Effet non linéaire d’auto-démodulation d’amplitude dans
les milieux granulaires: théories et expériences
Vincent Tournat
To cite this version:
Vincent Tournat. Effet non linéaire d’auto-démodulation d’amplitude dans les milieux granulaires:
théories et expériences. Acoustique [physics.class-ph]. Université du Maine, 2003. Français. �tel00004227�
HAL Id: tel-00004227
https://tel.archives-ouvertes.fr/tel-00004227
Submitted on 20 Jan 2004
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1
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0.5
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L‚[email protected] [email protected]Ÿ[email protected]᜝<Ñ¢qè?6‡˜›nžq< dBChœ<ف —œ<ٟd–¢„œ-?6‡˜‡•<„Á}<„@¿<hAžáƒ„ž–BŸd›ƒ[email protected]”—|?6Á}<„@¿áó ¢h<M?6–dë+”—@BŸ˜<hAŸ˜<Ù¢„›A ?6›nœnœ<„Á}<„@†3C
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@PDER"'B',D* (JLNEST)[email protected]"'LF)L,-/ &= 3V ;
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: Pef Rg +_\hB<"<$,C!E.DK!* iEjDL#$NOB<DLF *[email protected]"'B'D+* ,-
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6p [Ip
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o[ o
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0
Fc 5 &Œ# F ` F ` MF8 f16V^ \)B<C" $J/EB',K!E,)L(JLGEiB/"/"!Ž
[email protected]"'L/,)L#D+*OB<DL *QDB~D/!"'@PK3* ,"'~?"Jf"<C!E~)IH z D+C€JLO)B<$&"'BDL;$‘iLER"'B.z DB'EC3*’"%H'B,C!E,)L|-/
&= 3vV 6
n7
0mW
o o
q Y q
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e 38F _ &8 [ _ 5 _ e —[email protected] @JL&O•&H3)B<C"(CFDJLF/•aDL#$hH'B<D3O*’"˜@PDEER"'BR-/ &= 3
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o p
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œ
o'[
p 0o
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p [
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ˆ
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0
o 0 [ o+op o p
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o
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l‚;™
wo
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q qZƒ q+]ˆ
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o
Y q
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= 3FV 6
w
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Y+ƒƒ ˆ
+ 7 kMv4 :;\¯{DL;$)@´@ F* E z;*’"}'C/DEER"'B'iLOj)IHPF* EBID')L;$&±h²<²3±‰²[email protected]"}B<"'”"'B''D+*D~3"!* H'Ž
D”"'BIDOJLO^z#B<)C"'/hµh-/ &= 3 V ;
w#
+ o
02W
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X
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&= 3V0 ;Fc ™ p\[ + o
X ƒ ˆY+] ˆq
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C/)@fz#B/"'0 //,K3*’"(OBIDL *QDBz DC€,-/ &= 3FV ;
# o V
++
02W#W q Y
8 6>„)L* JL6"<DBk"!*QD/E,C!JEGDL;$%z#B/"'//B/"A$&"RzF"'L;$&"'LFEfSTD”" ,zF"/"<$%JLNOBIDL *QDB @A"<$,D- +6V^71 +
c 7 6n‚7o3 [/o o o ˆ Y
]] ef r \)B<C" C DJL/•[email protected]!B<)\"!*QD/E,C!JE•‰DL;[email protected]+C!B<)\"!*QD/E,C!JE-/ &= 3 V ; c ™
,Z +
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% % 7„+šFEB<DJL S‰D”"£"'”)+* E,)L "<!FDE,)L£H3)B†L;)L* JL6"<DB‹z#B<)z DODE,)L iL @PDER"'B',D* †SvJEi
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V^ = dkV^F \e iLNB<)C€GaDkL6"'S|z DB<[email protected]?-/ &= 3 V ;Fc ;œ  op o![
o
Yq] Yq+]q
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&= 3vV ;
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Y
Y ƒ
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0
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o ]]+Z V^ef7` F_ef79 _4‰7d 7;` 6 6V „`%78Fw` \x‰”)+* E,)L‡)IH?zF"'BIC/)+*QDEJL&O H3)BIC"
C DJL?JL‹C/)@?z#B<"'/3"<$POBIDL *QDB^@P"/$,D'-/ &= 3V ;Fc #™7# o ˆ+ˆ
[Iefh` :=[.
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++
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= 3V 3;™ 5 p [
W
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o o
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&= 3V ;
l o [ p ++
02W
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`%vM_8 T16wV% V^ 5 5 _¶"„zFC!G)–HOBIDL *[email protected]"'B',D* .-/ &= 3=& o![
o
Y Y+Z
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W
o [/o\p
o
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c #™
7o
[o
o
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]+]+] #cv#8 % 3F\¯{)+O\‹"!*QD/E,C^,zF"'B/"'^JL‹C/)LFE.DC!Ef[email protected]"'@k)B'U"Jf"<C!E%DL;$PEi"%EBIDL/”"'B!"H!)B<C"R-/
8 ; &= 31 ˜
o [/o
o p
Ÿ ,ƒ ˆ
Y+
ƒ
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= 3\¡aJL6"<DB(DL;$L#)L#* iL6"<DB
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c 7Q;1 n‚ o![
o
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V ;Fc #™ op \[
+
Ÿ
Y+ƒ+Y Y+ƒYˆ
s &d 3+\xhEF$&"^"<y'zF¥'[email protected]"'LFE.D+*’"^$fH'B<)EER"'@P"'LEh"'ET$&"'fBI)E.DE,)L„$+DL?$&"'@* ,"'y„OBIDL *QDiB/"'˜@P)$!*’"',-/ ¬ 9 „V (o+F9(4_VŒ1 Q¬ +#
o + o
5
8
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OBIDL *QDB^@kDER"'B'D+* -/ &= 3 V ;
l
p[
o
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Y
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o
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36¢f« o ]Z+ƒ 0
ãt* 6>*?"N-N54JjL2*?8 $&.' L(4Jj*,!#89
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o
0
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Y
Y q
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‚
p+[ p
o
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= 3FV 6
p\[ p+
o p
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q ]
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H'B<"/!"'L#C!."',- &= 3 Fr ™7# [– o p
0
q+ƒ ˆ
] q ` _ : + acv_8 c_M1 &b +7\{DC€JLOM)IHAC)@?z#B<"'/,K3*’"OB<DLFF*[email protected]"'B'D+* ,- &= 3V ;
c ;™F o o [ o
++
Z q ƒ q ƒ+
Y
Mv 7v16_1 w8F_` 66F³aB<)/JL&O‘)IHk,$&"'LFE,C/D+*T')+* iE.DB'NSTD”"'PiL†DMCFDJL†)IHU"!*QD/E,CGK"<D$,-/ &= 3
V ;
n7 o o
+
0 W
ƒ+ƒ q
16# w“ D”"hz#B<)z DODE,)LMJLUOB<DL *QDB D/!"'@PK3* ."',-/ &= 3V ;  ;
[– o o
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+
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(ω) 
τ (ω) Ω + i
τ (ω) Ω + i
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c
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Ve ∼
= V fe(Ω) ×

[email protected]<

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Ω .
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)
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m
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τ (ω) ∂f
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τ (ω)
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>0
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ω/ωc = 0.01, 0.1, 0.2, 0.4, 0.8
¹
LHAC<XB9Glš¤ijAC<XB€PbYV¡H„tESB9Glš‰ka‡^F£@£:[email protected]?:[email protected][email protected]<>ACLSLHGJPEsVPbYK[[email protected]<XijTcG}QSB9GJIV<>UJB9G[SGPbYŽ°[email protected]<>ACL­[SG}IKAM[[email protected]<>ACL ‡
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[SG€Pd\mACLH[SG%º; g•hcACLM?9GJLFEHGc=:[HYZLH=vPbY4[aESB9TcG,[SG,PbYIKA*[[email protected]<>ACL‰N*PbY[email protected]'YZLH=9<[email protected]<>ACLl[SG s Gc=9?¾B9TOYZPX<>=9Tc<G tFQ0ACESB
PbY2Ž¤B9TcDMEHGJLHhcG,[email protected]|YZPX<>=9TcG,[]\mACLH[SGw[SG,Q~ACIVQ.YZ„jG ω/ω = 0.1 GJL|PX<X„tLHG,„[email protected]<>=¦Ž°ACLHhcTtNO‹ŒEH=9DME‰f„\ s(θ)ω/ω = 0.4 GJLfPX<X„tLHG
„[email protected]<>=€hJPbYZ<XBO‡Hœ€ES?9ACESB:[SGPbY4Ž¤B9TcDMEHGJLHhcG ω/ω = 0.6 NHP>G%QSB9AZ¡HP [email protected]? [email protected][email protected]<>ACLSLHGJ,P s f (θ) ‡*ˆ Y[email protected]'[email protected]<[email protected]<>ACL‚[SG
s
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f (θ)
[SG%Q0ACIVQ.YZ„jGh; g˜‡
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[email protected]<>ACLf[SGc=¾ACLH[SGc= º; g˜NtIKJIKG:[HYZLH=vP>G:[SACI|YZ<XLHG,[SG ŽVeB9TcDFEHGJLHhcGc= Ω ‰š ka„‡ ktF£o‡ j GJQ~GJLH[HYZLM?ONM[HYZLH=¾Pd\[email protected]<>ACL
YZEgB9TJ„t<XIKG
[email protected]@?9ACES?YChchcACIVQ.YZ„tLHTcGQ.YZB%ESLHG
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[email protected]<[email protected]<>ACL‚[SG%Pd\^YZIVQSPX<[email protected][SG4[aE‚[email protected]<X„tL.τYZP='YZLHτ=˜[email protected](ω)
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∂f (θ)
∂θ
∂f (θ)
∂θ
c
c
∂f (θ)
∂θ
c
Ωb
m
GIHEH&G
1
τ− (ω)
+
m
+
u[UcdaIORT.¿‚L E ½‚a ]XOEQ ¿‚L'`RLfY E ½‚QÁOR½ >T.M½ QSOEQ ¿)a ]RORQÁORQÁ`ba c"`.LfeQ
ˆ [email protected]@<[email protected]<>ACL Ve [email protected]<X„tL.YZP?9AC?'YZP Ve = Ve + Ve Q.YZB[STJIKA*[[email protected]<>ACL[aE•h'[email protected][email protected]<>DMEHGK;hg
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š¤ijAC<XB
P>Gl=9GchcACLH[[email protected]=9ACEH=}Pd\^YChchcACPbYC[SGk[HYZLH=6P>Gc=[email protected]<>ACLH=†š‰ka‡„k’kF£}GJ?†š‰ka‡„k ¿ £@£o‡ ¥¦[email protected]<XPX<>=žYZLF?PbYƒLHAC?'[email protected]<>ACLš‰ka‡nm{yj£oN
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Ω
Ωd
Ωb
Ωd
„
Ωb
VeΩ = VeΩb





1+
τs (ω)
τa (ω)
+d
2
cg (ω) 2
c0



1

(Ωτs (ω))2 − iΩτs (ω) 
τs−1 (ω)
2
τa−1 (ω)+D(ω) Ω2 −iΩ
c
0
«
š‰ka‡„ktxj£
.
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h'R.YZIVQ•[email protected][email protected]<>DFEHG6h; g²[email protected]?,[SACLHhhcACIVQSP>UJ?9GJIKGJLM?2LHTJ„tPX<X„jGOYZeSP>G6=9< τ (ω) τ (ω) ACEg=9< τ ∼ τ (ω) š¤P>G
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cg (ω)
c0
a
s
m
1
Ω
s
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5
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š τ (ω) τ (ω)£[email protected]<¾PbY‚[a<¢§[email protected]<>ACLL‰\^Y†Q.YC=4PX<>GJEQ0GJLH[HYZLF?6Pd\^[email protected]<>ACL´[aEQ.YCDMEHGJ?4[]\mACLH[SGl;ºg=9ESBP>GKIV<XPX<>GJE
LHACL"PX<XLHTOYZ<XB9G‚š τ ∼ τ (ω)£o‡¯%YZLH=P>Gc=2[SGJEan"hOYC={[email protected]@<[email protected]<>ACL[[email protected][email protected]<>GVe.YZPX<>[email protected][email protected]<>DMEHGV[aEh9R.YZIVQ
[email protected][email protected]<>DMEHG4h; g²[SACIV<XLHGt‡0ˆYfhcACLH[a<[email protected]<>ACLgLHTchcGc='='YZ<XB9G}Q0ACESB%[email protected]@<XIKGJLF?'YZP>GJIKGJLF? Q~GJES?2YZP>ACB9=%[email protected]
Ž[email protected]>TcGhcACIVIKG τ (ω) ≤ τ (ω) GJ? τ (ω) ≤ ‡a¥L‚[email protected]='YZLF?:DME‰\mGJL†„jTJLHTJB'YZPdNaPd\ <XLHTJ„FYZPX<X?9VeT [email protected]?
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P>[email protected][email protected][SG2Pd\[email protected]<>ACL‰š ka„‡ ktxF£˜GJL†[email protected]<>=:B9TJ„t<>ACLH=€hOYZB'[email protected]<>[email protected][email protected]<>DFEHGc=

[email protected]< Ω ,


[email protected]<
‰š ka„‡ ktsj£
Ve = Ve ×
Ω
,

[email protected]<
 1
Ω .
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[email protected]<X?9=€hcACIVIKG

[email protected]< τ τ (ω) τ (ω) ,

V

‰š ka‡ ¿ j£
R
@
=
<
V =
τ
(ω)
τ
τ
(ω)
,
V
(θ
)dθ

[email protected]< τ (ω) τ (ω) τ .

V
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— g <>[email protected]=g[SG
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w
τ τ (ω)
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; g´[a<¢§[email protected]<XijGc=O‡
º
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[SGJEanfQSB9GJIV<>GJB9=€B9TJ„t<XIKGc=:[SGwPd\[email protected]<>ACL‰š ka‡ ¿ M£¾A i†PbY[email protected]@<[email protected]<>ACL­[aE†h9R.YZIVQ­[a<¢§[email protected]<¢Žº; g[SACIV<XLHGtNSQ0GJES?:[email protected]
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[email protected]‡
ˆ¦\ <XLHTJ„FYZPX<X?9T
[email protected][email protected]=9TcG´[email protected]<¢¡.TcdG s¸PbYµŽB9TcDFEHGJLHhcG²<XLS<[email protected]<bYZP>G N[email protected]<KhcGJPXP>Go_`hJ<[email protected]?
[email protected]_E r|='YZIVIKGJLM?†QSτB9AM(ωh'RHGg)[SG­PbYτŽ¤B9(ωTcDMEHGJ)LHhcGW[SG­hcACESQSESB9G ω pm“t<bYtstsCzKš¤Q.YZB9hcGWDMEHG‚P>Gg?9GJIVQH=kωhOYZB'[email protected]<>[email protected][email protected]<>DMEHG"[SG
[a<¢§~EH=9<>ACL τ (ω) ∼ [a<XIV<XLMEHG:QSPXEH=i*<X?9G:YcijGch:Pd\^YZES„tIKGJLF?'[email protected]<>ACLk[SG˜PbY,Ž¤B9TcDMEHGJLHhcG DFEHG˜P>G˜?9GJIVQH=vhOYZB'[email protected]<>[email protected][email protected]<>DMEHG
[]\^[email protected][email protected]<>ACL τ (ω) ∼ £o‡¾ˆ‰ACB9=9DMEHGƒPbY"hcACLH[a<[email protected]<>ACL τ (ω ) τ = n
[email protected][email protected]<¢¡.TcG
τ (ω )
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V ∼ V (θ)
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N˜Q~ACESB†P>GcDMEHGJPPd\ <XLHTJ„FYZPX<X?9T τ = n
hcACIVIKGJLHhcG s´[email protected]"[email protected]<¢¡.TcG•Gc=9?ƒ[SACLHh•Go§[email protected]‡
n
τ (ω )
j ACIVIKGPd\ <XLHTJ„FYZPX<X?9T
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τ (ω) ≤ min(τ (ω), τ (ω))
Pd\ <XLHTJ„FYZPX<X?9T τ τ (ω) ≥ τ (ω) [email protected][email protected]<¢¡.TcGt‡˜¯%ACLHhtN ='GJP>ACL½Pd\[email protected]<>ACL ‰š ka‡ ¿ F£oN [HYZLH=lP>Gc=f[SGJEanµB9TJ„t<XIKGc=
‡0ˆ[email protected]'YZLH=9<[email protected]<>ACL•[SG τ (ω ) τ τ (ω ) s τ τ (ω ) τ (ω ) =9G4I|YZLS<¢ŽGc=9?9G
V '2
YZP>ACB9= Q.YZB:ESLHG[[email protected]<[email protected]<>ACL‚[aEƒQSB9AZ¡HP<XLS<[email protected]<bYZP—wg
Ωd
a
s
1
Ω
m
s
s
a
1
τa (ω)
1
τs (ω)
Ω
τa (ω)
τs (ω)
1
−iΩτs (ω)
Ωd
Ωb
Ωd
1
τa (ω)
1
Ω
s
1
τs (ω)
4
1
τa (ω)
1
τa (ω)
1
τa (ω)
1
τs (ω)
1
τs (ω)
1
τs (ω)
Ωb
Ωd
Ωb
τa (ω)
τs (ω) Ωb
θ
τm
τs (ω) −∞
Ω
m
Ωb
0
0
Ωb
a
s
a
m
a
s
s
m
a
m
s
a
in
s
in
in
c
1
ω4
s
1
|ω|
a
a
in
m
in
θ
−∞
Ω
Ωb
0
2π
ωin
s
in
0
m
Ωb
in
f
m
f
2π
ωin
f
a
Ωb
in
m
τ (ω)
τm
+
∂f
∂θ
a
in
s
−
a
in
in
m
s
in
f
m
a
in
s
in
!"#%$%& '($)*+,&-. /* 1
0.8
0.6
0.2
0.4
0
−0.2
−0.4
−0.6
−0.8
−1
−5
−4
−3
|Zƒb~}Nz€‹p‚ƒ†‡‘„}‰Ž
−2
−1
0
1
2
3
4
5
t/τm
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¹
¹
n = 5, 100, 500, 5000 ω/ωc = 0.9
‰š ka‡ ¿ m{£
ˆ Y|[email protected]'[email protected]Ž[email protected]|[email protected]<>ACL•[aEWQSB9AZ¡HP¦[aE"[email protected]<X„tL.YZP[STJIKAM[aESP>TV[SAC<X?2[email protected][email protected]Q0ACESB n N0='[email protected]<>=ŒŽdYZ<>='YZLM?wPbYlhcACLH[a<[email protected]<>ACL
[email protected]?KYChchcACIVQ.YZ„tLHTcGš¤ijAC<XBVPd\[email protected]<>ACL š‰ka‡ ¿ m®£@£}Q.YZBESLHG†[a<XIV<[email protected]<>ACL·[SGlPd\^YZIVQSPX<[email protected][SG
n
∼ τ (ω )
[[email protected]<X„tL.YZP’[email protected][email protected]<>ACLSLHGJPXP>Gxs
‡0ˆ¦\[email protected]<>ACL"[[email protected][email protected]'YZLH=ŒŽ°[email protected]|[email protected]<>ACL•[SACLSLHG4[SGc=w<XLaŽ°[email protected]|[email protected]<>ACLH=
1
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[SGc=fACLH[[email protected][email protected]<>DMEHGc=l;hg˜‡¾ˆ‰G­BBvCP>G‚[aEµQ.YZB'[email protected] [[email protected]?9G
[email protected]'[email protected]Ž[email protected]|[email protected]<>ACL´šDFES<YPX<>GJE­GJτ[email protected](ω)
B9G}[SGc=:B9TJ„t<XIKGc=,[SACIV<XLHTc=€Q.YZB:PbYV[STJIKA*[[email protected]<>ACL­[]\ ESL‚h'[email protected][email protected]<>DMEHG}h; g
[a<¢§[email protected]<¢Žo£%[email protected]<XIV<XPbYZ<XB9GWš[aEQ0AC<XLF?[SGKi*EHGKQSRMŠ*=9<>DFEHG{£4YZEBBvCP>Gl[aEQ.YZB'[email protected] [HYZLH=}Pd\mGonShJ<X?'[email protected]<>ACL´[email protected]<>DFEHG
[]\ <XIVQSESP>=9<>ACLH=[email protected]<>DFEHGc=6Q.YZB}„jTJLHTJB'[email protected]<>ACL±[[email protected]|Y­GJP>[email protected][email protected]´[HYZLH=4P>Gc==9GJIV<¢_`hcACLH[aEHhJ?9GJESB9=†p A EH='stx®z`‡
 hJ<dN [email protected]?%PbYl[aESB9TcGV[SG6Pd\ <XIVQSESP>[email protected]<>ACL•PbYC=9GJB2GJ? τ [email protected]?%P>G6?9GJIVQH=[SG6B9GchcACI}eS<XL.YZ<>='ACL[[email protected]|YkTJP>[email protected][email protected]
„jTJLHTJτB9Tt‡
qMESBPbY,¡H„tESB9G}š‰ka‡ryj£[email protected]?QSB9Tc=9GJLM?9TcG:[email protected]'YZLH=ŒŽ°[email protected]|[email protected]<>ACLf[aEK=9<X„tL.YZPSQ.YZB'[email protected]@<>DFEHG€QSB9Tc[a<X?9G Q.YZB¦PbY%[email protected]<>ACL|[SG
Pd\[email protected]<>ACL"š‰ka‡„ktxF£oNM[HYZLH=vP>GwhOYC=vA ilPbY}[email protected]@<[email protected]<>ACL‚YZEl=9<X„tL.YZP~[STJIKA*[aESP>T€QSB9A{iM<>GJLM?˜[email protected]<XLHhJ<XQ.YZP>GJIKGJLF?€[aEkh9R.YZIVQ
;hg·[a<¢§[email protected]<¢Ž@‡8ˆ YfŽ[email protected]<>ACL"[SG6IKAM[[email protected]<>ACL•Gc=9?wESLHG6„FYZEH=9=9<>GJLSLHGV[SG6[aESB9TcGhOYZB'[email protected]<>[email protected][email protected]<>DMEHG
‡0ˆ‰Gc=
i*<X?9Gc=9=9Gc=|[email protected][email protected]<>hJESPbYZ<XB9Gc=†[[email protected]<X„tL.YZEan·[email protected]|YZPX<>=9TcGc={‡vˆY•ŽB9TcDFEHGJLHhcGg[SG‚Pd\mACLH[SGW[SG‚Q~ACIVτQ.YZ=„[email protected]?lh9RHAC<>[email protected]<>G
QSB9A*h9RHG4[SG2PbYŽB9TcDMEHGJLHhcG}[SGhcACESQSESB9GtN ω = 0.9ω ‡HˆGc=€?9GJIVQH=,hOYZB'[email protected]<>[email protected][email protected]<>DFEHGc=%[SG[a<¢§[email protected]<>ACLgGJ?,[]\^[email protected][email protected]<>ACL
='ACLF?:?9GJP>=:DME‰\ <XP>=€=ž[email protected]<>=ŒŽACLM?:Pd\ <XLHTJ„FYZPX<X?9T
Q0ACESB PbY6ŽB9TcDFEHGJLHhcG2<XLS<[email protected]<bYZP>G ω GJ?:Q~ACESB
P>GwLHACI4eSB9G2<XLS<[email protected]<bYZP‰[[email protected]<>A*[SGc=:h; g[HYZτLH=(ωP>GwQ.) YCDMEHτGJ?:(ω[]\mACLH)[SG nτ (ω‡aˆ‰Gc=) [email protected]?'YZLF?9Gc= C GJ? C [SG τ (ω)
GJ?
=
=9ACLM?€[SACLHh[[email protected]<XLHTcGc=,„tB Chc5G s6Pd\ <XLHTJ„FYZPX<X?9T}<XLS<[email protected]<bYZP>Gt‡.ˆ‰ACB9=9DMEHG2P>G2LHACI4eSB9G [[email protected]<>AM[SGc=€º; g´[email protected]?
τ (ω) =
YZES„tIKGJLM?9TtN.PbYK[aESB9TcG}[aEƒQ.YCDMEHGJ?,[]\mACLH[SG6YZES„tIKGJLF?9G6GJ?:¡HLS<X?:Q.YZBw[SGJijGJLS<XB,[email protected]@<>GJESB9G4YZEƒn ?9GJIVQH=w[]\^[email protected][email protected]<>ACL
‡ j GchJ<~IKUJLH–
G s}Pd\ <XLHTJ„FYZPX<X?9Tw¡HL.YZP>G τ (ω ) τ (ω ) τ (ω ) NaDFES<[email protected][email protected][ s}PbY4hcACLH[a<[email protected]<>ACL†Q~ACESB
τ (ω )
ACeH='[email protected] ESLlQSB9AZ¡HP‰[STJIKA*[aESP>T,[email protected][email protected]<>ACLSLHGJP s ‡*ˆ[email protected]'[email protected]<[email protected]<>ACLƒ[SG f (θ) s [email protected]? YZP>ACB9= eS<>GJL†ACeS?9GJLFEHG
VΩi ' VΩdi ' 2f (θ) ,
τa (ωin ) ∂f
VΩf ' VΩdf ' 2
τmf ∂θ
.
cr
cr
2π
ωin
a
in
f
τm
τa (ωin )
τa (ω)
τm
a
τR
τL
L
R
2π
ω
m
c
a
in
m
in
s
in
in
in
C2
ω4
s
a
1
in
m
a
f
∂f (θ)
∂θ
in
s
in
∂f (θ)
∂θ
2
a
C1
ω
¹ #º¨ª’µ ¦§¢5²´¢p¦5²’« º¥Œ¤B¢Œ¨U©¦»¤B¥ ;«¯¸¢p¦
NŸ
1
0.8
0.6
0.2
0.4
0
−0.2
−0.4
−0.6
−0.8
−1
−5
−4
−3
|ˆƒb~}Nz‹‡‚ƒ†p„„}‰Ž
−2
−1
0
1
2
3
4
5
t/τm
šh›™œ– fž ¤Zª¨¦ 9§¬¤p¯xª’©%«‰¬¨P²±4¦Œ« ;¨ ªµf²!¥Œ¯x¬ ²± µn¥b£¤B¥2²«%©:¢ £ª¤ µ ª<¦§¬µ ± ©%«‰¬¨ !² ¢ µB· ¥2°§±_ª’©%«‰¬¨
²« º¥Œ¤2¢—¨U©:¢p¦ 9Œ¤22¥ °§±¢Œ¨ ­‡¢p¦ ;,²± ;¤Œ«)¦¶­—µnª«¤5ª±l¨ ¬«&¤+%
¹
5
¹ 7
¬ —©:¢Œ¨±¢5ª .¢2­
ω/ωc = 0.9, 0.65, 0.3, 0.1
[HYZLH=[email protected]?9GVACQ0TJB'[email protected]<>ACL‰‡‰‘ ACESB
Q~ACESB n = 5000 ‡
G,HEHCB
n=5
P>[email protected]<X„tL.YZP¦[email protected][email protected][email protected]<>ACLSLHGJP.s
hYLN]RM½*E’½:a ] a ]RORQUMPORQÁ`ba c"`.LfeQSÀRLN¿:½™M E ½™\TRQUM
f (θ)
YZP>ACB9=}DFE‰\ <XP¦=9GB'YZQSQSB9A*h9RHG|[SG
∂f (θ)
∂θ
O.½ >T.M½ QUM
‘’ACESB|[email protected][email protected]'YZLH=9<[email protected]<>ACL·[aE±B9TJ„t<XIKGƒ[SACIV<XLHT†Q.YZBKP>[email protected]'[email protected]?|[a<¢§[email protected]<¢Ž2[SGkPd\[email protected]„t<>Gƒ[SGc=Kh; gijGJB9=
P>GVB9TJ„t<XIKGK[SACIV<XLHT|Q.YZB}P>[email protected]'[email protected]@?e.YZPX<>[email protected][email protected]<>DMEHGK[SGc=4;ºg˜N]<XP¾Gc=9?2Q~At=9=9<XeSP>GK[SGKhcACIVIKGJLHhcGJBVYcijGchKPbYƒhcACLH[a<[email protected]<>ACL
<XLS<[email protected]<bYZP>G τ (ω ) τ = n
‡ WYZ<>[email protected]?9G6ŽAC<>[email protected]_`hJ<dN8<[email protected]?%[email protected]<XP>G[SGV[a<XIV<XLFEHGJB}PbYKŽB9TcDMEHGJLHhcG
τ (ω )
[SGPd\mACLH[SGK[SGVQ0ACIVQ.YZ„jG ω ='YZLH=h'R.YZLS„jGJB}P>GVLHACI4eSB9G n [[email protected]<>AM[SGc=h; gºhcACLF?9GJLMEHGc=4[HYZLH=}P>GQ.YCDFEHGJ?}[]\mACLH[SG
š n = n = const£o‡0¯2YZLH=hcGKhOYC=ON τ (ω) GJ? τ (ω) ijACLM?4YZES„[email protected][email protected]<>ACLSLHGJPXP>GJIKGJLM? s N‰YZP>ACB9=DMEHG
itY:YZES„tIKGJLF?9GJBQSPXEH=‰B'YZQS<>[SGJIKGJLM?:š ∼ £oN®hcG¦DMES<FitY˜<XLHTJiM<X?'YZeSP>GJIKGJLM?IKGJLHGJB’YZE}B9TJ„t<XIKG τ (ω ), τ (ω ) τ (ω)
Q~ACESB,ESLHGŽB9TcDMEHGJLHhcG¡HL.YZP>G ω = ω ‡ q*GJP>ACLgPd\[email protected]<>ACL±‰š ka‡ ¿ M£oNHPbY|[email protected]'YZLH=ŒŽ[email protected]|[email protected]<>ACL"[aE­QSB9AZ¡HP
τ =n
[STJIKAM[aESP>T€[SG V = R V (θ )dθ ijGJB9= V ∼ V (θ) [SGJiMB'YZ<X?¾[email protected]€[email protected]‡j PSŽ°YZES?¦='G =9ACESijGJLS<XB¾DME‰\^YcijGch:PbY
hcACLH[a<[email protected]<>ACL
N.Pd\ <XLHTJ„FYZPX<X?9T
[email protected]?:[email protected]<¢¡.TcGtN~GJ?€P>GQSB9AZ¡HP
<XLS<[email protected]<[email protected]?€τ[STc(ω[email protected]<X?€)Q.YZB:PbτYQSB9=GJIVn<>UJB9G}GonaτQS(ω[email protected]<>)ACLƒ[SGPd\[email protected]τ<>ACL´‰š ka‡τ¿ m®(ω£o‡Sˆ‰AC)B9=9DMEHτG2Pb(ωY6ŽB9)TcDFEHGJLHhcG}[SG2Q~ACIVQ.YZ„jG}[email protected]?
[a<XIV<XLFEHTcGtN~<XP Gc=9?,[email protected]=9TDMEHGQ0ACESB,PbYVŽ¤B9TcDMEHGJLHhcG2¡HL.YZP>G ω PbYK[a<>[email protected][email protected]<>[email protected]?,[email protected]<Ž°YZ<XeSP>G4DFEHG4ht\[email protected]?wPd\ <XLHTJ„FYZPX<X?9T
QSPXESB? vC?:DFEHGwPd\ <XLHTJ„FYZPX<X?9T
DMES<][email protected]?˜[email protected]<¢¡.TcGt‡S‘’YZB:hcACLH='TcDFEHGJLM?ON
τ (ω ) τ τ (ω )
=9GJP>ACL†Pd\[email protected]<>ACL‰š ka„‡ kjyt£˜GJ? Pd\[email protected]<>ACL‰š ka‡ ¿ j£oNSP>G%τQSB9AZ(ω¡HP‰)¡HL.YZPτw— g(ωGc=9)?€[STcτ[email protected]<X?:Q.YZB
τ (ω ) ∂f
‰š ka‡ ¿ kj£
.
V 'V '
∂θ
τ
ˆ‰Gc= [email protected]<>ACLH=2‰š ka‡ j£¾GJ?w‰š ka‡ kF£[email protected]? DMEHG:PbY}[email protected]'[email protected]<[email protected]<>ACLk[aE|B9TJ„t<XIKGw[SACIV<XLHT,Q.YZBvP>Gc=vACLH[SGc=˜h; g"[a<¢§~EH=9<XijGc=
ijGJB9=•P>G·B9TJ„t<XIKG·[S¿ ACIV<XLHT½Q.YZ¿ BP>Gc=ACLH[SGc=h; g e.YZPX<>[email protected][email protected]<>DFEHGc={N4Y¸PX<>GJE¶Q~ACESB ω N}DMES<KQ~GJES?[email protected]·[email protected][email protected]<XIKTcG½[SG
‡ j [email protected]?9G†[email protected]'YZLH=9<[email protected]<>ACL·=9GkI|YZLS<¢Ž°[email protected]?9G†Q.YZBKPbYW[[email protected]<[email protected]<>ACL½[aE²[email protected]<X„tL.YZP,[STJIKAM[aESP>TƒGJ?|Q.YZB
.
τ ≡ n
∼ τ (ω )
Pd\^YZES„tIKGJLF?'[email protected]<>ACLg[SGw='ACL†YZIVQSPX<[email protected][SG%[email protected][email protected]<>ACLSLHGJPXP>5G s 1 ‡Sˆ¾\[email protected][email protected]<>ACLƒ[[email protected][email protected]'[email protected]<[email protected]<>ACLƒ[SACLSLHG
[SGc= <XLaŽ°[email protected]|[email protected]<>ACLH=€[email protected] P>G2?9GJIVQH=€hOYZB'[email protected]<>[email protected][email protected]<>DFEHG τ (ω) [SGc=€ACLH[SGc=:h; g˜‡
a
in
m
in
in
2π
ωin
s
in
s
f
m
in
a
1
ω4
in
m
a
2π
ωf
f
θ
0
−∞ Ωb
in
in
m
Ω
a
→
in
0
Ω
s
Ωb
in
m
in
+
in
−
f
+
f
f
m
−
+
f
f
Ω
cr
m
2π
in ωcr
f
Ωb
−
f
f
f
m
f
f
m
cr
s
cr
τ (ωf )
f
τm
s
in
1
ω
f
s
f
!"#%$%& '($)*+,&-. /* qMESB PbY:¡H„tESB9G2‰š ka‡^xF£oN{PbY€[email protected]'YZLH=ŒŽ°[email protected]|[email protected]<>ACLV[aE=9<X„tL.YZP*[STJIKA*[aESP>T˜[aEHG s€PbY:[email protected]'YZLH=9<[email protected]<>ACL[aEB9TJ„t<XIKG˜[SACIV<XLHT Q.YZB PbY
a[ <¢§[email protected]<>ACL|[SGc=¾ACLH[SGc=v;hggijGJB9=¦P>G B9TJ„t<XIKG,[SACIV<XLHT Q.YZB¾PbY%QSB9ACQ.YZ„[email protected]<>ACLfe.YZPX<>[email protected][email protected]<>DMEHG€[SGc=¦ACLH[SGc= ;º[email protected]?¾QSB9Tc=9GJLF?9TcGt‡
GJ?:PbYŽ¤B9TcDMEHGJLHhcG<XLS<[email protected]<bYZP>G
ˆ YVhcACLH[a<[email protected]<>ACL­<XLS<[email protected]<bYZP>G τ (ω ) = C /ω τ = n
τ (ω ) = C /ω
[SGPd\mACLH[SGf[SGVQ0ACIVQ.YZ„jG ω = 0.9ω [email protected][email protected]?9GJLM?6[SG|[[email protected]<XLHGJB}P>Gc=}hcACLH=9?'YZLF?9Gc= C GJ? C ‡]ˆYkŽB9TcDFEHGJLHhcG
[SG4Pd\mACLH[SG[SG}Q0ACIVQ.YZ„jG[email protected]?%[email protected]<X?9G[a<XIV<XLFEHTcxG s|[email protected][email protected]<XB%[SG
‹ EH='DFE‰„\ s
GJLg„FYZB9[HYZLF?2P>G
LHACI4eSB9G n [[email protected]<>AM[SGc=%hcACLF?9GJLMEHGc=}[HYZLH=wP>G6Q.YCDFEHGJ?2[]\mACLH[SGVωh; g·=hcAC0.9ω
[email protected]?'YZLM?O‡ j GchJ<IKωUJLH=xG sK0.1ω
PbYlhcACLH[a<[email protected]<>ACLW¡HL.YZP>G
‡vˆY•[email protected]'YZLH=9<[email protected]<>ACLµ[aEµQSB9AZ¡HP f (θ) YZE½QSB9AZ¡HP [email protected][email protected]‚PbY
τ (ω ), τ (ω ) τ = n
ŽB9TcDMEHGJLHhcG2[SG2Q~ACIVQ.YZ„jG ω = 0.9ω GJL†PX<X„tLHG„[email protected]<>=9GGJ? PbY6ŽB9TcDFEHGJLHhcG ω = 0.1ω GJLƒPX<X„tLHGLHAC<XB9Gt‡
[email protected]? [email protected]<¢¡.TcGtN
j GJQ~GJLH[HYZLM?ONSPbY[email protected]<[email protected][email protected]<>ACL‚[HYZLH= PbYCDMEHGJPXP>G2Pd\mTJ?'YZ?˜¡HL.YZPA i†Pd\ <XLHTJ„FYZPX<X?9T
τ (ω ) τ (ω ) τ
Q0GJES?­[email protected]<XLHhJ<XQ0GYZEH=9=9<6YcijAC<XBgPX<>GJE‰‡%¯2YZLH=­hcGhOYC=ON€PbY·[[email protected]<[email protected]<>ACL [aE QSB9AZ¡HPw— g¼DMES<}<[email protected]<>GJLM?"[HYZLH=­PbY
[email protected]'YZLH=9<[email protected]<>ACL‚[email protected][a<¢§[email protected]<>ACL‚GJ?€QSB9ACQ.YZ„[email protected]<>ACL‚e.YZPX<>[email protected][email protected]<>DMEHG|š¤QSB9Tc[a<X?9G[HYZLH=€Pd\[email protected]<>ACL‰š ka‡ ¿ M£@£ Q~GJES?:[email protected]I|YC=9DFEHTcG
Q.YZB,Pd\ <XLF?9TJ„tB'[email protected]<>ACL•[aEgQSB9AZ¡HP’w— g²[aEHGYZEan­Go§]GJ?9=,[SG6[a<>[email protected][email protected]<>ACL­[SG4i*<X?9Gc=9=9Gfš¤QSB9Tc[a<X?9=%[HYZLH=,Pd\[email protected]<>ACL²‰š ka„‡ kjyj£@£o‡
 P:[email protected]?K[SACLHh†e0GOYZEHhcACESQ²QSPXEH=f[a< r|hJ<XP>Gƒ[]\mGon*[email protected]'YZ<XB9G­[SGc=V<XLaŽ°[email protected]|[email protected]<>[email protected]>Gc=KQ.YZB'[email protected]=f[aE²IV<XPX<>GJE·P>ACB9=9DFEHG
[a<¢§]TJB9GJLF?9=†Go§]GJ?9=lQSRFŠ[email protected]<>DMEHGc=k<XLa³HEHGJLHhcGJLF?ƒ[email protected]<XI4ESPX?'YZLHTJIKGJLF?‚P>G‚Ž°[email protected]<>ACLSLHGJIKGJLF?‚[SGgPd\^YZLF?9GJLSLHGWQ.YZB'[email protected]@<>DMEHGt‡
‘’ACESB¾Pd\^YZL.YZPXŠa=9G:[SG [email protected]<[email protected][email protected]<>ACLH=ONt<[email protected]?LHTchcGc=9='YZ<XB9G,[SG [a<>[email protected]~At='GJB¾[SG IKA*[SUJP>Gc=¦QSPXEH=QSB9TchJ<>=¾Q~ACESB¦PbY[STJQ~GJLH[HYZLHhcG
GJLkŽB9TcDMEHGJLHhcG[SG%Pd\^[email protected][email protected]<>ACL­[SGc=€ACLH[SGc=,[email protected][email protected]<>DMEHGc=ONH[SG%PbYV[a<¢§[email protected]<>ACLƒGJ?€[SG%PbYViM<X?9Gc=9='G2[SG%QSB9ACQ.YZ„[email protected]<>ACL‰‡
ˆ‰Gc=fGo§]GJ?9=fQSB9Tc[a<X?9=l[SG­[[email protected]<[email protected]<>ACL¸ACEµ[]\ <XLF?9TJ„tB'[email protected]<>ACL[aE·QSB9AZ¡HP%[STJIKA*[aESP>T­P>ACB9='DFEHG‚P>Gc=fhOYZB'[email protected]<>[email protected][email protected]<>DMEHGc=
[SGƒPd\mACLH[SGƒ[SG†Q0ACIVQ.YZ„jGš ω N τ £[email protected]<>GJLM?ON¦Q0GJESijGJLF?lYZEH='[email protected]<:[email protected]†I|YC=9DMEHTc=KQ.YZBKP>Gc=fGo§0GJ?9=|[SGƒ[a<¢§~B'[email protected]<>ACLµDFES<
YZQSQ.YZB'YZ<>='=9GJLF?€[HYZLH=€ESLHGhcACLa¡H„tESB'[email protected]<>ACL‚„[email protected]@<>DMEHG[email protected]@<>[a<XIKGJLH=9<>ACLSLHGJPXP>Gt‡
ˆ‰Gc=,hcACLH=9?'YZLF?9Gc= C N C GJ? C N.DMES<’YZQSQ.YZB'YZ<>='=9GJLF?2[HYZLH=,P>Gc=,[email protected]<>ACLH=w[SGc=€Q.YZB'[email protected]=%hOYZB'[email protected]<>[email protected][email protected]<>DMEHGc=
[]\^YZeH='[email protected][email protected]<>ACLgACEg[SG}[a<¢§[email protected]<>ACL‰NHQ0GJESijGJLF?wQSB9GJLH[aB9G4[SGc=,itYZP>GJESB9=,[a<¢§]TJB9GJLF?9Gc=%=9GJP>ACL­PbYKhcACLa¡H„tESB'[email protected]<>ACL"[aE‚IV<XPX<>GJE‰‡
‘YZB"GonaGJIVQSP>GtN%[email protected]<4PbY²QSB9Gc='[email protected]<>ACL [email protected]?'[email protected]<>DFEHG´YZQSQSPX<>DMEHTcG=9ESB­Pd\^YC=9='GJI}eSPbYZ„jG„tB'YZLFESPbYZ<XB9G±[email protected][email protected]<>TcGtNwhcG=9ACLM?ghcGc=
hcACLH=9?'YZLF?9Gc=¾DMES<.=9ACLM?¾IKA*[a<¢¡.TcGc=¦Q0ACESB¦QSB9GJLH[aB9G€GJLfhcACIVQS?9G:PbY%[email protected]<[email protected]<>ACLK<XLH[aES<X?9G€[]\^YZeH='[email protected][email protected]<>ACLlACE|[SG€[a<¢§[email protected]<>ACL‰‡
¯\^[email protected]=%ŽdYChJ?9GJESB9=[email protected]<XIKGJLM?'YZEan"Q~GJESijGJLM?}<[email protected]<XB6[HYZLH=[email protected]?'YZLM?9Gc=}hcACIVIKGVPd\mTJ?'YZ?}[SGV=9ESBŒŽ°YChcGV[SGc=
eS<XPXP>Gc=:Q.YZB€GonaGJIVQSP>Gt‡
a
in
1
in
in
in
m
in
2π
ωin
s
f
s
f
f
m
4
in
1
c
c
+
2
c
∂f (θ)
∂θ
c
m
2
f
2π
ωf
in
1
2
c
in
a
in
f
−
f
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