - < P^M > < - kz00- GH7v<6M >6> ;'361V9yKm1#9 - Km147:5LM > [ ! \ / M&'3C7 - 7 -?> I < - G - ?B369 - G:GH5LM > G \ ! 1n@ -?/ γ00 = 0 - 7 \!$# >
k2z00 = k2 − k2
w00 = k2 = k20
[
1 +1 − j√
2
1√
k0
2
a(√
` ′v + (γ − 1)
√
`h )
]
, \! >
KwPR1V<'587H7U1 >6/=- < - 32149:MZ5 Yp00361V9_Km1b9 - Km17E5LM > [ !$#
Yp00 =S
ρ0c0ζz00 =
S
ρ0c0
1 + j√2
√
k0(γ − 1)√
`h , \! > >
- 7TKQPr1V< `5l7:7U1 >6/?- 5l7E;=9E147:58@ - Yi00321V9 ;eu !1\! > #
Yi00 =S
ρ0c0
kz00
k0(1 −Kv) ≈
S
ρ0c0
[
1 +1 − j√
2
1
a√
k0
(−√
` ′v + (γ − 1)
√
`h )
]
. \! > N
> @Z;=9:5 2- 1V5 > GH5uNI - KwP - ?B369 - G:GH5LM > \! > \ - GH75L< -=> 7E5uNI - C_KwP - ?B369 - GHG:5LM > < - KwPR1V<'587H7U1 >C/?- 1 / MZI6Gc7E5uNI -< - 7:9E1 > G t - 9c7 YT
<6I - G aGH7>= -d/ 1VK / I6K8; -k-?>D>6-d/ M > GH5L<6;=9E1 > 7 uaI - Km1y369:MV321 147:5LM > < P^M > < - Gh36KL1 >6- G 32149 - ? - '36K - KQP - ?B369 - G:G:58M > 369:MZ3MZGH; - 321V9h1VK8Km1 ^_ !
< : . # - - -%._ $ '/) c 2> < - '5 - I? uNI21 > 7:5 2- 9#KwP^5 > 2I -?>6/=- < - Km1 369HMZ321Z147E58M > < P^M > < - G9E1V<65L1VK - GgGHI69Km1x@41VK - I69D< -
KwPR1V<'587H7U1 >6/=- 1 / MZI6Gc7E5uNI - < - 7:9E1 > G t - 9c7 uNI - K uNI - Gk9:;=G:I6Kl7U147:GkGHM > 7k369:;=G -?> 7:;?Gd<61 > G /?- 321V9:1Z9:1V3 - ! - I6K8GYK - G 'Ma< - G % % - 7 % # c G:M > 7 Co369 -=> <69 -T-=>`/ M&`3B7 - <21 > G]K -_/ 1VK / I6K ! ;Y>'- ` - 7]Km1 / M > 7E9:5 6I H
7E58M > < - G `Ma< - G]G:IC3;=9:5 - I69:G CyKm1o@41VK - I69W< - KwPR1V<'587:7E1 >6/=- 1 / MZICGH7E5 uNI - < - 7E9:1 > G t - 9H7 YT
;Ru !/\! > - GH75 >#t ;=9:5 - I69 - C # % −3 < <CM >6/0> ; VKL5 - 16K - 5 / 5 K - Gf7 - 9 - G - ?C3C9:5`1 > 7e< - G / MZI636KL1 - G -=> 7:9 - <65 `;?9 -?> 7EG`Ma< - G_3MZIB@V1 > 7 A7E9 -e> ; ZK85Z;=G_<21 > GTK -o/ 1VK / I6K /Rt ! [ ! !.- G ZI69 - G\! - 75\ ! 9 - 369H;?G -=> 7 -?> 7`Km1 <C5 `;?9 -=>6/?- < - G 'MB<6ICK - Gv< - G'1V< `5l7:7U1 >6/?- Gb1 / MZI6GH7:5uNI - G< - 7:9E1 > G t - 9c7 YT
/ 1VK / I6K8; - GvG - KLM > I > `Ma<6=?K -`>C- 369 -=> 1 > 7 -=>)/ M&'3C7 - uNI - Km1369:MV321 147:5LM > < P^M > < - G36Km1 >6- G ;eu !\! > \ - 7yG - K8M > K - 'Ma<6=?K - <6;=@ - K8MZ363;q<21 > G /?-0/ 21436587:9 - ;Ru !\! > 1n@ -=/ uNI217E9 -0/ 1F@a5l7E;?G< - KLM > ZI - I69:Gf<C5 `;?9 -=> 7 - G K - G / 1n@a587E;=Gy< PR;A7U1VK8M >6> 1 - <658G:3M > 5 CK - Gd1VI ): ; 3MZI69d< - G '5 / 9HMZ36M >6- G K - GJ9:1 VM > Gq< - G - 69:1 >6- G ae
- 7 ar;A7U1 > 7q; 1VIG? - 7b@41VKL1 > 7 ae = ar = 0, 96 a ! j 1 > GJKm1 ZI69 - \! Km1 - 69E1 >C- < - KwP^; - 7:7 - I69 - GH7eG:IC363MVG:; - 1n@ZMV5L9oI > `MVIC@ - -=> 7 > M > I > 5 t MZ9 - <6M >6> ;
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C4EPaQ J J D 1[\m_ H J b J O J SNS J 0MS F b J b J S J Q^S%CLOFID b \GPaS b J Q%HNMQN\ R SYCZSNQ50M b PaF OH J Q D J QUCZSNPQStnCVP *QNCVDeS K CVQ2CV\ SUF R MHNCVDGO J Q K H J QNOHNPoS J Q K CZH R CLDmFIH6 J K FI\GH RJ Q QNP& K CVHNC SUH J Q b M&S J H60PD J H 9 QNC _IFIPaHRaJ Q b J \ ]_IF R \ J Q dRJ Q b J \ HNMQNPQpSYCVDmO J QWCVOFI\mQ^SUP u \ J Q J S RaJ Q b J \ PD b \GOSUCVDGO J QWCVOFI\GQpSUP u \ J QWH J O GJ H *O M J Q7 g J Q^S K FV\GH u \GFIP\GD J 0F b M R PaQUCZSNPFID KmR \GQ OF' KGR S J b J Rhg FIH J P RaRJ CZH^SUP32 OP JRaRJTJ S b J Q J QTM R M- J DeSNQOFIDGQ^SNPaSN\mSUP t`Q2C M&SUM b M&_ JR F KGK M J98-
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RJ OC b H J b J[R g M&SYC R FIDGD C J b J EPaOH^F K FID J Q K CVH R C 0MS F b J b J HNMOP K HNF OPaSNM # b J \ EPaOHNF K FID J Q \mDM- J S^S J \mH nJ&\ O CVOPaSUM Mpe
C b 0PaS^SYCVDmO J CVOFI\mQ^SUP u \ J Ye _ PaS J QNQ J b JR C0 J > HUCVD J ve
J S \GDHNMO JK S J \mHnJ&\ OCVOPoSUM Mpr C b 0PaSNSUCVDGO J CVOFV\GQ^SNP u \ J Yr
_ PaS J QNQ J b J J > HUCVD J vR QNFVDqS OFI\ KGR MQ CVOFV\GQ^SNP u \ J8* J DeS K CVH9\mD J O C _ PaSNM b J OFI\ KGR C J J SNS J O C _ PaSUM J Q^S OFIDGQ^SNPaSN\GM J b J b J \ K FIHpSUPaFIDGQ b J O R PD b H J
b J J b P C SUH J a J Q J > HNCVD J Q b J Q EPaOH^F K FID J Q FIDeS b J Q HUC IFIDmQ b P MH J DeSUQ b J O JR \GP b \O R PaD b H J J QM R M J DqSNQ OCVHUCVO&SUMHNPQ J H2QNFVDqSPDmQNMH^MQ R C QNM K CVHUCXSUPFVD b J QO R PD b H J Q-$ J Q M R M- J DeSUQ !u \ CXSUH JSU\ >dJ Q b g C b EPoSNSUCVDGO J PD b Pa_ P b \ JRaRJ Yt
FI\ \GD J t J DeS J CVDmD \ R CVPaH J EPaDGO J b g C b 0PaSNSUCVDGO J Yf
QNFVDqSWH JK HNM *Q J DqSNMQ K CVH R g C b EPoSNSUCVDGO J Q K MOP2 u \ J y1
QN\ KGK FVQNM J R F OC R PaQNM J QN\GH R C OPH^OFIDt`MH J DGO J, r = a b CVDGQ RJ
KGR CZDGQ b J QNM K CVHUCXSUPFVD z = `11 J Q J K H J QNQNPaFIDGQ b J Q C b 0PaS^SYCVDmO J Q Yt
J S Yf
QNFVDqSWO C R O\ R M J Q b CVDGQ RJ QK CVHNCIHUC KGJ Q * ) J S * b J[R C HNMt`MH J DGO J 8
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z = `1ibl
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J QTS J O DGP u \ J Q b J J QN\GH J QNFIDeS RaJ Q J Q u \ J O JRRaJ Q\mSUP R PQ^M J Q K FI\GH Rhg MSYC R FIDGDGC J[J D K H J Q^QNPFVDK CVH R C EM&S F b J b J H^MOP K H^F OPaSUM i) C u \ CVDeSNPaSUM J QN\GH^M J R g C b EPoSNSYCZDGO J M RJ O&SUH^P u \ J b J SUHNCVDGQSt J H^SYE = 1/ZE
b \ Q QpS J J Q^SfOF' K CVHNM J b J Ql_ZC RJ \GHNQ b J YE
b MS J H60PDmM J Q b g C K H Q R CWH JR CZSUPaFID //
YE =1
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MSUCVDeSb M&S J H60PDGM J CVD C R SNP u \ J J DqS b g C K H Q0\mD 0F b RaJ K QNP u \ J b \ Q Q^S J OFIDGQ^SNPaSN\GM K CVH RaJ Q0POHNF *K FID J Q RJ OFV\ KGRJ \GH J S R g C b EPoSNSYCZDGO J y1
%TJ \ EF b M R PQNCZSUPaFIDGQ b J O J Q QpS J H JK H J D CVDeS J D IHNCVD b J K CZH^SUP J RJ Q EF b RJ QTMSYC >GR PaQ b CVDGQ R C
H^MRt`MH J DGO J 98- QNFIDeS K HNMQ J DeSNM J Q b CZDGQ O J K CVHNCIHNC KGJ 5C K H J 0P H J EF b M R PQNCZSUPaFID QN\ KmK FIQ J u \ J R CK H^F K CICZSUPaFID b CVDGQ R C[O C _ PaSUM J QpS(O JRRaJ b g FID b J Q KGR CVD J Q b FIDeS R g C0FIHpSUPQ^Q J J DqS b CV\ J8 J SUQ_ PaQ u \ J \ J SES GJ HEP u \ J Q J Q^S K HNPaQ J D OF' K S J b CVDGQ RJ DGF' > H J b g FID b J b CVDGQ R g P K M b CVDGO J PoSUMHUCZSNPa_ J J S K CZHR g PaDqSNHNF b \GO&SUPFVD b g \mD J P K M b CVDGO J b J[K CVHNFVP CV\ J SUHNMEPoSUMQ-CEQ J OFID b J 0F b M R PaQUCZSNPFIDPaDqSNHNF b \GPoS RJ Q EF b J QHNC b P CZ\ nJ SWM&_ J DeSN\ JRaRJ J DeS M R POF < b CV\ b CZDGQR C]O C _ PaSNM b J OFI\ KmR C J RaRJ H JK FIQ J QN\GH R C t`FVH6 \ R CZSUPaFID PDeSUMIHUC RJ b J Q K HNF >GR J QLCV\ R P0PaS J Q RJ QJ8 J SUQ b J b PQ^QNP K CZSUPaFID9_ PQ u \ J \ J S%S GJ HEP u \ J Q2MSUCVDeS K HNPaQ J D OF' K S J b CZDGQ RJ DGF' > H J b g FID b J J D]O Ju \GP"OFIDGO J HND J O J \ QN\GH RaJ Q K CZHNFIPaQ R CXSUMHNC RJ Q r = a
J S K CZH RJ O FIP& b g \GD J t`FIDGO&SUPFVD b J @ H JJ D u \GPQNCZSUPaQ"tnCVPoSlQN\GH RaJ Q K CVH^FIPQ J D z = 0 J S ` \GD J OFID b PaSUPaFID 0P& S J b CVDGQ R C u \ JRRJR C t`FIDGO&SUPaFID J SQUC b MHNPa_VM JQ^FIDeS R PaM J Q K CZH R g P K M b CZDGO J _ PQ^OFVS GJ HEP u \ J b J>K CVHNFVP t`FIDGO&SUPFVD b J @[H JJ D O C R O\ R M J CV\ K CVHUCVHUC KmJ6 1+-, . , 249;BTU<LZAV3 AVC E TUL JE89ZJLNL 5VHU9ZM ;KP 9ZM:;<BA J TULZM<BA! !# !7-')(*!#"' 5! !05(8+'-$!3
J Q HNC IFIDGQ b J Q"EPaOHNF K FID J QQ^FIDeS(Q^\ KGK FIQNMQ%POP MeCV\ O JR \GP b JTR C[O C _ PaSNMC K H^F K CeCXSUPFVD b J QFID b J Q Q^\ KGK FIQNM J Q KGR CVD J Q b CZDGQ2O CVO\mD b J Q O R PD b H J Q J Q^S HNMIP J K CVH Rhg M u \ CZSNPFID 6
(
∂ 2
∂z2 + k2z
)
p(z) = 0 , 0 ≤ z ≤ ` # / F RaJ DGF' > H J b g FID b J kz
u \GP K H J D b J D OF' K S JWRhg PDG\ J DGO J b J Q J<J SUQ%_ PQ u \ J \ J S(S GJ HEP u \ J Q b CZDGQRaJ Q2OFI\mO GJ Q R PEPoS J Q R CZSNMHNC RJ Q J QpS b FIDGDGM K CVH R g M u \GCZSUPaFID I
k2z = k2
0
1 + (γ − 1)Kh
1 −Kv
≈ k20
[
1 +1 − j√
2
2
a√
k0
(√
` ′v + (γ − 1)
√
`h )
]
. # *)'C QNF R \mSUPaFID K FV\GH R C K H J Q^QNPFVD CVOFI\GQpSUP u \ J[J Q^S MOHNPaS J
p(z) = A(e−jkzz +Bejkzz) , ∀z ∈ (0, `1) , # C
p(z) = C(e−jkzz +Dejkzz) , ∀z ∈ (`1, `) , # >
RaJ Q OFVDGQ^SUCVDeS J Q A B C D MSUCVDeS b MS J H60PDmM J Q K CVH RJ QWOFID b PaSNPFIDGQ MOH^PH J CZ\ PaDqS J HStnCVO J Q z = 0 z = `1
J S z = ` J Q[OFVD b PaSNPFIDmQ[CV\ t`HNFVDqSNP H J Q J D z = 0 J S z = `
Q^FIDeS RaJ Q J Q u \ J b CVDGQ RJK CZHUCIHNC KGJ 6 )P C OFID b PaSNPFID CV\ st`HNFIDeSNP H J Q J D z = ` J Q^S b FIDmDGM J[K CVH R g M u \GCZSUPaFID
1
Zv
∂
∂zp = −(Yr + Yp) p ,
K FI\GH z = ` , # 6 C _ J O M u )'
Zv =1
S
j k0ρ0c01 −Kv
# #'
[ !"#$&% &# %#%#$ ' (#%#$ *)
J S M!+ 8 Yp =
S
ρ0c0
1 + j√2
√
k0(γ − 1)√
`h , #-,
O J +q\GP b FVDGD J
D = e−2j kz`Yi − (Yr + Yp)
Yi + (Yr + Yp), #
F Rhg C b 0PaS^SYCVDmO J PoSUMHNCZSUPo_ J Yi_ZCV\mS M.+ J S 6
Yi =j kz
Zv≈ S
ρ0c0
[
1 +1 − j√
2
2
a√
k0
(−√
` ′v + (γ − 1)
√
`h )
]
. #
PP % CVDmQ RaJ KGR CZD z = `1 R C OFIDeSUPaDq\GPaSNM b J Q K H J QNQ^PFIDGQ b FVDGD J
C = Ae−j kz`1 +Bej kz`1
e−jkz`1 +Dejkz`1. # 8
F' K S J S J Dq\ b \ b M > PoS σ1
ρ0c0y1p(`1)
J DeSUHNCVDeS b CVDGQ Rhg M R M J DeS CZOFI\GQpSUP/+ \ J5 O CVHNCVOSNMHNPaQ J H lR COFVDqSNPDq\GPoSUM b J Q b M > PoSUQ b FIDGD J
− 1
Zv
A∂
∂z(e−jkzz +Bej kzz) = − 1
Zv
C∂
∂z(e−j kzz +Dejkzz) + Y1A(e−jkzz +Bej kzz) ,
K FI\GH z = `1 ,
FY1 =
σ1
ρ0c0y1J QpS Rhg C b 0PaS^SYCVDmO J CVOFV\GQ^SNP$+q\ J b JLRhg M R M J DeS OCVHUCVO&SUMHNPQ J H D10nCVPaQUCVDeSW\mQUC J
b JER C]H JR CZSUPaFID # 8 J DeSUH JERJ QLOFVDGQ^SUCVDeS J Q C J S A O J SNS J H JR CZSNPFID b FIDmD J R C]OFVDGQ^SUCVDeS J B J D0`FIDGO&SUPaFID b J D OFIDGDq\ M!+ $#
B = −e−2j kz`1Y1 +De2j kz`1(Y1 − 2Yi)
(Y1 + 2Yi) +De2jkz`1Y1
. #3/
PPaP C OFID b PaSUPaFID J D20`HNFVDqSNP H J Q J D z = 0 b FIDGDGM J[K CVH M.+$ ' #
− 1
Zv
∂ p
∂z= Sve − Yp p ,
K FI\mH z = 0 , # /'/ KdJ H6 J S b g J K HNP J H RaJ b M > PaS Sve
#
Sve = A[Yi(1 − B) + Yp(1 +B)] . # / ) J S^S J J K H J Q^QNPaFID b \ b M > PoS Sve
KJ H6 J S b g J K HNP J H Rhg C b 0PaSNSUCVDGO J CVOFI\GQpSUP$+q\ J b J SUHUCZDGQ30 J H^S YTb \0Q Q^S J 0POHNF K FID J Q * OFI\ KGRaJ \GH eJ S b FIDmOTCV\mQNQNP Q^FID C b EPoSNSUCVDGO J M RJ OSUH^P$+q\ J b J SUHNCVDGQ30 J H^S YEb g C K H Q R C H JR CXSUPFVD /'3/ 1
!#"$%!#&')(*'+',"-(.&"')(/(0(1.'23456!78(59:;5 5
$ /((8!7"$ =(8*+ (8!#"'-1 8(C H JR CZSUPaFID /'/ b FIDmD J Rhg J K H J QNQ^PFID b \ K HNF b \GPoS b J Q J\ O CVOPaSNMQ b J Q b J \ 0POHNF K FID J Q J D
0`FIDmOSUPaFID b JR g C b EPoSNSUCVDGO J CVOFI\mQ^SUP/+q\ J b J SNHUCVDGQ 0 J HpS YT O C R O\ R M J K CVH^SNPH b \0EF b RaJlK QNP$+q\ J O FIPaQNP
J S b J R g C b EPoSNSUCVDGO J M RJ OSNHNP$+q\ J b J SNHUCVDmQ30 J HpS YE J Q^\GHNM J b \ Q Q^S J 0POHNF K FID J Q * OFI\ KGRaJ \GH #
MpeMpr =YT
YE
.
J Q 0POHNF K FID J QE\mSNP R PQNMQ K FI\GH0O J SNS J MSN\ b J J K MH^P J DeSYC RJ QNFIDeS9M&SYC R FIDGDGMQ ' RJ \mHNQ J&\ O CVOPaSUMQMpe
J S MprQNFVDqS b FIDGO OFVDGDq\ J Q g C b 0PaS^SYCVDmO J M RJ O&SUH^P$+q\ J b J SNHUCVDmQ30 J HpS YE
J Q^S b FIDmO b FIDGDGM J[K CVH R CH JR CXSUPFVD i # / #
YE =1
MpeMpr
YT =1
MpeMpr
Sve + Yep(0)
p(`). #3/
D20nCVPQNCVDeS2\GQUC J b CVDGQ2O J SNS J J K H J Q^QNPFVD b J QH JR CZSUPaFIDGQ #/ )' J S # 8 P R _ P J DeS
YE =1
MpeMpr
Yi(1 − B) + Yp(1 +B) + Ye(1 +B)
e−j kz`1 +Bej kz`1
e−jkz`1 +Dej kz`1
e−j kz` +Dej kz`, #3/ 6
+q\GP b FIDmD J OF' K S J S J Dq\ b J Q J K H J QNQNPaFIDGQ #3/ J S #
YE =1
MpeMpr
j sin kz`
[
Yi +(Ye + Yp)(Yr + Yp)
Yi+Y1
2
Ye + Yr + 2Yp
Yi
]
+ cos kz`
[
Ye + Yr + 2Yp +Y1
2
(
1 +(Ye + Yp)(Yr + Yp)
Y 2i
)]
+Y1
2
[
j sin kz(`− 2`1)Yr − Ye
Yi
+ cos kz(`− 2`1)
(
1 − (Ye + Yp)(Yr + Yp)
Y 2i
)]
. #3/ #'
% CZDGQ RaJ O CVQ K CZH^SUPaO\ R P J H F `1 = `/2 O J S^S J J K H J QNQ^PFID b J _ P J DeS
YE =1
MpeMpr
j sin kz`
[
Yi +(Ye + Yp)(Yr + Yp)
Yi
]
+ cos kz` (Ye + Yr + 2Yp)
+Y1
2
[
1 − (Ye + Yp)(Yr + Yp)
Y 2i
+ j sin kz`Ye + Yr + 2Yp
Yi+ cos kz`
(
1 +(Ye + Yp)(Yr + Yp)
Y 2i
)]
.
#3/, DmFVS J H+ \ g J D R g C > Q J DGO J b g M R M- J DeS OC K P RaR CVPaH J Y1 = 0
Rhg J K H J QNQ^PFID #3/ #' KdJ H J S b J H J SUH^FI\m_ J HR g J K H J Q^QNPaFID b J[Rhg C b 0PaS^SYCVDmO J CVOFI\GQpSUP$+q\ J b J SNHUCVDGQ 0 J HpS YT
+-, . , . 249;BTU< LNAD3=ADC:ETLo9XE >YC:TL 5VHU9ZM HUMS5V3 ZE8LZT< ;:C JE89 T >@<=LZC TUH
>@H 58<BA J EF b RaJ9K H^F K FVQNM b CVDGQ RaJK CVHNCIHUC KGJK H^MOM b J DeSED J SUP J DeS OF' K S J + \ J b J R C K HNF K CeCZSNPFID
b g FID b J Q KGR CVD J Q % CZDGQW\GD S JR Q Q^S J b J Q EF b J QQN\ K MH^P J \GHNQ EF b J QHUC b P CV\ J S M_ J DeSU\ JRRaJ J DeS M R PaOF < b CV\ KJ \m_ J DeSfC _IFIPaHc\GD J PD \ J DmO J DmFIDLDGM R P J C >GRJ Q^\GH R C2_ZC RaJ \GH b J YE
J Q EF b J Q-Q^FIDeS R PMQ
[ !"#$&% &# %#%#$ ' (#%#$
CV\ K HNF2 R Q b J _ PaS J QNQ J DGFID \GDmP/0`FIH J Q b J Q J > HNCVD J Q b J 0POH^F K FVD J Q CV\ LHUC VFIDGQ b J Q J > HNCVD J Q+q\GP QNFIDeS b P MH J DeSNQ b J O JR \GP b J[R C OC)_ PoSUM J S R C K H^MQ J DGO J b J Q M R M J DqSNQ OCVHUCZOSUMHNPaQ J H J D K CZHNFIP J EF b RJ K HNMQ J DeSNM b CVDGQO JWK CZHUCIHNC KGJ K H^F K FVQ J b JK H J D b H JWJ D9OF' K S JRJ Q"EF b J QQ^\ K MHNP J \GH^Qb CVDGQ R C OC _ PoSUM J S H JK FIQ J QN\mH R C 0`FIH6 \ R CZSNPFID PaDeSUM-VHUC RaJ b J Q K H^F >GR0 J Q CV\ R P0PaS J Q- R H JK H J D b J DIHUCZD b JLK CVH^SNP JLRaJ 0F b RJ K H^MQ J DeSUM b CVDGQ RJ HNC KGK FIH^S b \ OFVDqSNHUCZS ( D%3/ / 98 i # * / CVPDmQNP+q\ J b J DGF' > H J \ M R M J DeSUQ b \EF b RaJ b M&_ JR F KGK M b CVDGQ RJ O C K PaSNH J 6 ! !# !#&')(*!#"$ $"! /(81 ? '( ' 36 "
J Q K FVS 9 Q J Q QN\GH RaJ O C K CVOFI\GQ^SNP$+q\ J M&SYCVDeS RJ Q J Q PaOP +q\ J b CVDGQ RJ K CVHUCIHUC KGJ 6 3/ RJ QWM!+q\ CZSNPFIDmQ 0`FID b C J DeSUC RJ QWQ g J K H^P J DeS b J R C* J 0nCFID J S K CVH Q^\GPaS J RJ O C K b J K H J Q^QNPFVDCVOFV\GQ^SNP$+q\ J QUCZSNPQ 0nCVPaS 0R g M.+ \GCZSUPaFID b J[K HNF K CeCZSNPFID 6 3/ #
(∆ + k2)p(~r) = 0 , ∀~r ∈ D , #3/ C _ J O M.+ 6 / ,
k2 = k20
1 + (γ − 1)Kh
1 −Kv
≈ k20
[
1 +1 − j√
2
2
a√
k0
(√
` ′v + (γ − 1)
√
`h )
]
. #3/ O J S^S J M!+q\ CZSNPFID b J K HNF K CeCZSNPFID QNFVDqSTCVQ^QNF OPM J Q RJ QOFID b PoSUPFVDGQ2CV\ 20`H^FIDeSUP H J QQ^\GPa_ZCVDeS J Q CVDGD J J M.+ / #' #
ρ0c0vz = ρ0c0ve − ζz0p ,K FI\GH r ∈ (0, ae) , θ ∈ (0, 2π) , z = 0 , #/ 8 C
ρ0c0vz = −ρ0c0vR + ζz`p ,K FV\GH r ∈ (0, ar) , θ ∈ (0, 2π) , z = ` , #/ 8=>
ρ0c0vz = −ζz0p ,K FV\GH r ∈ (ae, a) , θ ∈ (0, 2π) , z = 0 , #/ 8 O
ρ0c0vz = ζz`p ,K FI\GH r ∈ (ar, a) , θ ∈ (0, 2π) , z = ` , #/ 8 b
Fζz0
J S ζz`
QNFIDeS RJ QfC b 0PaSNSUCVDGO J QlCVOFI\GQpSUP$+q\ J QfQ K MOP32+q\ J Q b J Q K CVHNFIPaQ J D z = 0 J S z = `SUHUC b \GPQNCVDeS
RJ Q J<J SUQ b J Q OFI\GO GJ Q R P0PaS J QS GJ HEF)_ PQ3+q\ J \GQ J QQN\GH RJ O C K CVOFI\GQ^SNP$+q\ J b CVDmQ R C O C _ PaSNM CVDGD J J M.+ / , GJ S
vr =y1
ρ0c0p K FI\mH r = a , θ ∈ (0, 2π) , z = `1 ,
#/ 8VJ vr = 0 K FI\mH r = a , θ ∈ (0, 2π) , z ∈ (0, `) , #3/ 8 0
RJ Q J8 J SUQ b J Q OFV\GO GJ Q R PEPoS J Q0S GJ H60F)_ PaQ+q\ J \GQ J Q R CZSNMHNC RJ QMSYCZDqS b M K H^PQ J D OF K S J b CVDGQ RJDGF' > H J b g FID b J OF' KGRJ J k C Q^\GH30nCVO J σ1
Q^\GH R C+q\ JRRaJTJ QpS R F O C R PQ^M JTR g C b EPoSNSUCVDGO J Q K MOP2+q\ J y1
O CVHNCVOSNMH^PQ J HfMSYCZDqSfSNH Q KdJ SNPaS J b J _ZCVDeS R C QN\GH 0nCVO J R CZSNMHNC RJ b J R C O C _ PaSUM eJRaRJ D g J QpS K CVQZSUM J b J O JRRaJ8* OPb CVDGQ R C OFID b PaSUPaFID # / 8 0 1 J Q2C b 0PaS^SYCVDGO J Q ζz0
J S ζz`
H J Q^SUCVDeS KJ SUPoS J Q OF K S J S J D \ b J Q2M!+q\ CZSNPFIDGQ 6 '6 3/ 6 O J Q2OFID b P *SUPaFIDGQ%CV\ 0`HNFIDeSNP H J Q # / 8 C * 0Q g MOHNPo_ J DeS
∂ p
∂n= −∂ p
∂z=
j k0ρ0c01 −Kv
vz ≈j k0ρ0c01 −Kv
ve − j k0ζz0 p # ) IC
K FV\GH r ∈ (0, ae) θ ∈ (0, 2π) z = 0
∂ p
∂n=∂ p
∂z= − j k0ρ0c0
1 −Kvvz ≈
j k0ρ0c01 −Kv
vR − j k0ζz` p # ) >
!#"$%!#&')(*'+',"-(.&"')(/(0(1.'23456!78(59:;5 5
K FI\GH r ∈ (0, ar) θ ∈ (0, 2π) z = `
∂ p
∂n= −∂ p
∂z= −j k0ζz0 p
# ) VO K FI\GH r ∈ (ae, a)
θ ∈ (0, 2π) z = 0 ∂ p
∂n=∂ p
∂z= −j k0ζz` p ,
# ) b K FI\GH r ∈ (ar, a)
θ ∈ (0, 2π) z = ` ∂ p
∂n=∂ p
∂r= −j k0y1p ,
# ) J K FI\GH r = a θ ∈ (0, 2π) z = `1
∂ p
∂n=∂ p
∂r= 0 , # ) 0
K FI\GH r = a θ ∈ (0, 2π) z ∈ (0, `) F'( J b M PaD b P/+ \mM CV\ O C K PaSNH J 6 M!+ 6 *) ' K FI\mH \GD J FID b J CVH60FIDGP/+ \ JRJ O C K b J
K H J Q^QNPFVD CVOFV\GQ^SNP$+q\ J p(~r) b CVDGQ>\GD b F CVPaD J D R PEPoSUM K CVH>\mD J Q^\GH30nCVO J 0 J HEM J S KdJ \mS SNH J MOH^PaSQ^FI\GQ R C 0`FIH J PaDeSUM-VHUC RaJ QN\GPo_ZCVDeS J: i , )*) nRaJ 0nCVOS J \GH ej ω t J QpSFEPaQ J S RaJ _ J O&S J \GH ~r J QpSH JK H^MQ J DeSUMPaOP K CZH RaJ Q(SNHNFIPaQ%OF FIH b FVDGDGM J Q r θ J S z #p(~r) =
∫∫∫
DG(~r, ~r0)f(~r0) dD0 +
∫∫
S
[
G(~r, ~r0)∂
∂n0
p(~r0) − p(~r0)∂
∂n0
G(~r, ~r0)
]
dS0 . # )/
C0`FIDGO&SUPaFIDf(~r0)
H JK HNMQ J DqS J Rhg CVOSNPFID b J Q QNFV\GHNO J Q(H^M K CVHpSUP J Q >Rhg PDeSNMHNP J \GH b \ b F'ECVPD J D # b CVDGQRaJ O CVQ K H^MQ J DeS f(~r0) = 0
D0nCVPQNCVDeS(\GQNC J b J Q%OFID b PoSUPFVDGQ CZ\ 0`HNFIDeSUP H J Q #*) IC0 K FI\mH RJ O C K b JWK H J Q^QNPaFID RJ O FIP&
b g \GD J 0`FIDmOSUPaFID b J @ H JJ D QUCZSNPQ30nCZPQUCZDqSWQN\GH Rhg J DmQ J >GRJ b J Q K CVHNFVPQ b J R C9O C _ PaSUM CZ\ OFID b PoSUPFVDGQ O JQ^FIDeS RaJ Q J Q +q\ J O JRaRJ Q%H J S J Dq\ J QTCV\ O C K PaSUH J 6 M!+ 6 ) )'
∂
∂nG(~r, ~r0) = −j k0ζz0G(~r, ~r0) ,
K FI\GH r ∈ (0, a) , θ ∈ (0, 2π) , z = 0 , # ) )VC'∂
∂nG(~r, ~r0) = −j k0ζz`G(~r, ~r0) ,
K FI\mH r ∈ (0, a) , θ ∈ (0, 2π) , z = ` , # ) ) > ∂
∂nG(~r, ~r0) = 0 , K FV\GH r = a , θ ∈ (0, 2π) , z ∈ (0, `) , # *) )XO
KdJ H J S b g J K HNP J H R g PaDqSNM-IHNC RJ b J QN\GH 0nCVO J C KmK CVHUCZPQNQNCVDeS b CVDGQ R C H JR CZSNPFID# *)/ +q\GPH JK HNMQ J DqS JLR CK H J Q^QNPFVD CVOFI\GQpSUP$+q\ J[J D SNFI\mS K FVPDeS b J[R C OC)_ PoSUM b J R C CZDGP H J Q^\GPa_ZCVDeS J#p(~r) =
j k0ρ0c01 −Kv
[∫∫
Se
ve(~r0)G(~r, ~r0) dS0 +
∫∫
Sr
vR(~r0)G(~r, ~r0) dS0
]
− j k0
∫∫
σ1
y1(~r0)p(~r0)G(~r, ~r0) dS0 , # )
SeJ S Sr
MSUCVDeS RaJ Q2QN\mH30nCVO J Q H J Q KJ OSNPa_ J Q b J R C J > HUCVD J b J Rhg M- J S^S J \GH J S b \ HNMO JK S J \GH RaJ Q J<J SNQb J Q OFI\GO GJ Q S GJ H60F)_ PaQ+q\ J \GQ J Q"MSUCVDeS K FIH^SNMQ K CVH RJ 0nCVOS J \GH Kv
+ \mP PaDeS J Hp_ P J DeS b CVDGQ Rhg P K M b CVDGO J b \0P R P J \ J S K CVH R C0`FIDGOSNPFID b J @[H JJ D JRRaJ8* J b CVDGQ RJ DGF' > H J b g FVD b J k K FI\GH RJ Q J<J SNQ R CXSUMHNCV\ J S b CVDmQ RJ Q(C b 0PaS^SYCVDmO J QQ K MOP2+q\ J Q ζz0J S ζz`
CV\9DmPa_ J CV\ b J Q5 J > HUCVD J Q b J Q.0POH^F K FVD J Q J S RaJ \GHK MH^P K MHNP J DGFZS J H R C[Q EM&SUHNP J M- J S^S J \GH * H^MO JK S J \GH b CVDGQ R g J K H J Q^QNPaFID b \0O C K b JK H J Q^QNPaFIDCVOFI\GQpSUP$+q\ J O J O C K b M KdJ D b]b J O J Q%_ PaS J Q^Q J Q +q\ JR +q\ g J D Q^FIPaS RaJ \GH%OCV\GQ J g J K H J Q^QNPaFID b J R C"0`FIDGOSNPFID b J @ H JJ D J QpS b FVDGDGM J CV\ K CZHUCIHNC KGJ:6 M.+$ 6 *)'# 96 /)C J S > 1
[ !"#$&% &# %#%#$ ' (#%#$ '!# !#&$)(84*!#"' 5! ! 5(
J O C K b J K H J QNQNPaFID]CVOFI\mQ^SUP/+q\ J b CVDGQ R C O C _ PaSNM J Q^SF > S J Dq\ J D H JK FIH^SUCVDeS R g J K H J QNQNPaFID 6 )'#'b J R C 0`FIDGOSNPFID b J @[H JJ D b CVDGQ R g M.+q\ CZSUPaFID # ) 1$ F' K S J S J Dq\ b J>R C 0`FIH J b J O J SNS J 0`FIDGO&SUPFVD b J@[H JJ D R C K H J QNQ^PFID p(r, θ, z) C KGK CVHNCSTC R FVHNQTQ^FI\GQ R C 0`FIH J b g \GD b M_ JR F KGKJ J DeS QN\GH RJ Q 0`FIDGOSNPFIDmQK HNF K H J Q ψt
µν(r, θ, z)#
p(r, θ, z) =∑
µ,ν=0,1...
t=c,s
[
j k0ρ0c01 −Kv
(
gµν(z, 0) < ve|ψtµν >e +gµν(z, `) < vR|ψt
µν >r
)
−j k0gµν(z, `1)∑
µ′,ν′,t′
pt′
µ′ν′(`1) < y1ψt′
µ′ν′|ψtµν >σ1
]
ψtµν(r, θ) ,
# ) 6
F R CEDGFVSUCZSUPaFID < ve|ψtµν >e
b MQNPID J>Rhg PDeSUMIHUC RaJ QN\GH Se
b \ K H^F b \GPaS ve(r, θ)ψtµν(r, θ)
< vR|ψtµν >rb MQ^PID J Rhg PDeSUMIHUC RaJ Q^\GH Sr
b \ K HNF b \GPoS vR(r, θ)ψtµν(r, θ)
J S < y1ψt′
µ′ν′ |ψtµν >σ1
Rhg PDeSUMIHUC RaJ Q^\GH σ1
b \K HNF b \GPoS y1(θ)ψ
t′
µ′ν′(r, θ)ψtµν(r, θ)
-C Q^F R \mSNPFID K H J D b]b FVDGO R C"0`FIH6 Jp(r, θ, z) =
∑
µ,ν,t
ptµν(z)ψ
tµν(r, θ) ,
# )'#'C _ J O
ptµν(z) =
j k0ρ0c01 −Kv
[
gµν(z, 0) < ve|ψtµν >e +gµν(z, `) < vR|ψt
µν >r
]
− j k0gµν(z, `1)∑
µ′,ν′,t′
pt′
µ′ν′(`1) < y1ψt′
µ′ν′|ψtµν >σ1
. # ),
% CVDGQ R C QN\GPoS J b J R g J K FIQ^M fRJ O C K b J _ PaS J Q^Q J ve O C K _ P > HUCXSUFIPaH J QNFV\GHNO J J Q^S QN\ KGK FIQ^M
PD b M KdJ D b CZDqS b JR g CZD RJ θ %J S K CVHQN\mPaS J OF K S J S J D \ b J R C Q EM&SUHNP J b J Q H^ RJ Q9S J Dq\GQ K CZH RJ Qb J \ &0POH^F K FVD J Q R C>_ PaS J QNQ J vR
MeC RaJ J DeS PDmQNP RJ O C R O\ R b J Q%PaDeSUM-VHUC RaJ Q(Q^\GPa_ZCVDeS J Q b FIDmD J( ve,Rb MQ^PIDGCVDeS2PD b P MH J J DeS veFI\
vR
< ve,R|ψtµν >e,r=
∫ ae,r
0
r0 dr0ve,R(r0)
∫ 2π
0
dθ0ψtµν(r0, θ0) ; # )
b g C K H Q RJ Q J K H J Q^QNPaFIDGQ 6 *) C J S > b J Q ψtµνXRhg PDeSUMIHUC RJ QN\GH θ0 J Q^S"D \ RRJK FI\GH µ 6= 0 ZJ S b J Q^\GHNOHNFaS JRaRJWJ Q^S M-eC RJ J DeS2Dq\ RaRJ K FI\GH t = s µ = 0 CVDmD J J i #3/ J S # *)'1
cFI\GH J K HNP J H R C _ PaS J QNQ J vR+q\GP b M KdJ D b JRaRJ J b \ O C K CVOFV\GQ^SNP$+q\ J p(r, θ, z) K CZH J *S J DGQNPaFID b J R C H JR CZSUPaFID SrvR = −Yrp(`)
H JR PCVDeS Rhg C b 0PaS^SYCVDmO J Yr
b \ 0POH^F K FVD J HNMO JK S J \GH0CV\ K H J QNQ^PFIDGQ J S>_ PoS J QNQ J Q CVOFV\GQ^SNP$+q\ J R FIHNQ3+q\ J RaJ O C K CZOFI\GQpSUP/+ \ J J QpS PD b M KdJ D b CVDeS b J r -R C_ PaS J QNQ JvR(r) J Q^SMOH^PaS J( M!+ 6 #
SrvR(r) = −Yr(r) < p(r, `) >θ , # )
FYr(r)
J QpS Rhg C b 0PaS^SYCVDmO J CVOFI\GQpSUP$+q\ J OPH^O\GPoS2FI\m_ J H^S b \ H^MO JK S J \GHDGFID * \GDGP/0`FVH6 J QN\GH R C Q^\GH30nCVO Jb J R C J > HUCVD J M!+ 6 8 1 -C K H J QNQ^PFID0F J DGD J Q^\GH R g CVD RJ θ < p(r, `) >θ
b JR C K H J QNQNPaFID p(r, θ, `)
!#"$%!#&')(*'+',"-(.&"')(/(0(1.'23456!78(59:;5 5
Q^\GH RJ HNMO JK S J \GH J Q^S b FIDGDmM J[K CVH
< p(r, `) >θ=1
2π
∫ 2π
0
p(r, θ0, `) dθ0 =∑
ν
pc0ν(`)ψ
c0ν(r) ,
# *) 8
R g PaDqSNM-IHNC RJ QN\GH θ0 b J Q 0`FIDGO&SUPaFIDGQ K H^F K H J Q ψtµν
M&SYCVDeSDq\ RRaJ Q K FI\GH t = s J S K FV\GH µ 6= 0 J S RaJ Q 0`FIDGO&SUPFVDGQK H^F K H J Q ψc
0ν
MSUCVDeSPaD b M KJ D b CVDeS J Q b J R g CZD RJ θ PaDGQNP XJ D QN\GPo_ZCVDeS RJ J HUCVPaQNFIDGD J J DeS +q\ J O JR \mP K H^MQ J DeSUM b CVDmQ RJK CVHUCIHUC KGJ 6 6XRJ K HNF b \GPaSQ^O C R CVPH J < vR|ψc
0ν >J QpS b FIDGDGM K CZH
−Sr < vR|ψc0ν >r=
∑
ν′
pc0ν′(`)Yr < ψc
0ν′ |ψc0ν >r .
#
J H JK FIHpS b J O J S^S J[J K H J QNQ^PFID b CVDGQ R g J K H J Q^QNPaFID # ) 6 b J[R C K H J Q^QNPaFID]OFID b \mPaS #
p(r, θ, z) =∑
µ,ν,t
[
j k0ρ0c01 −Kv
(g0ν(z, 0) < ve|ψc0ν >e
−Yr
Sr
g0ν(z, `)∑
ν′
p0ν′(`) < ψc0ν′ |ψc
0ν >r
)
δtcδµ0
−j k0gµν(z, `1)∑
µ′,ν′,t′
pt′
µ′ν′(`1) < y1ψtµν |ψt′
µ′ν′ >σ1
]
ψtµν(r, θ) ,
# /
F RJ QlS J H J Q < ψc0ν |ψc
0ν′ >r
b MQNPID J DqS RJ QfPDeSUMIHUC RaJ QlQ^\GH R C QN\GH 0nCVO J Sr
b J Q K H^F b \GPaSNQ ψc0ν(r)ψ
c0ν′(r)
OC R O\ R MQ b CVDGQ Rhg CVDGD J J i ) M!+ 3/'/)C > 1 J Q(S J H6 J Q < ψc
0ν |ψc0ν′ >r
J K HNP CVDeS b J Q2OFV\ KGR C J Q J DeSUH J b P MH J DeSUQ.0F b J Q ν 6= ν ′QNFVDqS SUH Q
KdJ SNPaSUQ b J _ZCVDeS RaJ Q%CV\ SUH J Q J S KdJ \ _ J DeS b FVDGO SNH J DGM R PIMQ nRaJ Q FVH b H J Q b J VHUCVD b J \GH(Q^FIDeS b FIDGDmMQ b CZDGQRaJ K CVHUCIHUC KGJ *)'1C K H J QNQ^PFID CVOFI\GQpSUP$+q\ J p(r, θ, z) J Q^S b FIDGO H J S J Dq\ J QNFV\GQ R C 0`FIH6 J&`J D10nCVO&SUFIH^PQUCZDqS Rhg PDe_ J HNQ J b JR C Q^\GH30nCZO J S = πa2 #
p(r, θ, z) =∑
µ,ν,t
[
j k0
1 −Kv
ρ0c0S
(g0ν(z, 0)S < ve|ψc0ν >e −Yr0νg0ν(z, `)p
c0ν(`)) δtcδµ0
−j k0gµν(z, `1)∑
µ′,ν′,t′
pt′
µ′ν′(`1) < y1ψtµν |ψt′
µ′ν′ >σ1
]
ψtµν(r, θ) ,
# )
FYr0ν = Yr
a2
a2r
< ψc0ν |ψc
0ν >r
7 ' /((8!#"' 5=(84*%' (8!#"$1 (%TJ J +q\ J b CZDGQ RJK CVHNCIHUC KGJ # ) > VR g J K H J Q^QNPaFID b J(R g C b EPoSNSYCZDGO J M RaJ OSNHNP/+ \ J b J SNHUCVDGQ 0 J HpS
b \ Q QpS J J Q^S MOHNPoS JYE =
1
MpeMprYT .
[ !"#$&% &# %#%#$ ' (#%#$
J O C K b J K H J QNQ^PFID J S RJ O C K b J _ PaS J QNQ J D g M&SYCVDeS K CVQ \GDGP 0`FIH6 J Q b CZDGQ R C O C _ PaSNM lRhg C b 0PaS *SYCVDmO J CZOFI\GQpSUP/+ \ J b J SNHUCVDGQ 0 J HpS HNC KGK FIH^S b \ b M > PaS b JWR g M J SNS J \GH J DOFI\mH^S * OPaHNO\mPaS CVOFI\mQ^SUP/+q\ J QN\mH R CK H J QNQ^PFID J D z = ` J QpS\mD J 0`FVDGOSNPFID b J r %TJ J +q\ J b CVDGQ RaJ]K CZHUCIHNC KGJ26 # R g C b EPoSNSUCVDGO J CVOFI\GQpSUP$+q\ J b J SNHUCVDGQ 0 J HpS DGFVH6 J ,/ , 6* ) ) ) J QpS b M2GDGP J9J D H J KGR C CZDqS RJ Q+q\ CVDeSNPaSUMQ"+q\GP b M KJ D b J DqS b J9R C _VCZHNP C >GRJ r K CVH RaJ \GH_ZC RJ \GH EF J DGD J Q^\GH R C]Q^\GH30nCVO J b \ SUHUCZDGQ b \mOS J \GH P KGR P$+q\GM `K CVH J S J DmQNPFVD b J R C]H JR CXSUPFVD ) / QUC _IFVPH #
YT =Se < ve >e + < Ye(r)p(r, 0) >e
< p(r, `) >r
, # F
Ye(r) =< Ye >e +δYe(r)J S p(r, 0) =< p(0) >e +δp(r, 0) .
CVH%Q^\GPaS J J D C b F K SYCZDqS R C J b M CZHNO GJ +q\ J b CVDGQ RaJ K CVHUCIHUC KGJ:6 # J S J D K FIQNCVDeS
Ve =< ve >e=2
a2e
∫ ae
0
ve(r0)r0 dr0 , # 6
J S
Ye =< Ye >e=2
a2e
∫ ae
0
Ye(r0)r0 dr0 , # #'
P R _ P J DeS
YT =SeVe + Ye < p(0) >e
< p(`) >r
. # , J O C R O\ R b J Rhg C b 0PaS^SYCVDmO J CVOFV\GQ^SNP$+q\ J b J SUHNCVDGQ 0 J H^S DGMO J Q^QNPaS J b FIDmO b J OC R O\ RJ H RaJ Q_ZC RJ \GHNQ EF J DGD J Qb J[R C K H J Q^QNPFVD CVOFI\GQpSUP$+q\ J Q^\GH R C J > HNCVD J b J Rhg M- J S^S J \GH z = 0
J S b \ HNMO JK S J \GH z = `1
2 D b J O C R O\ RaJ H RJ Ql_VC RJ \mHNQ EF J DGD J Q < p(0) >eJ S < p(`) >r
eR C _ZC RJ \GH 0F J DGD J < p(z) >e,rb J[R C K H J Q^QNPFVD p(r, θ, z) QN\GH R C Q J O&SUPaFID b J HNC IFID ae,r ae,r
b MQNPID CVDeSae
FI\ar
J QpSTMOHNPoS J< p(z) >e,r =
∑
µ,ν,t
ptµν(z) < ψt
µν >e,r
=∑
µ,ν,t
ptµν(z)
1
πa2e,r
∫ ae,r
0
r0 dr0
∫ 2π
0
dθ0ψtµν(r0, θ0) ; #
b g C K H Q RJ Q J K H J Q^QNPaFIDGQ 6 ) C J S > b J Q ψtµνeRhg PDeSNM-IHNC RJ Q^\GH θ0 J Q^SDq\ RRaJ%K FV\GH µ 6= 0 J S K FV\GH t = s
µ = 0 C _ZC RJ \GH 0F J DGD J < p(z) >e,r
b J _ P J DeSC R FIHNQ
< p(z) >e,r=∑
ν
pc0ν(z) < ψc
0ν >e,r , #
O J +q\GP b FIDmD J C R FIH^Q K FV\GH < p(0) >eJ S < p(`) >r
< p(0) >e=∑
ν
pc0ν(0) < ψc
0ν >e , # 8
< p(`) >r=∑
ν
pc0ν(`) < ψc
0ν >r . # 6
3 !#"$%!#&')(*'+',"-(.&"')(/(0(1.'23456!78(59:;5 5
J H JK FIHpS b J O J Q J K H J Q^QNPaFIDGQ b CVDGQ Rhg J K H J Q^QNPFVD b J2R g C b EPoSNSYCZDGO J CZOFI\GQpSUP/+ \ J b J SNHUCVDmQ30 J HpS YT
b FIDmD JC R FIHNQ
YT =
SeVe + Ye
∑
ν
pc0ν(0) < ψc
0ν >e
∑
ν
pc0ν(`) < ψc
0ν >r
. # 6 /
g C b EPoSNSYCZDGO J M RJ OSUH^P$+q\ J b J SUHNCVDGQ 0 J H^S J Q^STC R FVHNQ b FVDGDGM J K CVH
YE =1
MpeMpr
SeVe + Ye
∑
ν
pc0ν(0) < ψc
0ν >e
∑
ν
pc0ν(`) < ψc
0ν >r
. # 6 )'
J OC R O\ R b JTR g C b EPoSNSUCVDGO J M RJ OSUH^P$+q\ J b J SUHUCZDGQ30 J H^S YED J DmMO J QNQ^PaS J b J OFIDGD C SUH J + \ JTRJ QS J H J Q
K FI\GH RaJ Q+q\ JR Q t = c J S µ = 0 R H J Q^S J b FIDGO %J KGR POPoS J H RJ Q OF J\ OP J DeSUQ pc
0ν(z)K FI\mH*+ \ JR g C b EPoSNSUCVDGO J
YE
Q^FIPaS J DeSUP H J J DqS b M&S J H60PDGM J J H JK FIHpS b J R g J K H J QNQNPaFID b J[R C K H J QNQ^PFID # ) b CVDGQ Rhg J K H J Q^QNPFVD # b FIDGD J
pc0ν(z) =
j k0
1 −Kv
ρ0c0S
[g0ν(z, 0)S < ve|ψc0ν >e −Yr0νg0ν(z, `)p
c0ν(`)]
− j k0g0ν(z, `1)∑
µ′,ν′,t′
pt′
µ′ν′(`1) < y1ψc0ν |ψt′
µ′ν′ >σ1, # 6
+q\GP b J _ P J DeS J D 0nCZPQUCZDqS2\mQUC J b J[R g J K H J Q^QNPaFID 8
pc0ν(z) =
j k0
1 −Kv
ρ0c0S
[g0ν(z, 0)S < ve|ψc0ν >e −Yr0νg0ν(z, `)p
c0ν(`)]
− j k0ρ0c0g0ν(z, `1)Y1
∑
ν′
pc0ν′(`1)ψ
c0ν(a)ψ
c0ν′(a) ,
# 6 6
FY1 = σ1/(ρ0c0)y1
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QNPO J S9M R M J DeS J Q^S\GD J 0 J DeS J CVDGDq\ R CZPH J J S 4Yt
QNP Rhg M R M J DeS J Q^S\GD HNMQ J CV\ b J +q\ CZSUH J SN\ >J QO R PD b HNP$+q\ J Q CVDGD J J J S HNM.0`MH J DGO J$ 8 i * ) J S * J SNS J M!+q\ CZSNPFIDEOFID b \GPaS J D K FIQUCVDeSQN\GOO J Q *Q^Pa_ J J DeS z = ` J S z = `1
b J \ H JR CXSUPFVDGQ%PD b M KdJ D b CZDqS J Q J DeSUH J pc0ν(`)
J S pc0ν(`1)
$C HNMQ^F R \ SUPFVDb J O J Q QpS J b g M.+ \GCZSUPaFIDGQ b FIDGD J0RJ Q[OF J\ OP J DeSUQ pc
0ν(`)J S pc
0ν(`1) O J +q\GP KdJ H6 J S b g J D b M b \GPH J
RaJ Q(OF J&\ OP J DqSNQ pc0ν(z)
J S b FIDGO RJ QOF J\ OP J DeSUQ pc0ν(0) DmMO J QNQNCVPH J Q C _ J O RaJ Q pc
0ν(`)K FI\mH RJ O C R O\ R b J
R g C b EPoSNSYCZDGO J M RJ OSUH^P$+q\ J b J SNHUCVDGQ 0 J HpS YE
+-, . , 3BADCT 5VL 58A 5V73B9XE8HC<BA <65 JE8<>@H<E8A EF3BADCT 5VLX5VA < J 3EFH>@<BMS5VLZC J QlHNMQN\ R SYCZSNQ"S MFIHNP/+q\ J QfF > S J Dq\GQ K FI\GH RJ Q b J \ 0F b RJ Q b MS J H60PDmMQ b CZDGQfO J O C K PoSUH J 0F b RJ
> CZQNM[QN\GH R C K HNF K CeCZSNPFID b g FID b J Q KGR CVD J Q%M.+ $#/ # J S EF b RJ > CVQNM[QN\mH R C 0`FIH \ R CXSUPFVD PDeSNM-IHNC RJ M.+ # 6 )' Q^FIDeS F > Q J H^_VMQ J S OF' K CVH^MQH J Q KdJ O&SUPo_ J J DeS b CVDGQ RaJ Q K CVHNCIHNC KGJ Q]C J S > J Q HNMQN\ R SYCZSNQS MFIH^P$+q\ J QQ^FIDeSOF' K CVHNMQ b J Q K H J EP J H^Q%HNMQ^\ R SUCZSUQ J K MH^P J DeSYCZ\ b CVDmQ RaJ K CVHNCIHUC KGJ O
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SUP/+ \ J Q b J SUHNCVDGQ30 J H^S b J[R C OC _ PoSUM b J OFV\ KGR C J DmFVSUMQ VT
b M2GDGPQ2OF J QN\mPaS #
VT =γRPR
jωYT ,
FγR
J S PR
Q^FIDeSTH J Q KdJ O&SUPo_ J J DeS RJ HUC KGK FIHpS b J Q O C K CZOPaSNMQ O C R FIHNP32+q\ J Q CpJ S Cv
CZQNQNP/+q\ J Q b JRhg CVPH J S R C K H J Q^QNPaFID]QpSYCZSNP$+q\ J b CVDmQ RaJ Q%OFID b PoSUPFVDGQ%Q^SUCVD b CVH b]b J H^M.0`MH J DmO J
cFI\GH b J Q OF' 0F b PaSNMQ b g J KGR FIPoSYCZSNPFID b J Q H^MQ^\ R SUCZSUQ RJ _IF R \ J b J SNHUCVDGQ 0 J HpS VTJ Q^S DGFVH6EC R PaQNM
K CVH>QUC _ZC RJ \GH CZ\ > CVQNQ J Q"0`HNM.+q\ J DGO J Q J D R g C > Q J DmO J b g M R M- J DeSUQ CVOFI\mQ^SUP/+q\ J Q J D K CVHNFVPQ J S b g J<J SNQS GJ H60P$+q\ J Q b M2 DmP K CVH i # )' #
VTbf0 = V + Vbfe + Vbfr .
J Q OFV\GH >dJ Q b JR C2V\GH J # # H JK H^MQ J DeS J DeS RJ QL_IF R \ J Q DmFIH6EC R PaQNMQ VT/VTbf0
b CZDGQ RJ O CVQ F R C O C _ PaSNMLD J OF' K FVH^S J[K CZQ b J>KJ SUPaSM R M- J DeSTCVOFI\mQ^SUP/+q\ JLJ D K CVH^FIP g C KGK CVH J DeS J CV\' J DeSYCXSUPFVD b J_IF R \ JLJ D > CVQNQ J Q0`HNM.+ \ J DGO J Q[QN\GHTO J Q OFI\GH >J Q J Q^S[\mDGP$+q\ J J DqS b \ J CV\ J8 J SUQWS GJ HEP/+ \ J Q b CZDGQ R CO C _ PaSNM b J OFI\ KmR C J
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b J \ EF b RJ Q K Q^P$+q\ J QTC _ J O \GD OFV\ KGRJ \GH2QUCVDmQ M R M J DqSCVOFI\GQpSUP$+q\ J[J D K CZHNFIP J Q"OFV\GH >dJ Q b J Q 2V\GH J Q #-, # 8 H JK H^MQ J DeS J DeS RJ Qc_VF R \ J Q CVOFV\GQ^SNP$+q\ J Q b J SUHNCVDGQ 0 J H^S DGFIH C R PQ^MQ
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L2
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cH JK HNMQ J DqS J DeSH J Q KJ OSNPa_ J J DeS R C 0 J DeS J%J S RJH^MQ J CV\ b J +q\ CZSNH J SN\ >J Q GJ S RJ Q(O C K CVOPoSUMQ%CVOFV\GQ^SNP$+q\ J Q C2
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H JK HNMQ J DqS J DeS RJ QO C _ PaSNMQ Rhg CVH^HNP H Jb J Q KdJ SNPaSNQ%M R M J DeSUQ CVOFV\GQ^SNP$+q\ J Q
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#a j s ngibh oi
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b J \ 0F b RJ Q K Q^P$+q\ J QTC _ J O>\mD]OFI\ KGRaJ \GH2OF K FIH^SUCVDeS J D K CVH^FIP \GD J 0 J DqS J CVDGDq\ R CZPH J FV\m_ J H^S J
102
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0
5
10
15Ondes planesFormulation intégrale
#a j s ngibh oi
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b J \ &0F b RaJ Q K Q^P$+q\ J QC _ J O \GDOFI\ KGRaJ \GHOF' K FVH^SYCZDqS J D K CVHNFIPd\GD J 0 J DeS J CVDmD \ R CVPaH J O CVH6VM JWK CZH\GD _IF R \ J CVHNH^P H J
\GHfO C+q\ J 2I\GH J b J ## # 8eRaJ QfHNMQN\ R SYCZSNQF > S J Dq\GQ C _ J O RJ Q b J \ (0F b RJ QfQNFIDeSSUH Q K HNF O GJ Q J Q MOCVH^SNQ J DeSUH J O J Q HNMQN\ R SYCZSNQ Q^FIDeS K HNMOPQ^MQ b CVDGQ RaJ K CVHUCIHUC KGJA>
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50Ondes planesFormulation intégrale
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Q JR FID b J \ &0F b RaJ Q K QNP/+q\ J QC _ J O \mD OFI\ KGRaJ \GH%OF' K FIHpSYCVDeS J D K CVH^FIP+q\ CZSNH J SN\ >J Q%O R PaD b H^P$+q\ J QFI\m_ J H^SNQ-
102
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30Ondes planesFormulation intégrale
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Q JR FID b J \ &0F b RaJ Q K QNP/+q\ J QC _ J O \mD OFI\ KGRaJ \GH%OF' K FIHpSYCVDeS J D K CVH^FIP+q\ CZSNH J SN\ >J Q%O R PaD b H^P$+q\ J QO CVHIMQ K CVH2\GD_VF R \ J CVH^HNP H J
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D HUC K P b J OC R O\ R b \]_VF R \ J DGFIH C R PQ^M[CV\ > CZQNQ J Q 0`H^M!+q\ J DGO J Q VTbf/VTbf0
eKJ H6 J S b J 0P J \ OF' K H J D b H J O J SNS J PaD \ J DGO J g J K H J Q^QNPaFID #/ #' J D Q^\ KGK FIQUCZDqS kz`
KdJ SUPoS J S OF' K S J S J Dq\ b J O J +q\ JRJ Q C b 0PaS^SYCVDGO J Q b J Q 0POHNF K FID J Q Ye Yr
J SR g C b EPoSNSYCZDGO J b J K CVH^FIP Yp
Q^FIDeS KdJ SNPaS J Q OFVD b \GPoS MOHNPaH J[Rhg C b 0PaS^SYCVDGO J CVOFI\GQ^SNP$+q\ J b J SUHUCZDGQ30 J H^S b \Q QpS J CV\ > CVQNQ J Q 0`HNM!+q\ J DGO J QTQ^FI\GQ R C0`FIH J Q^\GPa_ZCVDeS J$#
YTbf ≈ j kz` Yi + 2Yp + Ye + Yr + Y1 ,
+q\GP OFIHNH J Q K FID b R CAQ^F'( J b J R g C b EPoSNSUCVDGO J CVOFV\GQ^SNP$+q\ J b J SUHNCVDGQ 0 J H^S YTbf0 =jω VTbf0
γRPR
QUCZDGQOF' K FIQNCVDeS[CVOFV\GQ^SNP$+q\ J QN\mH RJ Q K CVHNFVPQ b J R C OC)_ PoSUM M.+ J S b J Rhg C b 0PaS^SYCVDmO J b g J DeSUHNM J Y1
b J QKdJ SNPaSUQOF K FIQUCZDqSNQ J D K CVH^FIP J _VF R \ J b J SNHUCVDmQ30 J HpS(DGFVH6EC R PaQNM2CV\ > CZQNQ J Q 0`HNM.+q\ J DGO J Q VTbf/VTbf0KdJ \mSC R FIHNQ%Q g MOH^PH J
VTbf
VTbf0
≈ 1 +γRPR
jω VTbf0
Y1 .
J SNS J J K H J Q^QNPFVD b \ _IF R \ J b J SNHUCVDmQ30 J HpS DGFIH6EC R PQNM KJ H6 J S b J 0nCVPH JWKGR \GQNP J \GH^Q H J ECVH+q\ J QQ^\GH RJ QOFI\GH >dJ Q b J Q 2I\GH J Q # , # 8 #
^\mQ+q\ g ]K H Q b J / cR C b MOH^FIPQ^QUCVDmO J0R PDmM CVPaH J`J DAMO GJRaRJ R FeCVHNPoS 0P$+q\ J b \_IF R \ J b JSNHUCVDmQ30 J HpS(P K FIQNM J2K CVH R C K HNMQ J DGO J b JR C0 J DeS J FI\m_ J H^S J EFVDqSNHNM JK CZH R C 2I\mH J # , OFIH^H J Q K FID b> P J D (J D MO mJRRaJ R F'ICVHNPoS 0P$+q\ J CV\ OF' K FVH^S J J DeSHNMQNPQpSUP 0LCXSNS J D b \ b J O JRRaJ8* OP Y1
H^M JR F > Q J H^_ZCZSUPaFID +q\GPQ g C KGKGR P$+q\ J MeC RaJ J DeS b CZDGQ RJ O CVQ F RJ OF' K FIQUCVDeS(CVOFI\mQ^SUP/+q\ JWJ D K CVHNFIP J QpS\mD]H^MQ J CV\ b J +q\ CZSUH J SU\ >dJ Q CVPaQ ^\GQ3+ \ g +q\ JR + \ J Q2O J DqSUCVPD J Q b J [Q J \ RJ J DeS 2I\mH J #
+q\ CZD b RaJ Q KdJ SNPaSUQ M R M J DqSNQ0CVOFI\mQ^SUP/+q\ J Q b M > FI\mO GJ DeSEQN\GH \GD J OC)_ PoSUM J D > CVQNQ J Q 0`HNM.+ \ J DGO J QR g J8 J S b \ _VF R \ J b J O JRRaJ8* OP J Q^S K HNM K FID b MHNCVDeS b J _ZCVDeS R g J8 J S9HNMQNPaQ^SUP 0 b J R C 0 J DqS J FI\ b J QSN\ >dJ Q O J +q\GP J Q^S2H JK HNMQ J DqSNM[QN\GH RaJ Q(OFI\GH >dJ Q b J Q.2I\mH J Q # J S # 8 K CVH RaJ 0nCVPoS +q\ J RJ _IF R \ Jb J SNHUCVDmQ30 J HpS2DGFIH C R PQ^MS J D b b CZDGQ RJ Q > CVQNQ J Q 0`HNM!+q\ J DGO J Q _ J HNQ \GD J _ZC RJ \GH(OFIDGQ^SUCVDeS J +q\GP J QpSMeC RaJ CV\_IF R \ J b J O J S^S J O C _ PaSUM
CVHCVP RRJ \GHNQ P R J Q^ST_ PaQNP >mRJ +q\ J Rhg PD \ J DmO J b J O J QWM R M- J DeSNQTCVOFI\GQpSUP$+q\ J Q J D K CVH^FIPcQ^\GH R C _ZC RaJ \GHb \E_IF R \ J b J SNHUCVDmQ30 J HpS VT/VTbf0
J S b FIDGOQN\GHO JRRJ b JWR g C b EPoSNSUCVDGO J M RJ OSUH^P$+q\ J b J SNHUCVDGQ 0 J HpS YE =YT/(MpeMpr)
mJ Q^S%P K FIHpSYCVDeS J O CZH JRaRJ KJ \mS b M K CVQ^Q J H 6 b K FI\GH RaJ S KdJ b g M R M J DeSUQ%CVOFI\GQpSUP$+q\ J QM&SU\ b PaMQ2POP g PaD \ J DGO J PaD b Po_ P b \ JRRJ b J O C+q\ J M R M J DqS CVOFV\GQ^SNP$+q\ J J D K CVH^FIPM&SYCVDeSLD J S^S J J DeS ECVH3+ \mM JR C EPaQ J J D 1[\m_ H J b J R C(EM&S F b J K H^F K FIQNM J b CVDGQTO J O C K PaSNH J K FI\GHO CVHNCVOSNMHNPaQ J HWPD b Pa_ P b \ JRRaJ J DeSRaJ Q KdJ SUPoSUQ(M R M J DqSNQ2CVOFV\GQ^SNP$+q\ J Q b J R g FVH J P RRaJ CZH^SUP32 OP JRaRJ Q J >GRJ b FVDGOWSUFI\ S 0nCVPaS J De_ PQUC J C >mRJ b CZDGQR C K HUCZSNP$+q\ J R H J QpS J SUFI\ S J 0`FVPQ 9K HNF K FIQ J H \GD K HNFZSUF OF RaJLK FI\GH J S^SUH JLJ D K HUCXSUP$+q\ J O J S^S J 0MS F b JJ S RaJ DGPa_ J CV\ b J[K H^MOPaQNPaFID2+ \ g JRRJ K FI\GH^HUCVPoS KdJ H6 J S^SUH J b g CZS^S J PD b H J J SNS J M&SU\ b J[J K MH^P J DeSYC RJ D g CK CZQ K \ SUH J9K FV\GHNQ^\GPa_ P J Q K FI\GHLC RRaJ H ^\GQ3+q\ g R C J Q^\GH JEK HNF K H J J DeS b PaS JEJ D HUCVPaQNFID b J b P \ O\ R SUMQS J O DGP/+q\ J Q )":'"' 0,' 5 ' M!#' /((8!#"' 5=(84* '%(8!#"$1 (
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SYC( J DeSWOFIDGO J HND CZDqS RaJ K HNF2 R b J _ PaS J Q^Q J b J Q J > HUCVD J Q b J Q 0POHNF K FID J Q Q^\ KGK FIQNM \GDGP 0`FIH6 J +q\GP%OFID b \GPQ J DeS C b J SNSUH J +q\ J b CVDGQ R C O C _ PaSNM b J OFI\ KGR C J O R PD b HNP$+q\ JRaJ O C K b J]K H J Q^QNPFVDJ QpSPaD b M KJ D b CVDeS b J Q OF FIH b FIDGDGM J Q r, θ O J +q\GP b CVDGQ RJ OCVQ K HNMQ J DeSF]\mD J VHUCVD b J K HNMOPQ^PFID J QpSH J O GJ H^O M J KdJ \mSfQ J HNM_VM RaJ HfHNM b \GOS J \GH J S J D J D b H J H b J Q J HNH J \GHNQQNPIDGP32 O CZSNPa_ J Q b CZDGQ RJ QlM&SYC R FIDGD C J Qb J 0POHNF K FID J Q-$D J EF b M R PQNCZSUPaFID KGR \GQ 2GD J b \ O C K b JLK H J Q^QNPaFID b CVDmQ R C O C _ PaSNM b J OFI\ KGR C JK H J D CVDeS J DLOF' K S JfRJ Q 0F b J QQN\ K MHNP J \GHNQ nJ QNQ J DeSUP JRRaJ J DeS-HUC b PCV\ CVPDmQNP+q\ J RaJ Q J<J SNQ b J QcOFI\GO GJ Q_ PQ+q\ J \GQ J Q J S S mJ H60P$+q\ J Q b CVDGQ R C O C _ PaSNM b J OFV\ KGR C J J QpS b FIDGO K HNF K FIQNM J b CVDGQ RaJ O C K PaSUH J6 JSUHNC)_ZCVP R b J EF b M R PQNCZSUPaFIDEC KJ H60PQ b g M&SYC >GR PH\GD J H JK HNMQ J DqSUCZSUPaFID b \O C K b JWK H J Q^QNPFVD b CVDGQ R C>O C *_ PaSUM b J OFI\ KmR C J + \mP H JK FIQ J Q^\GH R C 0`FIH \ R CZSUPaFIDLPaDqSNM-IHNC RJ b \ K HNF >mR0 J CV\ R PEPoS J Q S J DGCVDeSfOF' K S Jb J Q 0F b J QfQN\ K MH^P J \mHNQ b CVDGQ R C[O C _ PaSNMCZPDGQ^P+ \ J b J Q J8 J SUQ b J QOFI\GO mJ Q R P0PaS J Q _ PQ^OFVS GJ HEP/+q\ J Q K CVHRJ O FIP& b g \GD J 0`FIDmOSUPaFID b J @ H JJ DC b C K SUM J 7 D J S JRRaJ 0F b M R PaQUCZSNPFID KJ H6 J S b J KGR \GQ b g FI\m_ HNPaH R C_IFIP J R C0F b M R PQUCXSUPFVD b g \GDAQ QpS J0K FV\GH R C J QN\GH J b g P K M b CZDGO J Q b J0KdJ SUPoSUQ[OF K FIQUCZDqSNQ J8*QN\GH J +q\GPlH JK FIQ J QN\mH RaJ J K H^PDGOP KdJ + \ J O JR \mP b J R C&EM&S F b J b J HNMOP K HNF OPaSNM J 0F b RJ K CVHCVP RaRJ \GHNQ H J D b K CVH 0nCVPaS J J DeS"OF' K S J b J R CTQ 0MSNHNP J b J QlHN RJ Q"S J Dq\GQ K CVH RJ Q b J \ 0POHNF K FID J Q b CVDGQR C O C _ PaSNM b J OFI\ KmR C J O J +q\GP J QpSWP K FIH^SUCVDeSOCVH O g J Q^SWQN\GHO J S^S J Q EM&SUHNP J +q\ J H JK FIQ JLR C(EM&S F b Jb J H^MOP K HNF OPoSUM JRRaJ8* J J Q HNMQ^\ R SUCZSUQ J K FIQ^MQ 9R C(2 D b \ O C K PoSUH J6 EFVDqSNH J DqS + \ J RJ Q 0F b J QQN\ K MH^P J \GHNQ KdJ \m_ J DeSWC _VFIPH%\GD J PaD \ J DGO J DGFID DGM R P J C >mRJ Q^\GH RJ Q H^MQ^\ R SUCZSUQ b g MSUC R FVDGD C J
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% CVDGQ RaJ K CVHNCIHNC KGJ TWT R C QNF R \mSUPaFID K H^F K FVQNM J J Q^S PaQNQ^\ J b J R CAHNMQNF R \mSUPaFID b J R g M.+q\ CZSUPaFID b Jb P\GQ^PFID b JER C O C RJ \GH T T RaJEK HNF O J Q^QN\GQ b J b P\GQNPaFIDSNHNFI\ _VCZDqS>QNFVD FIHNPIPD J b CVDGQ RJ O C K b JK H J QNQ^PFIDECVOFV\GQ^SNP$+q\ J p qRJ O C K b J S J K MHUCZSN\GH J τ1 b J _VCZDqS b J QN\GH^OH^FSQUCXSUPQ 0nCVPH J" b J QOFID b PaSUPaFIDGQCV\ R P0PaS J Q b J % PH^PO GRaJ S F'0F' D J Q J S b J Q2OFVD b PaSNPFIDmQ%PDGPoSUP C RJ QDq\ RRaJ Q$#
(
1
c0`h
∂
∂t− ∆
)
τ1 =γ − 1
βγ
1
c0`h
∂
∂tp K FI\GH ~r ∈ D , t > 0 , 3/)C
τ1 = 0 K FV\GH ~r ∈ S , t > 0 , 3/3 τ1 = 0 K FV\GH ~r ∈ D , t = 0 , 3/ O
F S J S D H JK HNMQ J DeS J DeSWH J Q KdJ O&SUPo_ J J DeS R C QN\mH30nCVO J J S RJ _IF R \ J b J[R C OC)_ PoSUM[OFIDmQNP b MHNM J -CH^MQNF R \mSNPFID +q\ J K H^F K FVQ J J H J H b CZDGQ[QNFID CVH^SNPO RJ / )#"lH J _ P J DeS " MOHNPaH J R C9QNF R \mSUPaFID K FV\GH RJO C K b J S J K MHUCZSN\GH J DGFVSUM J POP τ2(~r, t) QNFV\GQ R C&0`FIH6 J b \ K HNF b \GPoS b J OFIDe_IF R \mSUPaFID9Q^\GH R C[_ZCVHNPC RJtb \]O C K b J[K H J QNQ^PFID]C _ J O \GD J 0`FIDmOSUPaFID
g(~r, t)#
τ2 = g ∗t
(
γ − 1
βγ
∂
∂tp
)
,
F R C0`FIDmOSUPaFIDg(~r, t)
b FIPaS2QNCZSUPaQ30nCVPaH J CV\ K H^FF R J( T %#(
1
c0`h
∂
∂t− ∆
)
g = 0 K FI\GH ~r ∈ D , t > 0 , )VC /'/ #
) ) 46-?HL EGLO+.1 460 /0 JMEG/ N5/1 1 46N JG+54 0H1 @B/ / E / EL /D DF/N N5/IKJML N.+./ ?4 EN.L ?/ JM+5JM/DFL +%JMI
g = 0 K FI\GH ~r ∈ S , t > 0 , ) g = 1 K FI\GH ~r ∈ D , t = 0 . )ZO
cFI\mHc_IMHNP32 J H R g M.+q\GPa_ZC RaJ DGO J b J O J Q b J \ K HNMQ J DqSUCZSUPaFIDGQ P R Q^\ \ S b J H J ECVH+q\ J H +q\ g JRRaJ QcQNFIDeS R g \GD JJ S Rhg CV\mSNH J b J Q(OCVQ K CZH^SUPaO\ R P J HNQ b g \GD& J K H^FF R J IMDmMHUC R O J Q(QNF R \mSUPaFIDGQ K CVH^SNPO\ R P H J QCVQNQ^F OPM J QM&SYCVDeS RJ Q - J Q J[K HNFF R J IMDGMHNC R OFVDGQNP b MH^M J Q^S b FIDGDmM K CVH$#
(
1
c0`h
∂
∂t− ∆
)
τg(~r, t) = f(~r, t) K FV\GH ~r ∈ D , t > 0 , TIC
τg(~r, t) = 0 K FV\GH ~r ∈ S , t > 0 , WTτg(~r, t) = τi(~r)
K FV\GH ~r ∈ D , t = 0 . TVO J[K HNF R J CV\_ZC RJ \GHNQ K H^F K H J Q CVQNQ^F OPM J QpS b J[R C0`FIH J
(∆ + k2m)ψm(~r) = 0 K FI\GH ~r ∈ D , C
ψm(~r) = 0 K FV\GH ~r ∈ S . C 0`FVDGOSNPFID
τg(~r, t)KdJ \mSC R FIHNQ SUH J b MOF' K FIQ^M J b J[R C0nCFID QN\GPo_VCZDqS J
τg(~r, t) =∑
m
< ψm|τg > ψm(~r) , #'F
< ψm|τg > J Q^S R C K H^F J O&SUPaFID b J τgQ^\GH RJ _ J OS J \GH K HNF K H J ψm
#
< ψm|τg >=
∫∫∫
Dψm(~r)τg(~r, t) d~r .
!
C K HNF J OSUPaFID b J[Rhg M!+q\ CZSNPFID WTIC' QN\GH R C!GCVQ J b J Q%_ J OS J \GH^Q K H^F K H J Q b FIDGD J1
c0`h
∂
∂t< ψm|τg > − < ψm|∆τg >=< ψm|f > ,
F mJ D 0nCZPQUCZDqS \GQUC J b \9S MFIH J b J H JJ D J S%OF K S J S J Dq\ b J Q(OFID b PoSUPaFIDGQ(CZ\ 0`H^FIDeSUP H J Q T (QUCXSUPQ 0nCVPaS J Q K CZH ψmJ S τg OFID b PaSUPaFIDGQ b J0% PHNPaO GRJ S F'0F' D J Q < ψm|∆τg >=< ∆ψm|τg >= −k2
m < ψm|τg > ,O J +q\GP b FVDGD J C R FVHNQ RaJ K HNFF R J K H^F J SNM
(
1
c0`h
∂
∂t+ k2
m
)
< ψm|τg >=< ψm|f > K FI\GH t > 0 , -,IC < ψm|τg >=< ψm|τi > K FI\GH t = 0 , -,
b FVDqS R C Q^F R \ SUPFVDQ g MOH^PaS
< ψm|τg >= e−c0`h k2mt
[
< ψm|τi > +
∫ t
0
c0`h ec0`h k2mt0 < ψm|f > dt0
]
, U
! #$&# ( ) )
O J +q\GP b FIDmD J[Rhg J K H J QNQ^PFID b J[R C"0`FIDGO&SUPaFIDτg(~r, t)
#
τg(~r, t) =∑
m
e−c0`h k2mt
[
< ψm|τi > +
∫ t
0
c0`h ec0`h k2mt0 < ψm|f > dt0
]
ψm(~r) . '
( DGFVS J H +q\ J O J SNS J J K H J QNQ^PFID IMDGMHNC RJ>KJ H6 J S b J H J SNHNFI\m_ J H R C 0`FIDGO&SUPFVD b J H JJ D G(~r, ~r0; t)b \ K HNF R J( / J D K FIQUCZDqSf(~r, t) = δ(~r − ~r0)δ(t)
J S τi(~r) = 0 ,
O J +q\GP b FIDmD JG(~r, ~r0; t) =
∑
m
c0`h e−c0`h k2mtψm(~r0)ψm(~r) . 8
B $
-C Q^F R \mSNPFID τ1 b \ K HNFF R J( 3/ J QpS b M b \GPoS J b J[R C Q^F R \ SUPFVD&IMDGMHUC RaJ τg J D K FVQUCVDeS
f(~r, t) =γ − 1
βγ
1
c0`h
∂
∂tp(t) J S τi(~r) = 0 ,
O J +q\GP b FIDmD J b \20nCVPoS +q\ J>RaJ O C K b J[K H J QNQNPaFID J Q^S \GDmP/0`FIH J b CZDGQ R C O C _ PaSNM
τ1 =∑
m
e−c0`h k2mtψm(~r) < ψm >
γ − 1
βγ
∫ t
0
ec0`h k2mt0
∂
∂t0p(t0) dt0 ,
3/
F< ψm >=
∫∫∫
Dψm(~r) d~r
-C QNF R \mSNPFID g(~r, t) b \ K HNF R J )'%HNMQNF R \ K CVH J H J H J QpS b M b \GPoS J b JLR C QNF R \mSUPaFID VMDGMHUC RaJ
τg J D K FIQUCVDeS
f(~r, t) = 0 J S τi(~r) = 1 ,O J +q\GP b FIDmD J
g(~r, t) =∑
m
e−c0`h k2mtψm(~r) < ψm > . /'/
C Q^F R \ SUPFVD K FI\mH RJ O C K b J S J K MHUCXSU\GH J PDGQpSYCVDeSYCZDGM J τ2 J Q^SC R FVHNQ%MOHNPoS J
τ2(~r, t) = g ∗t
(
γ − 1
βγ
∂
∂tp
)
=γ − 1
βγ
∫ t
0
g(~r, t− t0)∂
∂t0p(t0) dt0
=∑
m
e−c0`h k2mtψm(~r) < ψm >
γ − 1
βγ
∫ t
0
ec0`h k2mt0
∂
∂t0p dt0 .
3/ )' J Q Q^F R \mSNPFIDmQ τ1 J S τ2 b J Q b J \ K H^FF R J QC b J SNS J DeS R C - J b MOF' K FIQ^PaSUPaFID QN\GH R C CVQ J b J Q_ J OS J \GHNQ K H^F K H J Q ψm
# JRaRJ Q%QNFIDeS b FIDGO[P b J DeSUP/+q\ J Q(TPDmQNP qRaJ Q(Q^F R \mSNPFIDmQOC R O\ R M J Q K CVH J H8 J H b CVDGQ RJ K CVHUCIHUC KGJ T )>QNFVDqS%CV\GQNQ^P Q^F R \mSNPFIDmQ b \ K HNF R J HNMQ^F R \ b CVDmQ RaJ K CVHNCIHUC KGJ T T ZKdJ SNPaS J 9O C _ PaSUM
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Y B& C Q^F R \mSNPFID 3/'/ b \ K H^FF R J *) H JK HNMQ J DqS J SUH Q!GP J D \GD J 0`FIDmOSUPaFID b J H JJ D ) RRaJ J QpS J D J J S M-eC RaJ J DeS2QNF R \mSNPFID b \ K H^FF R J(
1
c0`h
∂
∂t− ∆
)
g(~r, t) =1
c0`hδ(t) K FI\mH ~r ∈ D , t > 0 , 3/MTIC
g(~r, t) = 0 K FI\GH ~r ∈ S , t > 0 , /MT OF' J P RJ Q^S 0nCVOP RJ b JRJ _IMHNP2 J H J D MOHNPo_ZCVDeS b CVDGQ R C]QNF R \mSUPaFID IMDGMHNC RJ ' τi = 0 J S f =
1
c0`hδ(t)
6,
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5MN* ( ! D IE O, O*O " a @!,!P"!*)\0 Q* ( ,J - R ( ! !/+ BF ( T) ;D1,!2 '1?"* " !/+!#
(
∂ 2
∂r2 +2
r
∂
∂r− 1
σ
∂
∂t
)
g = 0"
r ∈ (0, a) , t > 0 , % )3/S
g = 0&"
r = a , t > 0 , % )3/3 g = 1
&"r ∈ (0, a) , t = 0 . % )3/31
T , 1Z,1 T 3<UZ<BTU90VWVX>YX6ZS5 >@9;:LNT5AE," !" O* [,F ( >\ % )3/^];=1!4%,":"0/" (
g(r, t) =1
r
+∞∑
n=1
(
Anej nπra +Bne−j nπra)
exp−σn2π2t
a2 ,
+!6*" _@1 ( H. ?[*G1&+g,, 2 !6
r = 0\An = −Bn
]
g(r, t) =1
r
+∞∑
n=1
an sinnπr
aexp
−σn2π2t
a2 .\ % )*)"]
5AE," !" O* "!.!30`! " :a1" *!4.!" O! !4.!\ % )3/313]
g(r, 0) =1
r
+∞∑
n=1
an sinnπr
a= 1 ,
/'/ 8
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+!M*" _@1 ( .H. *G ;="P"" !4.2* 0/" 1C.!" sin
nπr
a
_an = − 2a
nπ(−1)n .
\ % )WT"]5M 0/ 1!"
g(r, t);=1!2* 1
g(r, t) = −2a
πr
+∞∑
n=1
(−1)n
nsin
nπr
ae−σn2π2t
a2,
\ % ) ]1 +!6* G
τ\ T ) )T-] _
τ(~r, t) = g ∗t∂ θg
∂t,
\ % ) #]
θg =γ − 1
βγpg
\ T ) 8 ] )65M $ ( *τG>* ( ! D ;=1!43_61 ( E. *F1
+θg(t)
,2 ! 0/" ( H*@ D _< τ >=< g > ∗t
∂ θg
∂t,,"!_*@ B* ( ! W* !_
< τ >= jω < g(ω) > θg(ω) = jωγ − 1
βγpg < g > .
5M ( a@ .!:%G$"" ( V
*:>&" ,a! ( !" < g >
*@ W* ( ! * @",!+*" _@*<;D \ % ) ]C_
< g >=6
π2
+∞∑
n=1
1
n2
1
jω +n2π2
9
A2
V 2 σ
,
N&", 0`1 $"" ( >$ A/V = 3/a
) 5M ( < τ >
*τ $" (
V;D1!4%",_* B* ( ! H* @",!_
< τ >=γ − 1
βγpg
+∞∑
n=1
6
π2
1
n2
1
1 +n2π2
9X2
,\ % ) ,"]
X
I%* " @\σ = c0`h
]X2 =
A2
V 2
c0`hjω
.
T ,1N, 9ZTUC 5VHU99Z@O 9NC EV5 5XEYX LXE 5DE8L7Z:A o9XE@36X ;WX QL TULZO X"!4
gL = 1 − g_B F ( F\)^]#* #(
∂ 2
∂r2 +2
r
∂
∂r− 1
σ
∂
∂t
)
gL = 0&"
r ∈ (0, a) , t > 0 ,\) U -]
gL = 1&"
r = a , t > 0 ,\) U ]
gL = 0&"
r ∈ (0, a) , t = 0 ,\) U 1S]
) *)
"!43_*@ 2H* ( ! B*5M1(
∂ 2
∂r2 +2
r
∂
∂r− q2
)
TL = 0&"
r ∈ (0, a) ,\ )"-]
TL =1
s
&"r = a ,
\) @]
GL(r, s),%:. 0/" ( B* 5M :1B*
gL(r, t)_s,2:a$!H*5M:1B
q2 = s/σ)5AE," !" O* 1GF ( B;=1!4
GL =1
s
a
r
sinh qr
sinh qa.
\)-]5ME$ ( "
< GL >*
GLH$" (
V;=1!4
< GL >=3
sqa
[
cosh qa
sinh qa− 1
qa
]
.
,.G ,,!" [ G*$"&G*:0`" !4$
< GL >≈3
sqa
[
1 − 2+∞∑
n=1
(−1)ne−nqa
]
− 3
sq2a2 ,\) -]
!qa
;D,%@C.!43)9 !/+@ -W2.. 30/" ( H* E5M:1! -$">\ ))WT )"]E\ ) ) T )"]2* : .0/ 1 ". +*@ 2 ;D _ :a$ (
< gL >*
gL,H$" (
V*"
< gL >≈3
a(σt)1/2
[
2√π
− 4+∞∑
n=1
(−1)nierfcna
2√σt
]
− 3
a2 σt + O(
(σt)2)
,\) ^]
!30/1B*!P G ;=! -.P".W*G 0/ 1!" O*<;=,%* @ !G@ierfc x =
∫ ∞
x
erfc t dt ,3$1
erfc x =2√π
∫ ∞
x
e−t2 dt .
5A" .!" < g >
*O,F ( a\) ^]7,#** !.W*H1W*O,F ( a\ ) "]8@#,: .! g = 1 − gL
_@ ' P!P3 -++. ( '!30/1_
< g >≈ 1 − 2√π
(σt)1/2A
V+
1
3σtA2
V 2 + O(
(σt)2)
.\) )"]
59;= ,,! *G:$ ( " < τ >
*τ,:$" (
V
< τ >=< g > ∗t∂ θg
∂t,*" _@*@ 2B* ( ! H* @"!_
< τ >= jω < g(ω) > θg(ω) = jωγ − 1
βγpg < g(ω) > .
5ME$ ( " < τ >
;D1,!%_ *@ B* ( ! H* @" !_< τ >≈ γ − 1
βγpg
[
1 −X +3
4
1
3X2 + O
(
X4)
]
.\) MT-]
) * 46EGEG/DJG+54 0H1 J / E-+ * /1!?4 E @H+ IEG/0 JM/1 4)E#-/1 @H/CDFL +%JMI1 7 / E / EF:
K 56E* ( ! *<;DC.* >,!1! ;=,@1>1" -. ? -F* : B@. W!4.B z = 0
Cz = `
) 5M * !!" CJ -E !$+ ( -" P!. *! @_ >F ( \.T )-]G;D1,!a 1?"* " 1,! !#(
∂ 2
∂z2 − 1
σ
∂
∂t
)
g = 0&"
z ∈ (0, `) , t > 0 ,\) "]
g = 0&"
z = 0"
` , t > 0 ,\) ]
g = 1&"
z ∈ (0, `) , t = 0 .\) 13]
56" .!" #* 1 F ( H 2* %g(z, t) = ψ(z, t) ,&0/" 1.!"
ψ(z, t). -* @ !2* @.P". T )*)?)WTG1 (>( 2B" .!" F*N,F ( \.T ) ,-]C)5M ( " G 2H$"" (
V* "0/" 1!"
g;=1!4%* " 1_@* ; \.T )*) ^]C_
< g >=2
π2
+∞∑
n=0
1
(n+ 1/2)2 exp−(n + 1/2)2π2A2
V 2 σt ,\) ]
'B&",A/V
$ 2/`
_"!4*@ B* ( ! W* !_
< g >=2
π2
+∞∑
n=0
1
(n+ 1/2)2
1
jω + (n+ 1/2)2π2A2
V 2 σ
,\),"]
1 +!6* G+:E ;=1,! ,. -J B*B. ( &. .,τ_
< τ > = jωγ − 1
βγpg < g >
=2
π2
+∞∑
n=0
1
(n + 1/2)2
1
1 + (n+ 1/2)2π2X2 ,\) U ]
X
I%* " @\σ = c0`h
]X2 =
A2
V 2
c0`hjω
.
5MG,".! " 1"I9. ( + . 30/ ( +*5M1+* 2W$ ( < g >
I8* " *<;D 2 ;= ,,!" \ T)WT U ]C_
< g >= 1 − 2
`(σt)1/2
[
2√π
+ 4
+∞∑
n=1
(−1)nierfc n`
2
1
(σt)1/2
]
,\) "]
1 +!6& ( C*G** !< τ >
_ [ P"!P" -+H ( G '!30/1_< τ >≈ γ − 1
βγpg(1 −X) .
\ ) -]
# % % )
K 0 5M8* ( ! 7* ;DC.*, a1 ! * ! @ !*8. "
a) ( 7 E*2 ( C.!1 ! * !$+* * ( ! B*<;DC.* _@B F ( >\ T )-];D1!42 '1?"* 1?! *!/+ #
(
∂ 2
∂r2 +1
r
∂
∂r− 1
σ
∂
∂t
)
g = 0"
r ∈ (0, a) , t > 0 ,\ ) ) "-]
g = 0&"
r = a , t > 0 ,\ ) ) ]
g = 1&"
r ∈ (0, a) , t = 0 ,\ ) ) 1S]
W ( ( +G,1* (>( -3_ +" .!" +*G1 F ( B" -2*" %@g(r, t) = ϕ(r, t) ,
:0/" 1C.! ϕ(r, t)
CJ -8* !%*@ [email protected]". >T)*)) T1 (>( 2:G" .!" >*N,F ( a\.T ) "]C)5M ( " G2%$"" ( V
*G 0/ 1!" g;=1!2* 1_@*<;D \ T)*) "]C_
< g >=
+∞∑
n=1
4
λ2n
exp−λ2
n
a2
A2
V 2 σt ,\) ) ^]
B&",A/V
$ 2/a
_@"!4+*@ B* ( ! B* @" !_
< g >=+∞∑
n=1
4
λ2n
1
jω +λ2
n
a2
A2
V 2 σ
,\) ) )"]
1&+!6*" G" +:E ;=1,! ,. -J B* G ( .τ_
< τ > = jωγ − 1
βγpg < g >
=+∞∑
n=1
4
λ2n
1
1 +λ2
n
a2 X2
,\) )T-]
X
,%*" G@\σ = c0`h
]X2 =
A2
V 2
c0`hjω
.
5AB," !" " 1"I7 ( +@ .. 30/" ( +*25M:1#*W$ ( " < g >
,7*" *<;D ;= ,!" \ T ) T ^] _< g >≈ 1 − 2√
π
A
V(σt)1/2 +
1
4
A2
V 2 σt +1
24√π
A3
V 3 (σt)3/2 ,\) ) ]
1&+!6& ( * * *!G:$ ( " < τ >
< τ >≈ γ − 1
βγpg
[
1 −X +1
4X2 +
3
4
√π
1
24√πX3
]
.\) )"]
) * 46EGEG/DJG+54 0H1 J / E-+ * /1!?4 E @H+ IEG/0 JM/1 4)E#-/1 @H/CDFL +%JMI1 7 / E / EF:
K $ &
56* ( ! G*<;=. * D ,W . ! *B**! ( !" `x, `y, `z
)&56.&"I%$" ( ,30`1$V
A=
2
1/`x + 1/`y + 1/`z,
C%.,.R1
CR2
," -%* @ !@
R1 =`x`y
R2 =
`x`z
.
56BF ( >\ T) -] [1*" %1,! ;=1!(
∂ 2
∂x2 +∂ 2
∂y2 +∂ 2
∂z2 − 1
σ
∂
∂t
)
g = 0\)*),-]
&"x ∈ (0, `x)
_y ∈ (0, `y)
_z ∈ (0, `z)
t > 0
_g = 0
" x = 0
`x , y = 0
"`y , z = 0
"`z , t > 0 ,
\)*), ]g = 1
" x ∈ (0, `x) , y ∈ (0, `y) , z ∈ (0, `z) , t = 0 .
\) ),13]5MF,".!
gI21 ,1 " 0/ ( H*<;= &0/" 1!" " $!#,@#
g(x, y, z, t) = ψx(x, t)ψx(y, t)ψx(z, t) ,
ψx(x, t)
_ψy(y, t)
Cψz(z, t)
" -*" .!" G* F ( [\ T ) ,-]H&"3+ ;D#>**! ! ',2&1C.!$ ( -x_yz)
T ,:,1 T 3<U9XBT9 X YX6ZS5 @9;:LZT5M$ ( B+W$"" (
V* 0/" 1!"
ψx
_ψy
ψz
," -*" +@#H* $"" & ( - ( ?*@F\ T)*) S] #
< ψx,y,z >=2
π2
+∞∑
n=0
1
(n+ 1/2)2 exp−(n + 1/2)2π2 4
`2x,y,z
σt .\)*) U ]
56,* !%*ψx,y,z
*" a"2 ;= ,,! *g)&5M>$ ( %G$"" (
V*
g;=1,!%"_.131 0`!_
< g >=8
π6
+∞∑
m=0
+∞∑
n=0
+∞∑
p=0
1
(m + 1/2)2(n + 1/2)2(p+ 1/2)2 e−π2A2
V 2 σt
exp(m+ 1/2)2 + (n+ 1/2)2R2
1 + (p+ 1/2)2R22
(1 +R1 +R2)2,
\)*) "],"!*@ B* ( ! H* @",!_
# ( #$ &# 3 )
< g > =8
π6
+∞∑
m=0
+∞∑
n=0
+∞∑
p=0
1
(m + 1/2)2(n + 1/2)2(p+ 1/2)2
1
jω + π2A2
V 2
(m + 1/2)2 + (n+ 1/2)2R21 + (p+ 1/2)2R2
2
(1 +R1 +R2)2 σ
,\) )-]
1&+!6*" G" < τ >
< τ > =γ − 1
βγpg
8
π6
+∞∑
m=0
+∞∑
n=0
+∞∑
p=0
1
(m + 1/2)2(n + 1/2)2(p + 1/2)2
1
1 + π2 (m+ 1/2)2 + (n+ 1/2)2R21 + (p+ 1/2)2R2
2
(1 +R1 +R2)2 X2
.\) T-]
T , , 9ZTUC 5VHU97Z @O69ZC EV5 5XEYX LXE 5DEFL9Z:A o9XE@36X ;WX KL TULNOXW ( ( +29:H !" ( *@_:H," !" " 1,9. ( +@0. 0/" ( *25M1I%* *!4.G* *C$""& ( -a\ T ) T U ] )@56+$ ( "
< ψx,y,z >;D1!4$"
< ψx,y,z >≈ 1 − 2√π
2
`x,y,z(σt)1/2 .
\) T ^]5MG,* !+*
ψx,y,z
* #E$ ( " *< g >
#
< g > ≈ 1 − 2√π
(σt)1/2A
V+
4(R1 +R2 +R1R2)
π(1 +R1 +R2)2 σt
A2
V 2
+−8R1R2
π3/2(1 +R1 +R2)3
(σt)3/2A3
V 3 ,\) T#)"]
1&+!6*" G" +:a$ ( < τ >
_
< τ >≈ γ − 1
βγpg
[
1 −X +4(R1 +R2 +R1R2)
π(1 +R1 +R2)2 X2 +
−8R1R2
π3/2(1 +R1 +R2)3X3
]
.\) TFT-]
K
59;= ,,! '* B:$ ( " < τ >
& C.,H1!_*@ +W* ( ! W* @" !_:0/" ( BP .>\ T ) ))WT"] #< τ >= θg(ω)EP ,
\) T ]
Ep
$
) * 46EGEG/DJG+54 0H1 J / E-+ * /1!?4 E @H+ IEG/0 JM/1 4)E#-/1 @H/CDFL +%JMI1 7 / E / EF:
EP =+∞∑
n=n0
DNn
MnX2 ,
\)WT]" !
EP = 1 −X +G1X2 +
3
4
√πG2X
3 ,\)WT ,"]
O7. ( D_n0
_Nn
_Mn
_G1
CG2
*& * -*HP" ( C.!%* R* ( ! )59;=! *!1n&,, -.% !+! *!1)56aJ3 )>*" EH$,B*F1* ! -.B$,!:H&"W,!BP" ( .,!B** ( ! H*<;=*)
D n0 Nn Mn G1 G2
6/π2 1
n2
n2π2
9
1
3 ! #"$%&$"
2/π2 ' 1
(n + 1/2)2(n + 1/2)2π2
(*)+ #"$,%&$" - 1
λ2
n
λ2n
4
1
4
1
24√
π
(*)+ /.01328/π2 40',5 76 1
λ2
n(m + 1/2)2(m + 1/2)2π2 + λ2
nR
(2R + 1)28R + πR2
π(2R + 1)2R3 − 6R2
3√
π(2R + 1)3
8:9 9 ;; 2 ; 2"<=8/π6 40',5 ',5 ' 6 Nmnp > Mmnp >> 4(R1 + R2 + R1R2)
π(1 + R1 + R2)2−8R1R2
π3/2(1 + R1 + R2)3
> Nmnp =1
(m + 1/2)2(n + 1/2)2(p + 1/2)2>> Mmnp = π2
(m + 1/2)2 + (n + 1/2)2R2
1+ (p + 1/2)2R2
2
(1 + R1 + R2)2
X Y2[\& ) 7,2* * ! -.1" ,. -.%, [aP" ( !H*'* ( ! W*<;DC.* )
5M/" 1C.!" * , .!!,!1!," -+/" 1.!" #* ,</ 1!" 8*, ( *! @1?! *!*G, ( ! ,B, 1) K !
5A"/" 1C.! Jν(z)
_3$1z ∈ C
_, #/" 1.!" *$ 1 ! * !%7*, ( ! 7&1+*<;="*,ν_@ !" O*G ;D&?.!"
∂ 2f
∂z2 +1
z
∂ f
∂z+
(
1 − ν2
z2
)
f = 0 . ' +))(^]5A*/ 1!"
Iν(z)_-3$1
z ∈ C_"* !P" ++/" 1C.! E*",@1?! *! ( *! @*+ ( ! ,, 1*<;D*, ν !" O*G ;D&@.!"
∂ 2f
∂z2 +1
z
∂ f
∂z−(
1 +ν2
z2
)
f = 0 . ' +)-,"] . $/ 10 2 $3
4 "!#* 5/ 1!" +* 68,71" 8? Zν(z)
CBν(z)
* ;D",*ν_@1"/" 1C.! O", * -+" !C.2* ;D! P".!" O !$ -. (MT _:9);9<,)(_=9)-9 ))(C*,#"
∫
zν+1Zν(z) dz = zν+1Zν+1(z) , ' +)WT"]∫
zZν(az)Bν(bz) dz =bzZν(az)Bν−1(bz) − azZν−1(az)Bν(bz)
a2 − b2, ' +) ]
∫
z [Zν(az)]2 dz =
z2
2
[
[Zν(az)]2 − Zν−1(az)Zν+1(az)
]
. ' +);9"]5A>/" 1.!" *?,
Zν(z)&" * '" G:a,",!B!4$ -. (3T _:@ ) U (") T"∂ n
∂z(zνZν(x)) = zνZν−n(z) . ' +)-A"]
(B, U
2R EG4)?HE+5IKJMI1 JG+ N5/1 @H/1 4602DJM+.460H1@B/ /131 /N
K $/ 2 ! 56B*$ ( -H ( .!B*G:>/" 1C.!" O*?, ( * ! @
Iν(z);D1!4 ,_ U );9)-, U "
Iν(z) =ez
√2πz
[
1 − 4ν2 − 12
1!8z+
(4ν2 − 12)(4ν2 − 32)
2!(8z)2 − ...
]
+e−z±jπ(ν+1/2)
√2πz
[
1 +4ν2 − 12
1!8z+
(4ν2 − 12)(4ν2 − 32)
2!(8z)2 + ...
]
. ' +) U ];D ., @I3_2 *$ ( - ( ..!%* : /" 1C.! * 8,
Jν(z);=1! ,?_
U )-9))( A"
Jν(z) =
√
2
πz[Pν(z) cosϕ−Qν(z) sinϕ] , ' +) @-]3$"1
Pν(z) ≈ 1 − (4ν2 − 12)(4ν2 − 32)
2!(8z)2 +(4ν2 − 12)(4ν2 − 32)(4ν2 − 52)(4ν2 − 72)
4!(8z)4 − ... , ' +)-]
Qν(z) ≈4ν2 − 12
1!8z− (4ν2 − 12)(4ν2 − 32)(4ν2 − 52)
3!(8z)3 + ... , ' +) ]ϕ = z − (ν + 1/2)π/2 . ' +)13]
56B*$ ( - ' ) @-]& % !M;D1!Jν(z) =
1√2πz
[
ej ϕ[Pν(z) + jQν(z)] + e−jϕ[Pν(z) − jQν(z)]]
,
,"!Jν(z) =
1√2πz
[
ej ze−j (ν+1/2)π
2 Aν(z) + e−j zej (ν+1/2)π
2 Bν(z)
]
, ' +)("]
Aν(z) = Pν(z) + jQν(z)
Bν(z) = Pν(z) − jQν(z) .
K $/ 2 ! Kh
/ 0 +/ 3
4 "!4+ . &G*G. "" a_@H (
Kh
;=1!4 ' ,))( @-]Kh =
2
kha
J1(kha)
J0(kha), ' +)( (S]
kha =
1 − j√2
√
k0/`h a .
Kh! "$#&%' ( *) +
4 "!z = kha
_z " H1!4#,"+:?/" (
z = (1− j )α+ ;D !" ' +))(-]8* " *" 1
Jν(z) =1√2πz
[
eαej αe−j (ν+1/2)π
2 An(z) + e−αe−jαej (ν+1/2)π
2 Bn(z)
]
. ' ))(B,"] ( (
αI2P". *<_ ;= &" !
e−α ,%C.!B2 ;= ,,! ' +)(B,]+* $?! Jν(z) =
1√2πz
eαejαe−j (ν+1/2)π
2 [Pν(z) + jQν(z)] ,
"!43_ " ;D",*?9, ,
Jν(z) =ej z
√2πz
e−j (ν+1/2)π
2
[
1 − 4ν2 − 12
1!8j z+
(4ν2 − 12)(4ν2 − 32)
2!(8j z)2
]
. ' ))(MT-](W! !<$/" 1C.! +*?,
J0(z)CJ1(z)
;D1!4$"
J0(z) =ej z
√2πz
e−j π
4
[
1 +1
8j z+
9
128(j z)2
]
, ' ))( ]
J1(z) = −jej z
√2πz
e−j π
4
[
1 − 3
8j z− 15
128(j z)2
]
. ' ))(B9"]0 [,&",. -12*C$""& ( -._
z = khR_* ;= ,,!" '*
Kh
_@! $?! Kh =
2
j khR
(
1 − 1
2j khR− 1
8(j khR)2
)
. ' ))( A-]50;D ,!" [*
KvI2: ( ( ? 1H*
Kh ' ( -3
kh@
kv)
<
E aIH 1 ( ( -W* @.P". !T ) ,H ,"@P- .! [*<;=" * H )5A1 ( &". -N.*!: * R ;=" * ,F1" ,$_0 ,"@P- .! ,F C.,R1 *<;=" *@! )5M+4J#. $"2!1!&" - ( ( G1 N. $"2* [email protected]". T ) _1!6& ( * , ,.! @ 3! ( .!" ! O 2C.`!)
H ( !B. ( _6 @ .! G* "@P-.!" *> ;D1" ,.!F! 3!E 1!4.*@ G * ( ! D 1" ' ))(^]C)9B" - ,! ,* E13*<;= 1?! */ ( " B ?. ( !4.H@H* ( !1" W*<;D* ( !,J 1%&1!$
YeYr_ ;= * ( ! 1,"" B. - ! ( * ;D ( " $ ( -G*E$?!.,
ve ' ) T-] _ 11!0*@ B G*E* ( ! ;D* ( !,J 1H1"I.!%G*B. /I%* [ I ( ' ) ]C) 3 / 0 /
5MB @!" B! 3! ( "P HP" $" @ ( $" ( G* !*" -1H*" *@ [email protected]". T ))(#1
c0
∂ ~v
∂t+
1
ρ0c0~grad p = `v ~grad div ~v − ` ′
v~rot ~rot ~v ,
ρ0c0 div ~v +γ
c0
∂
∂t(p− βτ) = 0 ,
(
1
c0
∂
∂t− `h ∆
)
τ =γ − 1
βγ
1
c0
∂ p
∂t.
5M1 @ ( '*$?!,@,!1:!~v,%1!4%1 (>( G:F &"!4.! '* ;D [1 @ ( *G$?!,." ! @!
~vv
2*<;= [1 ( '*B$?!,G: ( ! @!~v`
#
~v = ~vv + ~v` , ~rot ~v` = ~0
div ~v` 6= 0 , div ~vv = 0C
~rot ~vv 6= ~0 . ' )(^]4 ! 1"P" * ( $" ( : ( ! @!,RC " !" @!OI P"!P"_+ ;D&?.!" *W3$?! 4 . ' T )(S]2& $C.,EE * &?.!" ' ) )-9 1! *,"J]2*.,.GB ?I ( * TF&@.!" ' T)("_ T );,aC>T)WT-]* $?! !#
1
c0
∂ τ
∂t− ρ0c0
γβdiv ~v` =
1
β
1
c0
∂ p
∂t, ' )-,"]
(MT 9
R EG46?HL)L JG+5460 @ 460H@H/1 LO13+ ?2N.L 0H/1
(
1
c0
∂
∂t− `h ∆
)
τ =γ − 1
βγ
1
c0
∂ p
∂t, ' )WT-]
(
1
c0
∂
∂t− `v ∆
)
~v` = − 1
ρ0c0~grad p , ' ) ]
(
1
c0
∂
∂t− ` ′
v ∆
)
~vv = ~0 , ' );9]3$"1 ~rot ~v` = ~0
div ~vv = 0
) + ?, ( +* &?.!" 0& ( *.," $0 @ .! *+,"@P".!" a&" $,!:
p_τ_~v`
C~vv
) 56G131<&"+a$!:τ,21C.*@ a&/ 1 @_ !,)-9))( " )50;D13I ! ,. -J O*O. ( &. .,
τ& " .,[ ! ( O,"F: /" ( R* [ (>( *<;= F. ( &. F1",!%
τa*<;= E. ( &. E -"!
τh? !0" -&1!$ ( -," !" +* @ .! ( "P
[
1
c20
∂ 2
∂t2−(
1 + `vh1
c0
∂
∂t
)
∆
]
τa = 0 , ' )-A"]
`vh = `v + (γ − 1)`h_
[
1
c0
∂
∂t− `h
(
1 + (γ − 1)(`v − `h )1
c0
∂
∂t
)
∆
]
τh = 0 ,
,"![
1
c0
∂
∂t− `h ∆
]
τh ≈ 0 . ' ) U ]50;D13I ! ,. -J 9J"*9,!"
p = pa+ph
C $?!.,8@,.!1!, ( ! @!~v` = ~va+~vh.! /" -% ( ( 2&@.!" ' ) A-] ' ) U ]G ;=1,! ,. -J G*B ( .)
. 0L 0L ?/ 5M 1 ,1 a*" .!" &"2$!
p = pa + ph
_~v` = ~va + ~vh
_τ = τa + τh
~vv
, ( " :E" .!" *<;D&@.!" 2*G@P-!" C%* *! ! '1:,!? ) 23 ( "! _1" 1 @ 80$!:01,!%77 " !% ' p _ ~v`_τ]C_-B" .!" E9* >* ;D 2& C.,a** !E*!,1 ( -B*EN !" [*E: "!! ( )60 C3_<,&",B* E:N," !"
τ = τa + τh* ;= @!" ' ) T-] _1 ( G '*2 @!" ' )-A-]C ' ) U ] _1 *!4 "
p = pa + ph , ' )-@-]3$"1
pa ≈ γβ
γ − 1
(
1 − `h1
c0
∂
∂t
)
τa ≈ γβ
γ − 1τa , ' ) "-]
ph ≈ γβ(`v − `h )1
c0
∂
∂tτh << pa . ' ) ]
( '
5MG,&",*G12,.. ' ) " _ ]*@ ;= @!" ' ) ]C_ 1 ( .H * @!" ' ) A-] ' ) U ]C_$",! @2+$?!4.,%@,.!1!, ~vaC~vh_* #
~v` = ~va + ~vh , ' ))(-]3$"1~va ≈ − 1
ρ0c0
γβ
γ − 1
[
(
1
c0
∂
∂t
)−1
+ (`v − `h )
]
~grad τa
≈ − 1
ρ0c0
γβ
γ − 1
(
1
c0
∂
∂t
)−1
~grad τa , ' )(<(S-]
~vh ≈ γβ
γ − 1`h ~grad τh . ' ))( (3 ]
(W! ! _&*@ H* ( ! /& -.! _ &"W E" * @ ( " ! ej ω t ' ω ,,, -.F:.!" *H ;=" * S]7,H,"@P" -#*@ #%* ( ! D _?# @ .! ' ))(^] " ' )( ( ]7* -$#
(∆ + k2a) τa(~r, t) = 0 ∀~r ∈ D , ∀t , ' ))(B,"]
(∆ + k2h) τh(~r, t) = 0 ∀~r ∈ D , ∀t , ' ))(MT-]3$"1
τ = τa + τh_@C
k2a ≈ k2
0
1 + j k0`vh
≈ k20[1 − j k0`vh ] , ' ))( ]
k2h ≈ − j k0
`h[1 − j k0(γ − 1)(`v − `h )] ≈ − j k0
`h, ' ))(B9"]
p = pa + ph ,3$"1pa ≈ γβ
γ − 1(1 − j k0`h )τa ≈ γβ
γ − 1τa , ' )( A"-]
ph ≈ j k0γβ(`v − `h )τh << pa , ' ))( A ]~v` = ~va + ~vh ,3$"1
~va ≈ j
k0ρ0c0
γβ
γ − 1[1 + j k0(`v − `h )] ~grad τa ≈ j
k0ρ0c0
γβ
γ − 1~grad τa , ' )( U -]
~vh ≈ γβ
ρ0c0`h ~grad τh , ' ))( U ]
(∆ + k2v)~vv(~r, t) = ~0 ∀~r ∈ D , ∀t , ' )( @"]3$"1
k2v = − j k0
` ′v, ' ))( -]
div ~vv = 0 . ' )-, -]
R EG46?HL)L JG+5460 @ 460H@H/1 LO13+ ?2N.L 0H/1
6/ / ! 0 / 'L
50;D" * @ ( " !%+,"@P+" Fz1"!. 9*@ 9 @!*+*!,!@.!) ' @ !*# ( $?! ]H1 *@ B E1 ?! *!? E @!W..H,!P"!*_* ; #
z_6* E" P
`B*E ""
a)656 E,/ ( " B ?., ( !.
z = 0Cz = `
@W* ( !1" ) 5M( !1" " F!4.>
z = 0/" 1C.! F ( 1!9
z = `/" 1C.!" F 1
' @P", T)(^] )-56 ( !1,"" # ( ,. 7! ( &",# z = 0
+$?!,~ve
* 7H1 ( . -.+*!,%&", P"!P")560. "" *B. 9IM. M&,! " ;D@!6*A1"1 ! ( !$?! AM ( !? _* . "I. O ;D *N* R,!" & .,R1 !*,R1 ( ( @! : # >I2@,! !/" ( *@ a1 <O1.!" * ' 1;=I "# *!N a ( -a$,!: 1/" 1C.! *R1",*" >.*!r] _6! * *@ - *>:O1",*" >! ( . θ @!," *> ( C.,!_6C,"@P".!4$"B !$ -2 ;D#
z* . )50;D C%* W1"1 W! ( !. ( !%2'$"!,! @P"G*2@,"!
z = 0z = `
IW,! 1 ( .G@+ ;D! ?*1C.! R*<;= * ( !4. 1B* "!Yp
*" G@Yp =
S
ρ0c0
1 + j√2
√
k0(γ − 1)√
lh , ' )-, (^]' ;=" *W1" ,.!H. -+ O! 1!* 1H ( #1+@!_? C.#*%$!1"!4.W," - !4$"[* 2 ?., ( !.+
z = 0Cz = `
]C)(W! ! _ [ @!" ' )( ,_ ))(MT C )( @-]C_&" -H,1!HW1" *!4.! 2 /" -.! ,W,! $ -.#
τa + τh = 0 , vaz + vhz + vvz = 0 , var + vhr + vvr = 0 ,&"
r = a , ∀z ∈ (0, `) , ' )-, ,"]C
S < vz(0) >S +Yp < p(0) >S= Sve ,
z = 0 , ' )-,T""]
S < vz(`) >S −(Yr + Yp) < p(`) >S= 0 ,&"
z = ` . ' );,T ];D .,G@,_ ( ,*<;=" * ka
_kh
Ckv
$" G1!4." 2>/" ( k2
a = k2ar + k2
z , ' )-, ]k2
h = k2hr + k2
z , ' )-, 9"]k2
v = k2vr + k2
z . ' )-,<A-]5M !" '* ;D&@.!" ' ))(B,"]# 1?"* " H1 ! * !%_ 1 ( * & ,,. _,2*G:>/" (
τa = AaJ0(karr)
J0(kara)(e−jkzz +Bej kzz) , ' )-, U ]
W1" ,J B
. -0* ( ! # :21" *!!" E 6/" -.! , ' )-,T ]M z = `) C+" .!" &"
τa ( *G**!
pa
C~va
_1 ( G '*2:!" ' )( A-]8 ' ))( U -] #
! %( ( ( ( +
pa ≈ Aaγβ
γ − 1
J0(karr)
J0(kara)(e−jkzz +Bej kzz) , ' )-,<@-]
vaz ≈ Aakz
k0ρ0c0
γβ
γ − 1
J0(karr)
J0(kara)(e−jkzz −Bej kzz) , ' )-,-]
var ≈ Aakar
j k0ρ0c0
γβ
γ − 1
J1(karr)
J0(kara)(e−j kzz +Bej kzz) . ' ) T-]
5A2" .!" G* ;D&@.!" ' )(MT"]C_S1 ( 7 a*8: ( ! ,81" *!4.! ?/" -.! , ' );, ,]C_;=1!
τh ≈ −AaJ0(khrr)
J0(khra)(e−jkzz +Bej kzz) . ' ) T8(^]
56:!" ' )( A ] ' ))( U ]& ( C -%",#*<;= [** !G+ ,,! %* ph
C~vh
#
ph ≈ −Aaj k0γβ(`v − `h )J0(khrr)
J0(khra)(e−j kzz +Bej kzz) ≈ 0 , ' ) T ,"]
vhz ≈ Aaj kzγβ`hρ0c0
J0(khrr)
J0(khra)(e−j kzz − Bej kzz) , ' ) TFT-]
vhr ≈ Aakhrγβ`hρ0c0
J1(khrr)
J0(khra)(e−jkzz +Bej kzz) . ' ) T ]
59;= @ .! ' ))( @-] _S 1*" 1 ! * !%_S1 ( .9. *8+.!" ' );, -] div ~vv = 0G*E1 ? vvθ = 0
_<* " ' a ( &" ∆,, -.N!1! a:1! !% " >/" 1C.! 1:!3]
(∆ + k2v)vvz = 0
(∆ − 1
r2 + k2v)vvr = 0 ∀r ∈ (0, a) ∀z ∈ (0, `) .
5MF,".! O*G ;= @!" ' ))( @-]&"+a1 ( . -.vvz
_ ,2* 1G*G:>/" (
vvz = AvJ0(kvrr)
J0(kvra)(e−j kzz − Bej kzz) , ' ) T 9"]
Av
,2*C. ( ! B@+E* ! ( B1" * !.!" /" -.! , ' );,<,"] #
Av = Aa−kz
k0ρ0c0
γβ
γ − 1. ' ) T A-]
5AE,: .! ' );, "] div ~vv = 0* " G&"
vvr
#
vvr = Avj kz
kvr
J1(kvrr)
J0(kvra)(e−j kzz +Bej kzz) . ' ) T U ]
9 @ <_ G,2*2 !" ' ) T _ ) T _ )WT U ]*@ 2:a.,"!,! ( B1" *!4.! 5/ ! ' );,<,"]8& ( * * . ( !
kz
#
R EG46?HL)L JG+5460 @ 460H@H/1 LO13+ ?2N.L 0H/1
k2z
1
kvr
J1(kvra)
J0(kvra)− kar
J1(kara)
J0(kara)− j k0`h (γ − 1)khr
J1(khra)
J0(khra)= 0 . ' )WT @"]
4 !kara
,9&.!43_^8. "I * /" 1C.! *, kar
& #* $"" + J1(kara)
J0(kara)≈
kara
2
) ( .2. ?O* W1*? k2
a = k2ar + k2
z
Ck2
h = k2hr + k2
z
_?:G.!" >1* -.G*" H"k2
z [1 +Kv + j k0`h (γ − 1)Kh] − j k0`h k2h(γ − 1)Kh = k2
a , ' ) T-]'#. ( Kv
Kh
" -%*" @Kv,h =
2
kv,hra
J1(kv,hra)
J0(kv,hra). ' ) -]
H > 1" *!4.!" E,. *@* * [,&/, 1_8F" P">13.1C.,!I.!%`h
C`v
$ - 3! ( .!4$" ( `h = 6.10−8 ( C
`v = 4.10−8 ( )#56 J 3 ))( *" $ R* ( k2
a/k20
k0`h (γ − 1)
/, 1% ?. ( ( GC>T ("_-A ) EGI / 0HDF/
k2a/k
20
( ("_-,)( −8 ( T_ U ))( −5
(γ − 1)k0`h _ ))( −10 ("_ ))( −5
XZY\[2& )(7,+*B. ( 2@! -%* :a,: .! ' )WT"]C)56$,a*" E* a1NJ 3 ( ,. -F",* , ,.! @,3 ! ( .! ,! $ -.#
k2a ≈ k2
0 , k2h ≈ −j
k0
`h, 1 ± j (γ − 1)k0`h ≈ 1 , ' ) (S]
F,.!" ' ) T-]*C$! -2*" 1k2
z
(
1 +2
kvra
J1(kvra)
J0(kvra)
)
= k20
[
1 + (γ − 1)2
khra
J1(khra)
J0(khra)
]
, ' ) ,"]H >: ( ' . ""
a*Q. &[CN P" ( ( O*5/&? 1 "! ' ! 8/,! "R ( ! 6/ 1F*E1" F* . &S]W," -B. F:R* !,!@ .! ! * @ !*E,, P"!P"32 O,P-,*O*H1%!H N ( $! ,B*$ -+*@ +81"1 ! ( !4._"
|kz| << |kh| ,* ;D
k2hr ≈ k2
h , |kz| << |kv| ,*<;=
k2vr ≈ k2
v .50;D ,!" ' ) ,"]* kz
*" * " 1k2
z = k20
1 + (γ − 1)Kh
1 −Kv, ' ) T-]
Kv,h =
2
kv,ha
J1(kv,ha)
J0(kv,ha), ' ) F ]
,"!_ " ;="*,2( , ,a*%" P"+1.1.!,.!
( (3 (*
k2z ≈ k2
0
[
1 +1 − j√
2
2
a√
k0
(√
` ′v + (γ − 1)
√
`h )
]
. ' ) 9"]
59;D* ( !,J 1F1",!%>*F.. /,* ?, ( IE.&"I* *!*>: ( (
Sve ' ",! ..!" P")(^]9 ;=13,+! ,. -J W*H,,!" ( " ',# B1C.!" O*!. " :"I.!< p(`) >S
H;D1!4YT =
Sve + Ye < p(0) >S
< p(`) >S
, ' ) A-] :$ ( " *
p;D1,!
< p(z) >S=2
a2
∫ a
0
p(r, z)r dr ≈ 2
a2
∫ a
0
pa(r, z)r dr .
( .% O*+, ! ( .! 9 +*@ 8W.P. W,1* - ' ,!" ) (^]F*<;D F ;=,,! * ;D13IF*O,!" Q -.,"!? ph ' ) ,"] _ !#@>? '1>1,*F,!" . * $, )650;=,,!" *> ;=13,G! ,J . F*F,,!" .J
p& *" 1C.,. ( " pa
*" ' );,<@"]C)5M ( G*p;=1!42",
< p(z) >S≈ Aaγβ
γ − 1(1 − j k0`h )
2
kara
J1(kara)
J0(kara)(e−j kzz +Bej kzz) ,
!6*" _@ '*C$""@ %B&",#*"/" 1C.! +*?, J1(kara)
J0(kara)≈ kara
2' kara < 1
] _
< p(z) >S≈ Aaγβ
γ − 1(e−j kzz +Bej kzz) . ' )"]
5AN$?!4.,ve
*a ( ( _ F !/" ( B E1C.!" G",! a,!$ -G ; z_@I2* ( ! G@+a1" * !.!" /" -.! , ' );,-]8 z = 0
#
Sve = S < vz(0) >S +Yp < p(0) >S , :$ ( " * $?!.," P"!4.*! @vz
;D1!4< vz(z) >S=
2
a2
∫ a
0
vz(r, z)r dr =2
a2
∫ a
0
[vaz(r, z) + vhz(r, z) + vvz(r, z)]r dr .
5MH,,! B&1!$G*%$!4.vaz
_vhz
vvz
_ ' ) );,"]C_ ' &&) ) -] ' ) ) 9_ ) A-] _01 ( [. * 1 (γ − 1)k0`h
IN NC.! *C$ ( ' ) ) 8(^]C_91 *! - "1" !* ? O$?!4.,> -.,"!? " P"!4.*! @vhz
I &!.E* $ -vaz
vvz
_61 ! ( *W P!PM*@ 11?*# ;=! -P"9,1* -)50;D!" F*#2$?!, ( < vz(z) >S
,%#* @
< vz(z) >S≈ Aakz
k0ρ0c0
γβ
γ − 1(1 −Kv)(e
−jkzz −Bej kzz) . ' ) @-]
50; * ( !4J 1H!4..!$"Yi
*'1?! *,B,2* @Yi =
S
ρ0c0
kz
k0(1 −Kv) =
S
ρ0c0
√
[1 + (γ − 1)Kh](1 −Kv) , ' )-],"!_ " ;="*,2( , ,a*%" P"+1.1.!,.!
Yi ≈S
ρ0c0
[
1 +1 − j√
2
2
a√
k0
(√
` ′v − (γ − 1)
√
`h )
]
. ' )-9 -]5MF1" IJ -
B& %" C.,* . ( ! G*<; :E1" *!!" /" -! , ' );,-]8 z = `
#
B = e−2j kz`Yi − (Yr + Yp)
Yi + (Yr + Yp). ' )-9 (^]
! ! _ 1 ( a. ? * ,!" ' );9 "]% ' )-9 (^] _< ;D* ( !,J 1a1"I.!%F*E. /IYT
;D1,!YT = j sin j kz`
[
Yi +(Ye + Yp)(Yr + Yp)
Yi
]
+ cos j kz` (2Yp + Ye + Yr) . ' )-9 ,"] ,.W!" O*H ; * ( !4. 121"I.!%%*%. /I#,+!* -.! " ;=,!" RCJ !*@ >1*<;D *: ' ) @ "] )A5M ( * E*<;=" * : ,*" 1> > -a&"F131*> ;D* ( !,J 1E1"I.!%>*F. /,* ?, ( ( !1," ! %1" G*@ F13
( " $ ( -2* ( ( +* ( !1" +IW! ( ! " 1!6*<;= [ !," : <)
5M <;= 2" * 1" ,.! @!* +*!,!.!), 1 !2,9 @,"!,!P"!*_0 C.9*1" 1 B! ( !.#$?!1 ( !%H& $ -.* !.%@% 1 1 G*<;D* ( !. 1,1! ? /" 1C.! * 1? >1! -*E$?!1"!4.>*>1!! ( W* 1? >1! -* 1" *1C.!" ( !_* !*$`-" >$2:H @!" * % ;D" *1,!%29 : F,!P"!*# @!*+$?!1 ( !#&"* ( ( 0131,!,!% ? 2H @! >,0 >: E*<; * ( !4. 1&1! :
ζ @!* % R*!,!@.!) ( _ )8,<9 8,<A , )50;= ,C#*21C.W _8!& *R RP". *2@,!2@P". );,))(W* H:,&/ 1 @! _?I*H,1!#1CH !" *<; * ( !4. 1%&1! :B*@,"!? !8 R 1 ( * ;D1,!. >*a1" *!4.!" / !" @!8 ! ( *<;= B$?!., ' _! ( !4J -8 ( !! @!*B*!,!@.!) )5M @!* +,9* 0 E@1 ( ! ! !! ( !7 F a! @ ! `!4. ( !P!*7 y = 0 ! ( H* ;D G$?!4., !)/" ( ~V0
( $#E1 : ' P" )(^] )
%x
&('*),+.-/'*01-2),+1'34+ &('*),+.5468796:04;,-2+
~V0
y
<(=?>.@ BA(DCFEHGHIBJLKNMLOHPQ8R%*OHISHTUJ!EHIR"EHKVR"GHIVE*Q8I$P8IW8X YXNK:Z[R"P8P8JLE,QFPOHW\OHES[TUJLEDWR"]^R_GHILA`aI SHT"JLEbGHX YEHRcSYJLW$T"JdGHR"WINKVQ8R"M^EeGgfhOHEHI M^E[GHIJLKNM^O[PQ1R*OHI SHTUJLE[IiZYJLW8jM^EHR*OHIkR"E[KNR"G[INE,Q1I IVQBTlfmJnI
OyE[M^W8joJLTJLOdSHT"JLEdIVPQ.KVM^E8pM^EHG[OgqnSHJLWrKVM^E,s^INE,Q8R"M^EgqHJtsLINKFT_IuSHT"JLE
xOyA vIFTUJBP8M^WwQ1ILqnT"IvSHW8M[T"NjIxGHI
W8X YInR"MLEyPSzXVKNOHT"JLR"WI$SzMLP8XINPwQuR_EHGHXNSINE[GYJLE,QxGHITUJQ1WM^R"PR"VjoIFKVMnMLW8GHM^E[EHXNIGHIT|fhINPSYJLKVIzA
`aISHW8M HT"VjoIBSzMLP8XIVPQuGHXYE[R~SYJLWxT"IVPuX&9OHJ!Q1R_M^EHPuG[R_zXNW8IVE,Q1R"IVT"T_INP\IVQuW8INT"J!Q1R_M^EHP|iA @,x#|iA;, ,pK&4Av A-,^qYJLO *OHINT_T"IVPxPM^E,QvJLP8PMnKVR"XNIVPvT_INP.KVM^EHGHRQ1R"MLEHP\JLO pW8M^E,Q1R_NWINP
τ = τa + τh = 0 , ~v = ~v` + ~vv = ~V0 , y = 0 , ∀x , A)((
IQvG[INPxKNM^EHG[R_Q1R_M^EHP G fmJLE[E9O[TUJ!Q8R"M^E # T|fhR"E HEHR|A` f XVK JLWwQ R_EHPQ1JLE,Q:JLEHXGHIQ8INjSzXVW1J!Q8OHW8I
τSINOnQ Q1W8IXNKVW8R_QoKNMLjojIT"J POHSINW8SM^PR_Q1R_M^EGgf O[EHI M^EHG[I
SHT"JLEHI$R_EHKNR_GHINE,Q1Iτ+ IQvWX YXVK:ZHR"I τ− pT"I JLKQ1INO[W ej ω t INPwQxM^joR_P:
τ = τ+ + τ− = (τ+a + τ+
h ) + (τ−a + τ−h ) . BA;,L
`aJ KNM^EHG[R_Q1R_M^Eτa = −τh
GHIsLJ!E*QvVQ8W8IBP1J!Q8R"P JLR_Q8I$INEy = 0
IVQxSM^OHW.Q1M^OnQxq T_INP.Q1IVW8jINP
τ+ IQ τ− PM^E,QXVKNW8RQ1Pτ+ =
[
ej kayy + A+h ej khyy
]
e−jkxx ≈ ej kayye−jkxx , BA ^J^
τ− =[
Rae−j kayy + A−
h e−j khyy]
e−jkxx , BA Y
MeTlfmJLjSHT_R_Q1O[GHIGHIiTlf MLEHGHIiR"E[KNR"G[INE,Q1Iτ+a
IVPQBK:ZHM^R_P8R"IiXV],JLT_Ii#DT|fhOHEHR_Q8XLqzMeT|f JLjoS[T"R_Q8OHGHIA+
h
GHIiT|fhM^EHG[IQ8ZHINWjoR9O[I.R"EHKVR"GHIVE*Q8I.INPwQ(E[XN]^T_R"]^XVI GHIVs!JLE,Q KNIVT"T_I GHI\T|fhM^EHGHI\J!KNM^OHPwQ1R9O[I.IVQcM
Ra
G[XNP8R_]^EHI\T"IrKNM9I KNR_INE,QGHIW8X HInR"M^EGHITlf M^E[GHIJ!KNM^OHPwQ1R9O[I |Ra|
R"E pXNW8R_INOHWx# Tlf O[EHR_Q8XBjoJLR_PxsLM^R"PR"EG[IKNIVT"T_I KVR?A `aIVPFEHM^j HW8IVPGgfhM^EHG[IP8J!Q1R_P pM^E,QxJLODW8INT"J!Q1R_M^EHP POHR_s!JLE,Q8INP
k2a = k2
x + k2ay , k2
h = k2x + k2
hy , BA ,
MDT_INP*OYJ!E*Q8R_Q8XNP SM^WQ1JLE,Q.T"IVPrR_EHGHR"KVINPxIQyW8INS[W8XNPINE,Q1IVE,QvT_INP KVM^jSzM^P8JLE,Q1IVP P8OHRs!JLE,Q
xIVQyqka
IVQkhXQ:JLE,QxGHM^EHE[XNPxW8INPSzIVKVQ8R_s^IVjoIVE,QuSYJ!W.T"INP\I[S[W8INPP8R_M^EHP|iA(,(IVQ|iA)(^A
`INPFPM^T"O[Q8R"M^E[PvGHObS[W8MHT_NjISM^OHWvT"Is9RQ1INPP8IiSYJ!WQ1R_KNOHT"JLR"WITUJLjR"EYJ!R"W8I!qHP8O[SzIVW8SM^P8RQ1R"MLE GgfhM^EHGHIVPFR"E KVR"GHIVE*Q8IdrIVQxW8X YXNK:ZHR_Io wq
~v` = (~v+`a + ~v+
`h) + (~v−`a + ~v−`h) BAL
PM^E,QvG[M^EHEHXVINPxSYJLW X&AiA)( LJiIVQ
v±`x = −j kx
[
j
k0ρ0c0
γβ
γ − 1τ±a +
γβ
ρ0c0`h τ
±h
]
, BA-A^J^
v±`y = ±[
j kayj
k0ρ0c0
γβ
γ − 1τ±a + j khy
γβ
ρ0c0`h τ
±h
]
, BA-A Y
IQvSM^OHW.T"JksnRQ1IVP8P8I$Q8M^OHW [R"T"T_M^EHEYJ!R"W8I~vv = ~v+
v + ~v−vSYJLWB X& A iA)( @i# iA-, ,
v±vx = ±A±v
kvy
kxe±jkvyye−jkxx ,
BA LJ^
v±vy = A±v e±jkvyye−j kxx ,
BA YJ s^IVK
k2v = k2
x + k2vy ,
BA-@^IQM gq#DT|fhR"EHPwQ:JLWuGHI TUJ s9R_Q8INP8PI INE,Q1WM^SHR%*OHI R_EHKNR_GHINE,Q1I!qTUJ s9R_Q8INPP8IiQ1M^O[W HR_T"T"MLEHEYJLR_W8IBR"EHKVR"GHIVE*Q8I
~v+v
IVPQNqGYJ!EHP\TUJiP8OHRQ1I!qHEHXN]LT"R"]LXNI$INEyW8IV],JLWGyG[IBT"JksnRQ1IVP8P8IBJLKVM^OHPwQ1R%*OHIR_EHKNR_GHINE,Q1I!A`INPuKVM^EHGHRQ1R_M^EHPx# TUJ2pW8MLE*Q8R"VW8I
y = 0SzM^O[WuT|fhXNKNJLWQvGHIBQ1INjSXNW1JQ1OHWI
τqzIQFSM^OHWxT"INPvKNM^jSM^P1JLE,Q8INP
SYJ!W1JLT_T"NT_INP IQxEHM^W8joJLT_I$JLO SHTUJ!Ey = 0
GHIT"Jis9RQ1INPP8ISYJLWwQ1R_KNOHT"JLR"WI~vqYPNfhXNKVW8R_sLINE,QvW8INPSINKVQ8R_sLINjINE,Q
1 +Ra + A−h = 0
A ,
(1 +Ra)kx
k0
1
ρ0c0
γβ
γ − 1− jA−
h
γβ
ρ0c0kx`h − A−
v
kvy
kx= 0
BA( ,
−kay
k0
1
ρ0c0
γβ
γ − 1(1 −Ra) − jA−
h
γβ
ρ0c0khy`h + A−
v = V0 . BA( (t
INPcQ1WM^R"P GHIVW8EHR_NWINP W8INT"J!Q1R_M^EHPKVM^EHGHO[R_s^IVE,Q q9SYJLW XNT_R"jR"EYJQ1R"MLEGHIVP(SYJ!W1JLjVQ8W8INPA−
h
IQA−
v
q*#$TUJ$W8IVTUJ!Q8R"M^EP8OHRs!JLE,Q1I
j (γ − 1)k0`h (1 +Ra)(k2x + khykvy) + (k2
x − kaykvy) +Ra(k2x + kaykvy)
= kvyk0ρ0c0γ − 1
γβV0 ,
A)(B,^
P8M^RQ qnKNM^jS[Q1I$Q8INE*OG[INPxW8INT"J!Q1R_M^EHP A*rIVQ A @,qYJ sLINKkx << khy
IQkx << kvy
q
khy =
√
−j k0
`h− k2
x ≈√
−j k0
`h
IVQkvy =
√
−j k0
` ′v
− k2x ≈
√
−j k0
` ′v
, A)( ,
kay1 − Ra
1 +Ra=
1 + j√2
√
k0k0
[(
1 −k2
ay
k2a
)
√
` ′v + (γ − 1)
√
`h
]
+k0ρ0c01 +Ra
γ − 1
γβV0 .
A)(*` J SYJLWM^R$INE
y = 0q\J!EHR"jXNIbGgfhOHEHIes9R_Q8INP8PI
V0
q.INPQJLT_M^W8PDKNJLW1JLKQ1XVW8R"PXNI SYJLW P8MLE JLGHjR_QQ:JLE[KNIP8SXNKVR :*OHI @[q ?(^A nA L
ζ = ρ0c0V0 − vay(0)
pa(0),
A)(^M X&AiA)( A^J[q iA)( LJ,
pa(0) ≈ γβ
γ − 1(1 +Ra)e
−j kxx ,
vay(0) ≈ − 1
ρ0c0
γβ
γ − 1
kay
k0(1 −Ra)e
−j kxx ,
KNI*OHR|q[KNM^jS[Q1IFQ8INE*OG[IBT"JkW8IVTUJ!Q8R"M^E A)(*q[KNM^EHG[OHR_Qx#y @nqHX A [A)(!
ζ ≈ 1 + j√2
√
k0
[(
1 −k2
ay
k2a
)
√
` ′v + (γ − 1)
√
`h
]
. A)( A,
5 g <
IVQ8Q8I JLEHEHInIDW8IN]LW8M^OHSI*OHIVT%*OHINPiKNJLT"KVOHT"PGHM^E,QiT_INPkWXNP8O[T_Q:JQ1PPM^E,Q O[Q1R_T"R_P8XNPBGYJLE[PiT"I K:ZHJLSHR_Q8W8I j J!R"P*OHREgfDPM^E,QvSHJLP\GHXVsLINT"MLSHSzXVPuJYEG[IBE[IBSHJLP\INEJLT"M^O[W8GHR_WrT"I$S[W8M^SM^PNA
/ 5/ 2 gµν(z, z0)
`aIDWINSM^WQ G[IDTUJ pM^WjoIdG[IDP8M^T_O[Q1R_M^E HA-, ^BGYJLEHPiTlf X&9OHJ!Q1R_M^E GHIDSHWM^SYJL],JQ1R"MLE GHO SHW8M[T"NjIdGHIWINIVE HA-,*rGHM^E[EHI
∑
µ,ν
ψµν(r0, θ0)
[
∆~w +∂ 2
∂z2 + k2
]
ψµν(r, θ)gµν(z, z0) = −δ(~w − ~w0)δ(z − z0) ."!:A$#
`IuSHWMnGHO[R_Q(P8K J!TUJLR_W8I\GHIFKVIVQ8Q8IuX&9OHJ!Q1R_M^EdSYJLWψµ′ν′(~w)
qnKVM^EHGHOHRQ q*KNMLjoS[Q8IvQ8INE*ODGHIuT"JWINTUJQ1R"MLEoGgfhM^WwQ1ZHM ]^M^EYJ!T"R_Q8XuGHIVP$pM^EHKVQ8R"M^E[PxS[W8M^SHWINPVq
∫ a
0
∫ 2π
0
ψµν(r, θ)ψµ′ν′(r, θ) dθ dr = δµµ′δνν′
"!:A;,^IVQ.G[I$T|fhX *OYJ!Q8R"M^EdG[I$SHW8M^SHJL],J!Q8R"M^E HA-,<A^J,cGHO SHW8M HT"VjoIuJLOds!JLT_INOHWP SHWM^SHW8IVPNqY#kTlf X&*OYJ!Q1R_M^EdPOHR_s!JLE,Q1ISzMLOHW.T"IVP.KNM9I KNR_INE,Q1P
gµν
GHOyGHXVsLINT"MLSHSzIVjoIVE,Q [
∂ 2
∂z2 + k2zµν
]
gµν(z, z0) = −δ(z − z0) , ∀z ∈ (0, `) ,%!:A ,
J s^INKk2
zµν = k2 − k2wµν .
"!:A * KNIVQQ1IeX *OYJQ1R"MLE "!:A ,q.R"TFKVM^E,s9R"INE,QGgf JLP8PMnKVR"INW T_INP KNM^E[GHR_Q8R"M^EHPDIVE pWM^E,Q1R_NW8IVP KNM^E,Q1IVE*OHINPGYJLEHPyT"ISHW8M HT"VjoI HA-,*
(
∂
∂z− j k0ζz0µν
)
gµν(z, z0) = 0 ,SM^OHW
z = 0 ,"!:A&!J,
(
∂
∂z+ j k0ζz`µν
)
gµν(z, z0) = 0 ,SzM^O[W
z = ` .%!:A'
#
IP 9PQ1VjoIBGgfhX *OYJ!Q8R"M^E[Pk"!:A [q !:A'^ INPwQuKVINT_OHR JLO *OHINTaP8J!Q1R_P JLRQ.OHEHIpM^EHKQ1R"MLEGHI$W8IVINEIVEIVP8SYJLKVIYE[Ra#OHE[IBG[R"jINEHPR"M^EDGYJ!EHP\T"I$GHM^joJLR_EHIFGHIYM^OHWR"INWF @[q+A[A #^A [|A`aJdP8M^T_O[Q1R_M^E GHIiKVIiP 9PQ8NjISINOnQVQ1WIiGHXVQ8INWjoR_EHXNIkIVEbQ1W8MLO[s!JLE,QGHINO P8M^T_O[Q1R_M^EHPFR_EHGHXVSzIVEHGYJLE,Q1IVP
g1(z)IVQg2(z)
G[IFT|fhX *OYJ!Q8R"M^EoZHM^jM^]^VEHIuJ!P8P8M9KNR_XNIu#T|fhX *OYJ!Q8R"M^E %!:A ,qn#\WM^EHP *R_INEWg = g1g
′2 − g′1g2pR"EHGHXVSzIVEHGYJLE,QxGHI
z EHM^EyE*OHT|qHIQxP1J!Q8R"P J!R"P1J!E*Q\K:ZHJLKNOHE[I#iOHEHIBKNMLEHGHR_Q8R"M^E J!OpW8MLE*Q8R"VW8INP"!:A&LJnq AY`aJ
pM^E[KVQ1R_M^Egµν(z, z0)
P8MLT"O[Q8R"M^EDGHOySHWMHT"VjoIi%!:A [q !:A&LrPNfhXNKNWR_QvJLT_M^W8P
gµν(z, z0) = −g2(z)
∫ z
0
g1(x)δ(x− z0)
Wgdx− g1(z)
∫ `
z
g2(x)δ(x− z0)
Wgdx .
"!:A-A^`INP P8M^T_O[Q1R_M^EHP
g1
IQg2
G[IxTlf X&*OYJ!Q1R_M^E ZHM^jM^]^VEHILq,P8J!Q1R_P JLR_P1JLE,QcW8IVP8SINKQ1R_sLINjINE,Q.J!O KNMLEHGHR_Q8R"M^E[P IVEpWM^E,Q1R"VW8IVP"!:A&!J[q YqHSINO[sLINE,QFQ1WIBXVKNWR_Q1IVP\P8M^OHP\T"J2pMLW8jIFP8OHRs!JLE,Q1I
g1(z) = cos(kzµνz + ϕ0µν)IQ
g2(z) = cos(kzµν(z − `) − ϕ`µν) ,
Mϕ0m
qϕ`m
P8M^E,QxGHM^EHE[XNPxSYJLW.T"IVP\KNM^EHG[R_Q1R_M^EHP.J!OpW8MLE*Q8R"VW8INP"!:A&LJnq
tanϕ0,`µν = − j k0ζz0,`µν
kzµν.
"!:A L`I\W8M^E[P *R"IVE
Wgs!JLO[QvJLT_M^W8P
Wg = kzµν sin(kzµν`+ ϕ0µν + ϕ`µν) ,
E YEgq\Ggf JLSHWNPyT|fhInSHW8IVP8PR"M^E %!:A A,dGHIgµν(z, z0)
q.PRz < z0
q\JLT"MLW8Pz0 /∈ (0, z)
IQTlf R_E,Q1XN]LW1JLT_I∫ z
0
g1(x)δ(x−z0) dxINPwQ~Q1MLO M^OHWP~E*OHT"T_ILA uIrjoVjoI!qPR
z > z0
q!JLT_M^W8P T|fhR"E,Q1XV]^W1J!T"I ∫ `
z
g2(x)δ(x−z0) dxIVPQxQ1M^O MLOHW8P.E*OHT_T"ILA R_EHP8Rlq[TUJ>pM^E[KVQ1R_M^E
gµν(z, z0)IVPQxGHM^EHE[XNIBSYJLW
gµν(z, z0) = −cos(kzµνz + ϕ0µν) cos(kzµν(z0 − `) − ϕ`µν)
kzµν sin(kzµν`+ ϕ0µν + ϕ`µν),
"!:A-@^J^PRz < z0
qYIQxSYJLW
gµν(z, z0) = −cos(kzµνz0 + ϕ0µν) cos(kzµν(z − `) − ϕ`µν)
kzµν sin(kzµν`+ ϕ0µν + ϕ`µν),
"!:A-@ YPRz > z0
A
/ / / `INPInSHW8IVP8P8R_M^EHPy HA;, LJIQ GHINP pMLEHKVQ8R"M^EHPBSHWM^SHW8IVPkSINW8jIVQQ1IVE*QGgfhXNKNWR"WI T"IVPI[S[W8INPP8R_M^EHPkGHIVP
SHWMnG[OHR_Q8P.P8K J!TUJLR_W8INP.G[IpMLEHKVQ8R"M^EHP.S[W8M^SHWINP< ψt
µν |ψt′
µ′ν′ >r
PM^OHPxT"IVPpM^W8jINP.POHR_s!JLE,Q1IVP
< ψcµν |ψc
µ′ν′ >r =< ψsµν |ψs
µ′ν′ >r=δµµ′
bµνbµν′
∫ ar
0
Jµ(kwµνr0)Jµ(kwµν′r0)r0 dr0"!:A^
< ψsµν |ψc
µ′ν′ >r =< ψcµν |ψs
µ′ν′ >r= 0 ,"!:A$# ,
( +
P8M^RQ< ψc
µν |ψcµν >r=
1
b2µν
a2r
2
[
J2µ(γµνar/a) − Jµ−1(γµνar/a)Jµ+1(γµνar/a)
]
,%!:A # #tJ,
IVQ
< ψcµν |ψc
µν′ >r=1
bµνbµν′
aar
γ2µν − γ2
µν′
[γµν′Jµ(γµνar/a)Jµ−1(γµν′ar/a)
−γµνJµ−1(γµνar/a)Jµ(γµν′ar/a)] ,%!:A # #Y
P8Rν 6= ν ′
A M^OHWOHEHIoW1JLSHSM^WwQ
ar/a = 0, 96qaT"INPBSHWMnGHO[R_Q1PBP8KNJLTUJ!R"W8IVPSM^OHWBT"IVP *OHIVT"P
ν = ν ′s!JLT_INE,QINE,s9R"W8MLE
[q ,nAg`IVP$SHW8M9GHOHRQ1PFI[S[W8R"joJLE,Q$GHIVPKNM^OHS[TUJL]^IVPFR"E,Q8INW8jM9GYJLOP8M^E,QBJLTQ1IVW8EHXVPFIVQGHIiTlf MLW8GHWIkGHI0, 075
qKNM^E[P8R"G[XNW8XVP PXNSYJLWXNjINE,Q q R"T_P EHIPM^E,QdGHM^E[KSYJ!P EHXV]^T"R_]^I J [T"INPVA INSINEHGHJLE,Q q TUJeKNM^E,Q1WR HOnQ1R"MLE GHIKNIVPQ1IVW8jINP$GYJLE[P$T"IkKNJLT"KVOHTT|f JLGHjR_QQ:JLE[KNIJLKNM^O[PQ1R*OHI GHIkQ1W8JLEHP pIVWQIVPQR"E pXNW8R_INOHWIi#
5.10−3 G:uq T"INP$S[W8M GHOHRQ1PkPK JLT"JLR"WINP "!:A$# # InSHWR"joJLE,Q GHINPiKVM^OHSHT"JL]^IVPiINE,Q1WIyG[R_zXNW8IVE,Q1PijMnGHIVPkP8MLE*QGHMLEHKDEHXN]LT"R"]LXNPkJ!OSYJLW8JL]^W1J!SHZHI HA W8IVTUJ!Q8R"M^E [A LA
`cfhInSHWINPP8R"MLE %!:A*q
k2zµν = k2 −
γ2µν
a2 ,
SzIVO[QxKNM^E[GHOHR"WIB# O[EHIBs!JLT_INOHW.R_j J!]^R"EYJ!R"W8IxSM^OHWkzµν
PRγ2
µν/a2 > Re(k2)
qYKNI?9O[RKNM^WW8INPSzMLEHG #iGHIVPM^EHGHIVP\XVs!JLEHIVP8KNIVE,Q1INPuP8OHRsLJ!E*Q
zA
`cf JLGHjR_QQ:JLE[KNIζzµν = ζz0µν = ζz`µν
P8O[WxT_INP.SYJ!W8M^R_P.P8R_Q8OHXNIVP\INEz = 0
IQz = `
INPwQvJLT"M^WP.XNKVW8R_Q8I
ζzµν ≈ 1 + j√2
√
k0
[
γ2µν/a
2
k20
√
` ′v + (γ − 1)
√
`h
]
SM^OHWRe(k2) ≥
γ2µν
a2
ζzµν ≈ 1 + j√2
√
k0
[
√
` ′v + (γ − 1)
√
`h
] SM^OHWRe(k2) ≤
γ2µν
a2 .
%!:A #B,^
! / 0 cT_OHP8R_INOHWPpM^W8jINP\SINO[sLINE,QvVQ1WIINE,s9R"P1J!]^XNIVPvSM^OHW.T_I$K:ZYJLjSGHI$s9R_Q8INPP8IBGHIT|fhXNjIVQ8Q8INOHW
ve
OHE[IBs9R_Q8INPP8IOHEHR)pM^W8jI
ve = Ve
q OHE[IBs9R_Q8INPP8IEHM^EyOHEHR)pM^W8jILqnR"jSzM^PXNI$SYJ!W.T"INP\KNJLW1JLKQ1XVW8R"PwQ1R9O[INP.jXNK J!EHR%*OHIVPxG[IBT"JijoIVj HW1J!EHILq
ve(r) = VeJ0(ker) − J0(keae)
J2(keae),
%!:A # ,
GHMLE*QuTUJis!JLT"IVOHWvjM ^IVEHEHIIVPQVe
IVQuMke = ω
√
MS/TMS
XQ:JLE,QvTUJ j JLPP8IP8OHW JLKNR9O[IIVQTT"JiQ8INEHPR"M^E GHIT"J jINj HW1JLE[I$GHOjR"KVW8M^S[ZHM^EHIFXVjoIQ8Q8INOHW1A
OHE[I s9R_Q8INPP8IcEHM^EBOHEHR)pM^W8jILqVXNKNWR_Q8I PM^OHPaTUJpM^WjoI G f OHEBGHXs^INT_M^SHSINjINE,Q P8OHWaT"J HJLP8IcGHIVP pM^E[KVQ1R_M^EHPSHWM^SHWINP
ψc0ν(r)
qHT"IVP.joM9GHIVP.P8OHSXNWR"INO[W8Pv# #XQ:JLE,QxEHXN]LT"R"]LXNPNqve(r) = Aψc
00 +Bψc01(r) ,
PM^R_Qve(r) =
A
a0b00+
B
a0b01J0(kw01r) ,
"!:A$# *T_INP\KNMLEHPQ1JLE,Q1INP
AIVQBXVQ1JLE,QxGHXVQ8INW8jR"E[XNINP\SM^OHW.P8J!Q1R_P JLR_W8IFT_INP\KNM^E[GHR_Q8R"M^EHPrP8OHRsLJ!E*Q8INP
ve(ae) = 0IQ
< ve >e= Ve ."!:A$# L
IVPvG[INOyKVM^EHGHRQ1R"MLEHP%!:A # ^rSINWjoIQ8Q8INE,QxGgf XVKNWR"W8IA
B= −b00
b01J0(kw01ae)
IVQVe =
A
a0b00+
B
a0b01
2
γ01
a
ae
J1(kw01ae) ,
KVI9O[R G[M^EHEHIA =
Ve
1 − αa0b00
IVQB = − Ve
1 − α
a0b01J0(kw01ae)
,"!:A$# A^
J sLINKα =
2
γ01
a
ae
J1(kw01ae)
J0(kw01ae).
"!:A$# L` fhInSHWINP8PR"M^EGHOyK:ZYJLjSyG[Is9RQ1INPP8I
ve
G[IBTlf XVjoIQ8Q8INOHW.IVPQxGHM^EHKGHMLEHEHXNISYJ!W
ve(r) =Ve
1 − α
[
1 − J0(kw01r)
J0(kw01ae)
]
."!:A$# @^
EHM!Q1INW*OHIuKVINPrQ8W8M^R_PrInSHWINPP8R"MLEHP SHWM^SzMLP8XNIVP SM^OHWrT_IvK:ZHJLjoS GHIvs9R_Q8INPP8Ive
E[IuSINO[sLINE,Q\SYJLP Q1WIKVM^EHP8R_GHXNWXNIVP$KNM^jjoI!I[JLKQ1IVP^A EbIVzIVQNqgTUJjoIVj HW1J!EHIXQ:JLE,Q nIkPOHW$T"IVP M^W8G[PNq PM^EbjM^O[s^IVjoIVE,QEHI$SINOnQ\VQ1WI$KNIVT"OHRzGgf O[EyS[R"PQ8M^EDSHT"JLE SHWINjR"VW8IFI[S[W8INPP8R_M^E A fmJ!O[Q1WIBSHJLWQNq[T"IuK:ZYJLjSDGHIFs9R_Q1IVP8PIBG[IT"JjINj [W1JLEHIGHMLR_Qv#oT"J pM^R_P\GHXNSINEHG[W8IG[INPuKNJLW1J!KVQ1XVW8R_PQ1R*OHINPxjoXVK JLE[R%*OHINPxGHIKNINT_T"I KNRIQuG[INPuK:ZHJLjoS[PJLKVM^OHPwQ1R%*OHIVPvPOHW\P8IVPvGHIVO JLKVINPVA
/ / / ! / 0
`INP\SHW8M9GHOHRQ1P.PK JLT"JLR"WINP< ve|ψt
µν >e
GHXNSINEHGHJLE,QuGHIT"JsnRQ1IVP8P8Ive
P8M^E,QxGHM^E[EHXNPxSYJLW
< ve|ψtµν >e=
∫ 2π
0
∫ ae
0
ve(r0, θ0)ψtµν(r0, θ0)r0 dr0 dθ0 .
`aJks9R_Q1IVP8PIve
IVPQ\P8O[SHSzMLP8XNIBR"E[GHXNSINEHGHJLE,Q1I$GHIT|f JLEH]LT"IθqHP8R HR"IVE*OHI$PINOHTT_INP Q8INW8jINP.KVM^W8WINPSzM^E[GYJLE,Q
#µ = 0
P8M^E,QxEHM^EyE*OHT"PBpX A HA-,LJkIVQ R"T s9R"IVE,Q
< ve|ψcµν >e =
2πδµ0
a0b0ν
∫ ae
0
ve(r0)J0(kw0νr0)r0 dr0%!:A # ^J,
< ve|ψsµν >e = 0 .
"!:A$# W8M^R_PFInSHWINPP8R"MLEHP$INE,s9R"P8JL]^I JHT"IVP$SzM^O[WFTUJ snRQ1IVP8P8I
ve
PM^E,QGHM^EHE[XNINP$GHJLEHP$T"ISYJLW8JL]^W1J!SHZHIkSHWXNKVXNGHIVE*QNqT_INPuInSHW8IVP8P8R_M^EHP$GHIVP$SHW8M9GHOHRQ1PxP8KNJLTUJLR_W8IVP
< ve|ψc0ν >e
SzM^O[WuK:ZYJ<*OHI>pMLW8jIGHIve
P8M^E,Q$G[M^EHEHXVINPFGYJ!EHPT_INP\SYJLW8JL]^W1J!SHZHINP\POHR_s!JLE,Q1PVA
( ' ( ( ( ' ' (
! #"$&%'%')(*+ -,/.1023
` J s9R_Q8INPP8IyGHI T|fhXNjIVQ8Q8INOHWiIVPQoP8OHS[SzM^PXNIyOHEHR)pM^W8jIve = Ve
A ` fhInSHWINP8PR"M^E %!:A # ^J,G[O SHWMnG[OHR_QP8KNJLTUJLR_W8I
< ve|ψc0ν >e
G[M^EHEHIBJLT_M^W8P
< ve|ψc0ν >e=
2π
a0b0νVe
∫ ae
0
J0(kw0νr0)r0 dr0 ,
P8M^RQ< ve|ψc
0ν >e=1
a0b0νSeVeV0ν ,
%!:A-, ,M
V0ν
IVPQxOHE5 JLKVQ8INOHW\P1J!EHP\GHR"jINEHPR"M^E[P$9O[RgsLJ!O[Q
V0ν =
1P8Rν = 0
a
ae
2
γ0νJ1(kw0νae)
P8Rν 6= 0 .
%!:A-, #tJ,
#4 56! #"7&%7%'8*+.*9(+* :,/.103; -<.%'=610?>:&%A@B10$@C"7=D02 :%E"7 :F(+&%3=&@B*+ -F(&%G)>:93H)I0$*
` Jks9R_Q8INPP8IBGHIT|fhXNjIVQ8Q8INOHW.IVPQxP8O[SHSzMLP8XNIBEHM^EyOHEHR)pM^W8jIFIQvS[W8INE[GT"J pM^W8jI X&A !:A # ,
ve(r) = VeJ0(ker) − J0(keae)
J2(keae).
E J!R"P1J!E*QFOHP1JL]LIiGHINPW8IVTUJ!Q8R"M^E[Pi'9A& ,9A$#Nlq @[A #^A-,lq nA'HA #NIQ @[A HA-,~GHIiTUJdWX&pXNWINEHKVIy # IVQGHIT|fhX *OYJ!Q8R"M^E HA-,<@,qzTlf I[S[W8INPP8R_M^E %!:A # ^J,vG[ObSHW8M9GHOHRQ$P8KNJLTUJLR_W8I
< ve|ψc0ν >e
IVPQXVKNW8RQ1IiPM^OHP$TUJojoVjoIpM^W8jI *OHIBT|fhInSHWINP8PR"M^E %!:A-, ,
< ve|ψc0ν >e=
1
a0b0ν
SeVeV0ν ,
J s^INK
V0ν =
1P8Rν = 0
a
ae
2
γ0ν
ke
k2e − k2
w0ν
kw0νJ1(keae)J0(kw0νae) − keJ0(keae)J1(kw0νae)
J2(keae)
P8Rν 6= 0 .
%!:A-, #
KJ ! #"$&%'%')=D&>:.1L=)%'(+0M>-&%63.!+&%?0$.+02&%GN>-O@PQ R"7=
`aIK:ZYJLjS GHI s9R_Q8INP8PIve
G[IoTlf XVjoIQ8Q1IVOHWBINPQGHXVsLINT_M^SHSX P8O[WT"IVPjM9GHINPBSHW8MLSHW8IVPGHIoTUJ K J s9R_Q1XIVQSHW8IVEHGyTUJ pM^WjoIpX A !:A$# @,
ve(r) =Ve
1 − α
[
1 − J0(kw01r)
J0(kw01ae)
]
.
`INPkjMnG[INPPOHSXNW8R_INOHWPi# #dXQ:JLE,QiEHXV]^T"R_]^XNPBGYJLE[PkKNIQ8Q1IDInSHW8IVP8P8R_M^EgqT_INPSHW8M9GHOHRQ1PPK JLT"JLR"WINPEHIdP8MLE*QK JLT_KNOHT_XNP$*OHISM^OHW.T"IVP\joM9GHINPB [q ,rIVQ nq$#^qhKtA
cE JLR_P1JLE,QaOHP8JL]^I GHINP W8IVTUJ!Q8R"M^EHP'nA' ,nA #Nlq*'9A&[A$#N^IVQ( @nA HA;,,GHIcTUJ.W8X pXNWINEHKVIF # !,IQ G[INP X *OYJ!Q8R"M^EHP HA-,LJxIVQ HA;, @,qT|fhInSHWINPP8R"MLE %!:A # ^J,aGHOkSHWMnG[OHR_Q P8KNJLTUJLR_W8I
< ve|ψc0ν >e
IVPQ XVKNWR_Q1I(P8M^O[P~TUJxjoVjoIpM^WjoI*OHIT|fhInSHW8IVP8PR"M^E %!:A-, ,
< ve|ψc0ν >e=
1
a0b0νSeVeV0ν ,
J s^IVK
V0ν =
1PRν = 0
−J0(kw01ae)
[
1 +1
1 − α
[
J1(kw01ae)
J0(kw01ae)
]2] PR
ν = 1 .
%!:A-, # K
/ / pc0ν(`)
pc0ν(0)
`I$KNM9I KNR_INE,Qpc
0ν(`)IVPQvGHM^EHE[XBSHJLWB HA'^
pc0ν(`) =
S < ve|ψc0ν >e
1 −Kv
j k0
S
ρ0c0
1
g0ν(0, `)+ Yr0ν
g0ν(`, `)
g0ν(0, `)
."!:A;, ,L
` fhInSHW8IVP8PR"M^E GHI T"J pM^EHKQ1R"MLEg0ν(z, z0)
INPwQ GHM^EHEHXVIbSYJLWdT_INPdWINT"J!Q1R_M^EHPb"!:A-@LJ[q YA MLjoS[Q8IQ1IVE*O G[ITlf XV],JLT"RQ1XyGHINP J!GHjoRQ8Q1JLEHKNIVP
ζz0µν
IVQζz`µν
T"INP 9OHJLE,Q1R_Q8XNPϕ0µν
IVQϕ`µν
P8M^E,Q XN]^JLT"IVPdIVQg0ν(0, `)
IQg0ν(`, `)
PNfhXNKNWR_sLINE,Q
g0ν(`, 0) = g0ν(0, `)
= − cos2 ϕ0ν
kz0ν sin(kz0ν`+ 2ϕ0ν),
"!:A;, ^g0ν(`, `) = g0ν(0, 0)
= − cosϕ0ν
kz0ν sin(kz0ν`+ 2ϕ0ν)cos(kz0ν`+ ϕ0ν) ,
"!:A;,,J s^IVK
ϕµν = ϕ0µν = ϕ`µν
GHM^EHE[XBSHJLW X A !:A ^JLR_EHP8R INEySYJLWQ8R"KVOHT"R_INW
1
g0ν(`, 0)= −kz0ν
[
(1 − tan2 ϕ0ν) sin kz0ν`+ 2 tanϕ0ν cos kz0ν`]
=
[
j kz0ν + jk2
0
kz0νζ2z0ν
]
j sin kz0ν`+ 2j k0ζz0ν cos kz0ν` ,"!:A;, L
IQg0ν(`, `)
g0ν(`, 0)= cos kz0ν`− tanϕ0ν sin kz0ν`
= cos kz0ν`+k0
kz0νζz0νj sin kz0ν` ,
"!:A;,<A^PM^R_Q
1 −Kv
j k0
S
ρ0c0
1
g0ν(`, 0)=
[
Yi0ν +Y 2
p0ν
Yi0ν
]
j sin kz0ν` + 2Yp0ν cos kz0ν` ,"!:A;,L
IQg0ν(`, `)
g0ν(`, 0)= cos kz0ν`+
Yp0ν
Yi0ν
j sin kz0ν` ,"!:A;,<@^
MYp0ν =
S
ρ0c0ζz0ν
IQYi0ν =
S
ρ0c0
kz0ν
k0
(1 −Kv) .
< ψc0ν >e,r
E JLR_P1JLE,Q.O[P1JL]^IGHIKVINPxGHINOyG[INW8E[R"NWINP\W8IVTUJ!Q8R"M^E[P GYJLEHP\Tlf I[S[W8INPP8R_M^E "!:A;,<,^rGHIpc
0ν(`)qHR_T s9R"INE,Q
pc0ν(`) =
S < ve|ψc0ν >e
YT00ν + Yr0ν
[
Yp0ν
Yi0ν
j sin kz0ν`+ cos kz0ν`
] ,%!:A-, ,
MYT00ν =
[
Yi0ν +Y 2
p0ν
Yi0ν
]
j sin kz0ν` + 2Yp0ν cos kz0ν` .%!:A ,
`cfhInSHWINPP8R"MLEGHIpc
0ν(0)IVPQvGHXNG[OHR_Q8I$GHIo HA'<@,
pc0ν(0) =
j k0
1 −Kv
ρ0c0S
g0ν(0, `)
[
S < ve|ψc0ν >e
g0ν(0, 0)
g0ν(0, `)− Yr0νp
c0ν(`)
]
,
*OHRGHM^EHE[ILqYINE5 JLR_P1JLE,Q\OHP8JL]^IGHIo%!:A-, ,^q
pc0ν(0) = pc
0ν(`)
[
g0ν(0, 0)
g0ν(`, 0)+ Yr0ν
j k0
1 −Kv
ρ0c0S
g20ν(`, `) − g2
0ν(`, 0)
g0ν(`, 0)
]
.%!:A #
`INP I[S[W8INPP8R_M^EHPB%!:A-, ,cIVQB"!:A;,* GHIVPpM^EHKQ1R"MLEHPg0ν(`, 0)
IQg0ν(`, `)
SzIVW8jIVQ8Q8INE,Q.GgfhXNKVW8R_W8IFT"JW8IVTUJ!Q8R"M^EP8OHRs!JLE,Q1I
g20ν(`, `) − g2
0ν(`, 0)
g0ν(`, 0)=
sin kz0ν`
kz0ν.
%!:A ,^ E J!R"P1J!E*QuOHP1J!]^IiGHIiKNIQ8Q8IiGHINWEHR"VW8IiW8IVTUJ!Q8R"M^E JLR"EHPR *OHIiGHIiT"J W8IVTUJ!Q8R"M^E %!:A-,<@,vGYJ!EHP$T|fhInSHWINP8PR"M^EeGHIpc
0ν(0)qHR"T s9R_INE,Q
pc0ν(0) = pc
0ν(`)
[
Yr0ν + Yp0ν
Yi0ν
j sin kz0ν` + cos kz0ν`
]
.%!:A ,
/ 5/< ψc0ν >e,r
`aIVP?pMLEHKVQ8R"M^EHP$S[W8M^SHWINPψc
µν
PM^E,QG[XVQ1IVW8jR"EHXVINPGYJLE[PT_ISHJLW1JL]LW1JLSHZ[I [A X& A [A;,LAa`INP$s!JLT"IVOHW8PjoM LINEHEHIVP
< ψc0ν >e,r
PM^E,QFJLT_M^W8PrGHM^EHEHXVINPxSYJLW
< ψc0ν >e,r=
2
a2e,r
∫ ae,r
0
1
a0b0νJ0(kw0νr)r dr ,
P8M^RQ
< ψc0ν >e,r=
1
a0b00
P8Rν = 0
1
a0b0ν
a
ae,r
2
γ0νJ1(kw0νae,r)
P8Rν 6= 0 .
%!:A *
/ 5/ / < p(`) >r
2 0
YEGHIQ8INEHR_WxKVM^jS[Q1IBGHOjM^O[sLINjINE,QFE[M^EOHE[RpM^WjoI$GHITUJ joIVj HW8JLEHIGHO WXNKNIVS[Q1IVOHWuIQuG[IP8MLER"E HOHINEHKVIoPOHWBTUJDP8IVEHP8R HR"T_R_Q1XiG[O jR"KVW8M^SHZ[M^EHI GYJLEHPBP8MLE jMnGHVT"IiGHIK:ZYJLjS JLKNM^O[PQ1R*OHILqxJLP8jkO[P8P8IVE
R_E*Q8W8M9GHOHRQkGYJLE[PiT"IdKNJLT"KVOHTrGHIDT"JSHWINP8PR"M^E jM ^INE[EHI< p(`) >r
POHWiT"IdWXNKVINS[Q8INOHW OHEHIDSM^EHG[XNW1JQ1R"MLEur(r)
GHITUJ pM^WjoI$POHR_s!JLE,Q1I ur(r) =
J0(krr) − J0(krar)
J2(krar),
"!:A LM
krINPwQvG[XYEHRaSM^OHW.T_I$W8XNKVINS[Q8INOHWxGHITUJkjNjI J-VM^E59O[I
keSM^OHW.Tlf XVjoIQ8Q1IVOHWB X& A !:A # ,A
`aJ jM ^IVEHEHIBSzM^E[GHXNWXNI< p(`) >r
IVPQuJLT"M^WP XNKVW8RQ1I
< p(`) >r =1
Sr
∫
Sr
ur(r0)p(r0, `) dS0
=2
a2r
∑
ν
pc0ν(`)
∫ ar
0
ur(r0)ψc0ν(r0)r0 dr0 ,
KVI?*OHR GHM^EHEHI!qHKNMLjoS[Q8IFQ1IVE9O GHIVP\InSHW8IVP8PR"M^EHP"!:A LrIVQB HA;, ^(GHIurIVQ GHINP
ψ0νqHIVQ\IVE5 J!R"P1J!E*Q OHP1JL]LI
GHIVPvWINT"J!Q1R_M^EHP$ nA' ,nA #N|qg @[A #LA;,lq nA'HA #NIVQB @[A HA-,G[IBT"JkW8X pXNW8IVEHKNI #!IVQxGHITlf X&9OHJ!Q1R_M^E HA-,<@,q
< p(`) >r=∑
ν
1
a0b0ν
pc0ν(`)U0ν ,
"!:A A^J s^IVK
U0ν =
1P8Rν = 0
a
ar
2
γ0ν
kr
k2r − k2
w0ν
kw0νJ1(krar)J0(kw0νar) − krJ0(krar)J1(kw0νar)
J2(krar)
P8Rν 6= 0 .
"!:A L
d 5
`cfhM IQ~G[IrKNIQ8Q8I JLEHE[InIrIVPQG[IrSHWXNP8IVE,Q1INW HW8R_Vs^IVjoIVE,Q~T_I(KNJLT"KVOHT*GHINP R"EHKVINWQ8R_Q8OHGHINP P8O[W T_INP W8XNPOHT_Q1J!Q1PGgfhXVQ:J!T"M^EHEHJL]^ILA SHW8VP *OHIVT%*OHINP W1JLS[SzIVT"P IVQ GHXYE[R_Q1R_M^EHPaP8O[W T_INP K JLT_KNOHT_P G f R_EHKNIVWQ1RQ1OHG[INPNqOHE[I JLSHS[T"R"KNJ!Q1R_M^EJLOWXNPOHT_Q1J!Q1P GgfhXVQ:J!T"M^EHEHJL]^INPaMnQ1INE*OHP SYJLWaW8XVKNR_SHW8M9KNRQ1X INPwQ~S[W8XNPINE,Q1XVILAL`aIVP R_EHKNIVWQ8R_Q1O[GHINP JLR"E[P8R,M[Q8INE*OHIVPJLOy` PM^E,QvIVEHP8O[R_Q1IGHXQ:JLR_T"T_XNINPrIVQvJLEYJLT 9P8XNIVPNA
. $/ / / / 0 `cfhR"E[KNINWwQ1RQ1OHGHI Q 9SIdINPwQiT|fhR"EHKVINWwQ1R_Q8OHGHIoGgf O[EHIDjoIVP8OHWIdInSHWR"jXNIdPM^OHPiT"JpM^W8jI G f OHE XVK JLWwQ |Q nSILA
` f R_EHKNIVWQ1RQ1OHG[I Q 9SIDGHI T|fhINPwQ1R"joJ!Q8R"M^EyGgfhOHEHI ]^W8JLEHGHIVOHW
Yq EHMLQ8XNI
u(y)IVPQTUJ W8JLKNR_EHIDK JLWW8XVIyG[IyT"J
s!JLW8R"JLEHKNI$G[Iyq[E[MLQ1XVI
V (y)A
`cfhR"E[KNINWwQ1RQ1OHGHI Q 9SIiKNM^jSzMLP8XNI IVPQT|fhR"EHKVINWQ8R_Q8OHGHI |Q 9SzIkG[O WXNP8O[T_Q:JQG f OHE[IjINPOHW8I T_M^W8P 9O[IKVIoWX P8OHTQ:J!QBIVPQkM [Q1IVE9O #ySYJ!WQ1R_WGHINPBsLJ!T"INO[W8PBGgfmJ!O[Q1WINP]LW1JLEHG[INOHWPNA cT_T"I INPwQXV],JLT_I #yTUJDW1JLKVR"EHIK J!W8W8XVI GHITUJP8M^jjoIBGHIVPus!JLW8R"JLEHKNIVPuM^OKNMts!JLW8R"JLEHKNIVPFGHIKNINPJ!O[Q1WINPF]^W8JLEHGHIVOHW8PVq SM^EHGHXVW8XVINPFSYJ!WuTUJ s!JLW8R"J!Q1R_M^EG[OW8XVP8OHTQ:J!Q\GHIjINPOHW8IINE5pM^EHKQ1R_M^EDGHIKNIVT"T"I$G[IBKVINPx]^W1J!EHGHINO[W8PNA
R"E[P8R|q9P8MLR_Q(OHEHI$]^W8JLEHGHIVOHWYG[XVQ1IVW8jR"EHXVI$#SYJ!WQ1R_WrGHI
nJLO[Q8W8INPr]^W1JLE[GHINOHWP
X1
qX2
qA_AAqXn
SYJ!WrTUJpM^EHKQ1R"MLE
fQ8INT_T"I*OHI
Y = f(X1, X2, ..., Xn) ,P8M^RQyIQxi
GHIVPINPQ8R"joJ!Q1R_M^EHPFWINPSzIVKVQ1Rs^IVjoIVE*QGHIYIVQXi
qgTlf R_EHKNIVWQ8R_Q1O[GHI |Q nSIiKNMLjoSM^P8XVIuc(y)
INPwQGHM^EHE[XNISYJLW
V (y) = u2c(y) =
n∑
i=1
(
∂ f
∂xi
)2
u2(xi) ,[A$#
M T"IVPu(xi)
P8MLE*QxT_INP.R"E[KNINWwQ1RQ1OHGHIVP |Q nSIGHINPxXVs!JLT_OYJ!Q8R"M^EHPxi
G[INPXi
A`cfhR"E[KNINWwQ1RQ1OHGHIXNT"JLW8]^R_I INPQdOHEHIb]^W8JLEHGHIVOHW GHX YEHR_P8P1J!E*Q O[E R"E,Q1IVWs!JLT_T"IJLO[Q1MLOHW GHO W8XNPOHT_Q1J!QdG f OHE[I
joIVP8OHWI GYJLE[PdT"I&9O[INTuG[M^R_Q P8I Q1WM^O[sLINWdOHE[IbSYJLWQ8R"I nXNI K 4ArR_E8pW1J,iGHIVPdsLJ!T"INO[W8P*OHRvSzIVO[s^IVE*QyQ1W8IQ1WM^O[s^XVINPxSzMLOHW\TUJi]^W1J!EHGHINO[WxjINPOHW8XVILA` f R_EHKNIVWQ8R_Q1O[GHI$XNT"JLW8]^R_I
UGHIBT|fhINPwQ1R"joJ!Q8R"M^E
yGHITUJi]^W8JLEHGHIVOHW
YINPwQvG[XNGHOHRQ1IGHIT|fhR"E[KNINWwQ1RQ1OHGHI Q 9SIu(y)
SYJLW\T"JkW8INT"J!Q1R_M^E
U = ku(y) ,[A-,^
MkINPQvJLS[SzIVT"X JLKQ1INO[WxG f XVTUJLW]^R"PP8INjINE,QNA
#
R_EHP8RlqKVM^EHPR"GHXVW1JLE,QyKVM^jojI TUJxjoIVR"T"T_INOHWIVPQ8R"joJ!Q1R_M^EG[I TUJ\sLJ!T"INO[W GHI
YqT|fhR"E,Q1IVWs!JLT"T_I
[y−U, y+U ]GHMLR_Q.KVM^E,Q1INE[R"W.OHEHI]^W8JLEHGHI pW8JLKVQ8R"M^EpG[INP.s!JLT"IVOHW8P$*OHRSINO[sLINE,QvVQ1WIFQ1WM^O[s^XVINPxSzMLOHW
YAY`aJi]^W8JLEHGHIVOHW
pINPwQxT"I$EHRs^I J!OGHIKVM^E JLE[KNIGHIBT|fhR"E,Q1IVWs!JLT"T_I
[y − U, y + U ]A
M^O[WBM [Q1IVEHR"WFO[E JLKQ1IVOHWGgf XVTUJLW]^R"PP8IVjoIVE*QkKVM^W8WINPSzM^E[GYJLE,QJLO E[R_s^INJLOeGHI KNM^E JLEHKVI
pqR"T J!O[Q
KVM^EHEYJ_Q1WI TUJrT_M^RLGHI S[W8MYJHR"T_R_Q1X GHIYK J!W1JLKQ1XNWR"PXNI SYJLWT_IW8XVP8OHTQ:J!QaG[I joIVP8OHWI IVQaT|fhXNKNJLWQ Q 9SI KNM^jSzMLP8XLA
IQ8Q1IT"MLR~INPwQBP8M^O[sLINE,QOHEHI T"M^R~EHM^W8joJLT_IDpQ1J HT_I JLO [A #uj J!R"PFOHE[IT_M^R~WINKQ:JLEH]LOHTUJLR_W8IiSINO[QBJ!OHP8PR Q1WIK:ZHMLR"P8R_ILA
xR_sLI JLOyGHIKNM^E JLEHKVIp
JLKVQ8INOHW\GgfhXNT"JLW8]^R_P8PINjINE,QkpSzMLOHW8KVINE,Q:JL]LIt
A @[q-, # #^q A #^q A
nq , ,nq'<A
[q
@ [A # JLT_INOHWPkGHIdTlf XVTUJLW]^R"PP8IVjoIVE*Qk*OHR G[M^EHEHIdO[E R"E,Q1IVWs!JLT"T_IoKVM^W8WINPSzM^E[GYJLE,QJLO EHRs^INJLO GHI
KVM^E JLE[KNIpqYINEyP8O[SHSzMLP1JLE,QxOHEHIT"MLRgEHM^Wj JLT_ILA
FJLEHP T"I\KNJLPcGgf O[EHIuT_M^RHGHIxSHW8M YJ HR_T"R_Q8X W8IVKVQ1JLEH]^OHT"JLR"WILq^T_I.EHR_sLI JLOoGHIxKNM^E YJLEHKNIpINPQ(GHI #SzMLOHW
OHE5 JLKQ1INO[WxG f XVTUJLW]^R"PP8INjINE,Qk ≥
√3A
`I SHW8M IVQDGHIbWX&pM^WjoIGHIbTUJ EHM^WjoI ! A # , #!FSHWM^SzMLP8I OHE[IbT"R"PwQ1IyGHINPDSHJLW1JLjVQ8W8IVPD#SHWINEHG[W8IBINEyKNM^jS[Q1I$GHJLEHP\T"I$KNJLT"KVOHTGHINP\R_EHKNIVWQ1RQ1OHG[INPNA`I$K JLT_KNOHTGHITlf R_EHKNIVWQ1RQ1OHG[IFP8OHW\Tlf I oKNJLKNRQ1X$IVPQvIVzINKVQ8OHXINEyGHINO ySYJLWQ8R"IVP Xs!JLT"OYJQ1R"MLE GHIFTlf R_E YOHIVEHKNIuGHINP SHJLW1JLjVQ8W8IVP INE,Q1W8JLE,Q.GHR_W8INKQ1IVjoIVE*Q INEDKNMLjoS[Q8IFGYJLE[P T"IvK JLT_KNOHTzGHOS[W8M9GHOHR_Q\G[INP\I K J!KNR_Q8XNP [A-,^q
R_E YOHIVEHKNIGHIVPSYJLW8JLjVQ1WINP*OHRcEHIW8INE,Q8W8INE,QSYJLPBGHR"WINKQ1INjINE,QBGYJLE[PT"JpMLW8jkOHT"J!Q1R_M^EbGHO K JLT_KNOHTG[OSHWMnG[OHR_Q\GHIVP\I K JLKVR_Q8XNP [A ,A
! / 2 / / / ! !
` f I oKNJLKNRQ1XMp
GgfhOHEjR"KVW8M^S[ZHM^EHIFE f IVPQvJLKNKVINPP8R [T"I*OHISYJLW.T"I [RUJLR_P GHINP.Q8W8M^R_P.SHW8M9GHOHRQ1P.GgfhI K J KVR_Q1XVP\GHXVQ8INW8jR"E[XNP\SYJLW.T"JkjoXQ1ZHM9GHIGHIW8XVKNR"S[W8M9KNR_Q8XLA
4 M^RQOHEkQ8W8R"S[T"IVQ~GHI.jR"KNWM^SHZHMLEHINPG f I oKNJLKNRQ1XVPMp1
qMp2
qMp3
IQcGHM^E,Q T_INPSHWMnG[OHR_Q8PGHINP I K JLKVR_Q1XVPPM^E,QvE[MLQ1XVP
M12
qM13
qM23
qYT|fhI KNJLKNRQ1XMp1
qYGHXQ1IVW8jR"EHXVI$SYJLW\TUJkjXVQ8ZHMnG[I$GHIW8XNKVR"SHWMnKVR_Q8XLqHPVf XVKNW8RQ
Mp1 =
√
M12M13
M23
.[A ^
f JLSHW8VP\T"INP\WINTUJQ1R"MLEHP #LA$# LrIVQ ,9A @^
MpeMpr = ZEYT ,JtsLINK
ZE =−1
jωCref
uA
uB
u′Bu′A
,
(* $ ( ( ( ( ( (
IVQ 9O[INT *OHI PM^R_Q T"I(joM9GHVT"I SHZ nPR%*OHI.K:ZHMLR"P8RnSzMLOHW~KNJLT"KVOHT"IVW T_INPJ!GHjoRQ8Q1JLEHKNIVPJLKVM^OHPwQ1R%*OHIVPGHIrQ8W1JLE[P pINWwQ qKNIVPxS[W8M9GHOHR_Q8P.GgfhI K JLKVR_Q8XNPxGHXNSINEHG[INE,Q
#^A.GHIiSHJLW1JLjVQ8W8IVP$R"EHGHXVSzIVEHGYJLE,Q8P$GHINPjR"KNWM^SHZHMLEHINPFOnQ1R"T_R"PXNPuIQGHIiTUJojoIVP8OHWIiW8X J!T"R"PXNI pEHMLQ8XNPXq
P8JLEHPxR"EHGHR_KNI Cref
TUJkKNJLSYJLKVR_Q8XBG[IBjINPOHW8I!q`c, a
T"JkT"M^E[]^OHINO[W.IVQxT"I$W8J LM^EyGHOKVM^OHSHT_INOHWVq`t, at
T"JiT"M^EH]LOHINOHW.IQ\T"I$W8J LM^EG[INP.Q1O INP\KNJLSHR"T_TUJLR_W8IVPNqfTUJ pW8X&*OHINEHKVILq
,nA.GHIeSYJ!W1JLjVQ8W8INP T_R"XNP OHEHR%*OHIVjoIVE,Q1PJLO jR"KVW8M^SHZ[M^EHINP pEHMLQ8XNPXi
qi = 1, 2, 3
XQ:JLE,QyT|fhR"EHG[R"KNIWINSXNW1J!E*QvT"IujoR_KNW8MLSHZHM^EHI `iTUJiSHW8M pM^EHGHIVOHW.GHO s^M^T_OHjoI pWM^E,Q:JLTlq
ai
T_I$W1J ^MLEGHIT"JkSYJLWQ8R"IFs9R [W1JLE,Q1IGHIT"JkjoIVj HW1J!EHILqVfi
T_IFs^M^T_OHjoI pWM^E,Q:JLTlqV +
i
T"IVP s^M^T_OHjoIVP !K JLK:Z[XNP^qA+
i
T"IVP\P8OHW JLKNIVP !KNJLK:ZHXVINP^qVbfi
T"IFsLM^T"O[joI$X&9O[R_s!JLT"IVE,QxJLO YJ!P8P8IVP#pWX *OHIVEHKNIVPNqF0i
TUJ pWX *OHIVEHKNIGHIW8XNPM^EHEYJ!EHKNI!qRai
TlfmJ!joM^WwQ1R_P8P8IVjoIVE,Q qCPi
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T"I$KNM9I KVR"INE,QxGHIKNMLW8W8IVKVQ8R"M^E GHI$Q1IVjoSXNW8J!Q1O[W8ILq[A.T_INPSYJLW8JLjoQ1WINPT"R_XNPJLObjoIVP8OHWINP EHMLQ8XNP
Xij
qT"INP$R_EHGHR_KNINPi = 1, 2, 3
IVQj = 1, 2, 3
WINSXNW1J!E*QWINP8SINKQ1Rs^INjINE,QvT|fhXNjIVQQ1INO[W.IVQxT"I$WXNKNIVS[Q1IVOHW: (ur/ue)ij
T_IdW1JLS[SzM^WwQiGHIVPkQ1INE[P8R"MLEHPiGHIdPM^WQ8R"IdIQ Ggf IVE,Q1W8XVIDGHO P 9PQ8NjI KNM^EHPwQ1RQ1OHXdG[INPijR"KNWM SHZ[M^EHINPxW8XVKNIVS[Q1IVOHWvIQ\XNjIVQQ1INO[W
iIQjq
Pij
TUJkSHWINP8PR"M^EJLj [RUJLE,Q1I$WINT_IVs^XVIGHOHW1J!E*QxT"J jINPOHW8IGHI(ur/ue)ij
qTij
TUJQ8INjSzXVW1J!Q8OHW8IBJLj [RUJLE,Q1IuW8INT_IVsLXNIBGHOHW1J!E*QxT"JkjoIVP8OHWIBG[I(ur/ue)ij
qHij
T"IFQ1JLOdGgf Z*OHjR"G[R_Q1X$WINT"J!Q1Rs^IFWINT"Is^XBGHOHW8JLE,QvT"JkjoIVP8OHWIGHI(ur/ue)ij
A fmJ!SHW8VPT_INPDW8IVTUJ!Q8R"M^E[Pe[A # IVQ #^A$# ^q(TUJ KNMLE*Q8W8R HO[Q1R_M^E GHINP SYJ!W1JLjVQ8W8INP
XqXi
IQXij
GYJLEHPT|fhR"EHKVINWwQ1R_Q8OHGHI Q 9SzIuGHIT|fhI K JLKVR_Q8X
Mp1
qHINPwQuGHMLEHEHXNISYJ!W
u2c(Mp1) =
∑
[
(
∂ Mp1
∂x
)2
V (x) +
(
∂ Mp1
∂xi
)2
V (xi) +
(
∂ Mp1
∂xij
)2
V (xij)
]
.[A *
`cfhR"E[KNINWwQ1RQ1OHGHI Q 9SIuKNMLjoSM^P8XVIu2
c(Mp1)SINO[QrGHM^EHKFQ1WIFInSHWR"jXNIuIVEpM^EHKQ1R"MLE GHIVP GHXNWR_sLXNINP.SHJLW
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KVM^joSM^PXNIu2
c(Mp1)PVf XVKNW8RQ\P8R_joSHT_INjINE,Q.PM^OHPxTUJ pM^WjoI
(
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Mp1 SHJLWQxT"IVPvKVMnI KNR"IVE,Q1PuG[IKVM^W8WINKVQ8R"M^EGHIBSHW8IVP8P8R_M^EIQFGHIQ1IVjoSXNW8J!Q1O[W8ILq T_INP\SYJLW8JLjoQ1WINPxT"R"XVP\OHEHR
*OHIVjoIVE*QJLO jR"KNWM^SHZHMLEHINPdPM^E,QDQ1M^O[PDKNM^jSHW8R_PdINE,Q8R"NWINjINE,Q GYJLE[PDT|fhInSHW8IVP8PR"M^E GHIbTlfmJLG[joRQ8Q:J!EHKNIJLKVM^OHPwQ1R%*OHIGHINPdjR"KVW8M^S[ZHM^EHIVPb X& A$,nA-,<A[q,nA-,<@, JLR"E[P8R#9O[I GYJLEHP T|fhInSHW8IVP8PR"M^E G[O s^M^T_OHjoIyIVQdGHIT"JT_M^EH]^OHIVOHW Q1M!Q:JLT_I$GHITUJkK J s9R_Q8XpM^W8jXNI$SYJ!W.T"INP\jR"KVW8M^SHZ[M^EHINP.IQ\T"I$KVM^OHSHT_INOHWVA
! T~WINPQ8Ii#dP1J sLM^R"WFPR~T"Js!JLT"IVOHWFGHIT|fhI K JLKVR_Q8XGgfhOHEbjR"KNWM^SHZHMLEHILqGHXVQ8INWjoR_EHXNIkSHJLWFTUJjoXQ1ZHM9GHIkG[IWXNKNR_SHW8M9KNRQ1X!qINPQBR_E YOHIVEHKNXVI SYJLWT_INPSYJLW8JLjoQ1WINPGHINPBGHIVO JLO[Q1WINPjR"KNWM^SHZHMLEHINPo KVI *OHRcEHI GHIs9W1JLRQSYJ!P VQ1WI T"IrK JLP JFS[W8R"MLW8R?A`cfhXVQ8OHGHI$*OYJLT_R_Q1J!Q1Rs^I W1JLSHR_GHI*OHRHPOHR_Q~GHI.T|fhI KNJLKNRQ1X
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Mp1
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Mp1 =
√
M12M13
M23=
√
ZE12ZE13
M23
√
YT12YT13 ,
MyT"IVP\JLGHjR_QQ:JLEHKVINP\JLKVM^OHPwQ1R%*OHIVP\GHIBQ8W1JLE[P pINWwQYT ij
P8M^E,QxXNKVW8R_Q8INP\Ggf JLSHWNP\T|fhInSHWINP8PR"M^E ,9A$#B,L
YT ij = jωVij
γijPij
CTHij + Yi + Yj ,[A-A^
MVij = V + Vfi + Vfj + V +
i + V +j
IQCTHij
GHXVSzIVEHGDGHINP.KVM^EHGHRQ1R_M^EHP(JLj HR"JLE,Q1INPVqnG[Ods^M^T_OHjoIVij
IQGHIT"J POHW J!KNI$Q1MLQ1JLT"I
Aij
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Mp1
SYJLW.W8JLSHSM^WQx#iTlfmJLG[joRQ8Q:J!EHKNIY1
GHM^EHE[IJ!T"M^WP
∂ Mp1
∂Y1=
√
ZE12ZE13
M23
1
2
[
√
YT12
YT13
∂ YT13
∂Y1+
√
YT13
YT12
∂ YT12
∂Y1
]
,
J s^INK ∂ YT12
∂Y1
=∂ YT13
∂Y1
= 1qHKVI?*OHRaKVM^EHGHOHRQx#
∂ Mp1
∂Y1=
1
2Mp1
(
1
YT12+
1
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)
.[A L
uIFT"JjoVjoI# J-NMLEgqnT"JGHXVW8R_sLXNIFSHJLWQ8R"INT_T"IxGHIMp1
SYJ!W W1JLS[SzM^WwQ.JLO DJLG[joRQ8Q:J!EHKNIVPY2
IVQY3
PNfhXNKVW8RQ∂ Mp1
∂Y2
=1
2Mp1
(
1
YT12
− 1
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) IVQ ∂ Mp1
∂Y3
=1
2Mp1
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− 1
YT23
)
.[A-@^
(* $ ( ( ( ( ( ( +
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YT ij
P8MLE*QP8OHSHSM^PXNINP(SzIVOoG[R_zXNW8IVE,Q1INP INE,Q8W8I\INT_T"INPVA,`INP GHINO GHINWEHR"VW8INPcGHXVW8Rs^XNIVP(SHJLWQ8R"INT_T"IVPP8M^E,Q(GHM^EHK"*OYJLPR E9O[T"T"IVPvIQFTlf I oKNJLKNRQ1X
Mp1
GHOjoR_KNW8MLSHZHM^EHI#SINOnQFGHM^E[KVQ8W8IKNM^EHPR"GHXVW8XVIKNMLjojIR"EHGHXVSzIVEHGYJLE,Q8IGHINPvJLGHjR_QQ:JLEHKVINP G[INPxjoR_KNW8MLSHZHM^EHIVP#,kIVQ [AvEHI JLS[SHT"R_K J!Q8R"M^EyE*OHjoXVW8R9O[IijoM^E,Q8W8I *OHIiT_INPFG[XNW8Rs^XVINPSYJLWwQ1R"IVT"T_INP ∂ Mp1
∂Y2
IQ ∂ Mp1
∂Y3
J!Q8Q8INR_]^EHINE,QJLObj J [R_jkOHj
10−3∂ Mp1
∂Y1
SM^OHWuGHINPGHR XVW8INE[KNINP I9Q1WNjINPbINE,Q8W8I T"IVP$JLGHjR_QQ:JLEHKVINPY1
qY2
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uc(Mp1)SINO[QDG[M^EHK Q1W8I
EHXN]^T_R"]^XVILAR"R|A ( (* (* ( `aIVPrKNJLW1JLKQ1XVW8R"PwQ1R9O[INPr]LXNM^jXVQ8W8R%*OHIVP(G[INPrjR"KVW8M^S[ZHM^EHIVP(EgfhM^E,Q*OgfhOHEHIuSYJLWQ8R"KVR"SYJ!Q8R"M^EiWXNGHO[R_Q1IvGYJLEHPT"INPW8XVP8OHTQ:J!Q8P Ggf XQ:JLT_M^EHEYJ!]^ILAc`aJ S[W8M<pM^E[GHINOHW IQ T_I sLM^T"O[joIpW8MLE*Q1JLT.IVQoEHIySHW8XVP8IVE*Q8INE,Q*OHISzIVOGHI s!JLWRUJ!Q8R"M^E[PdGgfhOHE jR"KVW8M^S[ZHM^EHIe# T|f JLO[Q8W8ILAvE PR"jSHT"IK JLT_KNOHT$SINWjoIQ GHI PVfmJ!P8P8O[W8INW*OHI T"INPK JLW8JLKVQ8XNW8R_PQ8R%*OHINPc]^XVM^jXVQ1WR%*OHIVP GHINPcjR"KNWM^SHZHMLEHINP,FIQ $M^E,QcOHE[IxKVM^E,Q1WR HO[Q8R"M^EiE[XN]^T_R"]^INJ HT"I(GYJLEHPT|fhR"EHKVINWQ8R_Q8OHGHIFKVM^joSM^PXNI
uc(Mp1)A
4 M^R_QX1
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KNINPSYJLW1J!joQ1W8IVP]LXNM^jXVQ8W8R%*OHIVPNqaT"IVPG[XNW8Rs^XVINPkSYJ!WQ1R_INT_T"INP$G[IMp1
SYJ!WW8JLSHSM^WQJLO IVPQ1R_j JQ1R"MLEHP
x1
qx2
qx3
GHINPX1
qX2
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PVf XVKNWR_s^IVE,Q
∂ Mp1
∂x1=
1
2Mp1
[
1
M12
∂ M12
∂x1+
1
M13
∂ M13
∂x1
]
,[A ^J,
∂ Mp1
∂x2=
1
2Mp1
[
1
M12
∂ M12
∂x2− 1
M23
∂ M23
∂x2
]
,[A
∂ Mp1
∂x3
=1
2Mp1
[
1
M13
∂ M13
∂x3
− 1
M23
∂ M23
∂x3
]
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XQ:JLE,Q P8O[SHSzMLP8XNPJ s^M^R_WdOHEHIeR_E YOHIVEHKNIbX&9O[R_s!JLT"IVE,Q1IbP8O[WDT"INPSHW8M9GHOHRQ1P
Mijq*T"IVP G[R_zXNW8IVE,Q1INP GHXVW8R_sLXNIVP(SYJ!WQ1R_INT_T"INP ∂ Mij
∂xi
P8MLE*Q(KNMLEHP8R_GHXNWXNINP(KNM^jjI.X *OHR_s!JLT_INE,Q1IVPNA`aIVPG[XNW8Rs^XVINPSHJLWQ8R"INT_T"IVPFSYJLWW1J!SHSzMLWQJLO SYJ!W1JLjVQ8W8INPGHIVPjR"KVW8M^S[ZHM^EHIVP ,DIVQ SzIVO[s^IVE*QGHM^EHKVQ1WIEHXN]^T_R"]^XVINP.GYJ!EHP\T"I$K J!T"KNO[TgGHIT|fhR"EHKVINWwQ1R_Q8OHGHI$KVM^joSM^PXNI
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Pij = P0 + ∆Pij , Tij = T0 + ∆Tij , Hij = H0 + ∆Hij ,
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uc(Mp1)PVf XVKNWR_QxJLT_M^W8P
[
(
∂ Mp1
∂∆X12
)2
+
(
∂ Mp1
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+
(
∂ Mp1
∂∆X23
)2]
V (∆Xij) .[A # #
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KNM^E,Q8W8R [O[Q1R_M^EoG[INP(ur/ue)ij
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! / 0 2 / / / !
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(* #
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X A#^A @,A `aJ GHXNWR_s^XVISHJLW Q1R_INT"T_IuGHI
Mp1SYJLW.W8JLSHSM^WQx#
U0GHMLEHEHIGHM^EHK
∂ M1
∂U0
=Mp1
U0
,
KNI*OHRGHM^E[EHIOHEHIKNM^E,Q8W8R [O[Q1R_M^E GHIUpol
GYJLEHP\Tlf R_EHKNIVWQ1RQ1OHG[I$GHIMp1
G[I(
∂ Mp1
∂U0
)2
V (U0) =M2
p1
U20
V (U0) ,[A #B,^
V (U0)XQ:JLE,QxTUJs!JLWRUJLE[KNI$GHI
U0
A fmJ!O[Q1WI SYJLWQNq GHIVP S[ZHXNEHMLjoVEHINPaSYJLW8JLP8RQ1IVPGgf MLW8R"]LR"EHI XNT"IVKVQ8W8R%*OHI MLOJ!KNM^OHPwQ1R9O[I R_E,Q1INWws^IVEYJLE,Q GHOHW1J!E*Q
TUJ\joIVP8OHWIrSYJ!WQ1R_KNR_SzIVE*Q #uTlf R_EHKNIVWQ8R_Q1O[GHIcP8OHW T"INP W8XVP8OHTQ:J!Q8P GgfhXVQ1JLT"MLEHEYJL]^I!A INPSHZ[XNEHM^jNE[INPPM^E,QT"I HWOHR_QJLj HR"JLE,Q qT"IVP pOHR_Q8INP$JLKVM^OHPwQ1R%*OHIVPNqT"JdGHR"PwQ1M^WP8R"MLEgq T"J GHRUJ!SHZHM^EHR_ILqzT"INPuIVzIVQ1PXVT"INKQ1WR%*OHINP$GHJLEHPTUJ ]^JLW8GHIGHINP\jR"KVW8M^SHZ[M^EHINPVA
cE YEgqgGHMLR_s^IVE,QVQ1WIiKNM^jS[Q1J HR"T_R"PXNPNqGYJLE[P$T"IKNJLT"KVOHTGHIVP$R"EHKVINWwQ1R_Q8OHGHIVPNqgT_INPFIVW8W8IVOHW8P$G fmJLWW8M^E[GHR"PVq TUJW8XVSzXQ:J HR_T"RQ1XFIQuT_INP\INWW8INO[W8PuG[OHINPuJLOjM9GHNT_IFQ1ZHXVM^W8R9O[IO[Q8R"T_R"P8X$SM^OHW\WINSHWXNP8IVE,Q1INWuT"IP 9PQ8NjIDpINW8WINOHWP8OHW\T"JkSHW8R_P8IINEyKNM^jS[Q8I$GHINP\IVzIVQ8PxQ8ZHINWjoR*OHINP\IVQ.s9R_P *OHINO IQ\GHINPxM^EHG[INP\W1JLG[RUJLT_INP:A`aIVP\R"EHKVINWQ8R_Q8OHGHINP\POHW\KNINP\SYJ!W1JLjVQ8W8INP\PM^E,QvPINOHT_INjINE,QxXVs!JLT_OHXNP\GHIjoJLEHR"VW8IFR_E,Q1OHRQ1R_sLILA
!
`aI ]^W8JLSHZHR *OHIiGHI TUJ Y]^O[W8I [A$# WINSHWXNP8IVE,Q1ILq INEpM^EHKQ1R_M^EeGHI TUJpW8X *OHINEHKVILq T|fhR"EHKVINWwQ1R_Q8OHGHIkXVTUJLW]^R"I!qJ s^INKBOHE JLKVQ8INOHWxGgfhXNTUJ!W8]^R_P8P8IVjoIVE,Qy X A [A^qPOHWxT"IjoM9GHOHT_I G .G[ITlf I oKNJLKNRQ1XGgfhOHEjR"KVW8M^S[ZHM^EHI` M[Q8INE*OHIVPJLO ` .A IVQQ1IkR_EHKNIVWQ1RQ1OHG[ISINO[QuVQ8W8IKVM^jSYJLW8XVI#oKNINT_T"IBM[Q8INE*OHIkJ!O c` pQ1J gA[ALA E YJ!P8P8IVPpWX *OHINE[KNINPIVQ OHP *Ogf #INE,s9R"WM^E !q Tlf R_EHKNIVWQ8R_Q1O[GHI GHO ` KNM^WW8INPSzMLEHG # KNINT_T"IGHO c` A FJLE[PFT"IVPvZYJ!O[Q1IVP pW8X 9O[INEHKVINPNq T|fhR"EHKVINWQ8R_Q8OHGHIGHM^EHE[XNIkSYJ!WvT_I` IVPQ$POHSzXVW8R_INOHWIGHISHT"OHPGgfhOHEG[R nR"VjoIFG[IBG #iKNIVT"T_I$GHO ` #! LA
102
103
104
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
"$#&%'(*),+$-,)/.10243
5 6798:<;=7 ><?;@ ABC D EF
<(=?>.@ [A$# G ! EHKNIVWQ8R_Q1O[GHIXNTUJ!W8]^R_IuP8O[WvT_I$joM9GHOHT_IFGHIT|fhI K J!KNR_Q8X$GgfhOHEjR"KVW8M^S[ZHM^EHI$` A
*G nq #*G # nq
#nq$# G # nq-A nq #*nq G # A nq # A[q # G nqnq$# G nq nq$#
~@ [AG ! EHKVINWQ8R_Q8OHGHINPxXNT"JLW8]LR"INP JLEHEHM^E[KNXNIVPvSHJLW\T"I c` SzM^O[W.GHINPxjoR_KNWM^SHZHM^E[INP\` A
vEHIJ!EYJLT$9PI$GHITUJkSYJLWwQ.W8IVTUJ!Q8R_sLI$MnKVKNOHSXNI$SYJ!W.T"INP.G[R_zXNW8IVE,Q1P\SYJLW8JLjoQ1WINP.GgfhR"E!HOHINEHKVIGYJLEHP.KVIVQ8Q8IR_EHKNIVWQ1RQ1OHG[IyXVTUJLW]^R"I!qJLO YJLPP8INPoIVQoJLO ZYJLO[Q8INP pW8X 9O[INEHKVINPNq(INPwQoR_T"T_OHPQ8W8XNIDSHJLWiT"IVP]LW1JLSHZ[R *OHINP GHIVPY]LOHW8IVP [A IVQ [A [A INPd]^W8JLSHZHR *OHINPoSzIVW8jIVQ8Q8INE,QDGHIGHXVQ8INWjoR_EHINWoT"INPoSYJLW8JLjoQ1WINP SM^OHWoTUJ joIVP8OHWIGHIVP *OHIVT"PxOHEPM^R"E SYJ!WQ1R_KNOHT_R"IVW GHM^RQ.VQ8W8IjoR_PNA
P: 4%T: 5%
H: < 1%
CP,C
T :2%
Vbf
:12%
F0,Q :< 1%
Vf,Vsup :21%
Param. Cav. :17%
Param. Elec :13%
Cth :2%
Fuites, Bruit :< 1%
Repet., Arr. :8%
Diaphonie, Disto. :10%
Ondes Rad., Visco. :4%
<(=?>.@ [AG JLWQ8R"KVR"SYJ!Q8R"M^EiG[INP SYJLW1J!joQ1W8IVPcGgf R_E!YOHIVEHKNIxGYJLE[P(Tlf R_EHKNIVWQ1RQ1OHG[I.XNTUJ!W8]^R_IU(Mp)
###"%$LA E YJLP8PINPpWX *OHINE[KNINPVqTUJSYJLWQFTUJoS[T"OHPvR"jSzM^WwQ:JLE,Q8IGYJ!EHPFT|fhR"E[KNINWwQ1RQ1OHGHIXNT"JLW8]^R_IIVPQ$M9KNKVOHSXNIkSYJ!W
T_INPxSYJLW1J!joQ1W8IVP\]^XNM^jXVQ8W8R 9O[INPNq SHT_OHP\SYJLWwQ1R"KVOHT"R_NWINjINE,Q\T"I$s^MLT"OHjI pW8M^E,Q:J!TGHOjR"KVW8M^SHZ[M^EHILA cEIVzIVQNqT_I\sLM^T"O[joI/pW8MLE*Q1JLT GgfhOHE jR"KVW8M^SHZ[M^EHI\INPwQ(Q8W8NP GHXVT"R"KNJ!Q #G[XVQ1IVW8jR"EHIVWNq,T_IxG[RUJLjVQ8W8I.R"E,Q8INW8E[IvEgfhXVQ1JLE,Q SYJ!PjINPOHW1J [T"IJ s^IVKSHWXNKVR"P8R_M^E nA [A #A
M^O[WGHINP pWX *OHIVEHKNIVP S[T"OHPXVT"Is^XNIVPNq,T_INPR_EHKNIVWQ1RQ1OHG[INPPOHWT"IVPSHZHXVEHM^jNEHIVP GHI.GHR_PQ1MLW8P8R_M^EkIQ P8OHWwQ1M^OnQGHI$G[RUJLSHZ[M^EHR"IvSHW8IVEHEHIVE*QxGHIVP.SHW8MLSzM^WwQ1R_M^EHPrQ8W8NP.XVT"Is^XNIVP\GYJLEHP Tlf R_EHKNIVWQ8R_Q1O[GHIuXVTUJLW]^R"I!A[`aJkGHR"JLSHZHM^E[R"IuIVPQNqIVE I IQ OHE SHZHXVEHM^jNEHIbGHXVT"R_K J!Qd# KNJLW1J!KVQ1XVW8R_P8INW K JLWDR_TvSINO[QyJ sLM^R"WdG[R_zXNW8IVE,Q1INPDMLW8R"]LR"EHIVPb KNJLSYJLKVR_Q8XSYJ!W1JLPR_Q1IINE,Q1WI T_INPDjR"KVW8M^SHZ[M^EHINP IVQ INE,Q8W8IeT"IVPdK& [T"INP1A.`aJ SYJLWwQDR"jSM^WQ1JLE,Q1IS[W8R"PIbSYJLWdKVINP IVzIVQ8PSYJ!W1JLPR_Q1IVPFjM^E,Q1W8I 9O f OHE[IiXVQ1O[GHIiInSXNW8R_joIVE,Q:JLT_IkSHT"O[P$JLSHSHWM pM^EHG[R"Is9R_P1JLE,Q#djR"IVOT"INPuK JLW8JLKVQ8XNW8R_P8IVWIVPQvEHXNKVINP8P8JLR"WILA
E YEgqnR"TzJLSHSYJLW8J"Q *OHI$T|fhR"E!HOHINEHKVIuGHI$Tlf R_joSXNGHJLEHKNIxGHODjR"KVW8M^SHZ[M^EHIuPOHW.T|fhI KNJLKNRQ1Xvs!JLOnQ.P8OHWwQ1M^OnQ
P: 3%T: 4%H: < 1%CP,C
T :< 1%
Vbf
:3%
F0,Q :2%
Vf,Vsup :5%
Param. Cav. :6%
Param. Elec :5%
Cth :< 1%
Fuites, Bruit :< 1%
Repet., Arr. :21%
Diaphonie, Disto. :30%
Ondes Rad., Visco. :19%
<(=?>.@ [A G ~J!WQ1R_KNR_SYJ!Q1R_M^EbGHIVPSYJ!W1JLjVQ8W8INPBGgfhR"E!HOHINEHKVIoGHJLEHPBT|fhR"EHKVINWQ8R_Q8OHGHIiXNT"JLW8]LR"IU(Mp)
# #"%$!A
SYJLWT_Irs^MLT"OHjI JLO YJLPP8IVP pWX *OHIVEHKNIVPNA9`aJ pW8X *OHIVEHKNI\GHI.WXNPM^EYJLEHKVI\IVQ TlfmJLjM^WwQ1R"PP8IVjoIVE*Q~EgfhM^E,Q 9O f OHE[IQ1WNPcSzIQ1RQ1I\SYJLWwQ(GHJLEHPcT|fhR"EHKVINWwQ1R_Q8OHGHI.XVTUJLW]^R"I!A vEHI\KVM^EHEYJLR_P8P8JLEHKNI\Q1WNP SHW8XVKNR_P8IxGHI\KNINP SYJLW8JLjVQ1WINPcEgfhINPwQGHM^EHKiSHJLP$EHXNKVINPP1JLR_W8I IVQT_INP$jXVQ1Z[MnGHIkJLKVQ8OHINT_T"INjINE,Q$IVEeOHP1JL]LILq HR_INE *OgfhR"jSYJLW J!R_Q1IVPuPINj HT_INE,QBJLT_M^W8PP8O P8JLE,Q1INPxR"KVR|A
E[MLQ1IVW *OHIBKNIQ8Q1IXVQ8OHGHIBGHINPxR"EHKVINWwQ1R_Q8OHGHIVPuJ XVQ8XI IVKVQ1O[XNIkJ s!JLE,Q 9O[IEHIBP8MLR"INE,QvW8IVjoR_P.INEKNJLOHPIT"IVPoI[S[W8INPP8R_M^EHPdEHMLW8joJLT"R_P8XNIVP SzM^O[WoTlfmJLG[joRQ8Q:J!EHKNI JLKVM^OHPwQ1R *OHIGHI Q1W1J!EHP pIVWQ
YT
IQ T_INPjoM9GHNT_INPP8OHWT"IVP *OHINT_P$INT_T"INPuW8INSM^PINE,Q A uIVSHOHR_PNqR"T INPwQBJLSHSYJLWO 9O[IT_INPFI IQ1P$Q1Z[INW8jR *OHIVP$P8M^E,QSHWR"P$IVE KVM^jS[Q1IiGHI JNMLE R"EHIHJ!KVQ1I IQ 9O[IyT_INPiI IQ1Pis9R"P *OHIVOEHIDGHMLR_s^IVE,QoSHJLPiVQ8W8IDEHXV]^T"R_]^XNP ,A `IVPiR"EHKVINWQ8R_Q8OHGHINPP8OHW.T"JiSHW8R_P8I$INEyKNMLjoS[Q8I$GHI$KNIVPvG[INO I IQ1PxINPwQvG[M^EHK$EHIQ8Q1IVjoIVE,QvPM^OHP XVs!JLT_OHXNIGYJ!EHP\KNIVQQ1IXVQ8OHGHIGHIVPR"EHKVINWwQ1R_Q8OHGHIVP P8OHW.T_INP.XQ:JLT_M^EHEYJL]LINP IVEyS[W8INPP8R_M^E GHI$joR_KNWM^SHZHM^E[INP SYJ!W.TUJjXVQ8ZHMnG[IFGHI$W8XVKNR_SHW8M9KNRQ1XLA ~JLWInINjSHT_ILq[Tlf IVW8WINOHW(KNM^jjoR_P8IvP8OHW Tlf InSHW8IVP8P8R_M^E GHIFT"JKNM^WW8IVKVQ1R_M^EdQ1Z[INW8jR *OHI
CTH
INPwQ\R"KVRzXsLJ!T"OHXVI# # GHI P8Js!JLT"IVOHWdpP8M^RQkIVE*s9R_W8M^E nA<A yG # " $dSzMLOHWOHE joR_KNW8MLSHZHM^EHI ` q KNI *OHR(INPwQkQ1WNPkIVEGHINPP8M^O[PGHIoT|fhINW8WINOHWKVM^jojR"PI INE WX JLT_R_Q8Xy SHT_OHPG[I [A GPOHWT"J s!JLT_INOHWGHIVPI oKNJLKNRQ1XVPM[Q8INE*OHIVPT"M^WP GgfhOHEyXVQ:J!T"M^EHEHJL]^I$SYJLW.WXNKVR"SHWMnKVR_Q1X!qHK4A [A'^A
g
\
\ d
IVQ8Q8IJ!EHEHInIGHXVsLINT"MLSHSzIOHE KNJLT"KVOHT.GHIySHW8M9GHOHRQP8K J!TUJLR_W8IDGHM^E,QoT"I W8XVP8OHTQ:J!QoINPwQ OnQ1R"T_R"PXDGYJLEHPT"IK:ZYJLSHRQ1WI y nA! nA! ,rjoJLR_P *OHREgf INPQxSYJ!PxG[XVs^IVT"M^S[SzXBJ YEyGHIEHISYJLP\IVEJLT_M^OHWGHR"W(T"I$SHWM^SzMLPNA`aI SHW8M9GHOHRQkP8KNJLTUJ!R"W8I
< y1ψt′
µ′ν′ |ψc0ν >σ1
q~GHJLEHPiT|fhInSHWINPP8R"MLE nA#"$ ,GHINPiKVMnI%oKVR"IVE*Q8Ppc
0ν(z)qINPwQ
GHM^EHE[XBSHJLW < y1ψ
t′
µ′ν′|ψc0ν >σ1
=
∫
σ1
y1(θ0)ψc0ν(a)ψ
t′
µ′ν′(a, θ0) dS0 ,
M& T|f JLGHjR_QQ:JLEHKVIFP8SXNKVR$ *OHIy1
GHIT|fhXNT_XNjINE,Qx# K JLW8JLKVQ8XNW8R_P8IVW pINE,Q1IM^O Q1O('zIVP: INPwQpM^E,Q8R"M^E GHIT|f JLEH]LT"IOHEHR 9O[INjINE,QxIVQ\T"IVP/pM^EHKVQ8R"M^E[P.SHW8M^S[W8INP
ψc0ν
PM^E,QuR_EHGHXVSzIVEHGYJLE,Q1IVP\GHITlfmJ!EH]^T"Ik X A)"HA*LJkIQ+' A M^OHW(GHXNKVW8R_W8IuT"JPOHW JLKVI
σ1qθ0sLJ!W8R"IxGHI k#
2πIVQz0G[I`1#`1 + δz
qδzXQ:JLE,Q(Q1W8VPrSIVQ8R_Q G[IVs!JLE,Q
`1qYKNI *OHRGHM^E[EHI
< y1ψcµ′ν′|ψc
0ν >σ1= a δz ψc
0ν(a)
∫ 2π
0
y1(θ0)ψt′
µ′ν′(a, θ0) dθ0 .-, A #
`aIVPSHJLW1JL]LW1JLSHZ[INP.,iA$#IVQ/,iADGHXQ:JLR_T"T"IVE,Q$T"I K J!T"KNO[T GHI KNISHW8M9GHOHRQP8K J!TUJLR_W8IiGYJLE[PBT_I K JLPBM& T|f JLG10joRQ8Q1JLEHKNIF# KNJLW1J!KVQ1XVW8R_P8INW\IVPQxW8IVP8SINKVQ8R_sLINjINE,QuO[EHI pINE,Q1IJLEHE*OHT"JLR"WIFGgf JLGHjR_Q8Q1JLEHKVI$Ggf IVE,Q1W8XVI
Yf
M^OO[EW8XVP8I J!OGHI *OYJ!Q8W8I$Q1O('zIVPvK nT_R"EHG[W8R *OHIVP.Ggf JLGHjR_Q8Q1JLEHKVI$Ggf IVE,Q1W8XVIR_EHGHRsnR_GHOHIVT"T"I
Yt
A
2 3 54 6 FJLEHPoT"IKNJLP M& T|fhXNT_XNjINE,Q # K JLW8JLKVQ8XNWR"P8XyINPwQDOHEHIpINE,Q1I JLEHE*OHTUJ!R"W8I Ggf JLGHjR_QQ:JLE[KNI
Yf
q W8XVSYJLWQ8R"IOHEHR1pM^W8jXNjINE,Q POHWcTUJ$P8OHW, JLKNI
σ1 = 2πa δzq,T|f JLGHjR_QQ:JLE[KNI PSzXVKNR$ 9O[I
y1
INPwQ(R_EHGHXVSzIVEHGYJLE,Q1IxGHI\T|f JLEH]LT"IθIVQxINT_T"IFIVPQvXNKVW8R_Q8I
y1 =ρ0c0σ1
Yf =ρ0c0
2πaδzYf .
`cfhInSHWINPP8R"MLE 7,iA #rGHM^E[EHIBJLT"M^WP
< y1ψt′
µ′ν′ |ψc0ν >σ1
= ρ0c0Yfψc0ν(a)
∫ 2π
0
ψt′
µ′ν′(a, θ0) dθ0 ;-, A^
# A
"!#$% !'&& (&)! " * +& ,-
".( *!' /0 12
Ggf JLSHWNPT"IVP$InSHWINP8PR"M^E[PdpX A"HA*LJdIVQ 'YuGHINP pM^EHKQ1R_M^EHP$SHWM^SHW8IVPNqTlf R_E*Q8XN]^W8JLT"IP8OHWθ0INPwQBE9O[T"T"ISzM^O[W
µ 6= 0IVQ\SM^OHW
t′ = sqµ′ = 0
qHKNI *OHRGHM^E[EHIBJLT"M^WP.SzMLOHW.T"I$SHWMnG[OHR_Q.PK JLT"JLR"WI
< y1ψt′
µ′ν′|ψc0ν >σ1
= ρ0c0Yfψc0ν(a)ψ
c0ν′(a)δt′cδµ′0 .
-,iA! ,
2 3 54 4 4 76 6 84 FJLEHPuT"IkKNJLPFM&bTlf XVT"XNjINE,QF#dK JLW8JLKVQ8XNW8R_P8IVWFINPwQOHEbW8XVP8I J!O G[I *OYJ!Q1WIkQ1O 'INPuGgfmJ!GHjoRQ8Q1JLEHKNIR"EHG[R_s9R 0
GHO[INT"T_IYtqYW8XN]LOHT"R_NW8IVjoIVE,Q\W8XNSHJLWQ8R"P.INE
θn = nπ/2n 9 # ^qYP8O[W *OYJQ1W8IPOHW JLKVINPxXN],J!T"INP.K:ZHJLKNOHE[I#
a δθδzq T|f JLGHjR_Q8Q1JLEHKVIuP8SXNKVR$ *OHI
y1
INPwQuXVKNW8RQ1I
y1 =ρ0c0a δθδz
Yt
P8Rθ = θn
IVQy1 = 0
P8Rθ 6= θn ,
IQvT"JkP8OHW, JLKNIσ1
sLJ!O[Q4δθa δz
A` f I [S[W8INPP8R_M^E -,iA$#rG[M^EHEHIJLT"MLW8P
< y1ψt′
µ′ν′|ψc0ν >σ1
= ρ0c0Ytψc0ν(a)
3∑
n=0
ψt′
µ′ν′(a, θn) ,-,iA "*
PM^R_Q< y1ψ
t′
µ′ν′|ψc0ν >σ1
= ρ0c0Ytψc0ν(a)
Jµ′(γµ′ν′)
aµ′bµ′ν′
3∑
n=0
cosµ′θn
PRt′ = c
sinµ′θn
PRt′ = s .
-,iA&^ M^jS[Q8IQ8INE*OyGHINPxW8IVTUJ!Q8R"M^EHPrP8OHRsLJ!E*Q8INP
3∑
n=0
cosµ′θn =(
1 + (−1)µ′
)(
1 + cosµ′π
2
)
=
4P8Rµ′ = 0
jM9GHOHT"M4
0P8R_EHM^E
,
-,iA-A,
IQ3∑
n=0
sinµ′θn =(
1 + (−1)µ′
)
sinµ′π
2= 0 ,
7,iA *L
IQcGHO JLR_Q *OHI.PINOHT_PcT"INP~joM9GHINP SM^OHW~T"INP *OHINT_Pµ < 3
P8M^E,QcSHWR"P INE KVM^joSnQ1IpT"INPcJ!O[Q1WINP XVQ1JLE,QcP8OHS[SzM^PXNPJ s^MLR"W.OHEHIBKNMLE*Q8W8R 'HO[Q1R_M^EDEHXV]^T"R_]^I J'[T"Itq[Tlf I [S[W8INPP8R_M^E 7,iA&LrGHIVs9R"IVE,Q
< y1ψt′
µ′ν′ |ψc0ν >σ1
= ρ0c04Ytψc0ν(a)ψ
c0ν′(a)δt′cδµ′0 .
-,iA,
2 ;:=< > 6 f JLSHW8VPFT"IVPvI nSHW8IVP8P8R_M^EHP-, A ,.IVQi7,iA^q T"IBSHW8M9GHOHRQuPK JLT"JLR"WI
< y1ψt′
µ′ν′ |ψc0ν >σ1
pX A ,iA #xSINOnQJLT_M^W8P.Q1WIBI nSHW8R_joX$PM^OHP\TUJ pM^WjoIF]LXNEHXVW1JLT_IBPOHR_s!JLE,Q8I
< y1ψt′
µ′ν′|ψc0ν >σ1
= ρ0c0Y1ψc0ν(a)ψ
c0ν′(a)δt′cδµ′0 ,
-,iA,M&
Y1
s!JLO[QYf
PRrT|fhXNT_XNjINE,Q#K JLW8JLKVQ8XNWR"P8IVWkINPwQiOHEHI pINE,Q1IDJ!EHE*OHTUJLR_W8IoGgfmJ!GHjoRQ8Q1JLEHKNIoJLKVM^OHPQ8R *OHIYfIQ
4Yt
P8RuKVIVQDXVT"XVjoIVE*QdINPQdOHE W8XVP8I J!O G[I *OYJ!Q1WIQ1O('zIVPdK9T"R_EHGHWR *OHINP GgfmJ!GHjoRQ8Q1JLEHKNIVPdJLKVM^OHPwQ1R *OHIVPR_EHGHR_s9R_GHOHINT_T"IVP
Yt
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