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Transkript

Kapitoly 1-4

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PRQRNOLQLFNi
SURVWi
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SURVWiKFS
D≠E≠F
α=γ=90ο≠β
D E≠F α=β=90ο,
γ=120ο
D E F
α=β=γR≠R
D E≠F
α=β=γ R
SURVWiVF
D E=F
SURVWRURYFHQWUEFF α=β=γ R
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6KUQXWt
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NGHXYZMVRXFHOiþtVODD E F MVRXPtåNRYpWUDQVODþQtYHNWRU\
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SRORKiFKU = [ D + \ E + ] F
M = V
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D  E × F  MH PLQLPiOQt D MHVWOLåH NU\VWDORYRX VWUXNWXUX O]H Y\WYRLW
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Y]WDKHP ( = Kν = KF
K -V MH 3ODQFNRYD NRQVWDQWD 9
λ
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Q
3LSRPHPHåHSODWt( = 0 λ = KS 9ODERUDWRUQtFKMHGQRWNiFKMHλ(QP) =
S
( (H9)
3URQHXWURQ\VHQHUJLtH9Y\FKi]tYOQRYiGpONDQP1HXWURQ\GtN\
VYpPX PDJQHWLFNpPX PRPHQWX PRKRX LQWHUDJRYDW V PDJQHWLFNêPL
PRPHQW\HOHNWURQ$YSHYQpOiWFH8åtYDMtVHSURWRSLVWUXNWXUQtFKVWXGLtFK
PDJQHWLFNêFKNU\VWDO$9QHPDJQHWLFNêFKOiWNiFKLQWHUDJXMtQHXWURQ\SRX]H
VMiGU\DWRP$
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VWHMQêP Y]WDKHP MDNR MH WRPX X QHXWURQ$ 8YiåtPHOL åH KPRWQRVW
HOHNWURQX MH P NJ GRVWDQHPH Y ODERUDWRUQtFK MHGQRWNiFK
λ(QP) =
( (H9)
(OHNWURQ\ VLOQ LQWHUDJXMt V OiWNRX SURWRåH PDMt HOHNWULFNê QiERM 'R
NU\VWDOXSURQLNDMtQDSRPUQPDORXY]GiOHQRVW
%UDJJ$Y]iNRQ
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REU
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SRGPtQNX
G VLQ Θ = Qλ %UDJJ$Y]iNRQ
Θ
G VLQ Θ
Θ
ΘΘ
G
2EU %UDJJRYD GLIUDNFH QD VRXVWDY URYLQ Y]GiOHQêFK G 'UiKRYê
UR]GtOMHG VLQ Θ
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PRåQp XUþLW PH]LURYLQQp Y]GiOHQRVWL Y GDQpP NU\VWDOX %UDJJ$Y ]iNRQ MH
G$VOHGNHP SHULRGLþQRVWL PtåN\ 1HVRXYLVt V XVSRiGiQtP DWRP$ Y Ei]L
XPtVWQpGRNDåGpKRPtåNRYpKRERGX%i]HRYOLYXMHLQWHQ]LWXGLIUDNFtQD
U$]QêFKURYLQiFKYL]VWUXNWXUQtIDNWRU
([SHULPHQWiOQtGLIUDNþQtPHWRG\
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E\O\ Y XUþLWpP Y]iMHPQpP Y]WDKX 0RQRFKURPDWLFNp UWJ ]iHQt R YOQRYp
GpOFH ODPEGD GRSDGDMtFt QD NU\VWDO SRG OLERYROQêP ~KOHP VH Y REHFQpP
StSDGQHEXGHY$EHFRGUiåHW$E\NGLIUDNFLGRãORPXVtEêWYåG\VSOQQD
%UDJJRYD SRGPtQND ([LVWXMH QNROLN NODVLFNêFK H[SHULPHQWiOQtFK PHWRG
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1D SHYQê NU\VWDO GRSDGi ~]Nê VYD]HN ]iHQt VH ãLURNêP RERUHP
YOQRYêFKGpOHN2EY\NOHRGGRQP.U\VWDOViPY\EtUiDGLIUDNWXMH
]iHQt SUR WDNRYp KRGQRW\ YOQRYêFK GpOHN NG\ MH QD QNWHUpP V\VWpPX
PtåNRYêFKURYLQVSOQQD%UDJJRYDSRGPtQND'LIUDNþQtREUD]HF]DFK\FHQê
QD SORFKpP ILOPX VHVWiYi ] DG\ VNYUQ MHMLFKå UR]PtVWQt Pi V\PHWULL
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1D NU\VWDO GRSDGi PRQRFKURPDWLFNp UWJ ]iHQt 2WiþHQtP NU\VWDOX VH
PQt ~KHO GRSDGX SUR U$]Qp URYLQ\ -HOL QD QNWHUpP V\VWpPX URYLQ
VSOQQD %UDJJRYD SRGPtQND GRMGH N GLIUDNFL )LOP MH XPtVWQ YH YiOFRYp
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WpWRPHWRG\1DSILOPMHQDKUD]HQYKRGQêPGHWHNWRUHPUWJ]iHQtNWHUêVH
SRK\EXMHRNRORNU\VWDOX
3UiãNRYiPHWRGD
0RQRFKURPDWLFNpUWJ]iHQtGRSDGiQDSUiãNRYêQHERSRO\NU\VWDOLFNê
Y]RUHN 5R]GOHQt RULHQWDFt MHGQRWOLYêFK NU\VWDON$ MH QDKRGLOp SURWR YåG\
H[LVWXMt PtåNRYp URYLQ\ QD NWHUêFK MH VSOQQD %UDJJRYD SRGPtQND SUR
Y]QLN GLIUDNFH 'LIUDNþQt REUD] VH ]DFK\FXMH QD ILOP XPtVWQê YH YiOFRYp
NRP$UFH NROHP Y]RUNX QHER GHWHNWRU UWJ ]iHQt VH RWiþt NROHP Y]RUNX
GLIUDNWRJUDI $QDOê]RX GLIUDNWRJUDP$ O]H XUþLW QDS PtåNRYê SDUDPHWU
VOHGRYDQpOiWN\
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SRSLVRYDW ]DYHGHQtP W]Y UHFLSURNp PtåH -HMt ]iNODGQt YHNWRU\
 E E E  MVRX GHILQRYiQ\ SRPRFt ]iNODGQtFK WUDQVODþQtFK YHNWRU$
 
(D D D ) PtåHStPp
E = 9π (D × D )
E = 9π (D × D )
π
E = 9 (D × D )
9 = D (D × D ) MHREMHPHOHPHQWiUQtEXN\StPpPtåH
=iNODGQtYHNWRU\StPpDUHFLSURNpPtåHVSOXMtSRGPtQNXD L E M = πδ LMNGH
δ LM = SUR L M D δ LM = SUR L ≠ M MH .URQHFNHURYR GHOWD .DåGp PtåNRYp
URYLQ StPp PtåH FKDUDNWHUL]RYDQp 0LOOHURYêPL LQGH[\ KNO MH Y
UHFLSURNpPtåLSLD]HQERGRSRORKRYpPYHNWRUX * KNO = KE + NE + OE 9]GiOHQRVW URYLQ KNO StPp PtåH PH]LURYLQQi Y]GiOHQRVW GKNO VH Gi
Y\MiGLWSRPRFtRGSRYtGDMtFtKRYHNWRUXUHFLSURNpPtåHY]WDKHP
G KNO = π
*
KNO
'LIUDNþQtSRGPtQN\
3RMHP UHFLSURNp PtåH XPRåXMH SHIRUPXORYDW %UDJJRYX SRGPtQNX
GLIUDNFH0tVWRDE\FKRPGRSDGDMtFt]iHQtFKDUDNWHUL]RYDOLYOQRYRXGpONRX
ODPEGD ]DYHGHPH SRMHP YOQRYpKR YHNWRUX N MDN MH WR REY\NOp Y RSWLFH
6PUYHNWRUX N MHXUþHQVPUHPãtHQtYOQ\MHKRYHOLNRVW N = λπ 9OQRYê
YHNWRUSULPiUQtGRSDGDMtFtYOQ\EXGHPHR]QDþRYDWN GLIUDNWRYDQpYOQ\N
3URWRåH Y\ãHWXMHPH SUXåQê UR]SW\O EH] ]PQ\ HQHUJLH MVRX YHOLNRVWL
RERX
YHNWRU$
VWHMQp
=
REU
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SODWt
π * KNO
∆N = N − N = N Q = λ
VLQ Θ -HOL λ Θ D GKNO WDNRYp åH %UDJJRYD
* KNO
SRGPtQNDMHVSOQQDSDNPXVtSODWLW
N − N = * KNO
'LIUDNFHQDVWDQHWHKG\MHOLYHNWRUUR]SW\OXN − N URYHQOLERYROQpPX
YHNWRUXUHFLSURNpPtåH
*→
Q
→
→
N
Θ
→
N
Θ
Θ
→
→
∆
N
KNO
N
2EU%UDJJRYDSRGPtQNDYUHFLSURNpPtåFH
(ZDOGRYDNRQVWUXNFH
-HGQRGXFKi JHRPHWULFNi NRQVWUXNFH Y\FKi]HMtFt ]H Y]WDKX QDSRPiKiOpSHSRFKRSLWSULQFLSMHGQRWOLYêFKGLIUDNþQtFKPHWRG
9UHFLSURNpPSURVWRUX]YROtPHSRþiWHþQtERGDXPtVWtPHYHNWRU
N NRQFRYêPERGHPGR6HVWURMtPHNRXOLRSRORPUX N VHVWHGHP
YSRþiWNXYHNWRUXN REUD
%RG\ UHFLSURNp PtåH OHåtFt QD NRXOL XUþXMt YOQRYp YHNWRU\
GLIUDNWRYDQp YOQ\ 2EHFQ QHPXVt H[LVWRYDW åiGQê WDNRYê ERG GLIUDNFH Y
GDQpP XVSRiGiQt QHQDVWDQH QHER QDRSDN QNROLN ERG$ P$åH OHåHW QD
(ZDOGRYNRXOLSDNEXGHVRXþDVQQNROLNPtåNRYêFKURYLQ-DNRStNODG
Y\XåLWt (ZDOGRY\ NRQVWUXNFH VH YUD"PH N /DXHKR GLIUDNþQt PHWRG 1D
.U\VWDOLFNêVWDY
NU\VWDO GRSDGi SRO\FKURPDWLFNp ]iHQt V YOQRYêPL GpONDPL RG λ GR λ 6HVWURMtPHGY(ZDOGRY\NRXOHRSRORPUHFK N DN REUE'LIUDNFH
QDVWDQH QD YãHFK URYLQiFK MHMLFKå ERG\ Y UHFLSURNpP SURVWRUX OHåt YH
Y\þiUNRYDQp REODVWL 1D GLIUDNWRJUDPX P$åHPH VOHGRYDW VRXVWDYX
GLIUDNþQtFKVWRSMHMLFKåUR]PtVWQtNRUHVSRQGXMHVHV\PHWULtNU\VWDOX
-DN E\OR XNi]iQR VSOQQt %UDJJRY\ SRGPtQN\ MH QH]E\WQêP
SHGSRNODGHP SUR Y]QLN GLIUDNFH 7DWR SRGPtQND YãDN MHãW QHGRVWDþXMH
,QWHQ]LWDGLIUDNFHQDURYLQiFKKNO]iYLVtQDDWRPRYpPUR]SW\ORYpPIDNWRUX
DQDVWUXNWXUQtPIDNWRUX
*KNO
→
→
N
→
N
→
→
N
*KNO
→
N
E
D
2EU(YDOGRYDNRQVWUXNFHD-HGiQYOQRYêYHNWRUGRSDGDMtFtKR]iHQt
N .ROHP SRþiWNX WRKRWR YHNWRUX VH RStãH NRXOH R SRORPUX N 'LIUDNFHMHSR]RURYiQDSURWDNRYpN
NWHUêPRGSRYtGiQNWHUêERG
UHFLSURNp PtåN\ OHåtFt QD NRXOL E (YDOGRYD NRQVWUXNFH SUR
/DXHKRPHWRGX
$WRPRYêUR]SW\ORYêIDNWRUPtUDUR]SW\ORYpVFKRSQRVWLMHGQRKRDWRPX
=iYLVtQDVORåHQtHOHNWURQRYpKRREDOXDWRPXDQDGLIUDNþQtPYHNWRUX N − N -HKRPD[LPiOQtKRGQRWDMHURYQDDWRPRYpPXþtVOX=
6WUXNWXUQt IDNWRU FKDUDNWHUL]XMH UR]SW\ORYRX VFKRSQRVW MHGQp
HOHPHQWiUQtEXN\NU\VWDOX3ODWt
6(KNO) = Σ I M H[S(−L* KNO U M )
NGH I MH DWRPRYê UR]SW\ORYê IDNWRU MWpKR DWRPX EXN\ YêUD] * KNO U M
SHGVWDYXMH Ii]RYê UR]GtO YOQ\ UR]SW\ORYDQp MWêP DWRPHP Y]KOHGHP N
DWRPX3URWRåHSODWt
M
* KNO = KE + NE + OE D U M = [ M D + \ M D + ] M D O]H VWUXNWXUQt
IDNWRU]DSVDWYHWYDUX
6(KNO) = Σ I M H[S [−πL(K[ M + N\ M + O] M )]
$PSOLWXGD YOQ\ GLIUDNWRYDQp QD URYLQiFK KNO MH StPR ~PUQi
VWUXNWXUQtPX IDNWRUX LQWHQ]LW\ GLIUDNWRYDQp YOQ\ MHKR þWYHUFL -HOL SUR
QMDNRX NRPELQDFL LQGH[$ KNO VWUXNWXUQt IDNWRU 6KNO SDN VH GDQi
GLIUDNFH QD GLIUDNWRJUDPX QHREMHYt SHVWRåH %UDJJRYD SRGPtQND MH SUR
GDQRXVRXVWDYXURYLQKNOVSOQQD9OQQtVHLQWHUIHUHQFtY\UXãt
9\MiGHPH VWUXNWXUQt IDNWRU PtåN\ NXELFNp SURVWRURY FHQWURYDQp
7XWRPtåNXO]HFKiSDWMDNRSURVWRXNXELFNRX6&VEi]tWYRHQRXDWRP\Y
SRORKiFK [ \ ] D [ \ ] 3HGSRNOiGiPH åH YãHFKQ\
DWRP\YEXFHPDMtVWHMQêDWRPRYêUR]SW\ORYêIDNWRU'RVD]HQtPGRY]WDKX
Y\FKi]t
6(KNO) = I M { + H[S [−πL(K + N + O)]} -HOL KNO OLFKp þtVOR SDN 6KNO MHOL KNO VXGp þtVOR SDN
6KNO I = WRKR SO\QH åH QD GLIUDNWRJUDPX VH QHREMHYt GLIUDNFH QD
URYLQiFKPDMtFtFKMHGHQQHERYãHFKQ\WL0LOOHURY\LQGH[\OLFKiþtVODQDS
DWG
3RUXFK\NU\VWDORYpPtåH
5HiOQpNU\VWDO\MVRXYåG\YXUþLWpPVP\VOXQHGRNRQDOp
3RUXFK\NU\VWDORYpPtåN\VHREY\NOHGOtQD
D ERGRYp FKHPLFNp StPVL QHREVD]HQp SRORK\ Y PtåFH DWRP\ PLPR
UHJXOHUQtSRORK\
EþDURYpGLVORNDFH
F SORãQp SRYUFK SHYQp OiWN\ YQLWQt SRYUFK\ KUDQLFH ]UQ YUVWHYQp
FK\E\
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NU\VWDOLFNêFK OiWHN 1DS HOHNWULFNi YRGLYRVW QNWHUêFK SRORYRGLþ$ MH
]S$VREHQD PDOêP PQRåVWYtP StPVt ]DEDUYHQt PQRKêFK NU\VWDO$ MH
]S$VREHQR SRUXFKDPLOXPLQLVFHQFH MH G$VOHGNHP StWRPQRVWL StPVt
GLIX]H DWRP$ VH Gi XU\FKOLW StPVHPL QHER SRUXFKDPL PHFKDQLFNp D
SODVWLFNp YODVWQRVWL SHYQêFK OiWHN MVRX XUþHQ\ REY\NOH SRUXFKDPL ,RQWRYi
YRGLYRVWMHXPRåQQDH[LVWHQFtYDNDQFt
9DNDQFHDLQWHUVWLFLiO\
9DNDQFH 6FKRWWN\KR SRUXFKD SHGVWDYXMH DWRP QHER LRQW FK\EMtFt Y
PtåFHREU
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,QWHUVWLFLiOMHDWRPXPtVWQêPLPRPtåNRYRXSRORKX3LY\VXQXWtDWRPX
] PtåNRYp SRORK\ GR LQWHUVWLFLiOQt Y]QLNQH SiU YDNDQFH LQWHUVWLFLiO
QD]êYDQê)UHQNHO$YSiUREU
9 WHSHOQp URYQRYi]H MH Y GRNRQDOpP NU\VWDOX YåG\ XUþLWê SRþHW Q
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)
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−
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G[ 8[ SRWHQFLiOQtHQHUJLLYURYQRYiåQpSROR]HSRNOiGiPHURYQRXQXOH
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R
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VHEH S$VREt HODVWLFNêPL VLODPL HODVWLFNi NRQVWDQWD β 3HGSRNOiGHMPH åH
QDVHEHS$VREtYåG\MHQVRXVHGQtDWRP\DåHSODWt+RRN$Y]iNRQ3RK\ERYi
URYQLFHQWpKRDWRPXHW]FHSDNPiWYDU
GX
P GW Q = −β(X Q − X Q− ) − β(X Q − X Q+ ) = β(X Q+ + X Q− − X Q ) X MHYêFK\ONDQWpKRDWRPXMHKRåURYQRYiåQiSRORKDMH[
HãHQtURYQLFHO]HKOHGDWYHWYDUXSRVWXSQpYOQ\ãttFtVHSRGpOHW]FH
X $H[S>LωWN[@
ω
N = F MHYOQRYpþtVORFMHU\FKORVWãtHQtYOQ\
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Q
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Q
Q
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ω(N) = ω PD[ VLQ ND ω PD[ =
β
P
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π
π
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D
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WM H[LVWXMH
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]yQ
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ω
ND
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IUHNYHQFtFKRGGRωPD[ 3URWRåHVHHODVWLFNpUR]UXFK\ãttYNU\VWDOXU\FKORVWt]YXNXPOXYtVHR
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X UHVS X 3RK\ERYp URYQLFH SUR VXGp UHVS OLFKp DWRP\ VH ]DStãt YH
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0 G X Q = β(X
Q+ + X Q + X Q )
GW Q
Q
Q
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P G X Q+ = β(X
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GW %XGHPHOLKOHGDWHãHQtURYQLFYHWYDUXURYLQQêFKYOQW\SX
X $H[S>L^ωWQND`@
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0ω β $β FRVND% β FRVND$Pω β % NWHUi Pi QHWULYLiOQt HãHQt SL QXORYpP GHWHUPLQDQWX VRXVWDY\ = WpWR
SRGPtQN\REGUåtPHGLVSHU]QtY]WDKSURIUHNYHQFLNPLW$
Q
Q
ND
ω = β  P + 0  ± β  P + 0  − VLQ
P0 1D UR]GtO RG MHGQRDWRPRYpKR HW]FH PiPH Q\Qt GYIUHNYHQFH ω D
ω RGSRYtGDMtFtMHGQpKRGQRWYOQRYpKRYHNWRUXN
9JUDIXωNREUWRYHGHNHY]QLNXGYRXYWYt
9WHY ω RGSRYtGi DNXVWLFNêP NPLW$P MHMLFKå IUHNYHQFH QDEêYDMt KRGQRW
RGω GRω PD[ =
0 -HGQiVHRVRXIi]RYpNPLWiQtVRXVHGQtFKDWRP$
β
)UHNYHQFH YH YWYL ω QDEêYDMt KRGQRW RG  
P GR β  P + 0  D
β
NPLW\DWRP$MVRXSURWLIi]RYp-GHOLRLRQW\MHNPLWiQtVSRMHQRVH]PQDPL
GLSyORYêFKPRPHQW$DYOQDPiFKDUDNWHUHOHNWURPDJQHWLFNpKR]iHQt3URWR
VHYWHYω QD]êYiRSWLFNi
9\ãHWHPHI\]LNiOQtYODVWQRVWLRERXYWYt
3URN WMYHOPLGORXKpYOQ\λ → ∞ YWHYDNXVWLFNiω DDPSOLWXG\$ %
RSWLFNiω + = β  P + 0  DDPSOLWXG\$ %P0
9StSDGYHOPLNUiWNêFKYOQλ → N → πD GRVWDQHPH
β
YWHYDNXVWLFNiω − = 0 $ ≠ % = β
YWHYRSWLFNiω + = P $ = % ≠ 9 DNXVWLFNp YWYL MVRX OHKNp DWRP\ P Y NOLGX % ]DWtP FR Y
RSWLFNpYWYLMVRXYNOLGXWåNpDWRP\0$ 3R]QiPND -H ]QiPR åH LRQWRYp NU\VWDO\ DEVRUEXMt LQWHQ]LYQ ]iHQt Y
LQIUDþHUYHQpREODVWLFRåMHG$VOHGHNY\EX]HQtNPLW$PtåN\RGSRYtGDMtFtFK
RSWLFNpYWYLGLVSHU]QtUHODFH
'LHOHNWULFNpDRSWLFNpYODVWQRVWL
ÒYRG
0H]LSHYQêPLOiWNDPLH[LVWXMHVNXSLQDQHYRGLYêFKOiWHNQD]êYDQêFK
GLHOHNWULND 6YRX StWRPQRVWt MVRX VFKRSQD ]HVLORYDW ~þLQN\ HOHNWULFNpKR
SROH]YWãRYDWNDSDFLWXNRQGHQ]iWRUX
= PDNURVNRSLFNpKR KOHGLVND SRSLVXMHPH YODVWQRVWL GLHOHNWULN MHMLFK
SHUPLWLYLWRXDGLHOHNWULFNRXVXVFHSWLELOLWRX
= DWRPRYpKR KOHGLVND MVRX RE YHOLþLQ\ XUþHQ\ VFKRSQRVWt DWRP$ D
PROHNXO SL]S$VRELW VH HOHNWULFNpPX SROL =iNODGQtPL VWDYHEQtPL
MHGQRWNDPLGLHOHNWULNMVRXDWRP\LRQW\QHERPROHNXO\NWHUp
LPDMtWUYDOêGLSyORYêPRPHQWSROiUQtOiWN\
LLQHPDMtWUYDOêGLSyORYêPRPHQWQHSROiUQtOiWN\GLSyO\Y]QLNDMtWHSUYH
~þLQNHPYQMãtKRSROH
%XGHPH VH ]DEêYDW WtP MDN YQMãt HOHNWULFNp SROH RYOLYXMH
HOHNWURQRYRXVWUXNWXUXGLHOHNWULNDDMDNpYODVWQRVWL]WRKRSO\QRX
9]RUHN GLHOHNWULND XPtVWtPH GR VWDWLFNpKR HOHNWULFNpKR SROH QDS PH]L
GHVN\ NRQGHQ]iWRUX =DMtPi QiV QD þHP ]iYLVt VWDWLFNi GLHOHNWULFNi
NRQVWDQWD
9\ãHWtPH RSWLFNp YODVWQRVWL GLHOHNWULN WM MHMLFK UHDNFL QD VWtGDYp
HOHNWULFNpSROHVSRMHQpVHOHNWURPDJQHWLFNRXYOQRX
=iNODGQtY]WDK\
9 VRXVWDY 6, YHNWRU HOHNWULFNpKR SROH ( D YHNWRU HOHNWULFNp LQGXNFH
SRVXQXWt' YStSDGL]RWURSQtKRSURVWHGtVRXYLVHMtY]WDK\
' = ε ( + 3 = ε ( + η H )( = ε ε U ( NGH3 = ε η H ( MHYHNWRUSRODUL]DFHSRODUL]DFHREMHPRYpMHGQRWN\
η H MHGLHOHNWULFNiVXVFHSWLELOLWD
ε U MHUHODWLYQtSHUPLWLYLWDSURVWHGt
ε )PMHSHUPLWLYLWDYDNXD
9StSDGDQL]RWURSQtKRSURVWHGtYHNWRU\3 D( QHMVRXURYQREåQpDη H Dε U
MVRXWHQ]RU\
3LSRPHPH åH SODWt L DQDORJLFNp Y]WDK\ PH]L YHNWRUHP LQWHQ]LW\
PDJQHWLFNpKRSROH+ DPDJQHWLFNRXLQGXNFt% % = µ + + 0 = µ ( + η P )+ = µ µ U + =-DQiþHN33RQtåLOÒYRGGRI\]LN\SHYQêFKOiWHN
0 = µ η P + MH YHNWRU PDJQHWL]DFH PDJQHWLFNê PRPHQW REMHPRYp
MHGQRWN\
η P MHPDJQHWLFNiVXVFHSWLELOLWD
µ U MHUHODWLYQtSHUPHDELOLWD
µ +PMHSHUPHDELOLWDYDNXD
9HNWRU\HOHNWULFNpKRDPDJQHWLFNpKRSROHY\KRYXMt0D[ZHOORYêPURYQLFtP
YGLIHUHQFLiOQtPUHVSLQWHJUiOQtPWYDUX
GLY' = ρ URW( = − G%
GW ∫∫ 'G6 = ∫∫∫ ρG9 = 4*DXVVRYDYWDHOHNWURVWDWLN\
GLY% = URW+ = M + G'
GW ∫∫ % G6 = ∫ + GO = ∫∫ M G6 = ,
ε L = − GGWΦ
M
ρ
εL
)DUDGD\$Y]iNRQHOHNWUR
PDJQHWLFNpLQGXNFH
*DXVVRYDYWDPDJQHWLVPX
$PSpU$Y]iNRQFHONRYpKR
SURXGX
MHYHNWRUKXVWRW\HOHNWULFNpKRSURXGX
MHKXVWRWDHOHNWULFNpKRQiERMH
MHLQGXNRYDQpHOHNWURPRWRULFNpQDSWtQH]DPVSHUPLWLYLWRX
0DNURVNRSLFNiDPLNURVNRSLFNiWHRULHGLHOHNWULFNêFK
YODVWQRVWt
(OHNWULFNp SROH Y\WYiHQp NROHP ERGRYpKR QiERMH MH GiQR
T
&RXORPERYêP]iNRQHP( =  πε   UU  'LHOHNWULFNpYODVWQRVWLMVRXXUþHQ\GLSyO\QLNROLYL]RORYDQêPLDWRP\
'HILQXMPHGLSyORYêPRPHQWGLSyOXS = T5 REU
5→
T
=
T
S
→
2EU'LSyORYêPRPHQW
+
2
+
S→
2EU3HUPDQHQWQtGLSyORYêPRPHQWPROHNXO\YRG\
(OHNWULFNpSROHY\WYiHQpYHYDNXXGLSyOHPMHGiQRY]WDKHP
[(SU )U −U S]
((U) = πε U (
)
/LERYROQiDV\PHWULFNiPROHNXODPiWUYDOêGLSyORYêPRPHQWREU
MHGQRWNRX MH GHE\H &P 3RODUL]DFH 3 GLHOHNWULND MH URYQD
GLSyORYpPXPRPHQWXREMHPRYpMHGQRWN\
3 = Σ S L = Σ TL5L
L
L
3RGtYHMPHVHEOtåHQDPHFKDQLVPXVSRODUL]DFHGLHOHNWULNDREU
4
4
9
9 9 η
4
9 9 η
4
4
4
2EU'LHOHNWULNXPPH]LGHVNDPLQDELWpKRNRQGHQ]iWRUX
'HVN\ NRQGHQ]iWRUX QDELMHPH QiERMHP 4 PH]L GHVNDPL VH Y\WYRt
4
KRPRJHQQt HOHNWULFNp SROH R LQWHQ]LW ( = ε 6 D UR]GtO SRWHQFLiO$ PH]L
GHVNDPLMH8 = (G 9ORåtPHOL PH]L GHVN\ NRQGHQ]iWRUX GLHOHNWULNXP LQWHQ]LWD SROH L
SRWHQFLiOVHε U NUiWVQtåt1DSRYUFKXGLHOHNWULNDVHREMHYtLQGXNRYDQêQiERM
R SORãQp KXVWRW σ L = 3 &KFHPHOL XUþLW MDN EXGRX MHGQRWOLYp GLSyO\
UHDJRYDW QD YQMãt SROH PXVtPH Y\SRþtWDW LQWHQ]LWX SROH Y PtVW NGH VH
GLSyOQDFKi]tMHWRW]YORNiOQtHIHNWLYQtSROH0XVtPHSLWRPXYiåLWYQMãt
HOHNWULFNp SROH SROH Y]QLNOp RG SRYUFKRYêFK LQGXNRYDQêFK QiERM$ L
HOHNWULFNpSROHY\WYRHQpRVWDWQtPLGLSyO\7HQWRYêSRþHW]iYLVtMLåQDW\SX
PtåN\NWHURXGLHOHNWULNXPY\WYit2EHFQMH]QDþQVORåLWê6SRNRMtPHVH
V/RUHQW]RYêPYêVOHGNHPNWHUêSODWtSURNU\VWDO\NXELFNpPtåN\
( ORN = ( + 3ε 3URYHNWRUSRODUL]DFHSODWtY]WDKNWHUêSHSLãPHGRWYDUX
3 = ε (ε U − )( 3RGRVD]HQt]tVNiPHY]WDKSURORNiOQtSROH
(ε +)(
( ORN = U
9 GDOãtFK ~YDKiFK S$MGH R QDOH]HQt Y]WDKX PH]L SHUPLWLYLWRX MDNR
PDNURVNRSLFNRX YHOLþLQRX D SRODUL]RYDWHOQRVWPL DWRP$ MDNR YHOLþLQDPL
PLNURVNRSLFNêPL
3RODUL]RYDWHOQRVW α DWRPX MH GHILQRYiQD SRPRFt ORNiOQtKR
HOHNWULFNpKRSROHNWHUpQDWHQWRDWRPS$VREt
S = α( ORN
3RODUL]RYDWHOQRVWMHYODVWQRVWtDWRP$]DWtPFRSHUPLWLYLWD]iYLVtQD]S$VREX
MDNMHNU\VWDO]DWRP$VHVWDYHQ
3RODUL]DFHFHOpKRNU\VWDOX
3 = 1 M S M = Σ 1 M α M ( ORN (M)
M
1 M MH NRQFHQWUDFH D α M SRODUL]RYDWHOQRVW DWRP$ M ( M MH ORNiOQt SROH
S$VREtFtYDWRPRYêFKSRORKiFKM
'RVDGtPH]DORNiOQtSROH]/RUHQW]RYDY]WDKX
ORN
Σ 1 α (ε )(
3 =
6XVFHSWLELOLWD
M
M
M
U
Σ 1 (ε +)
M
U
ηH =
= ε 5HODWLYQtSHUPLWLYLWDVHY\MiGtXåLWtPY]WDKX3RMHGQRGXFKêFK
~SUDYiFKGRVSMHPHNH&ODXVLRY0RVVRWLKRY]WDKX
ε −
= ε Σ 1 M α M
ε +
M
3
ε(
U
U
M
=QiPHOL]PLNURVNRSLFNpWHRULHKRGQRW\SRODUL]RYDWHOQRVWL α DWRP$
SDNYStSDGNXELFNêFKNU\VWDO$P$åHPHY\SRþtWDWSHUPLWLYLWXGLHOHNWULNDD
SHGYtGDWRSWLFNpYODVWQRVWLOiWN\
&HONRYi SRODUL]RYDWHOQRVW DWRP$ VHVWiYi ]H Wt StVSYN$
HOHNWURQRYpKRLRQWRYpKRDGLSyORYpKR
(OHNWURQRYê StVSYHN Y]QLNi SRVXQXWtP HOHNWURQRYpKR REDOX Y$þL
MiGUX ,RQWRYê StVSYHN SRFKi]t RG SRVXQXWt QDELWpKR LRQWX Y]KOHGHP N
RVWDWQtP LRQW$P 'LSyORYi SRODUL]RYDWHOQRVW Pi S$YRG Y PROHNXOiFK V
SHUPDQHQWQtP HOHNWULFNêP GLSyOHP NWHUp PRKRX ~þLQNHP SLORåHQpKR
YQMãtKRSROHPQLWVYRXRULHQWDFL
9 SHYQp OiWFH NWHUi MH WYRHQD WUYDOêPL GLSyO\ SLVStYDMt N
SRODUL]RYDWHOQRVWL YãHFKQ\ WL PHFKDQLVP\ -HMLFK StVSYHN MH DOH U$]Qê
1DS SL U\FKOH SURPQQêFK HOHNWULFNêFK SROtFK VH SURMHYt MHQ WDNRYê
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þDVWR WDNp S\URHOHNWULFNêP MHYHP $YãDN QH YãHFKQD SLH]RHOHNWULND QHER
S\URHOHNWULNDPDMtIHURHOHNWULFNpYODVWQRVWL
-DNRQDGMQpVHMHYtY\XåLWtHOHNWURRSWLFNêFKMHY$YHIHURHOHNWULFNêFK
OiWNiFKSURSDP"RYp]REUD]RYDFtV\VWpP\QDS3E=U7L2 =-DQiþHN33RQtåLOÒYRGGRI\]LN\SHYQêFKOiWHN
2SWLFNpYODVWQRVWLGLHOHNWULN
9 REODVWL RSWLFNêFK IUHNYHQFt VH SURMHYXMH SRX]H HOHNWURQRYi
SRODUL]DFH2VWDWQtGUXK\SRODUL]DFHPDMtUHOD[DþQtGREXQDWROLNY\VRNRXåH
QHVWDþt VOHGRYDW U\FKOp ]PQ\ HOHNWULFNpKR SROH Y RSWLFNp YOQ +] 9]WDK\SURHOHNWURQRYRXSRODUL]RYDWHOQRVWDSURUHiOQRXDLPDJLQiUQt
þiVW UHODWLYQt SHUPLWLYLW\ GLHOHNWULND MVPH RGYRGLOL Y SHGFKR]t
NDSLWROH
= 0D[ZHOOORYêFK URYQLF D O]H RGYRGLW åH Y QHYRGLYpP
SURVWHGt (σ = ) EH] YROQêFK QiERM$ (ρ = ) Y\KRYXMH YHNWRU HOHNWULFNpKR
SROH( YOQRYpURYQLFLW\SX
JUDG GLY ( = µε ∂∂ W( NGHU\FKORVWãtHQtYOQ\Y = µ ε =DYHGHPHOL LQGH[ ORPX SURVWHGt REY\NOêP Y]WDKHP Q = FY NGH F MH
U\FKORVWYOQ\YHYDNXX F = µ ε GRVWDQHPH
Q = µ U ε U
3UR GLHOHNWULND V Y\MLPNRX IHULW$ MH µ U ≈ D GRFKi]tPH NH ]QiPpPX
Y]WDKX
ε U = Q .G\åMHGLHOHNWULNXP]WUiWRYpMHUHODWLYQtSHUPLWLYLWDNRPSOH[QtYHOLþLQRX
ε U = ε U + L ε L DURYQåLQGH[ORPXEXGHNRPSOH[Qt
Q∗= Q − L N =HY]WDK$DVHVQDGQRRGYRGt
ε U (∞) = Q − N ε L (∞) = Q N
6\PERO ∞ SRXåtYiPH SURWR DE\FKRP GDOL QDMHYR åH VH MHGQi R
Y\VRNRIUHNYHQþQtSHUPLWLYLWX
9]WDK\ UHVS D MH GiQD VRXYLVORVW PH]L RSWLFN\
PLWHOQêPL YHOLþLQDPL Q N D HOHNWULFNêPL YHOLþLQDPL NRPSOH[Qt
SHUPLWLYLW\
(OHNWURPDJQHWLFNRXYOQXSRX]HMHMtHOHNWULFNRXVORåNXãttFtVHYSURVWHGt
YHVPUXRV\[O]HQDOp]WMDNRHãHQtYOQRYpURYQLFHYHWYDUX
∗
( = ( H[S [Lω(W − [Y )] = ( H[S  Lω  W − QF[  


8YiåtPHOL åH LQGH[ ORPX Q MH NRPSOH[Qt SDN YLGtPH åH
SRNOHV DPSOLWXG\ Y ]iYLVORVWL QD WORXã"FH [ MH GiQ IDNWRUHP H[S ωN[F
7RN HQHUJLH MH ~PUQê NYDGUiWX DPSOLWXG\ YOQ\ D MHKR SRNOHV V WORXã"NRX
SDNMHH[S  − λN[  'LHOHNWULFNpDRSWLFNpYODVWQRVWL
3RURYQiQtPY]WDK$DGRVSMHPHNHY]WDKXPH]LLQGH[HP
DEVRUSFH N LPDJLQiUQt þiVW LQGH[X ORPX D ]WUiWRYêP þLQLWHOHP WJ δ ε WJδ
GLHOHNWULNDN = Q 'LHOHNWULFNê ]WUiWRYê þLQLWHO WHG\ FKDUDNWHUL]XMH DEVRUSFL
HOHNWURPDJQHWLFNpHQHUJLHYSURVWHGt
'RSDGiOL URYLQQi HOHNWURPDJQHWLFNi YOQD QD UR]KUDQt GYRX
SURVWHGtGRFKi]tNRGUD]XDORPXYOQ\=H]iNRQD]DFKRYiQtHQHUJLHSO\QH
åH
57 NGH 5MHRGUD]LYRVWWMUHODWLYQtHQHUJLHRGUDåHQpYOQ\
7MHSURSXVWQRVWWUDQVPLWLYLWDWMUHODWLYQtHQHUJLHYOQ\SURFKi]HMtFt
SURVWHGtP
&KDUDNWHUL]XMHPHOL RE SURVWHGt LQGH[\ ORPX Q Q SDN SL
NROPpPGRSDGXSODWtSURRGUD]LYRVWDSURSXVWQRVWY]WDK\
Q∗− Q∗
5 = Q ∗ + Q ∗ 7 = Q∗ + Q ∗ 8YDåXMPH UR]KUDQt Y]GXFKX Q
LQGH[HPORPXQ QLN
3URRGUD]LYRVWGRVWDQHPH
(Q−) +N 5 =
Q∗ Q∗
D GLHOHNWULND V NRPSOH[QtP
(Q+) +N
7DWRURYQLFHVHGiSHSVDWMDNRURYQLFHNUXåQLFH
N +  Q − +−55  = 5 (−5)
1D REU MH ]REUD]HQD RGUD]LYRVW GLHOHNWULND V NRPSOH[QtP
LQGH[HP ORPX Y ]iYLVORVWL QD UHiOQp þiVWL LQGH[X ORPX Q D QD LQGH[X
DEVRUSFH N 8 EH]]WUiWRYêFK GLHOHNWULN N O]H GRViKQRXW PLQLPiOQtKR
RGUD]XQDUR]KUDQtMVRXOLLQGH[\ORP$RERXSURVWHGtYHOPLEOt]NpSULQFLS
LPHUVQtFKOi]Qt
N
5
Q
2EU 2GUD]LYRVW SURVWHGt V NRPSOH[QtP LQGH[HP ORPX Y ]iYLVORVWL QD
UHiOQpþiVWLQDLQGH[XDEVRUSFHN
7DE 6WDWLFNi SHUPLWLYLWD RSWLFNi SHUPLWLYLWD D IUHNYHQFH
WUDQVYHU]iOQtFKRSWLFNêFKIRQRQ$
ε
ε∞
Kω7N%
.
/L)
1D)
.)
5E)
&V)
/L&O
1D&O
.&O
5E&O
&V&O
/L%U
1D%U
.%U
5E%U
&V%U
/L,
1D,
.,
5E,
&V,
(OHNWURQ\YNRYHFK
=HYãHFK]KUXED]QiPêFKSUYN$MHSLQRUPiOQtFKSRGPtQNiFK
NRY$ 9HGOH þLVWêFK NRY$ H[LVWXMH PQRKR NRYRYêFK VOLWLQ D NRYRYêFK
VORXþHQLQNWHUpVHY\]QDþXMtWPLWRYODVWQRVWPL
DY\VRNiHOHNWULFNiYRGLYRVWσ H NOHVDMtFtVURVWRXFtWHSORWRX
EY\VRNiWHSHOQiYRGLYRVWσ W QH]iYLVOiQDWHSORWQDG7' σ
F SODWQRVW :LHGHPDQQRYD)UDQ]RYD ]iNRQD σHW = /7 NGH / MH /RUHQ]RYR
þtVOR
GY\VRNiNRQFHQWUDFHYROQêFKHOHNWURQ$QH]iYLVOiQDWHSORW
H Y\VRNi DEVRUSFH HOHNWURPDJQHWLFNpKR ]iHQt NRYRYê OHVN Y\VRNi
RGUD]LYRVW
IWYiUQRVW
9ODVWQRVWL Y\SOêYDMt ] NRYRYp YD]E\ DWRP$ $WRP\ WYRt NU\VWDORYRX
PtåNX LRQW$ YDOHQþQt HOHNWURQ\ MVRX YROQ SRK\EOLYp D WYRt HOHNWURQRYê
SO\Q9D]HEQpVtO\NRYRYpYD]E\MVRXQH]iYLVOpQDVPUX3RþHWQHMEOLåãtFK
VRXVHGQtFK DWRP$ MH RPH]HQ SRX]H JHRPHWULFNêPL PRåQRVWPL 3URWR NRY\
PDMt VWUXNWXU\ V QHMY\ããtPL NRRUGLQDþQtPL þtVO\ SUR )&& QHER
KH[DJRQiOQtSUR%&&
.ODVLFNêPRGHOHOHNWURQRYpYRGLYRVWL
'UXGH SHGSRNOiGDO åH Y NRYX MVRX YDOHQþQt HOHNWURQ\ MHGQRWOLYêFK
DWRP$ XYROQQ\ D SRK\EXMt VH YtFH PpQ YROQ NU\VWDOHP .DåGê DWRP
SLVStYi=HOHNWURQ\RVWDWQtFK=$=NGH=$ MHDWRPRYpþtVOR]$VWiYiSHYQ
Yi]iQRNDWRPX+XVWRWDYROQêFKHOHNWURQ$W]YHOHNWURQRYpKRSO\QXVHGi
XUþLWWDNWR
.DåGi OiWND REVDKXMH 12 DWRP$ Y MHGQRP NLORPROX
ρ
$YRJDGURYR þtVOR D $ NLORPRO$ Y P ρ MH KXVWRWD OiWN\ NJP $ MH
DWRPRYi KPRWQRVW NJNPRO 3URWRåH NDåGê DWRP SLVStYi =HOHNWURQ\MH
SRþHWYROQêFKHOHNWURQ$YREMHPRYpMHGQRWFH
=ρ1
Q = $ 3URYWãLQXNRY$MHQiGX P FP +RGQRW\HOHNWURQRYêFK
KXVWRWY\EUDQêFKNRY$MVRXYWDEXOFH
7\WR KXVWRW\ MVRX SLEOLåQ NUiW YWãt QHå KXVWRW\ DWRP$ Y
NODVLFNpP SO\QX ]D QRUPiOQt WHSORW\ D WODNX 3HVWR DSOLNDFH NODVLFNp
NLQHWLFNpWHRULHSO\Q$QDHOHNWURQRYêSO\QYNRYHFKSRVN\WXMHDGXFHQQêFK
YêVOHGN$
(OHNWURQ\YNRYHFK
7DE+XVWRW\YROQêFKHOHNWURQ$YHY\EUDQêFKNRYRYêFKPDWHULiOHFK
=
Q
/L.
1D.
..
5E.
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&X
$J
$X
%H
0J
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6U
%D
1E
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0Q
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+J.
$O
*D
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7O
6Q
3E
%L
6E
α
=iNODGQtSHGSRNODG\'UXGHKRWHRULH
FP L (OHNWURQ\ VH VUiåHMt QDY]iMHP D WDNp V LRQW\ PtåN\ 0H]L VUiåNDPL QD
VHEH QHS$VREt D SRK\EXMt VH YROQ DSUR[LPDFH YROQêFK HOHNWURQ$ -HMLFK
SRK\EVHtGt1HZWRQRYêPL]iNRQ\
LL 6UiåN\ HOHNWURQ$ MVRX RNDPåLWp MDNR X WXKêFK NRXOt D PQt U\FKORVW
HOHNWURQ$+ODYQtPVUiåNRYêPPHFKDQLVPHPMVRXLQWHUDNFHHOHNWURQDWRP
REU(OHNWURQHOHNWURQRYpVUiåN\KUDMtPHQãtUROL
LLL3HGSRNOiGiPHåHSUDYGSRGREQRVWVUiåN\]DMHGQRWNXþDVXMH τ NGHτ
MHW]YUHOD[DþQtGREDQHEROLVWHGQtGREDPH]LGYPLVUiåNDPL
2EU'UiKDYRGLYRVWQtKRHOHNWURQXUR]SW\ORYDQpKRLRQW\SRGOH'UXGHKR
WHRULH
6WHMQRVPUQiYRGLYRVWNRY$
3RGOH 2KPRYD ]iNRQD 8 = 5, RGSRU 5 ]iYLVt QD UR]PUHFK YRGLþH
$E\VHY\ORXþLODWDWR]iYLVORVW]DYiGtVHSURXGRYiKXVWRWDM = 6, 0UQê RGSRU µ MH NRQVWDQWRX ~PUQRVWL PH]L LQWHQ]LWRX HOHNWULFNpKR
SROH NWHUp Y\YROiYi SRK\E HOHNWURQ$ D SURXGRYRX KXVWRWRX ( = µ M 3DN
5 = /6 NGH/MHGpONDYRGLþHD6MHKRSU$H]
-HVWOLåH Q HOHNWURQ$ Y REMHPRYp MHGQRWFH VH SRK\EXMH U\FKORVWt Y MH
SURXGRYiKXVWRWD
M = −QHY H &
%H]StWRPQRVWLYQMãtKRHOHNWULFNpKRSROHVHHOHNWURQ\SRK\EXMtQDKRGLOHD
MHMLFKVWHGQtU\FKORVWY MHQXORYi
8YDåXMPH HOHNWURQ Y þDVH QXOD 1HFK" W MH GRED NWHUi XSO\QXOD RG
SHGFKR]t VUiåN\ 8U\FKORYiQtP Y SROL ( ]tVNi HOHNWURQ U\FKORVW − H(W
P 6WHGQtU\FKORVWNWHURX]tVNiPH]LGYPDVUiåNDPL
− τ
〈Y 〉 = H(
P DRGSRYtGDMtFtSURXGRYiKXVWRWD
M = QHP τ ( = σ( QH τ
9HOLþLQDσ = P = µ VHQD]êYiPUQiYRGLYRVW
5HOD[DþQtGRED τ DWtPLPUQêRGSRUNRY$VLOQ]iYLVHMtQDWHSORW
3URQNWHUpPDWHULiO\MVRXStVOXãQpKRGQRW\XYHGHQ\YWDE
3URGDOãt~YDK\RSRK\EXHOHNWURQRYpKRSO\QXEXGHYKRGQp]DYpVWEH]
RGYR]RYiQtSRK\ERYRXURYQLFLHOHNWURQXNWHUêVHSRK\EXMH~þLQNHPYQMãt
VtO\I (W) DSRGOpKiVUiåNiPVLRQW\PtåHVUHOD[DþQtGRERXτ
GS(W)
S(W)
GW = I (W) − τ
(OHNWURQ\YNRYHFK
7DE(OHNWULFNêRGSRUDUHOD[DþQtGREDQNWHUêFKSUYN$
(ρ 7) .
3UYHN
RGSRU
UHOD[DþQtGRED
/L
1D
.
5E
&V
&X
$J
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%H
0J
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6U
%D
1E
)H
=Q
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+J
$O
*D
,Q
7O
6Q
3E
%L
6E
. . .
(ρ 7) .
.
. .
+DOO$YMHYDPDJQHWRUH]LVWDQFH
8YDåXMPH YRGLþ SURWpNDQê SURXGHP D XPtVWQê Y StþQpP
PDJQHWLFNpP SROL 0DJQHWLFNp SROH XU\FKOXMH HOHNWURQ\ NROPR NH VPUX
MHMLFK SRK\EX RGSRU YRGLþH E\ PO Y]U$VWDW =iNODGQt VFKpPD +DOORYD
H[SHULPHQWXMHQDREU
]
+
([
M[
(\
\
[
Y[
2EU6FKHPD+DOORYDH[SHULPHQWX
0DJQHWLFNpSROHS$VREtQDHOHNWURQ/RUHQW]RYRXVLORX
) = −H  Y × %  DY\FK\OXMHKRYHVPUX]iSRUQpRV\\REU9URYQRYi]HStþQpSROH(
+DOORYRSROHNRPSHQ]XMH~þLQHN/RUHQW]RY\VtO\DSURXGSURFKi]tSRX]H
SRGpORV\[
([LVWXMtGYYHOLþLQ\FKDUDNWHUL]XMtFt+DOO$YMHY
PDJQHWRUH]LVWDQFH SRGtO LQWHQ]LW\ SROH ( SRGpO YRGLþH D SURXGRYp
(
KXVWRW\(%) = M [[ +DOORYDNRQVWDQWDLQWHQ]LWDStþQpKR+DOORYDHOHNWULFNpKRSROHY]WDåHQD
QD MHGQRWNRYRX SURXGRYRX KXVWRWX D MHGQRWNRYRX PDJQHWLFNRX LQGXNFL
(\
5 + = M [ % +DOORYRSROH( MHRULHQWRYiQRYHVPUX]iSRUQpRV\\5 MH]iSRUQp
.G\E\QRVLWHOpQiERMHPO\NODGQp]QDPpQNR+DOORYRSROHE\PORRSDþQê
VPU D 5 ! 7R XPRåXMH XUþRYDW ]QDPpQNR QRVLWHO$ QiERMH X
SRORYRGLþ$
([LVWXMt YãDN L NRY\ X NWHUêFK 5 ! MDNRE\ SURXG E\O
]SURVWHGNRYiQNODGQêPLQiERML9\VYWOHQtEXGHSRGiQRYNDSLWROH
8YDåXMPH VWiOH NODVLFNê PRGHO NRYX V HOHNWURQRYêP SO\QHP 3UR
YêSRþHW +DOORY\ NRQVWDQW\ D PDJQHWRUH]LVWDQFH Y\SRþWHPH QHMSUYH
SURXGRYpKXVWRW\M DM YStWRPQRVWLHOHNWULFNpKRSROHVHVORåNDPL( ( D
PDJQHWLFNpKRSROH% YHVPUXRV\]
\
[
\
+
+
+
[
\
[
\
(OHNWURQ\YNRYHFK
S
1DHOHNWURQS$VREtVtODI = −H  ( + Y[%  Y = P 3RK\ERYiURYQLFHHOHNWURQX
GS
S  S

GW = −H  ( + P[%  − τ
GS
9 XVWiOHQpP VWDYX SURXG QH]iYLVt QD þDVH WM GW = D SRK\ERYi URYQLFH
UR]HSVDQiGRVRXDGQLFSRGOHREUPiWYDU
S
= −H( [ − ω F S \ − τ[ S\
= −H( \ − ω F S [ − τ H%
NGH ω F = P VH QD]êYi F\NORWURQRYi IUHNYHQFH ~KORYi U\FKORVW URWDFH
YROQêFKHOHNWURQ$YPDJQHWLFNpPSROL
9\QiVREHQtPIDNWRUHP− QHPτ D]DYHGHQtPSURXGRYpKXVWRW\GRVWDQHPH
σ ( [ = ω F τM \ + M [ σ ( \ = −ω F τM [ + M \ QH τ( [
NGH M [ = P = σ ( [ σ MHPUQiYRGLYRVWEH]StWRPQRVWLPDJQHWLFNpKR
SROH+DOORYRSROH( MHXUþHQRSRåDGDYNHPåHQHH[LVWXMHStþQêSURXGWM
M \ = 3DN
ω τ
%
( \ = −( σ[ )M [ = − QH
M[ +DOORYDNRQVWDQWD5 + = − QH
]iYLVt MHQ QD NRQFHQWUDFL YROQêFK HOHNWURQ$ -HOL M SDN M [ = σ ( [ D
PUQêRGSRUQH]iYLVtQDPDJQHWLFNpPSROL
\
\
6WtGDYiHOHNWULFNiYRGLYRVWNRY$
3URYêSRþHWHOHNWULFNpKRSURXGXY\YRODQpKRYNRYXþDVRY]iYLVOêP
HOHNWULFNêPSROHPSHGSRNOiGHMPH( YHWYDUX
((W) = 5H ((ω)H[S(−LωW) 3RK\ERYiURYQLFHSURHOHNWURQYHOHNWULFNpPSROL
GS
S
GW = − τ − H(
HãHQt KOHGiPH YH WYDUX S(W) = S(ω)H[S (−LωW) D SR GRVD]HQt GR REGUåtPH
S(ω)
−LωS(ω) = τ − H((ω)
3RGOHY]WDKXMHM = − QH
P S DSRGRVD]HQt]DS
QH τ (
M (ω) = P −Lωτ 3RGOH 2KPRYD ]iNRQD MH M (ω) = σ(ω)((ω) NGH VWtGDYi PUQi
YRGLYRVW]iYLVtQDIUHNYHQFL]PQHOHNWULFNpKRSROH
σ
σ(ω) = −Lωτ σ = QHP τ
σ MHVWHMQRVPUQiPUQiYRGLYRVW9LGtPHåHSURVWtGDYiHOHNWULFNiSROH
MHYRGLYRVWNRY$NRPSOH[QtYHOLþLQRX3URQXORYRXIUHNYHQFL ω GRVWiYiPH
YêVOHGHN 'UXGHKR 1HMG$OHåLWMãt DSOLNDFt WRKRWR YêVOHGNX MH REMDVQQt
ãtHQtHOHNWURPDJQHWLFNêFKYOQYNRYHFK
3R]QiPND 9 UiPFL FYLþHQt RGKDGQWH SL MDNêFK IUHNYHQFtFK VH
UHiOQi þiVW VWtGDYp YRGLYRVWL EXGH OLãLW R RG YRGLYRVWL VWHMQRVPUQp
QDS SUR P -DNê SDN EXGH SRPU LPDJLQiUQt D UHiOQp þiVWL VWtGDYp
YRGLYRVWL"
(OHNWURPDJQHWLFNpYOQ\YNRYHFK
(OHNWULFNp SROH ( Y HOHNWURPDJQHWLFNp YOQ MH SURYi]HQR SROHP
PDJQHWLFNêP % 9]KOHGHPNPDOêPU\FKORVWHPHOHNWURQ$YNRYXYF
MH MHKR YOLY Y$þL SROL HOHNWULFNpPX ]FHOD ]DQHGEDWHOQê 'iOH EXGHPH
SHGSRNOiGDW åH HOHNWULFNp SROH VH StOLã QHPQt QD Y]GiOHQRVWHFK
VURYQDWHOQêFKVHVWHGQtYROQRXGUiKRXHOHNWURQX
9 SRK\ERYp URYQLFL VH SHGSRNOiGDOR åH YQMãt VtOD S$VREtFt QD
HOHNWURQ MH PH]L GYPD VUiåNDPL NRQVWDQWQt 7DNRYpPX SRåDGDYNX
Y\KRYXMHYOQQtRYOQRYpGpOFHRG GR QPWHG\LYLGLWHOQpVYWOR=
0D[ZHOORYêFKURYQLF
GLY' = ρ GLY% = ' = ε( % = µ+
∂%
∂'
URW( = − ∂W URW+ = M + ∂W SRSLVXMtFtFKHOHNWULFNpDPDJQHWLFNpSROHYHYRGLþLVHRGYRGtYOQRYiURYQLFH
SURYHNWRULQWHQ]LW\HOHNWULFNpKRSROH
∇ ( = F ∂∂W( + µ σ ∂∂(W
ýOHQ V SUYQt GHULYDFt ( GOH þDVX XUþXMH WOXPHQt YOQ\ -HOL YRGLYRVW
SURVWHGt σ QXORYi GLHOHNWULND SDN YOQD ãttFt VH Y WDNRYpP SURVWHGt MH
QHWOXPHQi 3tWRPQRVW YROQêFK QRVLWHO$ QiERMH σ MH QHQXORYp ]S$VREXMH
DEVRUEFLHOHNWURPDJQHWLFNpKRYOQQtYNRYHFK
F = µε
MHþWYHUHFU\FKORVWLãtHQtYOQ\YSURVWHGt
3HGSRNOiGiPHOL þDVRYRX ]iYLVORVW LQWHQ]LW\ HOHNWULFNpKR SROH YH
WYDUX((W) = ( H −LHωW P$åHPHYOQRYRXURYQLFLSHSVDWGRWYDUX
∇( + N( = µ σ F 
ϖ 
NGHN = F  + L ϖ 
9NRYHFKMHYRGLYRVWSURVWtGDYpHOHNWULFNpSROHNRPSOH[QtYHOLþLQRX
GDQRXY]WDKHP'RVD]HQtPGRY]WDKXSURN REGUåtPH
(OHNWURQ\YNRYHFK
 −
µ σ F 
L
+
 ω(+ω τ ) 

3L GRVWDWHþQ Y\VRNêFK IUHNYHQFtFK NWHUp VSOXMt SRGPtQNX ωτ〉〉 τMHiGX VVHY]WDKSRQNXG]MHGQRGXãt
µ σ
µσ
N =  ωF − τ  + L ωτ
$E\ HãHQtP YOQRYp URYQLFH E\OD KDUPRQLFNi YOQD PXVt EêW
UHiOQi þiVW N NODGQi 3DN GRVWDQHPH YOQX SRVWXSXMtFt OiWNRX 9 RSDþQpP
StSDG MH HãHQt DSHULRGLFNp SROH NOHVi H[SRQHQFLiOQ VH Y]GiOHQRVWt
åiGQiYOQDYSURVWHGtQHH[LVWXMH-HOLYHY]WDKXVSOQQDSRGPtQND
ω µσ
〉 τ
F
SDNVHOiWNRXHOHNWURPDJQHWLFNiYOQDãtLWEXGH3RGRVD]HQt]D σ D~SUDY
GRVWDQHPHSRGPtQNXSURRERUIUHNYHQFtSURNWHUpMHOiWNDSU$KOHGQi
N =
ω  
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