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 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. 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Îpq¢h<ÔA|?6WV¿B<Ô?6nWVBh< Aq-?J¢|è?#_@B<C§< <Ô-?¢h<Á}<@¿ÙA|?6WV¿B< ?AqAqm¢s¤[email protected]<é<hAnÁ7?6@ <F-?$|?6< K f£ @¢h@¿|?¢Ú R = 0.5 cm C Fδ = 10 N C E = 6 10 P a < ν = 0.21 @@B< CF¢h< VÔ¢hq<hAÚ@B @#Á}md<øf£ Ù@ó/< ff<h¢ ÎJ< E ' 5 10 P a CqùhA[email protected]Îp< K ' 5 10 N/m ¢h<nd1Á7?6q-?6#n Õ®ÁÁ}<s;?q<-?6@#<dAÚ<qA@1<@¿q<<[email protected]Á+q<&f£¥@B< <+-?A|?6@ ω k [email protected]¢h<q<?6q3ëmnÁ7?6@A3£¥<@¿J<@ABAn[email protected]¿@-?¢h<Á}<@ < -?wÎÏÁ}< u = Ae [email protected] ®£ ?6q3ëmnÁ7?6@Ên[email protected]3?6nq<+<U®£¥ V¿?6@#Ð T g Ó 67 0 0 10 0 6 8 iωt−k2Rn n ω=2 r Ð T T Ã§Ó K | sin(kR) | m 1 ω/ωc 0.9 %& 0.8 0.7 $ !#" 0.6 ! " 0.5 0.4 0.3 0.2 0.1 0 0 0.2 0.4 0.6 0.8 1 9 ÉWq iL M!&.I / NOo.6 h%TYM!hQ75Q0R7 ¢hq<H<Ô-?U¢qè?#_ @B<CA|?6@}Á7?×ëdnÁ+ÁR?67< <é-? V B<n<Ô<hAÑÁ}¿<hAÙ?¢hBAqWVB<hA k/kc ;?JA|?6@7< @ónBd[email protected]?6dë¢h@BAqqhA <¦¿<@@B<@¿J¦[email protected]<hAq¢h<@¿qA3C¦?6H?6qA3C Ð T TsT Ó ;?ÊÎ qV¿B<@B¢h<ø<ø¢hq<<²-?$¢|è?6[email protected]<?AqAqm¢h<C<ÖëmnÁ}h<<@aÎÏ@B¢@ <hAw?6|?6Á}ùq<hA&?¢h¢h<hAqAn<hA <ÖëdÚnÁ}<@|?6<Á}<@U<hA @@Bh<>?6 ωc = 2 r K m Ð T T Â§Ó G ¢h<q<øÎ qV¿B<@B¢h<CÑ<@BÁ+q<f£¥@B<<¢hq<¢hq<hAÚ@B $-?$nnÁwnq<<ø-?$q<Áwùq< [3@B<< êMnn[email protected]<+¦?6 k = ¤[email protected]<w<hAnÁ7?6@#<w-?&ÎÏqVB<@B¢h<7<}¢hq<7f£ @B<}¢|è?#_@B<7 R = 0.5 cm C C C < ν = 0.21 @@B< f ' 16 kHz â¿U-?=Bóq< T ÄCÚ<hA F = 10 N ρ = 4000 kg/m E = 6 10 P a q<qhA|<@qh<-?Ú<@B[email protected]¢h<&<7-?øA|?6@aA<}@BÁ+q<²f£¥@B<&[email protected]+-?q<Áwùq< [3@B<&<7êMnn[email protected] ;<|?6Ú ω/k @@B<-?/¦¿nq<hA|Aq<<è?A|<<ø®£¥@B<$?¢hBAWV¿B<[email protected]-?/¢qè?#_@B<<}-?/|[email protected]ó§<@q< -? ¢hÚ<q<qhAq<@qh<wAJ-? Bóq< T Ä ∂ω/∂k @@B<-?}¦mnq<hAqAq<<+óqÚ<sKM<hAJ¦¿nq<hAqA|<hAJA|@Uq<qhAq<@qh<hA <@²Îp@B¢@<>-?A ?6@A -? Bóq< T d0 ;?}¦mnq<hAqAq<<è?Aq<4?¢hBAWVB<4[email protected]-?&¢qè?#_@B<CÚ[email protected]-?7nnÁwnq<?A|Aq< ÎÏq V¿B<@B¢h<C<hAq<@$nBAÎp?6n< V¿B< -?7¦mnq<hAqAq<+<+q?6ó?6@/<hA@B<hA>?¢hBAWV¿B<hAU[email protected]< Á7?6q-?6¢h@BAn[email protected]><hAnn<hAUdή?6nJ< 1 fc = π c 0 π 2R r 1/6 K 3F0 = 3/2 1/2 m 4π ρ R4/3 4E 3(1 − ν 2 ) 1/3 67 3 10 c Q07Y p '!#XV0 P0,.$% D4$?y0= T, 9 ;[email protected] 909 c ω = 0) 1 % & 0.9 0.8 0.7 0.6 $ !#" 0.4 0.3 & & cφ (ω) 0.5 cg (ω) 0.2 0.1 0 0 0.2 0.4 0.6 0.8 1 9É S M1A"! 9 1+4UWV^$I!T5H$TAO*'4O P5/!4H PQ07Q0R07 ω/ωc -?q<-?6n¦§<Á}<@Îp?6n< |?6<H<hA ¢h@|?¢qAJ¢hÁw?6qh<+?6Á7?6q-?6<hAÔnn<hA3CB< <n<+A3£¥h¢n Ð T T Ä§Ó ;?¦mnq<hAqAq<U<Uè?Aq<><U®£¥@B< ?¢hBAWV¿B< <@&Îp@B¢@<U-?Î qV¿B<@B¢h<>[email protected]é-?4¢qè?#_@B<+Aq<>hdn <U-? q<-?6@< dAÚ<qA@Ð T T ÃÓ Ð T T d¿Ó ω ω c (ω) = = | sin(kR) | G`-?Î qV¿B<@B¢h<>@n<C ¢h<q<>¦mnq<hAqAq<J¦?6H<@k k Ð T T À§Ó c (0) = Rω f£¥ @B< ?6q< <Öëdq<hAqA@n< Ô-?4¦mnq<hAqAq<U<>è?A|< <@²ÎÏ@B¢@Ê<>-?Î q V¿B<@B¢h< Ð T T Y§Ó ω/ω c (ω) = c (0) arcsin(ω/ω ) á<hAnÁ7?6@Cf-?²¦mnq<hAqAq< c (0) ¦?6+<@¿¦¿nq@ 500 m/s Ð R = 0.5 cm C F = 10 N C ρ = 4000 kg/m C < Ó E = 6 10 P a ν = 0.21 ;?4¦mnq<hAqAq<U<>óqÚ<7C V¿[email protected] 4<n<C ?4-?ÎÏÁ}<+An¦[email protected]¿q< s Ð T T ^Ó ω c (ω) = c (0) 1 − ω ;qA V¿B< q<@B¦§<qA Cf<hA ¦mnq<hAqAq<hA <4óqÚ<7<<4è?Aq<}Aq@¿ó?6<h!A ;qA VB< ω q<@B¦§<qA ω -? ¦¿nq<hA|Aq<U<>óqω< q<@Bø¦§<0qA 0 C®£¥@B<ó<>@B< A|<Uq?6ó§<>nBAH[email protected]Ô-?}¢qè#? _@Bs< KM<hAq<-?6@BA+<@a¢h@@B<hA4<@è¿ÇmAqW VB<d#Aqn<Aq<q@nnAqh<hA[email protected]+-?Anq<7d#¿¢Á}<@¿!FL@ < fÚ<3C-? ¢h@@?6AqA [email protected]¢h<J<¢h<hAÙqhAqn|?6qA[email protected]<H¢3?AMf£ @&qhAq<3?67@dnÁ}<@BA@@B<<hAqÙnÁwÚ|[email protected]q<M@B< ?6n¢3?6@¦§<@B<n<+?6dëÁwnn<dëdnÁ}<@BA@@B<!A <<JÁ}¿ù<hAÔ²f£¥<Öëd<@B¢h<hA <JdAÚ<qA@ø< ¦¿nq<hA|Aq<é<ÖëmAq<@¿Ù[email protected]Ñ¢h<é Ç¿Ú< <éÁwnn<dëfC?6[email protected]®C-?UdAÚ<qA@7<é¦mnq<hAqAq<MÁ}mnAqh< <@w@B< dnÁ}<@BA@[email protected]BA @@B<>@øÚ[email protected]H< ?6 4®£¥B< <U®£ [email protected]q<@@B<?6|?6Á}W V¿B<+< ff<h¢Bh< [email protected]Ô<hA ¢|è?6nq<hAAn¦[email protected]qA+ ¤@B<+q<-?6@<dA<qAq@ ¢hÁwùq< |CÚq<@[email protected]¿><@$¢hÁwq<<hAJÁ}m<hAJ¦[email protected]<hAq¢h<@¿qAV?6?6|?6AqAq<@¿ ?6dÕ < w<>-?ÎÏq VB<@B¢h<+<>¢hq<C<hA |?6n< [email protected]<>¢qè?6nq<Äw< <hA[email protected]?6nÇdAqh< <@|?6n c1D ω = lim = 2R k→0 k r 1/6 K 3F = Rωc = √ 0 1/3 m 2 πρR 4E 3(1 − ν 2 ) 1/3 c φ c φ 67 φ 10 c φ c 3 0 φ 2 g φ 2 c c 9i 9 :ñ 81;=<*>6>*?81&(@A-0#47%&B6?47*?#81;C')"0<D<E8F<D*>6?*?8G&[email protected]<*?')5"JIK*>BL("<E"0-NM1283;O$QPR/3+&($B8 %5) *L c [email protected] [email protected]',47 IM. t/!#*%4, y9 )F&+D4O$t%V%"$#6tV&@ F' 9 4$#:M%"E[..d 'D41$?4&,H00 i+d &$>%@Fq, 9 U;5<$%*[10,Hoq/! 5 9 +$%D4?$?J B9É S KM<hA <@ùq<hA[email protected]@Bh<hA@ ¦mø@ø@BÁ+q<¢qAqA|[email protected]f£¥<ÖëmÚ<@B¢h<hAH[email protected]Ô<+Á7?6[email protected]< <>-?q?6ó?6@ ?¢hBAqWVB<[email protected] <hA qhA|<3?6dë@dnÁ}<@BA@@B<AU<>nn<hA ¡U<hêÔÃÃC KMAKa)^mCKMA asa Ln<hAHdRffùq<@J<hA @B<hA <hA?6q<hA?6-?#@?6q<<hAAnó@?6dë¯BdhA1Ð ÎÏqVB<@B¢h<C ?6ÁwnnB<C ÎÏÁ}<jÓ&<Aqø<hAø< fÚ<qA<$-? q?6ó?6@#¢h@BAqhA²Ð @B@dÕ®n[email protected]3?6nqCá¢h@¿¦§<qA@BA4<wÁ}¿<hAjCdAÚ<qA@<w¦mnq<hAqAq<jÓ áKM<|?6[email protected]<hA<}¢h<hA <ÖëdÚ<@B¢h<hA31C V¿Ú<¦§<@¿MÎÏ@n <hAéÁ}<@qAH<Hq w<Öëd@<M<U¢hÁwqèB<@BA@AqM<hAé¢3?6|?¢qAWV¿B<hA <U-?wq?6ó?6@dnÁ}<@BA@@B<n<C ¦§@q<?6qh<hA [email protected]-?4Anq<s ;Ñ£ h<+q<Áwùq<<+¢h<+ Çm<hAf£¥<ÖëmÚ<@B¢h<hAJ<hAf£ 3?6nAq<J< qùÁ}<< -?}q?6ó?6@$?¢hBAqWVB< [email protected]<hA}Áwnn<dëPó|[email protected]-?6nq<hA²<@]nnÁw[email protected][email protected]&@¢h<|?6[email protected]@BÁq<Ê<¢hÁwÚq<Á}<@qA&?6¢n<qA&nhA?6 ¢3?6|?¢qùq</ Õ dnÁ}<@BA@@B<+<hAøÁwnn<dë ó|[email protected]-?6nq<hA ?6|[email protected]ó§<Á}<@?63?6qnq<<hAøó|?6[email protected]Ô¢qè?6ó§<Á}<@¿qA dR ffq<@qA<hA ¢h@|?¢qAjC<q¢s + * 3 *,"-( 3 , ( ! , .0! * - 3 $ - * * ( $'& *,"%$ , ( 0! 9 ÉWqfÉ i É>9 ÝC($B83;t$B8F;+&2 4J')8 =DÚ ¡U<4ênnnÇ ¡U<hêÔÃà Ù?&Á}<@Bw<hA><ÖëmÚ<@B¢h<hA [email protected] <hA>¢|è?#_@B<hA+¢hÁwÚ|[email protected] Ç¿WV¿B<Á}<@+<4@B< @B<wd j[ ?6[email protected]<<4nn<hA3CfÚ <hAUÎÏqVB<@B¢h<hAf£¥<Öëd¢n|?6@/<@1A<q<hA &-?Î qV¿B<@B¢h<}<}¢hq< f <hA+Á}¿<hA?¢hBAW¿V B<hA @ónBd[email protected]?6dë]Ð nn<hAÁ}|?6nnWV B<hA<}d-?6Á}ùq< CÚÎÏq¢h<7A|?6WVB<< N C ÎÏq¿V B<@B¢h<Hf£¥<Öë¢n|?6@ 2.2 M Hz Ó m ¡[email protected]¢h<q<J¢h@ZB ó|?6@ 7nBA<qAM8 mm @óB<qAÔf£¥@B<hAÔ?¢hBA30 WV¿B<hA A|@¢h@q<@¿B<hAÙ[email protected]@wd-?6Á}ùq< <Mnn<C<hA@B<hA<ÔAÆή?¢h<é<MEJ?hÇm<nóè}Aq@ó§@Bqh<hAÑ?6F¢h@¿¦§<qA@ <Á}m< ²¢qè? ¿V B<w¢h@|?¢+<@¿q<4nn<hA!f nBA<qA>Á}jǧ<@BA>Ú<Á}<q<@+<wA3£ ?A|Aq<><-?&@?6q<w<4¢h<hA @B<hAjC¢hÁwÁ}< -?4Aq<@BAnnnnq+dAnó@?6 ®£¥|?6<>AqÆÎp?¢h<><hAHnn<hA <>è?A|?6ó§<< <hAH@B<hA ¢|è? 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Bóq< T T!g <hAÔq<qhAq<@¿q <>èB@BÁ}ù@B<< ¢-?6<Á}<@¿ ®£¥h¢|èB<n<+f£ @B<+m< ?¢hBAWV¿B<s KM<Ù¢hÁwÚq<Á}<@ÑAÚh¢RNV¿B<<hAáÁ7?6q-?6dëó|[email protected]¿-?6nq<hA<hA?6q6Îp@Bd[email protected]á<¢qè?6nq<d7×L@?6¢n<3C [email protected]@wÁwnn<}ó|[email protected]-?6nq<HhAqq@@BCnB<ÖëdAq<M@B<éq@}nÁwÚ|[email protected]q<H<Ô¢h@|?¢qAÙdnqA ®Î®?6n<hA|CsV¿ Aq@éή?6n<Á}<@HqÖÕ ¢h@|?6[email protected]qAjC< V¢-?6Ú<@ÔÚ <hA ?6ÁwnnB<hAHf£¥<Öë¢n|?6@Ê?¢hBAWV¿B<UÁ}mqh<h!A Ò Ú<hAÔ?6qAé@Bh¢h<hAqA|?6nq<U<Jq<@Bdq<><@&¢hÁwq<>¢h<hAé¢h@¿|?¢qA |?h¦§<qAé@B<UdAn@<JqÖÕ ÖÎÏÁ7?6@ <hA ¢h@|?¢qA[email protected] <UÁwnn<H 0 2 a q 9 :ñ 81;=<*>6>*?81&(@A-0#47%&B6?47*?#81;C')"0<D<E8F<D*>6?*?8G&[email protected]<*?')5"JIK*>BL("<E"0-NM1283;O$QPR/3+&($B8 Z GRqq<hA<hAh¢qèB<n<hA3CdÁw¢qAq¢hWV¿B<²?6Á7?¢qAq¢hWV¿B<C<hA Áwnn<dëó|[email protected]¿-?6nq<hAqhA|<@q<@w<hA ¢hÁwÚq<Á}<@qAÑ?6¢n<q!A ;?Jd 7¢nqÔ<ᮣ [email protected]¿q ÑÚh¢nq<é-?Hq?6ó?6@?¢hBAWV¿B<é[email protected]<Mq<A Áwnn<dëøqj¦¿<@@B<@¿J[email protected]¢n?6<Á}<@¿>dή?6nuVB<+¢h<hAHdR ffq<@qAJ¢hÁwq<Á}<@¿qAJÊèB@BÁ}ù@B<hAJ?6óA|Aq<@ ?6ÆÎpA AnÁn|[email protected]Á}<@ [email protected]@dÕ®n[email protected]3?6nqC dR fÚBA@CBdAq<qA@<U¦mnq<hAqAq<CB?6BA|D @ ?6²¢h@BAq V¿B<@3Cé<hA7èB@BÁ}ù@B<hA²@B¦§<3?6dë]A7-?q?6ó?6@ ?¢hBAWVB<<[email protected]|?6ÁwÁ}<@¿A7-? q?6ó?6@]@B@ n[email protected]3?6nq<Aq@ABAq¢h<n<hA&f£ q<ÁwA7<@]¦¿<@B¢hs< ;<7Á}mnA|?6@][email protected]h¢h<hAqA|?6 Õ q<Á}<@¿ q<@n¢hÁwq<7<hA dR fÚq<@¿qA?Aq<h¢qA Aqh¢R NV¿B<hA <hA>Áwnn<dëó|[email protected]-?6nq<hA²ÐÏ/nBA+ó§@B|?6<Á}<@ Á}hA|Aq¢hW VB<hAÖÓM@ V¿B<W V¿B<hAÆÕ®@BA @q>qhAq<@qhA[email protected]¢h<>¢|è?6nqs< ¡[email protected]<+¢|è?6nq<wÂdC®£ [email protected] wBB<@B¢h<<+-?7dR fBAq@A<>Îp@B¢@@B<Á}<@ <+®£ [email protected]q<@@B<?6|?6Á}W VB<4<hA Bd[email protected]?6nÇ¿WV¿B<Á}<@J<H<ÖëmÚnÁ}<@|?6<Á}<@¿! ¡[email protected]<+¢|è?6nq<ÄdC®£ [email protected] wBB<@B¢h<<+-?wdAÚ<qA@Ê<+¦¿nq<hAqA|< <d¢3?6|?¢qùq<¦[email protected]<hAq¢h<@U<hA Á}m<hA< q?6ó?6@1@ónBd[email protected]?6dëA <4Îp@B¢@@B<Á}<@4<w®£ [email protected]¿q<@@B<}?6|?6Á}W V¿B<7[email protected] @B<7¢qè#? _@B< T Õ ¡ < nn<hAH<hA Bdh<[email protected]?6nÇmWVB<Á}<@¿<Ô@¿Á}W VB<Á}<@¿! L[email protected] [email protected][email protected]á<Ñ¢qè?6nq< dBCh<Ù <Ùd¢-?6<Á}<@¿<hAáBd[email protected]|?6Á}<@¿áó ¢h<M?6dë+@B<hA<Ù¢A ?6nn<Á}<@3C qÅqAH?3¦§<h¢U<U[email protected]¢n< <U®£ [email protected]¿q<@@B< ?6|?6Á}W V¿Bs< B;?4Aq<@BAqnnnnq <U-?Á}èB¿< ?6dëø¢qè#? _@B<hA<JÎÏq¢h< < ®£ [email protected]A|q<+dÁwnn<<hAq Á}@qhs< D + 5 5 5 ]q 5 ] p\ 5 ]Y 5 +o! 5 +\ e e e e e ]Z p\ ] ]Z ]] ]+ e ]+ e ] p\ e ]] 5 + ]] o \ o !o Z b !#"%$&"'(')+* ,$"'.-/ 21 34 ,0 16 5 #9 5 ': ;1 <:&=&>[email protected]"'B,C/D+*/EF$G)IH(EB<"'/$J/EB',K!E,)LMJLNI"<DB<"<$POB<DLF *QDB @PDER"'B',D* (JLNEST)[email protected]"'LF)L,-/ &= 3V ; \[ + 02W#X ,Y YZ+Z Y+Z] V^ _ 5 a` b Tcad +T16 : Pef Rg +_\hB<"<$,C!E.DK!* iEjDL#$NOB<DLF *[email protected]"'B'D+* ,- = ml n#n7o![/o\p o ]+]] r'as 5 t= _s vu TM 7wxvy'zF"'[email protected]"'LFE.D+*{EF$|)IH}L#)L#* iL6"<DB~"!*QDECz#B<)'zF"'BE."')IH OB<DL *QDB^@A"<$,DkSvJEiGL;)LF,$&"<D+* z DCJLO'0 -/ &= 3 ; 6p [Ip o , Z+] ]+Y ]+]q r'^s 5 t= ^s u ;^>)L#* iL6"<DB "!*QDECjzB<)'zF"'BE."' )IH@,C!BI)JL )@P)/O&"'L;"/) ."'BIDB<CCD*J* /EB'FC!EB/"<[email protected]"/$,D'-/ = 3 6n o[ o o p . ]] 5 ` 3 T\J')BI$&"'BMDL;$OBIDL *[email protected]"<$D'-/V 3{` 1 != = 1 3 0 (1 o ]+]Y 0 0 Fc 5 &# F ` F ` MF8 f16V^ \)B<C" $J/EB',K!E,)L(JLGEiB/"/"! [email protected]"'L/,)L#D+*OB<DL *QDB~D/!"'@PK3* ,"'~?"Jf"<C!E~)IH z D+CJLO)B<$&"'BDL;$iLER"'B.z DB'EC3*"%H'B,C!E,)L|-/ &= 3vV 6 n7 0mW o o q Y q M 8F 5 : D"z#B<)'z D3ODE,)LF/EB<"'?B/"!*QDyDE)LTDL#$^OB<DJL.E.)JOBIDJL<F"/DB'JL&OPiLG'DEB<DER"<$L;C/)LF!)* $DER"<[email protected]"!"/[email protected]"'LFE.-/#8F 3#1 + Q6o ;+p [ o + Z ] Z M e 38F _ &8 [ _ 5 _ e [email protected] @JL&O&H3)B<C"(CFDJLF/aDL#$hH'B<D3O*"@PDEER"'BR-/ &= 3 V ;0 c #7l#7o o o Zq ]+]Z _+e 8F [ _ 5 6\E.DE,C?DD*QDL;C"'DL;$fO,DLE7/EB/"'/FC!EDE,)LAJLP/*Q),-/ &= 3\V 6c o![ o p Y Yq ]+] _Fe 8F [ _ 5 6FM Fe 8 a& +)$&"!* ()Hf/EB/"'/vFC!EFDE,)LF?JL}OBIDL *[email protected]"<$,D- = 3FV 6 o'[ p 0o 0 q q+qq qq& ]+]Z _;e 7 ¡wDzF!#" $&"'(E.D%$&"('DK3*"R-/ &= o X ]+]+] 16 %e ef [ 7c 7_1; \g ¢ 3= ? # 76\M)$&"!*H!)B{H3)B<C"{C!EFDE)LkiL K"<D$%z DC,-/ &= 3FV ; n p [ o 02 . q Y qZ ]]+ efe + v4 P1;a4 7\hBI)'z DODE)L£$w¤ )L;$&"'AL#)L* JL6¥<DJB<"'}$DLFP L;"GCFD¦iL6"G$&"UK!*J*"'}"'L§C/)LE.D+C!E $&"^¨^"'B'Eª©-/ 0 « « I-/ ^ef g <[e 8F7M;V != ` P¬ 3 7 YY o ]] efTe 4 P16v4 ;)+* JE.DB'STD"'AJL2DMCFDJL2)IHGK"<D$AL;$"'BA¨^"'B'Eª©~C/)LE.D+C!Et- &= 3_V ; 0 o 0 [ o+op o p #W . q ]+] ef#e 5 3 FhLNEi"(D+* ,$JEA)IH¨^"'B'Eª© C)LE.DC!E{*QDSH3)BOB<DLF *QDB@PDER"'B,D+*DC/)/E,C!,-/ = 3F8F 5 0 67o [/o o Z ]+]] M 5 t=&7\xvy'zF"'B'[email protected]"'LE.D+*/E$)IHk!) L#$kz#B<)'z D3ODE,)LJLDNCFDiL)IHk.z "'B'CD*hK"<D$,-/_8F a1 + l; wo [/o + q qZ q+] _ 3 F®hB<DL *QDB@PDEER"'BkaDPER"'LE.DEi"%."'S6-/V ; 7 &= 36;l#;1 p o Y q ]+]] + aM+4 F:6'¯{DL;$+)@°@ * E z;*"T'C/DEER"'BJLO )IHT * EB<D')L#$&±²3±6³)\"'B/"'LFEaDL#$^KD*J* J/E,CSTD"',-/ = 3FV 6 w +o 0 W Y+ + 7 kMv4 :;\¯{DL;$)@´@ F* E z;*"}'C/DEER"'B'iLOj)IHPF* EBID')L;$&±h²<²3±²[email protected]"}B<"'"'B''D+*D~3"!* H' D"'BIDOJLO^z#B<)C"'/hµh-/ &= 3 V ; w# + o 02W Y+ _#8 =&&¶""Jf"<C!EJ"k"!*QD/E,C^@P)$ * T)IHhz )BI)fOB<DLFF*QDB%BI)C,-#8F # 7#67 [ o o Z Y Z Z ]Z ]+] 7#8 ÄT qi ãt* F \p b 0 0 ]] \ ] \ / = ] ` ]] `/ ]+Z ` ` ` + ` Z ` + 8 g o' ] ] 8 ]+] 8 \p ] 8 ]Z 8 +Z 8 \q 8/+\ d o! d c c ] ]+ Z+ 6>*?"N-0#4JjLB*,8 $&.' L24JjB*,!58E9 8 ;= V^ # +FEB/"'/./EB<DJLB<"!*QDE,)LF^DL;$P,K!BIDE,)LF^)IH^D%OB<DLFF*QDB%@P"<[email protected]?-/ 8F 7 w X [ o p Z ]+ Y ]+ 1;;4 b ; +FE.DEJ/E,C/D+*[email protected]"<CFDL,C!%)IH/EB/"'/%EB<[email protected] J//,)LiL$J')B<$&"'B<"<$POB<DLF *QDBkDB'BID,-/ &= 3V0 ;Fc p\[ + o X Y+] q ]]+] 8 aM : 6 w F` Mw8 1;7V% 6\)B<C"U$J/EB',K!E,)LjJL|EiB<""![email protected]"'L/,)L#D+* C/)@fz#B/"'0 //,K3*"(OBIDL *QDBz DC,-/ &= 3FV ; # o V ++ 02W#W q Y 8 6>)L* JL6"<DBk"!*QD/E,C!JEGDL;$%z#B/"'//B/"A$&"RzF"'L;$&"'LFEfSTD" ,zF"/"<$%JLNOBIDL *QDB @A"<$,D- +6V^71 + c 7 6n7o3 [/o o o Y ]] ef r \)B<C" C DJL/[email protected]!B<)\"!*QD/E,C!JEDL;[email protected]+C!B<)\"!*QD/E,C!JE-/ &= 3 V ; c ,Z + Zq+Y % % 7+FEB<DJL SD"£"')+* E,)L "<!FDE,)L£H3)BL;)L* JL6"<DBz#B<)z DODE,)L iL @PDER"'B',D* SvJEi @P"'')'C)z#,C @P"/C DL,C/D+*_"!*"'@A"'LFE.-/#8F #1 + ;ll ; [ p + X , + ] 5 ¨?ER"'B<"'/J/h$i!C!B<"'ER"^@P"'@P)BADL#$L;)L* JL6"<DB(STD"hz#BI)'z DODE,)L V^ = dkV^F \e iLNB<)CGaDkL6"'S|z DB<[email protected]?-/ &= 3 V ;Fc ; op o![ o Yq] Yq+]q ]] v` ` a8F` c F_ )CGzB<)'z DODE,)L£JL DNOB<DL *QDBNC DJL+-/ &= 3vV ; 0 +\[ 0 ,Y o Y Y ]]+] ` v` +8F` 3 )C{JLPL;)LR*Q)D+$&"<$K"<[email protected]#B'iE."',- = 3+8F 5 l&w o [/o 0 + 0 ZY ] `%;8F` 78 [ _#` ?16#c 3 F\T/,C!%)IHfOB<DLF *QDB @P"<$,D'-/ d &b 3 o ]]+Z V^ef7` F_ef79 _47d 7;` 6 6V `%78Fw` \x)+* E,)L)IH?zF"'BIC/)+*QDEJL&O H3)BIC" C DJL?JLC/)@?z#B<"'/3"<$POBIDL *QDB^@P"/$,D'-/ &= 3V ;Fc #7# o + [Iefh` :=[. ` 4e F[ g , ~M 5 Q=&ThL Ei"|"<yiER"'L#C"|)IH~/FK!B/"'')L#DL;C" O&"'L;"'B<DER"<$UiLDU)L;"![email protected]"'LF)L;D+*wCFDJLM)IH(,$"'LE,C/D+*wK'"<D$.-/#8F #1 + ;ll [ ++ X ,Y Z Z ;8F` 7 \"'@P)B'TB<"!*QDyDE,)LDL#[email protected] ,C!B<)H'B<DC!EBJLOUiL$*QDE.DLFEhB<)C-/68F & = 3V 3; 5 p [ W Y+ o o q+Z ]+Z 8 ` `% d f8F [ _ `fb F³wFDB<DC!ER"'B©DE,)L|)IH(')+* JE.)[email protected]#JL&OPJLNEiF"?OB<DL *QDB%CFDiLL#$&"'BfOB<DiE'-/ &= 3V ; l o [ p ++ 02W ]q ] `%vM_8 T16wV% V^ 5 5 _¶"zFC!G)HOBIDL *[email protected]"'B',D* .-/ &= 3=& o![ o Y Y+Z ]+] `%vM_8 T16V^ V% a 5 w®B<DLFF*QDBU')+* ,$* ,!,$?DL#$GOD!"',-/aV ;a 6 &= 3 7 W o [/o\p o ] Y ]]+ %78 _efwe 5 :&=&w* EBID')L;$Pz#B<)'z D3ODE,)LJLj"<yER"'B'L#D+*J* /EB/"'/!"<$OBIDL *[email protected]"<$,D'-/ &= 36V ; c # 7o [o o X ,] Z+Y Z+ ]+]+] #cv#8 % 3F\¯{)+O\"!*QD/E,C^,zF"'B/"'^JLC/)LFE.DC!Ef[email protected]"'@k)B'U"Jf"<C!E%DL;$PEi"%EBIDL/"'B!"H!)B<C"R-/ 8 ; &= 31 o [/o o p , Y+ ]] c 8 cvFMF1 &b 8F 5 <= 5 ` & = 3\¡aJL6"<DB(DL;$L#)L#* iL6"<DB "!*QDEC!JEP)IHOB<DLF *[email protected]"/$,DP6/EB<"',0 JL;$C"<$GDLJ')EBI)'z#P)H(D B<DL;$)@m,zF"'B/"hz DC-/8F ,& [ W# Y+Z Y+ZZ o ]+]Z dkc8 ³)LE.D+C!ET"<CFDLF,C!/ e (9 = 36e o ]Z dk;c68 F+F BiH3DC" JLFER"'B<DC!E)LK"'ES""'L"!*QD/E,C/D+*J* U*Q)D$&"<$K/)$."'^L;$&"'B%E.DL&O&"'LE,D+* H3)B<C"',- +6V^71 + c 7Q;1 n o![ o X Y q+Z ]\q ef&8 6 % h: : F MF `% M8 "'@k)BA"Jf"/C!EJLkOB<DLF *[email protected]"'B'D+* ,- &= 3 V ;Fc # op \[ + Y++Y Y+Y s &d 3+\xhEF$&"^"<y'zF¥'[email protected]"'LFE.D+*"^$fH'B<)EER"'@P"'LEh"'ET$&"'fBI)E.DE,)L$+DL?$&"'@* ,"'yOBIDL *QDiB/"'@P)$!*"',-/ ¬ 9 V (o+F9(4_V1 Q¬ +# o + o 5 8 ;d 6ef ;4 77ef Ac 7;` ;M68 1;;V^ \ "'LF/JEUB<"!*QDyDE,)L|JLMDU,K!BIDER"<$ OBIDL *QDB^@kDER"'B'D+* -/ &= 3 V ; l p[ o 02 . Y] Y+]+ Y Tc gI: 5 ¯"'')L;[email protected] ]+* E]+ z; *"G'CDEER"'B'JL&O )IH* O\&Et- &= 3V X ,Y o q+Y [o o ]+] cv #c M#c t h¶"<)B})IH%"!*QD/E,C!JE 36¢f« o ]Z+ 0 ãt* 6>*?"N-N54JjL2*?8 $&.' L(4Jj*,!#89 ]q c c \ ] c ]+] c \p ] 3 d%p q : +\ ]q+q ] o' ]+Z 3\ ]Z Z+Y t« ]+] p ] b ]Z ¢ + ++\ V V ] p ] ] ]Z V 3+ V \ = \p Z V #p Z 1 Zq q0q M&c FM)$"!*H3)B{Ei"TzB<)'z DODE,)L~)IH{')L;$ iLkOBIDL *QDB@PDER"'B',D* .-/ &= 3V ; #o p\[/o o 0 q ]]q ef ` c ?1; V^ F;) L#$AiLN!DL;$'-/ &= 3V ;Fc 7# o'[I o W Y Y q ]+] c 7dk78F = 0 74 3:: =&\)B<C"[email protected]"<D/B/"'@A"'LFEA)L|/E.DECOBIDL *QDB @PDER"'B,D+* ,-/ &= 36V 6 0 p+[ p o W . Z Z Z ]+]] 16wc 7EB<"'/}$J/EB'K! E)L§iL|/E.DE,CUEST)[email protected]"'LF/,)L;D*_OBIDL *[email protected])$"!*@P"/$,DJLjEi"UD+K!3"'L#C")IHH'B'C!E,)L+-/ = 3FV 6 p\[ p+ o p 0 # q q q ] ]+] 5 ( 3d 4 ?1 !=&{\"<CFDLCD*/ER"'B/"'/J|iL B<)CDEU*Q)S /EB<DJL [email protected]?z#* JEF$&"'jDL#$ !"'J/@,C H'B<"/!"'L#C!."',- &= 3 Fr 7# [ o p 0 q+ ] q ` _ : + acv_8 c_M1 &b +7\{DCJLOM)IHAC)@?z#B<"'/,K3*"OB<DLFF*[email protected]"'B'D+* ,- &= 3V ; c ;F o o [ o ++ Z q q + Y Mv 7v16_1 w8F_` 66F³aB<)/JL&O)IHk,$&"'LFE,C/D+*T')+* iE.DB'NSTD"'PiLDMCFDJL)IHU"!*QD/E,CGK"<D$,-/ &= 3 V ; n7 o o + 0 W + q 16# w D"hz#B<)z DODE,)LMJLUOB<DL *QDB D/!"'@PK3* ."',-/ &= 3V ; ; [ o o 0 ,Y YY Y ]]q V^ 7 F³)@fz;* ,DL#C'"%)IH%"!*QD/E,CK/)$,"'JLC/)LE.DC!Et-/#8F ;6l## [ o ] Z ]q+] ; 5 e ¬ ;a{K!3"'BDE,)L)HUL;)L* JL6"<DBGJLER"'BIDC!E,)L )IHND+C) /E,CNSTD"'GiL 0 OB<DL *QDB^@PDER"'B,D+* ^v$"'@P)$ *QDE,)L}zB<)C"'/.-/ &= 3#c #n66o [+ +o X Y ;M6 #`%6M#8 1;#V^ ,F\w)B<C"k$J/EB,K!E,)LiLDOB<DLF *QDB^@A"<[email protected]?-/ &= 3V ; 02 ,Y o [ o o Y q Y ] ]+]Z 7 _ F` wM_8 16V% ,7+FEB/"'/PEB<DL/@J//,)LEiBI)+O\jEiB<"/"![email protected]"'L/,)L;D+*{)B<$&"'B<"/$ OB<DL *QDB DBB<D,- &= 3FV ; # o + 02WW Y Y q dk &= : cv &¢ <:&=&+>^)L#* JL;"/DB} D"'B<)C"'/!"'JL C/)/E,C!/ e h9 = 3 e o ]+]Z +4 : \\B<)'z DODE,)LN)H{L#)L#* iL6"<DBfC/)@?z#B<"'/)L zF* !"'{JL OBIDL *QDB@P"/$,D'-/8 + = 3 p\[ p o X q ]Z+Y 16 t« %` 7 _e 6LjEi"^z;*QDEC!JEDL;$M"!*QD/E,C!JE)H}$*QDE.DLFEOB<DL *[email protected]"'B,D+* ,-/w8Fv = 3#1 66o p o Y+] ]+]] 5 dk : ;f¢ V : % r : >)L* JL6"<DB L#$&"'B'STDER"'B C/)EC! 1 'b{[.s /: o p ]Z 5 [ 7` _Mw8 16V% \ "'LF/JE%FC!EDE,)LPiLj,K!B<DER"<$ _V^ b :;_8 5 d 0OB<DL *QDB^@PDER"'B,D+* ,-/ &= 3V ;0 7o p&o![/o o 02 ] ]Z ]]+Z F¢ ++E.DE,!" "'EaB<¥<)+*Q)/O."^$w¤ [email protected] * ."'%OBIDL *QDiB/"^C) L L;¥R-/ ¬ 9 ?r 9(4_V1 ¢ 3=&# g 8F ` +F` _#1 0 q 5 != & +&xvy'zF"'B'[email protected]"'LE.D*ER"'/E{)IH(Ei"%$ {)L~D'zzB<)[email protected],)L H3)B @ * E z#* ."<$P'C/DEER"'B<"/$}!) L#$'-/ &= 3FV ; o [ o+o o 0§#X ,Y Y Y q 8F ` +` _1 _r'_8 3 aT1 F a ]+]+ _³v*QD//,C/D+*{STD"z#B<)z DODE,)L£iL /EB<)L&O+* 'C/DEER"'B'[email protected]"<$D'-/ &= wl [po o p X q ]+] 38F#8 5 '= 5 ;1 ![.V '=&+¨?ER"'B<"'/JJLN,K!BIDER"<$kOB<DL *QDB@A"<$D'-/ &= Fw# pp\[ X Z Z]+Z + 4V \g ,M&8 8 [R8F {1;+V «6w)BIC'"$iEB,K!E,)LUJLU$&"'L!"fEST)+[email protected]"'L/,)L#D+* OB<DLF *QDBf//ER"'@,-/ = 3FV 6Fc 6 #+p [+pp o q ]+] 4{V \g h tfM{8 8F [8 h [email protected])$D+*CFDB<DC!ER"'B|)IH~/EB<"'EB<DL/@J//,)L JL OB<DLF *QDB 0 z DCJL&O,- &= 3 V 6c #F o o![ o q ]+]Z __V 77xEF$&"G$&"}*QDUO&¥<)@P¥'EB',"$&"[email protected] * ."'+yGOBIDL *QDJB<"'[email protected])$!*"'PEB,[email protected]"'L/,)LL6"!* z DB}/[email protected] *QDE,)L[email protected]¥! B,!"R- F¬ 9 V o79(4_V 1 Q¬ + g Y + V = 8 s F1 +F_z#ECD*O&"'L;"'B<DE)LMDL;$P$&"'ER"<C!E,)L)H?BiH3DC"^DC/)EC^STD"'()L~D ,zF"'B<"R- 0 &= 3c #p [p o p #X ,] Z ]+Z ¢ %V : (16Fr';1 = 7¶"<)B<"'E,C/D+*H3)L#$DE,)LF%)IHL;)L* JL6"<DB DC/)/E,C!/ e 5 !b{[ s /: o p ] Z c61 b ?;8F #* EB<D')LF,C(z#B<)'z D3ODE,)L|iL|C3*Q)!"!Qz DC"<$~$J')B<$"'B/"<$,zF"'L/,)L,-/w8 &= 3 oo p&o![ p+p o # Y+] Y+] ]Zq q ãt* 1 ]Z 1 1 ]q ]+ 1 ª+\ 1 ]+Z #o! +\ Z p ]+ u ]+ u ] Zq 6>*?"N-0#4JjLB*,8 $&.' L24JjB*,!58E9 1;v1 aM k8 w ;)+* iE.)L#* ª"z# * !"'PJL~zF"'B'EB!K'"<$jDL#$$BJ"'Lj¨%"'BEª©!,DL2CFDJLF}DL;$NEi"'JB z ) /,K3*"%D'zz;* CDE,)LiL$&"'ER"<C!EJLOGK!B."<[email protected]?z#BJE."',-/ &=3FV ; 6 [ p o 0§ Y+Z+ Y+] ]+]Z MM+1 3F®hB<DJLUC)LE.DC!EvD+$\"'/,)LPER"'B<"'/Jf;D^@P"/C DLJ/@H3)BDEER"'LFDE,)L)IHh!"'J/@C STD"',-/ &= 3 V 3#c l o 6 [I o X Y Y Y ]+]q _1 ²LFEB<)$FC!E,)LE.)STD"('C/DEER"'B'iLOT*Q)CD* ©DE,)LDL#[email protected]"'')'C/)'z#,C{zF"'L#)@A"'L#D'-/ 3 'b{[ s /: o ]] cv _1 w1; %8 wc <=&#+FE.DEJ/E,C!})IHEi"}C/)LE.DC!E?L6"'EST)B<NJL}H'B,C!E,)L;D+*DL#$ H'B,C!E,)L#*"'/ OBID0 L *QDBz DCiLO,-/ = 3 V ; o ++ 0 W#W Y Y 8 w161 ; "'B<DO&"P/EB/"'/!"'ADL;$H!)B<C"C!EFDE)L JL|L;)L;C/)\"'/J"%OB<DLF *[email protected]"'B',D* .-/ = 3;V ; 0 0 [ &o o ,Y Y q Y ]]+Z % C/)/E,!"L;)L°* JL6¥<DiB/"Gz )Bj*QD£C/DB<D+C!ER¥'B'J'DE)L $"|@kDER¥'B,D+y'-/(V « o o+ o + + p p 36¢ + o Yq q & Z cv , _e \8F [ _ 5 7M \e 3 \e a8 _! +FEB/"'/!"'JL J*Q)FC/)@?z DB'J')L K"'ES"/"'LMEiF"/)B/"'E,C/D+*[email protected])$&"!* ^DL#$AL;"'S "<y'zF"'B0 [email protected]"'LFE.-/ 0 &= 3FV ;c F p wo [/o + qY+] q+q dk& 7+¶"^"J?"<C!EJ"%"!*QD/E,C(@P)$ * v)HfB<DL#$)@ z D+CJLOP)IH?,zF"'B<"',-F8FF 7 &= 3 1 n Fo [+ Y o p ]+Z V^ c }{1 +( /,)L£)IHF* EBID')L;$JL DO+*QD/~K"/D+$* B'B-h8F h1 + ; q [o3 o ]q ]] dk : T4& &=&;C/DEER"'BJLOJLO+*QD/K"<D$?"Jf"<C!Ef)IHwH'[email protected]"DL;$z )B/"_,$%C/)@?zB/"'//,K!* JE."',-/ 8 #1 + # \[ o W ,Y Z Z ]Zq %Ts u ;T\>)L,$"/D*J* z DC"<$MOB<DLFF*QDB@P"<$,D[email protected]"'B,C/D+*@P)$&"!* iLO)IH"!*QD/E,CL#)L#* JL;"<DBkz#BI)'zF"'B'E,"'.-/ &= 3 wl [ o o ,Y Y+Z+ Y+] ]] %_s vu ;_ @P)$&"!*{)HGDL#)@PD+*Q) G"!*QD/E,CGL#)L#* iL6"<DBJE|)H}@C!B<)JLF)@k)/O&"'L6"<)[email protected]"<$,D'-/ vc l o\po'[/o\p o ,] q ]+] !" %# $'& (*),+.-0/214365+879/:) ;,<>[email protected][email protected]<>DFEHGJIKGJLM?ON0P>G6QSRHTJLHACIKUJLHGVLHACLWPX<XLHTOYZ<XB9GV[]\^YZES?9AZ_`[STJIKAM[[email protected]<>ACL[]\^YZIVQSPX<[email protected][[email protected]%P>GcDFEHGJPPd\^YZLF?9GJLSLHG Q.YZB'[email protected]@<>[email protected]?fe.YC=9TcGWYTJ?9Tg[[email protected]?kP>ACB9=l[]\[email protected]<>GJLHhcGc=fIKGJLHTcGc=k[HYZLH=lPd\mGOYZEprq8ijGOstsSNvuwAOiSxjy{z`NvI|YZ<>= Pd\^YZLF?9GJLSLHG}Q.YZB'[email protected]@<>DFEHG}Q~GJES?[email protected][email protected]<XPX<>=9TcG[HYZLH=:ESLIV<XPX<>GJERHACIKACjUJLHGLHACLPX<XLHTOYZ<XB9GtN8GJ? 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U~ NGJ? [email protected]?P>G?9GJLH=9GJESB4[SGc=}[email protected]'YZ<XLM?9Gc=OY[[email protected]<[email protected]<>ACL[SGc=ACLH[SGc=P>ACLSt<[email protected][a<XL.YZP>[email protected]?ACeS?9GJLFEHGlGJLYZQSQSPX<>DME.YZLF? σ Pd\mACQ0TJB'YZ?9GJESB[a<[email protected]jGJLHhcG<s6Pd\[email protected]<>ACL´ka =M£o ka À £ ∂ σ ∂ ~ , ρ div U = ∂t ∂x ∂x Y}[SGJLH=9<X?9T,[]\[email protected]t<>G%[email protected][email protected]<>DMEHG,Gc=9?¾IKA*[STJPX<>=9TcGwQ.YZB =9ACLlGon*Q.YZLH=9<>ACL}EH=9DME\^[email protected]<>DFEHGwGJLk[STo[email protected]|Y®_ [email protected]<>ACL t ka^j£ Γ ~ 1 ~ W =ρ c div U − div U , 2 3 c G 9 = ? b P k Y * i X < 9 ? c G 9 = 9 = K G a [ E ' = C A L J G ? > P K G . Q Z Y ' B Z Y K I J U @ ? 9 B | G S [ K G H L C A a L d _ X P X < [email protected]<X?9TlDME.YC[aB'[email protected]<>DFEHGK[aEIV<XPX<>GJEtB'YZLFESPbYZ<XB9GtY A i [email protected]<>ACL[email protected]'YZ<XLF?9G}_¾[STo°[email protected]|[email protected]<>ACΓL[email protected][email protected][HYZLF? s6Pd\[email protected]<>ACL´ ka^j£ Y6PbY6°[email protected] t karyt£ ~ ~ δ . div U − Γ div U σ =ρ c `PM°YZES?LHAC?9GJBDMEHG [HYZLH=P>[email protected]<>ACLH= ka^j£ GJ?: karyj£oNZPbYLHACLa_dPX<[email protected]<X?9T hJ<XLHTJI|[email protected]<>DMEHG:[email protected][email protected]<>[STJB9TcG hcACIVIKG LHTJtPX<XjGOYZeSP>G,[SGJiCYZLM?¾PbYwLHACLa_dPX<[email protected]<X?9Tw[SGc=¦hcACLF?'YChJ?9= [SG:;[email protected]? >}prq0YZ<bststYSNtµRS<bx ¿ NtGJPbs kZz`j Y%[email protected]<>[email protected]'YZ<XLF?9G _¾[STo°[email protected]|[email protected]<>ACLgYTJ?9TQSB9Tc=9GJLF?9TcG}YZEh9R.YZQS<[email protected]QSB9TchcTc[SGJLM?Q0ACESB:P>Gc=:hcACLM?'YChJ?9=[SG;[email protected]? >xt ka^xj£ ~ σ = −B −div U δ , YOijGch%PbY[email protected]?'YZLF?9G [email protected][email protected]<>ACLSLHGJPXP>G4YZEanfIKA*[aESP>Gc=:[email protected][email protected]<>DMEHGc=:[aEI|[email protected]<bYZE[email protected]<[email protected]?:P>Gc= eS<XPXP>Gc=2p <XL =FsSN FACR.x À NFµRS<bx ¿ N* GJBPbs ¿ z`NMGJ? PbY[email protected]?'YZLM?9G κ hcA [email protected]<>=9G,A [email protected] 3/2 GJ? 2 p^t<bYtstsSN j At='stsaN A A*[Hs ®z`NF?9ACES?9Gc= P>Gc=[SGJEan [STJQ0GJLH[HYZLF?9Gc=[SG Pd\^[email protected]'YZLSjGJIKGJLM?:[SGc=¦eS<XPXP>Gc=wp A*[Hs aN ESSs ®z`FY}[email protected]'YZ<XLM?9G,<XLS<[email protected]<bYZP>G,ACEf[SG:°Y cACLlTcDFES<XitYZP>GJLF?9G PbY6QSB9Gc='[email protected]<>ACLRFa[[email protected]?'[email protected]<>DMEHG2<XLS<[email protected]<bYZP>G P =9ACLF?,[[email protected]<X?9Gc=:Q.YZ¶B t ka^sj£ ~ σ = −B −div U δ = −P δ . ¹L·[STJijGJP>ACQSQ0GJIKGJLF?k[SG[email protected]?9G[email protected]<>ACLµ[email protected]'YZ<XLF?9G_6[STo°[email protected]|[email protected]<>ACLµYZES?9ACESBk[SGPbYhcACLH[a<[email protected]<>ACLµ<XLS<[email protected]<bYZP>G[SG [email protected]'YZ<[email protected]?'[email protected]<>DFEHGtN.GJL[email protected]<>[STJB'YZLF?w[SG%QSPXEH=:ESLHG%[email protected][email protected]@[email protected]<>[email protected][email protected]<>DMEHG[email protected]?:[email protected]<X?9G}=9ACEH= PbY6°[email protected] t kan mj£ ~ − κ − 1B P ~ , σ = σ − σ ' κB P div U div U 2 A i YhcACIVQ.YZB'YZ<>='ACL[[email protected]<>ACLH=k karyj£%GJ?k kan mF£,°[email protected]<X?}P>Gc= [STJQ0|div GJLH[HUYZ~LH|hc≡Gc=%|div [email protected]U~PbY|QSB9−[email protected]<>ACLgU~ RFa|[[email protected]|div ?'[email protected]<>DMU~EHG4|[SG4PbY|iM<X?9Gc=9='G}[SGQSB9ACQ.YZ[email protected]<>[email protected]<>DFEHG4e.YC=9=9G¤B9TcDMEHGJLHhcG4GJ? [SGPbYWLHACLa_dPX<[email protected]<X?9TDME.YC[aB'[email protected]<>DMEHGGo§[email protected]<XijGtv¥L·[email protected][email protected]<>hJESPX<>GJBONv[HYZLH=KP>GhOYC=f[]\ ESLµ[email protected]'YZLSjGJIKGJLM?fQ.YZBdYZ<X?f='YZLH= [email protected]?9Gc=k[SGhcACLM?'YChJ?9=[email protected]>Gc=leS<XPXP>Gc=WpmGJPbs ¿ N¾µRS<bx ¿ bN j At='sts®z κ = 3/2 N c ∝ P N Γ ∝ P w q*ACEH= [email protected]?'YZ<XLHGc=6hcACLH[a<[email protected]<>ACLH=Gon*[email protected]<XIKGJLF?'YZP>Gc={NESLHGf[STJQ~GJLH[HYZLHhcGfQSB9AMh'RHG|[SG c ∝ P Gc=9?}[email protected]t<bYtstsZz` j GJPbY Q0GJES?}QSB9A{ijGJLS<XBV[SGKPbY[STJQ~GJLH[HYZLHhcGl[[email protected]<¢w[SG|hcACLF?'YChJ?9=V=9ESB}PbYQSB9Gc='[email protected]<>ACL=9?'[email protected]<>DFEHGA p A AM[Hs ®zACE[SG PbY|[STJiM<[email protected]<>ACL"YZELS<XijGOYZEW[aEWhcACLF?'YChJ?%=9GJESP[SG}PbYVP>AC<[SG};[email protected]? >4[aEHGYZEan[email protected]@<X?9Tc=,[SGc=,eS<XPXP>Gc=4p AM[Hs Zz`f j G}hOYC= [email protected]=9Q~ACLHP[ s κ = 2 GJ? 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Ve = V f(Ω)τ (ω) τ (ω) Ω + i τ (ω) Ω + i Yk=9TJQ.YZB'[email protected]<>ACL[aE"?9GJIVQH=hOYZB'[email protected]<>[email protected][email protected]<>DMEHG|[]\^[email protected][email protected]<>ACL τ (ω) GJL[SGJEanW?9GJIVQH=hOYZB'[email protected]<>[email protected][email protected]<>DMEHGc= τ (ω) [email protected]? [aEHÁG s²PbY²QSB9Tc=9GJLHhcG[SGPbY·[a<>[email protected]~GJB9=9<>ACL[SGiM<X?9Gc=9='G[SGQSB9ACQ.YZ[email protected]<>ACL[HYZLH=P>GIV<XPX<>GJE tB'YZLMESPbYZ<XB9G c (ω) 6= £±pm;%YC='stsaN gACE.stxZz`4¯%[email protected]?9G½YZL.YZPXa=9GtNPd\ <XLHTJFYZPX<X?9T [email protected]=9TcG±[email protected]<¢¡.TcGµ[HYZLH=?9ACES?9G c Pd\mTJ?9GJLH[aEHG|[email protected]<>GJPXP>[email protected][a<>TcGt¥L±YChchcACB9[´YcijGchfPd\[email protected]<>ACL ka k <=*£oN 0τ (ω) ≥ τ (ω) [email protected]=9T|DMEHG DME.YZPX<X?'[email protected]<XijGJIKGJLF? =9GB'YZQSQSB9A*h9RHG%[SG P>ACB9='DFEHG ='GB'YZQSQSB9A*h9RHG2[SGwPbY¤B9TcDMEHGJLHhcG%[SG,hcACESQSESB9G NMGJ? 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GkB9TJt<XIKGk<[email protected][a<bYZ<XB9G Ω [email protected]?|YZeH='GJLF?VP>ACB9=9DMEHGkPd\ <XLHTJFYZPX<X?9T L\[email protected]?VQ.YC= [email protected]<¢¡.TcGt qM< f (θ) ≡ f (t/τ ) [email protected]?VESLHGl[email protected]<>ACL½[SGkIKAM[[email protected]<>ACL·[SGl? 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[email protected][email protected]<>ACLSLHGJP s4PbY6[[email protected]<XijTcG2[SG%Pd\mGJLFijGJP>ACQSQ0G [email protected]< τ (ω) τ a[email protected]<>[email protected]<>UJIKG%B9TJt<XIKG τ (ω) τ [SGPd\[email protected]<>ACL± ka kjyj£L\mGona<>[email protected]?9G}Q.YC==O\ <XP L\ YQ.YC=,[SG4[a<>[email protected][email protected]<>ACL[SG}iM<X?9Gc=9='G|¤P>ACB9=9DMEHG c (ω) = c N τ (ω) = 0 V0 Ωb 0 Ωb Ωb 1 τ− (ω) − 1 τ+ (ω) + ± g ∂cg (ω) ∂ω 0 + + − − c g c 1 τ+ (ω) 1 τ− (ω) 1 τ+ (ω) 1 τ+ (ω) 0 Ωb τ (ω) τ− (ω) 1 τ+ (ω) 1 τ− (ω) 1 τ− (ω) 1 τ+ (ω) 1 τ− (ω) 1 τ− (ω) 1 τ+ (ω) 1 τ− (ω) m m Ωb 0 τ (ω) ∂f τm ∂θ τ (ω) ∂f τm ∂θ τ (ω) τ− (ω) m + + − m + − − m m − m ∂f (θ) ∂θ − − Ωb m g 0 − ¹ #º¨ªµ ¦§¢5²´¢p¦5²« º¥¤B¢¨U©¦»¤B¥ ;«¯¸¢p¦ 5 GJ?Pd\ <XLHTJFYZPX<X?9T L\[email protected]? 9YZI|YZ<>=[email protected]<¢¡.TcG{£oI jw\[email protected]?P>GKhOYC=2Q0ACESBPd\^YZLM?9GJLSLHG|Q.YZB'[email protected]@<>DMEHGf[HYZLH=}Pd\mGOYZE GJLPd\^YZeH=9GJLHhcG[SτGeS(ω)ESPXP>Gc=[]τ\^YZ<XB}pmuwAOiSxjy®z`HYZB,hcACLH=9TcDMEHGJLF?ON.P>Gc=:[STJQ0GJLH[HYZLHhcGc=,[email protected]<XtLS<¢¡[email protected]<XijGc=w[aEQSB9AZ¡HP [aE[email protected]<XtL.YZP [STJIKAM[aESP>[email protected]>Gc=}Q.YZB'[email protected]=V[aEIV<XPX<>GJEtB'YZLMESPbYZ<XB9Gk=9ACLF?6ESLHG|<[email protected]?'YZLF?9GlI|YZLS<¢°[email protected]?'[email protected]<>ACL±[SGc=}[email protected]<>TJ?9Tc= [a<>[email protected][email protected]<XijGc=:[SG2hcG2IV<XPX<>GJE ¹,LHGlACeH='[email protected][email protected]<>ACL²[email protected]<XIKGJLM?'YZP>G[SGkhcGJ?KGo§0GJ?Q0GJES?V[email protected][email protected]<XPX<>[email protected]'YZ<XB9G[SGc=<XLa[email protected]|[email protected]<>ACLH=|[email protected] P>Gc=lQ.YZB'[email protected]=[aE¸IV<XPX<>GJEºtB'YZLFESPbYZ<XB9Gt:Gc=hcACLH[a<[email protected]<>ACLH=[email protected]_E r|='YZLF?9Gc=Q~ACESB[email protected][email protected]'YZLH=°[email protected]|[email protected]<>ACL[SG GJL V ∼ f (θ) Q0GJESijGJLF?[email protected][email protected]>TcGc=hcACIVIKGg=9ES<X? t [email protected]=9ACLH=|Q0ACESBhcACIVIKGJLHhcGJBDFEHG V ∼ 2ijGchPd\^YZL.YZPX*='GtN%QSB9Tc=9GJLF?9TcG´Q.YZBWPbYµ[email protected]<X?9GtN%[SGPbY·[email protected]@<[email protected]<>ACL Ve ijGJL.YZLF?[aE h'R.YZIVQ τ (ω) τ (ω) [email protected][email protected]<>DMEHG4[a<¢§[email protected]<¢¾[email protected]>G}[email protected]<XtL.YZP?9AC?'YZPdN~<XPYZQSQ.YZB'YZ<X?2DMEHG}[email protected][email protected]@<[email protected]<>ACLGc=9?,hcACIVQSP>UJ?9GJIKGJLF?2LHTJtPX<XjGOYZeSP>G |Ve | |Ve |b£ s?9ACES?9Gc=P>Gc= B9TcDFEHGJLHhcGc=,=9ACEH= PbYVhcACLH[a<[email protected]<>ACL τ (ω) τ (ω) HACESB,ESLHG4YZL.YZPX*=9G}DFE.YZPX<X?'[email protected]<XijGtN Gc=9?YC=9='AMhJ<>4T sESLHGk[a<¢§[email protected]<>ACL[SGl? MQ0Gf«wYc*P>GJ<XtR τ (ω) ∼ £|pmt<bYtstsCz`NGJ?4ESLHGk[STJQ0GJLH[HYZLHhcGlPX<XLHTOYZ<XB9G τ (ω) GJL"B9TcDFEHGJLHhcG B9TcDMEHGJLHhcGKhJ*hJPX<>DMEHG{£%[email protected]?}[email protected]<XPX<>=9TcGVQ0ACESBPd\^[email protected][email protected]<>ACL ∼ p µRS<bx ¿ Nq0YZ<bstsZeSz`¥LH=9ES<X?9GtN Pd\ <XLHTJFYZPX<X?9T τ (ω ) τ (ω ) [email protected]? [email protected]~At='TcG,[email protected]<¢¡.TcG%Q0ACESBESLHG2ACLH[SG2τYChc(ω) [email protected][email protected]<>DFEHG2h; [email protected]<8PbY4B9TcDMEHGJLHhcG,<XLS<[email protected]<bYZP>G [email protected][email protected]_E r|='YZIVIKGJLM?¦TJP>AC<XtLHTcG [SG PbY:¤B9TcDMEHGJLHhcG[SGhcACESQSESB9G ω ω ω £oC²[email protected]?9GvB9TcDFEHGJLHhcGtNZPbY,[a<>[email protected][email protected]<>ACL ω [email protected]?[email protected]<>[STJB9TcG°YZ<XeSP>G c (ω) ' c GJ?w[SACLHhtN τ (ω ) ' τ (ω ) 1 − .¥Lgh9RHAC<>[email protected]<>='='YZLF?,ESL τ (ω ) LHACI}[email protected]_E r|='YZIVIKGJLM? tB'YZLH[ n [SG,[email protected]<>A*[SGc=¾[]\mACLH[SGc=h; g[HYZLH=¾P>GQ.YCDFEHGJ?[]\mACLH[SGwº; gW<XLS<[email protected]<bYZP τ ≡ n N <[email protected]?k?9ACEC[email protected]<XeSP>GW[SGgB9TOYZPX<>=9GJBPbYhcACLH[a<[email protected]<>ACL τ τ (ω ) qM<YZP>ACB9=PbY¤B9TcDMEHGJLHhcG"[SGgPd\mACLH[SG"[SG Q~ACIVQ.YZjG ω Gc=9?6YZEStIKGJLF?9TcGGJL´FYZB9[HYZLM? n ¡SnaT{£oN P>Gf?9GJIVQH=V[SG|IKA*[[email protected]<>ACL τ [a<XIV<XLFEHGlQ~GJLH[HYZLM?VDFEHGfP>G ?9GJIVQH=hOYZB'[email protected]<>[email protected][email protected]<>DMEHG τ (ω) YZEStIKGJLM?9Gt®¥¦LdYZ<[email protected]<jQ0ACESB [email protected][email protected]<XI|[email protected]<>ACLN®PbY[email protected]<>ACL[SG¦[a<>[email protected][email protected]<>ACL4r[]\ ESLHGvh9R.Y >LHG [email protected]<>A*[a<>DMEHGK[SGKeS<XPXP>[email protected]@<>[email protected][email protected]<XPX<>=9TcGpm;%YC='sts®zw¤ijAC<XB6P>G|h'R.YZQS<[email protected] m{£oNYZP>ACB9= c (ω) = c 1 − GJ? Q~ACESBw?9ACES?9Gc=wP>Gc=B9TcDMEHGJLHhcGc=O8 P GJLW[STchcACESP>GDFE\mGJL"YZEStIKGJLF?'YZLM? ω EH=9DME\s >0 = ESLHGB9TcDFEHGJLHhcGK¡HL.YZP>G ω [email protected]_E r|='YZIVIKGJLF?6R.YZES?9GtNPbYhcACLH[a<[email protected]<>ACL τ = n τ (ω ) Gc=9?B9TOYZPX<>=9TcGt q*GJP>ACL Pd\[email protected]<>ACLW ka kjyt£oNtPbY%[email protected]'YZLH=°[email protected]|[email protected]<>ACLk[aEKQSB9AZ¡HP8[STJIKA*[aESP>T:w g"[SG s f (θ) [SGJiMB'YZ<X?¾[email protected],[email protected]YZES?9ACESB [SG PbYwB9TcDFEHGJLHhcG,[email protected]<[email protected]<>DFEHG ω NjA i τ (ω ) ∼ τ (ω ) ∼ τ (ω ) 1 − Gc=9?¦[email protected]<¢¡.TcGtF¦\mACeH='[email protected][email protected]<>ACLk[SG [email protected]?9G[email protected]'YZLH=°[email protected]|[email protected]<>ACLVQSB9AMhJESB9G:[SGc=<XLa°[email protected]|[email protected]<>ACLH=¦[email protected]\^[email protected][email protected]<>ACLfGJ?PbY,[a<>[email protected]<>ACLK[]\mACLH[[email protected][email protected]<>DFEHGc=¾º; g `P dYZES?wLHAC?9GJB2DMEHG}[email protected]'YZLH=°[email protected]|[email protected]<>ACL[aEWQSB9AZ¡HP¦w g·[email protected]?YChchcACIVQ.YZtLHTcGQ.YZB%ESLHGVYZEStIKGJLF?'[email protected]<>ACL=9<XtLS<¢¡[email protected]<XijG [SG%Pd\^YZIVQSPX<[email protected][SG}[aE=9<XtL.YZP[email protected][email protected]<>ACLSLHGJPXP>G<s − Ωb m ∂f (θ) ∂θ a Ωb s Ωd Ωd a Ωb s s 1 ω4 s 1 ω a a in s in in c g 0 − in a in in c cg (ωin ) c0 a in in m in m − 2π ωin in m − g ∂τ− (ω) ∂ω τa (ω) r 1 “ ”2 |ω| 1− ωω 0 ω ωc 2 c f m f cr m cr − cr a 2π ωf − f ∂f (θ) ∂θ cg (ωcr ) c0 cr τ (ωf ) τm (ωin ) 1 τm (ωin ) . = 1 c (ω ) τ− (ωf ) τ (ωin ) 1 − g c0in τ (ωin ) . w¯ EQ0AC<XLF?V[SGliMEHGfQSRF[email protected]<>DMEHGtNPd\ <XLF?9TJtB'[email protected]<>ACL½YC[S[a<[email protected]<>ACLSLHGJPXP>G[aE±[email protected]<XtL.YZP wgN [HYZLH=P>GlhOYC=AiPbYg[a<>[email protected][email protected]<>ACL´[SG iM<X?9Gc='=9G|[email protected]?6<[email protected]?'[email protected]?4PX<>TcG sPd\^YZeH=9GJLHhcGk[SGf=9MLHh'RSB9ACLS<>='[email protected]<>ACL²[email protected]=9ACESB9hcG|LHACL´PX<XLHTOYZ<XB9Gk[aE´[email protected]<XtL.YZP [STJIKAM[aESP>T hW i ACLH[SGc= ;hgDMES<8=9G:QSB9ACQ.YZjGJLF?>s2PbYiM<X?9Gc='=9G c (ω)£oNMGJ?vP>G[email protected]<XtL.YZP0[STJIKA*[aESP>TPXES<¢_dIKJIKG V DFES<*=9G¦QSB9ACQ.YZjwG sPbY i*<X?9Gc=9=9G £o®¥L}dYZ<X?ON [email protected]¾[SGvPbY:=9TJQ.YZB'[email protected]<>ACL?9GJIVQ0ACB9GJPXP>[email protected] P>Gc=B9ACLF?9= []\mACLH[SGc= [email protected][email protected]<>DFEHGc=h; gGJ?vcw gDFES<[email protected]?vTJτ?'YZeS(ω)PX<>G2YcitYZLF?Pd\^[email protected][email protected]<>ACL[aE|Q.YCDMEHGJ? []\mACLH[SG%h; g·ht\[email protected]?_ s®_`[a<XB9G2YZE e~ACES?:[]\ ESL?9GJIVQH=[SG%Pd\mACB9[aB9G[aE?9GJIVQH=,hOYZB'[email protected]<>[email protected][email protected]<>DMEHG[]\^[email protected][email protected]<>ACL τ (ω)£o qMESB}PbYk¡HtESB9Gg ka^j£[email protected]?QSB9Tc=9GJLM?9TcG|PbY[email protected]'YZLH=[email protected]|[email protected]<>ACL´[[email protected]<XtL.YZP [STJIKAM[aESP>TfQSB9Tc[a<X?9GKQ.YZB4PbY[email protected]<>ACL[SG Pd\mTcDFE.YZ[email protected]<>ACL ka k À £o*Gc= QSB9AZ¡HP>= ACeS?9GJLFEH=[email protected]<>DMEHGJIKGJLF?,='ACLF? [email protected]'YChcTc=:GJLl°[email protected]<>ACL[aE?9GJIVQH=[email protected]|YZPX<>=9T t/τ Q~ACESBwhJ<XLHDliCYZP>GJESB9=w[SGPbYV¤B9TcDMEHGJLHhcG}[email protected]|YZPX<>='TcG4[SG}Pd\mACLH[SG4[SG}Q~ACIVQ.YZjG t = 0.01, 0.1, 0.2, 0.4, 0.8 Y [email protected]<>ACL[SGwIKA*[[email protected]<>ACL f (θ) [email protected]? ESLHG A [email protected]<>GJLSLHG%[SG%[aESB9TcGwhOYZB'[email protected]<>[email protected]<>DFω/ω EHG2TJFYZP>G s4m[email protected]<>A*[SGc= [SGPd\mACLH[SG ; g[email protected]?'YZLM?9G C [aE´?9GJIVQH=KhOYZB'[email protected]<>[email protected][email protected]<>DMEHG[]\^[email protected][email protected]<>ACL τ (ω) = C/ω [email protected]?Kh9RHAC<>=9<>GkQ~ACESBV='[email protected]<>=dYZ<XB9G º Pd\ <XLHTJFYZPX<X?9TK<XLS<[email protected]<bYZP>G τ τ ]ACESB}ESLHGB9TcDMEHGJLHhcG°YZ<XeSP>GV[]\mACLH[SGf[SGQ~ACIVQ.YZjGf[SG 0.01 NP>GVQSB9AZ¡HP¾GJLPX<XtLHG ∂ ∂x g ωb 0 Ωb − m c a m − 5 !"#%$%& '($)*+,&-. /* 1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 −5 −4 −3 −2 |Zb−1~}Nzp01} 2 3 4 5 t/τm h N n¤2ª¨¦ 9§¬¤¯xª©%«%¬¨Á²±(¦« ;¨ ªµE²´¥¯x¬ ²± µn¥»£ ¤2¥2²«©:¢£_ª¤ nµ ªl¦¬µ ± ©%«¬¨ ²´¢xµ2· ¥2°±ª©%«¬¨ ¤B¢¨U©:¢¦ 9p¤B¥2°±¢¨¢¦³²´¢b£_¬¯º£_ª ;´¢ ;,²´¢¶µ ª B¬±!¤ B¢<¨¬«&¤B¢:3µnª¬±!¤ B> ¢ ;¤p«&¦5µnª«&¤+% 5 ¹ < 7 £_¬± ¤¸²« º¥ ω/ωc = 0.01, 0.1, 0.2, 0.4, 0.8 ¹ LHAC<XB9Gl¤ijAC<XBPbYV¡HtESB9Glka^F£@£:[email protected]?:[email protected][email protected]<>ACLSLHGJPEsVPbYK[[email protected]<XijTcG}QSB9GJIV<>UJB9G[SGPbY°[email protected]<>ACL[SG}IKAM[[email protected]<>ACL ACB9=9DFEHG PbYB9TcDFEHGJLHhcG[SG Pd\mACLH[SG[SGvQ0ACIVQ.YZjG [email protected]?¦YZEStIKGJLF?9TcGtNFGJL6FYZB9[HYZLM?¦[email protected]?'YZLF?P>G LHACI4eSB9G n [SG [email protected]<>A*[SGc= [SGPd\mACLH[SG%º; ghcACLM?9GJLFEHGc=:[HYZLH=vPbY4[aESB9TcG,[SG,PbYIKA*[[email protected]<>ACLN*PbY[email protected]'YZLH=9<[email protected]<>ACLl[SG s Gc=9?¾B9TOYZPX<>=9Tc<G tFQ0ACESB PbY2¤B9TcDMEHGJLHhcG,[email protected]|YZPX<>=9TcG,[]\mACLH[SGw[SG,Q~ACIVQ.YZjG ω/ω = 0.1 GJL|PX<XtLHG,[email protected]<>=¦°ACLHhcTtNOEH=9DMEf\ s(θ)ω/ω = 0.4 GJLfPX<XtLHG [email protected]<>=hJPbYZ<XBOHES?9ACESB:[SGPbY4¤B9TcDMEHGJLHhcG ω/ω = 0.6 NHP>G%QSB9AZ¡HP [email protected]? [email protected][email protected]<>ACLSLHGJ,P s f (θ) * Y[email protected]'[email protected]<[email protected]<>ACL[SG s [aEHhG sPbY[a<>[email protected][email protected]<>ACLl[SGi*<X?9Gc=9=9G:Gc=9?v[SACLHhwACeS?9GJLFEHG2YcijGch,ESLHGwYZEStIKGJLM?'[email protected]<>ACL[SG:PbY2¤B9TcDMEHGJLHhcG,[SGPd\mACLH[SG f (θ) [SG%Q0ACIVQ.YZjGh; g ACESB hcACLHhJPXESB9G%[email protected] Pd\^YZL.YZPX*=9Gw[SG,PbY}[email protected]@<[email protected]<>ACLYZEl=9<XtL.YZP~[STJIKA*[aESP>TwQSB9AOijGJL.YZLF?,[SGPbY}[email protected][email protected]<>Gwe.YZPX<>[email protected][email protected]<>DFEHG [aElh9R.YZIVQ[email protected][email protected]<>DMEHG,h; gNj<XPHdYZES?v=9<XtL.YZP>GJB DMEHG [email protected]<>GJLF?:[email protected]?'YZ<XLHGJIKGJLF? 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V q*GJP>ACL Pd\[email protected]<>ACL¼ ka F£oN,<XP} Y²ESLHG[a<¢§]TJB9GJLHhcG<[email protected]?'YZLM?9G±[email protected]>Gc=QSB9AZ¡HP>=W[STJIKA*[aESP>Tc="w g <>[email protected]=g[SG PbY²QSB9ACQ.YZ[email protected]<>ACL e.YZPX<>=9¿ [email protected]<>DFEHG´ACE[SGPbY½[a<¢§[email protected]<>ACL [SGc=gh'R.YZIVQH=[email protected][email protected]<>DMEHGc="º; g ESLS<>DMEHGJIKGJLF?=9< τ (ω) j [email protected]?9Gg[[email protected]<>UJB9Gg<XLHTJFYZPX<X?9Tg[SACLSLHGgPbYhcACLH[a<[email protected]<>ACLºLHTchcGc=9='YZ<XB9G[email protected][SGc=f=9<XtL.YZEan½w g w τ τ (ω) [email protected][email protected]<>ACLSLHGJP>= [email protected]<XIV<[email protected]<XijG4[SGPd\mGJLFijGJP>ACQSQ0G6[aEQ.YCDFEHGJ?w[]\mACLH[SG4GJ?GonShJ<X?9Tc=,Q.YZBPbYV[STJIKA*[[email protected]<>ACL"[]\mACLH[SGc= ; g´[a<¢§[email protected]<XijGc=O º ¯wEQ0AC<XLF?%[SGi*EHG}[email protected]<XIKGJLM?'YZPdN~<XP [email protected]?,<[email protected]?'YZLM?%[SG4hcACIVQSB9GJLH[aB9G6< º£ j [email protected]'[email protected]<[email protected]<>[email protected]>Gc= [SGJEanfQSB9GJIV<>GJB9=B9TJt<XIKGc=:[SGwPd\[email protected]<>ACL ka ¿ M£¾A iPbY[email protected]@<[email protected]<>ACL[aEh9R.YZIVQ[a<¢§[email protected]<¢º; g[SACIV<XLHGtNSQ0GJES?:[email protected] [email protected]S<X< º£ j ACIVIKGJLF?wPbY[email protected]'[email protected]<[email protected]<>ACL[email protected]P>Gc=,[email protected]<XtL.YZEan[SACIV<XLHTc=:Q.YZBP>G}h9R.YZIVQh; g±[a<¢§[email protected]<¢GJ?P>Gc=[email protected]<XtL.YZEan [SACIV<XLHTc=6Q.YZB}P>Glh9R.YZIVQ´º; g¸e.YZPX<>[email protected]<>DFEHGW¤P>GKQSB9GJIV<>GJBVGJ?}P>Gf=9GchcACLH[B9TJt<XIKGf[SGfPd\[email protected]<>ACL¸ ka ¿ F£@£2Q~GJES?[email protected] [email protected] ¦\ <XLHTJFYZPX<X?9T [email protected][email protected]=9TcG´[email protected]<¢¡.TcdG s¸PbYµB9TcDFEHGJLHhcG²<XLS<[email protected]<bYZP>G N[email protected]<KhcGJPXP>Go_`hJ<[email protected]? [email protected]_E r|='YZIVIKGJLM?QSτB9AM(ωh'RHGg)[SGPbYτ¤B9(ωTcDMEHGJ)LHhcGW[SGhcACESQSESB9G ω pmt<bYtstsCzK¤Q.YZB9hcGWDMEHGP>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\^YZEStIKGJLF?'[email protected]<>ACLk[SGPbY,¤B9TcDMEHGJLHhcG DFEHGP>G?9GJIVQH=vhOYZB'[email protected]<>[email protected][email protected]<>DMEHG []\^[email protected][email protected]<>ACL τ (ω) ∼ £o¾ACB9=9DMEHGPbY"hcACLH[a<[email protected]<>ACL τ (ω ) τ = n [email protected][email protected]<¢¡.TcG τ (ω ) YZP>ACB9=ON P>GWQSB9AZ¡HP[aE¸[email protected]<XtL.YZP[STJIKAM[aESP>T"Gc=9?[email protected][email protected]<>ACLSLHGJ5P s R V (θ )dθ [email protected]'YZLH=9<[email protected]<>ACLijGJB9=kP>GgQSB9AZ¡HP Q~GJES?}YZP>ACB9=2[email protected]§][email protected][email protected]<XIVQSP>GJIKGJLM?}GJLYZEStIKGJLF?'YZLM?}PbYk[aESB9TcG [aE"Q.YCDFEHGJ?[]\mACLH[SG|º; g V ∼ V (θ) GWQ.YC=9='YZjG"[]\ ESL¸LHACI4eSB9G n [[email protected]<>AM[SGc=h; g»hcACLF?9GJLMEHGc=[HYZLH=P>GWQ.YCDMEHGJ?[]τ\mACLH[SG sESL¸LHACI}eSB9G¡HL.YZP NQ~ACESBP>GcDMEHGJPPd\ <XLHTJFYZPX<X?9T τ = n hcACIVIKGJLHhcG s´[email protected]"[email protected]<¢¡.TcGGc=9?[SACLHhGo§[email protected] n τ (ω ) j ACIVIKGPd\ <XLHTJFYZPX<X?9T [email protected]?gB9TOYZPX<>=9TcGtNwYZP>ACB9=W[HYZLH=gP>Gc=W[SGJEan B9TJt<[email protected]<>[STJB9Tc=ON τ (ω) ≤ min(τ (ω), τ (ω)) Pd\ <XLHTJFYZPX<X?9T τ τ (ω) ≥ τ (ω) [email protected][email protected]<¢¡.TcGt¯%ACLHhtN ='GJP>ACL½Pd\[email protected]<>ACL ka ¿ F£oN [HYZLH=lP>Gc=f[SGJEanµB9TJt<XIKGc= 0[email protected]'YZLH=9<[email protected]<>ACL[SG τ (ω ) τ τ (ω ) s τ τ (ω ) τ (ω ) =9G4I|YZLS<¢Gc=9?9G V '2 YZP>ACB9= Q.YZB:ESLHG[[email protected]<[email protected]<>ACL[aEQSB9AZ¡HP<XLS<[email protected]<bYZPwg 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 |Zb~}Nzp} −2 −1 0 1 2 3 4 5 t/τm h NC ¤Zª¨¦ 9§¬¤p¯xª©%«¬¨l²±¦p« ;¨ªµf²´¥¯x¬ ²± µ ¥.£¤B¥2²«%©:¢.£_ª¤ºµnª<¦¬µ ±_©%«%¬¨P²´¢hµB· ¥2°§±_ª©%«¬¨ 5 ¹ 7¹8 ¢p¦º¨¬¯ ¤B¢p¦ ²!¢º£¥¤«¬ ²´¢p¦¸²´¢³¦p«¨±´¦$! ?XB¬¯º£¤«&¦§¢p¦x²ª¨¦³µ ¢ ©:¢ ¯º£U¦ ²´¢ ¯x¬ ²±!µnª©%«¬¨u¦¬¨U©>²´¢ µ ª µ « ;¨ ¢³¨ ¬&« ¤B¢ 3lµ ª µ « ;¨ ¢:;¤p«&¦ §µ ª«¤¤B¢p¦£¢B§©%«/.¢¯¸¢¨U© ; ¹ ¹ n = 5, 100, 500, 5000 ω/ωc = 0.9 ka ¿ m{£ Y|[email protected]'[email protected][email protected]|[email protected]<>ACL[aEWQSB9AZ¡HP¦[aE"[email protected]<XtL.YZP[STJIKAM[aESP>TV[SAC<X?2[email protected][email protected]Q0ACESB n N0='[email protected]<>=dYZ<>='YZLM?wPbYlhcACLH[a<[email protected]<>ACL [email protected]?KYChchcACIVQ.YZtLHTcG¤ijAC<XBVPd\[email protected]<>ACL ka ¿ m®£@£}Q.YZBESLHG[a<XIV<[email protected]<>ACL·[SGlPd\^YZIVQSPX<[email protected][SG n ∼ τ (ω ) [[email protected]<XtL.YZP[email protected][email protected]<>ACLSLHGJPXP>Gxs 0¦\[email protected]<>ACL"[[email protected][email protected]'YZLH=°[email protected]|[email protected]<>ACL[SACLSLHG4[SGc=w<XLa°[email protected]|[email protected]<>ACLH= 1 =9ESB|P>G?9GJIVQH=k[]\^[email protected][email protected]<>ACL [SGc=fACLH[[email protected][email protected]<>DMEHGc=l;hg¾GBBvCP>G[aEµQ.YZB'[email protected] [[email protected]?9G [email protected]'[email protected][email protected]|[email protected]<>ACL´DFES<YPX<>GJEGJτ[email protected](ω) B9G}[SGc=:B9TJt<XIKGc=,[SACIV<XLHTc=Q.YZB:PbYV[STJIKA*[[email protected]<>ACL[]\ ESLh'[email protected][email protected]<>DMEHG}h; g [a<¢§[email protected]<¢o£%[email protected]<XIV<XPbYZ<XB9GW[aEQ0AC<XLF?[SGKi*EHGKQSRM*=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]<>ACLPbYC=9GJB2GJ? τ [email protected]?%P>G6?9GJIVQH=[SG6B9GchcACI}eS<XL.YZ<>='ACL[[email protected]|YkTJP>[email protected][email protected] jTJLHTJτB9Tt qMESBPbY,¡HtESB9G}karyj£[email protected]?QSB9Tc=9GJLM?9TcG:[email protected]'YZLH=°[email protected]|[email protected]<>ACLf[aEK=9<XtL.YZPSQ.YZB'[email protected]@<>DFEHGQSB9Tc[a<X?9G Q.YZB¦PbY%[email protected]<>ACL|[SG Pd\[email protected]<>ACL"kaktxF£oNM[HYZLH=vP>GwhOYC=vA ilPbY}[email protected]@<[email protected]<>ACLYZEl=9<XtL.YZP~[STJIKA*[aESP>TQSB9A{iM<>GJLM?[email protected]<XLHhJ<XQ.YZP>GJIKGJLF?[aEkh9R.YZIVQ ;hg·[a<¢§[email protected]<¢@8 Yf[email protected]<>ACL"[SG6IKAM[[email protected]<>ACLGc=9?wESLHG6FYZEH=9=9<>GJLSLHGV[SG6[aESB9TcGhOYZB'[email protected]<>[email protected][email protected]<>DMEHG 0Gc= i*<X?9Gc=9=9Gc=|[email protected][email protected]<>hJESPbYZ<XB9Gc=[[email protected]<XtL.YZEan·[email protected]|YZPX<>=9TcGc={vYB9TcDFEHGJLHhcGg[SGPd\mACLH[SGW[SGQ~ACIVτQ.YZ=[email protected]?lh9RHAC<>[email protected]<>G QSB9A*h9RHG4[SG2PbYB9TcDMEHGJLHhcG}[SGhcACESQSESB9GtN ω = 0.9ω HGc=?9GJIVQH=,hOYZB'[email protected]<>[email protected][email protected]<>DFEHGc=%[SG[a<¢§[email protected]<>ACLgGJ?,[]\^[email protected][email protected]<>ACL ='ACLF?:?9GJP>=:DME\ <XP>==[email protected]<>=ACLM?:Pd\ <XLHTJFYZPX<X?9T Q0ACESB PbY6B9TcDFEHGJLHhcG2<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 (ωaGc=) [email protected]?'YZLF?9Gc= C GJ? C [SG τ (ω) GJ? = =9ACLM?[SACLHh[[email protected]<XLHTcGc=,tB Chc5G s6Pd\ <XLHTJFYZPX<X?9T}<XLS<[email protected]<bYZP>Gt.ACB9=9DMEHG2P>G2LHACI4eSB9G [[email protected]<>AM[SGc=º; g´[email protected]? τ (ω) = YZEStIKGJLM?9TtN.PbYK[aESB9TcG}[aEQ.YCDMEHGJ?,[]\mACLH[SG6YZEStIKGJLF?9G6GJ?:¡HLS<X?:Q.YZBw[SGJijGJLS<XB,[email protected]@<>GJESB9G4YZEn ?9GJIVQH=w[]\^[email protected][email protected]<>ACL j GchJ<~IKUJLH G s}Pd\ <XLHTJFYZPX<X?9Tw¡HL.YZP>G τ (ω ) τ (ω ) τ (ω ) NaDFES<[email protected][email protected][ s}PbY4hcACLH[a<[email protected]<>ACLQ~ACESB τ (ω ) ACeH='[email protected] ESLlQSB9AZ¡HP[STJIKA*[aESP>T,[email protected][email protected]<>ACLSLHGJP s *[email protected]'[email protected]<[email protected]<>ACL[SG f (θ) s [email protected]? YZP>ACB9= eS<>GJLACeS?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¨ªµ ¦§¢5²´¢p¦5²« º¥¤B¢¨U©¦»¤B¥ ;«¯¸¢p¦ N 1 0.8 0.6 0.2 0.4 0 −0.2 −0.4 −0.6 −0.8 −1 −5 −4 −3 |b~}Nzp} −2 −1 0 1 2 3 4 5 t/τm h f ¤Zª¨¦ 9§¬¤p¯xª©%«¬¨P²±4¦« ;¨ ªµf²!¥¯x¬ ²± µn¥b£¤B¥2²«%©:¢ £ª¤ µ ª<¦§¬µ ± ©%«¬¨ !² ¢ µB· ¥2°§±_ª©%«¬¨ ²« º¥¤2¢¨U©:¢p¦ 9¤22¥ °§±¢¨ ¢p¦ ;,²± ;¤«)¦¶µnª«¤5ª±l¨ ¬«&¤+% ¹ 5 ¹ 7 ¬ ©:¢¨±¢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]<XtL.YZP¦[email protected][email protected][email protected]<>ACLSLHGJP.s hYLN]RM½*E½:a ] a ]RORQUMPORQÁ`ba c"`.LfeQSÀRLN¿:½M E ½\TRQUM f (θ) YZP>ACB9=}DFE\ <XP¦=9GB'YZQSQSB9A*h9RHG|[SG ∂f (θ) ∂θ O.½ >T.M½ QUM ACESB|[email protected][email protected]'YZLH=9<[email protected]<>ACL·[aE±B9TJt<XIKG[SACIV<XLHTQ.YZBKP>[email protected]'[email protected]?|[a<¢§[email protected]<¢2[SGkPd\[email protected]t<>G[SGc=Kh; gijGJB9= P>GVB9TJt<XIKGK[SACIV<XLHT|Q.YZB}P>[email protected]'[email protected]@?e.YZPX<>[email protected][email protected]<>DMEHGK[SGc=4;ºgN]<XP¾Gc=9?2Q~At=9=9<XeSP>GK[SGKhcACIVIKGJLHhcGJBVYcijGchKPbYhcACLH[a<[email protected]<>ACL <XLS<[email protected]<bYZP>G τ (ω ) τ = n WYZ<>[email protected]?9G6AC<>[email protected]_`hJ<dN8<[email protected]?%[email protected]<XP>G[SGV[a<XIV<XLFEHGJB}PbYKB9TcDMEHGJLHhcG τ (ω ) [SGPd\mACLH[SGK[SGVQ0ACIVQ.YZjG ω ='YZLH=h'R.YZLSjGJB}P>GVLHACI4eSB9G n [[email protected]<>AM[SGc=h; gºhcACLF?9GJLMEHGc=4[HYZLH=}P>GQ.YCDFEHGJ?}[]\mACLH[SG n = n = const£o0¯2YZLH=hcGKhOYC=ON τ (ω) GJ? τ (ω) ijACLM?4YZES[email protected][email protected]<>ACLSLHGJPXP>GJIKGJLM? s NYZP>ACB9=DMEHG itY:YZEStIKGJLF?9GJBQSPXEH=B'YZQS<>[SGJIKGJLM?: ∼ £oN®hcG¦DMES<FitY<XLHTJiM<X?'YZeSP>GJIKGJLM?IKGJLHGJBYZE}B9TJt<XIKG τ (ω ), τ (ω ) τ (ω) Q~ACESB,ESLHGB9TcDMEHGJLHhcG¡HL.YZP>G ω = ω q*GJP>ACLgPd\[email protected]<>ACL± ka ¿ M£oNHPbY|[email protected]'YZLH=[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 PS°YZES?¦='G =9ACESijGJLS<XB¾DME\^YcijGch:PbY hcACLH[a<[email protected]<>ACL N.Pd\ <XLHTJFYZPX<X?9T [email protected]?:[email protected]<¢¡.TcGtN~GJ?P>GQSB9AZ¡HP <XLS<[email protected]<[email protected]?τ[STc(ω[email protected]<X?)Q.YZB:PbτYQSB9=GJIVn<>UJB9G}GonaτQS(ω[email protected]<>)ACL[SGPd\[email protected]τ<>ACL´ kaτ¿ m®(ω£oSAC)B9=9DMEHτG2Pb(ωY6B9)TcDFEHGJLHhcG}[SG2Q~ACIVQ.YZjG}[email protected]? [a<XIV<XLFEHTcGtN~<XP Gc=9?,[email protected]=9TDMEHGQ0ACESB,PbYV¤B9TcDMEHGJLHhcG2¡HL.YZP>G ω PbYK[a<>[email protected][email protected]<>[email protected]?,[email protected]<°YZ<XeSP>G4DFEHG4ht\[email protected]?wPd\ <XLHTJFYZPX<X?9T QSPXESB? vC?:DFEHGwPd\ <XLHTJFYZPX<X?9T DMES<][email protected]?[email protected]<¢¡.TcGtSYZB:hcACLH='TcDFEHGJLM?ON τ (ω ) τ τ (ω ) =9GJP>ACLPd\[email protected]<>ACL ka kjyt£GJ? Pd\[email protected]<>ACL ka ¿ j£oNSP>G%τQSB9AZ(ω¡HP)¡HL.YZPτw g(ωGc=9)?[STcτ[email protected]<X?:Q.YZB τ (ω ) ∂f ka ¿ kj£ . V 'V ' ∂θ τ Gc= [email protected]<>ACLH=2 ka j£¾GJ?w ka kF£[email protected]? DMEHG:PbY}[email protected]'[email protected]<[email protected]<>ACLk[aE|B9TJt<XIKGw[SACIV<XLHT,Q.YZBvP>Gc=vACLH[SGc=h; g"[a<¢§~EH=9<XijGc= ijGJB9=P>G·B9TJt<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<¢°[email protected]?9GQ.YZBKPbYW[[email protected]<[email protected]<>ACL½[aE²[email protected]<XtL.YZP,[STJIKAM[aESP>TGJ?|Q.YZB . τ ≡ n ∼ τ (ω ) Pd\^YZEStIKGJLF?'Y[email protected]<>ACLg[SGw='ACLYZIVQSPX<[email protected][SG%[email protected][email protected]<>ACLSLHGJPXP>5G s 1 S¾\[email protected][email protected]<>ACL[[email protected][email protected]'[email protected]<[email protected]<>ACL[SACLSLHG [SGc= <XLa°[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:¡HtESB9G2 ka^xF£oN{PbY[email protected]'YZLH=°[email protected]|[email protected]<>ACLV[aE=9<XtL.YZP*[STJIKA*[aESP>T[aEHG sPbY:[email protected]'YZLH=9<[email protected]<>ACL[aEB9TJt<XIKG[SACIV<XLHT Q.YZB PbY a[ <¢§[email protected]<>ACL|[SGc=¾ACLH[SGc=v;hggijGJB9=¦P>G B9TJt<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¤B9TcDMEHGJLHhcG<XLS<[email protected]<bYZP>G YVhcACLH[a<[email protected]<>ACL<XLS<[email protected]<bYZP>G τ (ω ) = C /ω τ = n τ (ω ) = C /ω [SGPd\mACLH[SGf[SGVQ0ACIVQ.YZjG ω = 0.9ω [email protected][email protected]?9GJLM?6[SG|[[email protected]<XLHGJB}P>Gc=}hcACLH=9?'YZLF?9Gc= C GJ? 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[SACLHh,QSB9AM[aES<[email protected] GJ? _qM<~PbY2B9TcDMEHGJLHhcG [SG:Q~ACIVQ.YZjG6°[email protected]<XIV<XP>TcG:<>hJ< s2PbYwB9TcDFEHGJLHhcG[email protected]?9GJEH='G:ACEKB9TcDFfEHGJLH=hcG60hcGJkHz [email protected]'YZP>Gw[afE|Q.=YCDM160EHGJ?¾kHz []\mACLH[SG,[SG:Q~ACIVQ.YZjG{£ Gc=9? GJLHhcACB9GYZEStIKGJLM?9TcGtNaESLHGw[[email protected]<XijTcG%[email protected]<>=9<>UJIKG2[SG A [email protected]<>GJLSLHG2YZQSQ.YZB'Y >?Q~ACESB ¤[email protected][email protected]<>GKh{£v[SG PbY%¡HtESB9ºG kan mF£oj¥¦La¡HLN*YZES?9ACESB [SG:PbYwB9TcDFEHGJLHhcG f = 300 kHz NjESLHG[[email protected]<XijTcG='GchcfACLH=[SG[email protected]? kHz [email protected]}¤[email protected][email protected]<>G[.£ [SG%PbY¡HtESB9¶G kan mF£oS,<[email protected]<dNS[SGJEan[[email protected]<[email protected]<>ACLH=,[email protected]<XiM<>Gc=[]\ ESLHG2<XLF?9TJtB'[email protected]<>ACLW[aE[email protected]<XtL.YZP[STJIKA*[aESP>T}=9ACLM?[email protected]= GJLYZEStIKGJLM?'YZLF?wPbY6B9TcDFEHGJLHhcG2[SG2Q0ACIVQ.YZjG[SG f = 60 kHz s f = 300 kHz :¡HLk[SG%hcACLa¡[email protected]:[email protected]?9GwACeH='[email protected][email protected]<>ACLDME.YZPX<X?'[email protected]<[email protected]'YZ<X?9GJIKGJLM?:[email protected]<>[email protected]<XIVQSP>G2Y6TJ?9Tw<XIVQSP>TJIKGJLF?9Tt ¾\mACB9[aB9Gg[SGW[[email protected]<[email protected]<>ACLº?9GJIVQ~ACB9GJPXP>[email protected][email protected]<>GJPXP>G p [email protected]<XPX<>=9TgQ0ACESBlACeS?9GJLS<XBESLµQSB9AZ¡HP2?9GJIVQ0ACB9GJPwIKA*[STJPX<>=9T [[email protected]<[email protected][email protected]<>GJPXP>G±[]\ ESLHG A YZEH=9=9<>GJLSLHG{£P>GQSPXEH=gQSB9A*h9RHGQ~At='[email protected]<XeSP>G´[SGc=QSB9AZ¡HP>=W?9GJIVQ0ACB9GJP>="[email protected]<XIKGJLM?'YZEan0 ACESB}hcGdYZ<XB9GtN][email protected]'[email protected][email protected][S¸G [email protected]<>GJB4[SGVPd\mGJLMijGJP>ACQSQ~Gf[SGVIKA*[[email protected]<>ACLQ0GJES?[email protected]|[email protected]<X?9GK=9ACEH=PbYl[email protected] A i [email protected]?kP>GW?9GJIVQH=[SGWijACP[[email protected]<XtL.YZPGJ? τ P>GW?9GJIVQH= S(f ) = T F [s(t)] = (−j2πf ) e e τ hOYZB'[email protected]<>[email protected]<>DFEHG,[SG Pd\mGJLFijGJP>ACQSQ0G[SG IKA*[[email protected]<>ACLg[aESB9TcG,hOYZB'[email protected]<>[email protected][email protected]<>DMEHG[aE|[email protected]<XtL.YZP8[STJIKAM[aESP>T{£oMG:Q.YZB'[email protected] hOYZB'[email protected]<>[email protected]<>DFEHG τ [email protected]<XPX<>=9Tf[HYZLH=6P>GfhOYZP>hJESP[SG|IV<XLS<XIV<>='[email protected]<>ACL±[email protected]?GJLdYZ<X?}ESLQ.YZB'[email protected]¡SnaT|Q.YZB4PbY[email protected] [aE[email protected]<XtL.YZP[SG%Q0ACIVQ.YZjGTJIV<>=Oa P[email protected]? [SACLHh[email protected]<>[email protected]<hcACLSLFE[email protected]<bYZeSP>G[email protected]<XIKGJLM?'YZP>GJIKGJLF?ON8GJ?:=9GJP>ACLP>G%IKAM[SUJP>G [STJijGJP>ACQSQ0T4QSB9TchcTc[SGJIVIKGJLF?[HYZLH=%hcG4h'R.YZQS<[email protected]~<XP[email protected]? sKPbYVAC<>=whOYZB'[email protected]<>[email protected][email protected]<>DMEHG[[email protected]<XtL.YZP[SG4Q~ACIVQ.YZjG6GJ?w[aE =9<XtL.YZP~[STJIKA*[aESP>TtU j GJQ~GJLH[HYZLM?ONS[]\^[email protected]=[email protected]<>=GJLkhcACIVQS?9G%[HYZLH=P>G,IKAM[SUJP>G%[STJijGJP>ACQSQ0T2=9ACLF? [email protected]<XeSP>Gc=:[SG=9G%I|YZLS<¢°[email protected]?9GJBONHGJ?:ht\mGc=9?:Q~ACESB:[email protected]?9GB'YZ<>=9ACLDMEHG2hcG2Q.YZB'[email protected]}[email protected]? [email protected]<>=hcACIVIKG}Y® EH=9?'YZeSP>Gt Gc=B9Tc=9ESPX?'YZ?9=g[[email protected]'YZ<X?9GJIKGJLF?"[email protected]<>DFEHG´=9ACLM?QSB9Tc='GJLF?9Tc=W[HYZLH=gP>G?'YZeSP>GOYZE kan mQ0ACESSB m k±itYZP>GJESB9=g[SG B9TcDMEHGJLHhcGc=6[SGfQ0ACIVQ.YZ[email protected] f = 60 kHz GJ? f = 300 kHz A P>ACe.YZP>GJIKGJLF?ONESLHGf°YZ<XeSP>Gl[STchJB9AC<>=9='YZLHhcG [SGc=}Q.YZB'[email protected]=[SGK?9GJIVQH= GJ? [email protected]?}[email protected][email protected]<[email protected]<>ACLH=6[SGKPd\mACB9[aB9Gf[SG|[[email protected]<[email protected]<>ACL hcACLa¡[email protected]? P>Gc=[email protected][email protected]<>ACLH=wDME.YZPX<X?'[email protected]<XijGcτ=B9TOYZPXτ<>=9TcGc=,[email protected]:PbY6¡HtESB9<G kan mSHACESB f = 60 kHz NHPd\mACB9[aB9G}[SG}[[email protected]<[email protected]<>ACL p Gc=9? QSB9A*h9RHGw[S¶G mZ mt k À £oNMQSES<>=¾<XP]YZEStIKGJLF?9G2YcijGch,PbY¤B9TcDMEHGJLHhcG[SG,Q~ACIVQ.YZj<G t!s f = 160 kHz N p ' 2.05 [[email protected]<XijTcG ='GchcACLH[SG{£ EH='DFE\ s f = 220 kHz A i p ' 2.7 ¤QSB9AMh'RHG}[]\ ESLHG4[[email protected]<XijTcG[email protected]<>[email protected]<>UJIKG{£oH¥¦La¡HLNH<XP[a<XIV<XLMEHG}Q0ACESB:=9G =9?'YZeS<XPX<>=9GJBYZES?9ACESB,[SG 2 Q0ACESB f = 300 kHz [[email protected]<XijTcG=9GchcACLH[SG{£o qMESB6PbY¡HtESB9G kan mmjNP>Gc=[email protected]<XtL.YZEan[STJIKA*[aESP>Tc=V<>=9=9EH=6[[email protected]<>=4B9TcDFEHGJLHhcGc=V[SGlQ0ACIVQ.YZjG[a<¢§0TJB9GJLM?9Gc=°YcijGch PbYfIKJIKGGJLFijGJP>ACQSQ0GK[SG6IKA*[[email protected]<>ACL8£[email protected]@Q~At='Tc=%Y®¡HL[SGhcACIVQ.YZB9GJB%dYChJ<XP>GJIKGJLF?P>GJESB9=%?9GJIVQH=2[SG6ijACP>=O Gc=,QSB9AZ¡HP>=2[SG6hcGc=%[email protected]<XtL.YZEang='ACLF?%QSB9AMh'RHGc=2[SG}PbYl[[email protected]<[email protected]<>[email protected]<>UJIKG[]\ ESLHG A YZEH=9=9<>GJLSLHG4GJ?%[email protected]<>GJLM?2Q.YC=w[SG p p p p p p − 2 f 2 τm 4 p −2πjf τ0 0 m m p 0 p m p p p p ¹ ¡¥¦p± µ ©ª©¦<¢ )p£¥¤«&¯¸¢¨©ª± ) 0.8 zE|Z~~|I{|,&$ |ZzZ| 1 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 0 2 |Zb~},n} 4 6 −4 x 10 h f%UU« ;¨ª± )5²´¥¯ ¬*²±!µ ¥p¦,£_¬±!¤©%¤2¬«&¦ 9¤B¥2°±¢¨¢p¦²´¢,£_¬¯º£_ª1;´¢h²« º¥¤B¢¨U©:¢p¦ ; ¢§© ¹ fp = 170, 200 230 kHz 8 ¢p¦l©:¢¯º£U¦P²´¢ .¬µº²´¢p¦ ¦« ;¨ ª± )S²«¯³«&¨U±¢¨©µ ¬¤¦§°±¢µnª 9p¤B¥2°±¢¨¢#²´¢³£_¬¯º£_ª1;´¢ª± ;¯x¢¨U©:¢ ¹ 8 ª¯x¬ ²± µnª©%«%¬¨ ²± ¦« ;¨ªµh²!¢x£_¬¯º£_ª1;´¢ ¢p¦©³©%¤2ª¥B¢4£_¬± ¤(²´¬¨¨ ¢¤±!¨f¢¤2¥C9 ¥¤B¢¨¢*¹ % ¢© 5ª± ©%¤B¢ P0 ' 300 kP a d ' 12 cm B¬1¨ 2 ;±!¤2ª©%«¬¨#¢ )p£¥¤p«¯¸¢¨U©ªµ ¢ °±¢<¢µµn¢£¤B¥p¦ ¢¨U©:¥B¢5¦±!¤¶µ ª 2G;± ¤2¢ ¹.7¹ °Y cACL<[email protected]?'YZLF?9Gt¦[email protected]<XtL.YZP [STJIKA*[aESP>TYC=9=9A*hJ<>T4sPbY¤B9TcDMEHGJLHhcGk[SGfQ0ACIVQ.YZjG f = 230 kHz GonaRS<Xe~GlESL ¤B9ACLM?IKACLF?'YZLM?VDFES<¾[email protected]@<XijGfGJLQSB9GJIV<>GJBON=9ES<XiM<¾[[email protected]<XtL.YZP¾[STJIKA*[aESP>TK<>[email protected][aEQ.YCDFEHGJ?6[]\mACLH[SGskPbYk¤B9TcDMEHGJLHhcG QSES<>=[aE¸[email protected]<XtL.YZP}YC='=9AMhJ<>uT s j [email protected]?9GJIKGJLF?IKGJ?eS<>GJLGJLµitYZP>GJESBPbY > f = 200 kHz [email protected] G¾[HYZLH=P>G¦?'YZeSP>GOYZ<E kan mw[email protected]?'YZ?TJ?'YZLF?hcGJQ~GJLH[HYZLM? [STchJB9AC<>=9='YZLHhcGv[SGc=Q.YZB'[email protected]= ?9GJIVQ~ACB9GJP>= τ GJ? fτ =ACeH170 ACeS?9GJLFE[HYZLH=ESLHGhcACLa¡HtESB'[email protected]<>ACL[email protected]<XIKGJLF?'YZP>G}[a<¢§]TJB9GJLF?9G{£o j [email protected]?9GJIKGJLF?[SGK[[email protected]<[email protected]<>ACLH=}GJ?<XLM?9TJtB'[email protected]<>ACL[aE=9<XtL.YZP¦?9GJIVQ~ACB9GJP [STJIKAM[aESP>T|L\^YkQ.YC=[email protected] QSB9TchcTc[SGJIVIKGJLF?O~`PGc=9?:[email protected]?'YZ<XLHGJIKGJLF?2YC=9=9A*hJ<>TYZEan[a<XijGJB9=IKTchOYZLS<>[email protected]=:QSRM*=9<>DFEHGc= QSB9Tc='GJLF?9=w[HYZLH= P>Gc=:IV<XPX<>GJEan tB'YZLFESPbYZ<XB9Gc=ONSDMES<~<XLa³HEHGJLHhcGJLF? P>G,[email protected]<>ACLSLHGJIKGJLM?:[SGwPd\^YZLF?9GJLSLHGtS¹,LHG,[[email protected]<[email protected]<>ACLDME.YZPX<X?'[email protected]<XijG%[SG%hcGc= Go§]GJ?9=GJ? [SG}P>GJESBw<XLa³HEHGJLHhcG[email protected]\^YZLF?9GJLSLHGQ.YZB'[email protected]@<>DMEHG6Gc=9?,B9TOYZPX<>=9TcG6[HYZLH=wPbY|='[email protected]<>[email protected]<XitYZLF?9GGJLgiMEHG6[]\ <[email protected]?9GJB P>Gc= [email protected]?'YZ?9=:[email protected]<XIKGJLF?'YZEanACeS?9GJLFEH={ p p p 0 m !"#%$%& '($)*+,&-. /* #2$ )<96 FU373+ 167BF.535FF.79/) ¹[email protected]*[email protected]=9Gk[SGc=VGo§]GJ?9=6QSRF[email protected]<>[email protected]<XeSP>Gc=K[]\ <XLa³HEHGJLHhcGJBVP>Gf[email protected]<>ACLSLHGJIKGJLM?f[SGlPd\^YZLF?9GJLSLHGGJ?[SG P>GJESB9=VhcACLH='TcDFEHGJLH=9Gc=|[email protected]>GlQSB9AZ¡HP[aE²[email protected]<XtL.YZP:[STJIKA*[aESP>T[email protected]?VI|YZ<XLF?9GJL.YZLM?|LHTchcGc=9='YZ<XB9GY®¡HL½[]\ <[email protected]?9GJBONvYZE IKAC<XLH=2DME.YZPX<X?'[email protected]<XijGJIKGJLF?ONP>Gc=%[email protected]'[email protected]<[email protected]<>[email protected][email protected]<XIKGJLM?'YZP>GJIKGJLF?O j [email protected]?9GV=9[email protected]=9G[email protected]?%B9TOYZPX<>=9TcG6<>hJ<GJ? 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[]\ <XIVQ0Tc[HYZLHhcG[email protected][email protected]<>DMEHG[email protected]>GlIV<XPX<>GJE±tB'YZLFESPbYZ<XB9GGJ?6PbYW=9ESB°YChcGk[aE´[email protected]'YZLH=9[aEHhJ?9GJESBKPXES<¢_dIKJIKGkL\[email protected]?Q.YC= LHTchcGc='='YZ<[email protected]<X?[HYZLH=P>Gc=hcACLa¡HtESB'[email protected]<>ACLH=¦Gon*[email protected]<XIKGJLF?'YZP>Gc={NjGJ??9ACES?[email protected][email protected]<>hJESPX<>UJB9GJIKGJLM? s,e.YC=9=9GvB9TcDFEHGJLHhcGt ¹,[email protected]dYChcGQ.YZB°YZ<X?9GJIKGJLM?}PX<[email protected]'Y >LHGJB'YZ<X?}ESLHGK[[email protected]<[email protected]<>ACL[aE[email protected]<XtL.YZP¦[STJIKA*[aESP>Tt j [email protected]?9GVhcACLH[a<[email protected]<>ACLQ~GJES? [email protected][email protected][email protected]<>GJPXP>GJIKGJLF?:B9TOYZPX<>='TcGP>ACB9=9DMEHG:PbY}P>ACLStEHGJESB [SG:Pd\^YZLM?9GJLSLHG2=9G:B'YChchcACESB9hJ<X?ONaGJ?vDMEHG,P>Gc=vGo§][email protected]<¢= [SG QSB9ACQ.YZ[email protected]<>ACLLHGw=9ACLM? 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[email protected][a<>[email protected][email protected]<XLStEHTc=='YZLH=YZI}eS<XtES<X?9T%tB ChcG%YZEanfhcACLH[a<[email protected]<>ACLH= [email protected]<XIKGJLM?'YZP>Gc=Q~ACESB P>Gc=9DMEHGJPXP>Gc=¾<XP>= <[email protected]*<>GJLSLHGJLF?:ACE LHACL0YZBGonaGJIVQSP>GtN0P>G6B9TJt<XIKGV[SG[a<¢§[email protected]<>ACL"[SGc=2ACLH[SGc=2[SGQ~ACIVQ.YZjG[email protected]?%?9AC?'YZP>GJIKGJLF?}<[email protected]<XeSP>G|YcijGchP>Gc= Go§]GJ?9=:[SG2[a<>[email protected][email protected]<>ACLYC='=9AMhJ<>Tc= s[SGc=,Go§0GJ?9=hJESI}[email protected]<¢°=[]\mACLH[SGc=:QSB9ACQ.YZ[email protected]<XijGc={ ¯%YZLH=4P>Gc=Gon*[email protected]<>GJLHhcGc=6QSB9Tc=9GJLF?9TcGc=QSB9TchcTc[SGJIVIKGJLM?ONP>[email protected]?9GJIKGJLF?K[SGfPd\^YZLF?9GJLSLHG[HYZLH=4P>GfB9TJt<XIKGl[SG [email protected]?ON,='YZLH=<XLa³HEHGJLHhcG´[SGc=Go§]GJ?9=QSRM*=9<>[email protected][email protected]<>ACLSLHTc={Nw='G[email protected]'YC[aES<X?WhcACIVIKG[HYZLH=Pd\mGOYZENQ.YZB ESLHG[[email protected]<XijTcG=9GchcACLH[SG[aE±QSB9AZ¡HP[]\ <XLF?9GJLH=9<X?9T[SGkQ0ACIVQ.YZjGQ~ACESBVP>G=9<XtL.YZP:[STJIKAM[aESP>TACeH='[email protected][email protected]\^Y®naG[SG QSB9ACQ.YZ[email protected]<>ACL jw\[email protected]?2P>GVhOYC=[HYZLH=2P>Gc=[email protected]<>GJLHhcGc=2QSB9Tc=9GJLF?9TcGc=}Q~ACESB f ' 160 kHz A i p ' 2 j GJPbY[email protected]<XtLS<¢¡.G DME\ sQSPXEH=Ve.YC=9=9GfB9TcDFEHGJLHhcG[SGkQ0ACIVQ.YZjGtNP>GkQSB9AZ¡HP:[email protected]?<XLF?9TJtB9TtN¾QSES<>=9DME\ <XP[email protected][email protected]<>ACLSLHGJºP sPbYW[[email protected]<XijTcG QSB9GJIV<>UJB9G|[]\ ESLHG A [email protected]<>GJLSLHGt¦[email protected]<>GJESB9=Go§]GJ?9=2QSRF[email protected]<>DMEHGc==9ACLF?[SACLHhVhOYZLH[a<>[HYZ?9=}Q~ACESB[email protected]?9GK<XLF?9TJtB'[email protected]<>ACL[aE =9<XtL.YZPd]¬ACES?4[]\^YZe0ACB9[]N]PbY[a<¢§[email protected]<>ACL[SGc=ACLH[SGc=4[SGVQ0ACIVQ.YZjGtNDFES<¾YZQSQ.YZB'Y >?4P>ACB9=9DMEHGPbYkP>ACLStEHGJESB4[]\mACLH[SGK[SG Q0ACIVQ.YZjGK[SGJi*<>GJLF?}hcACIVQ.YZB'YZeSP>4G skESLHGK[a<[email protected]<>ACLhOYZB'[email protected]<>[email protected][email protected]<>DFEHGf[]\ <XLSRHACIKACjTJLHTJ<X?9Tc=[HYZLH=2P>GVIV<XPX<>GJEN]L\[email protected]? Q.YC=|P>Ge0ACL·hOYZLH[a<>[HYZ?Ov Y"I|YZLS<¢°[email protected]?'[email protected]<>ACL·[email protected]<XeSP>G[SGPbY"[a<¢§[email protected]<>ACL½[SGc=fACLH[SGc=f[SGQ0ACIVQ.YZjG=9ESBKP>G[email protected]<XtL.YZP [STJIKA*[aESP>T,YTJ?9TYZL.YZPX*='TcG[HYZLH=¦PbY[email protected]<>AC4L ka =H ¿ M`[email protected]?¾QSB9Tc[a<X?vDMEHG PbY2[a<¢§[email protected]<>ACLf[SGc=vACLH[SGc=¾[SG Q0ACIVQ.YZjG:IKUJLHG s}Pd\ <XLF?9TJtB'[email protected]<>ACLg[aEkQSB9AZ¡HP0[STJIKA*[aESP>T%P>ACB9=9DMEHGwPbYB9TcDMEHGJLHhcGw[SGwQ0ACIVQ.YZ[email protected]?:YZEStIKGJLF?9TcG|¤PbY4P>ACLStEHGJESB:[]\mACLH[SG [SGwQ0ACIVQ.YZjG2=9GwB'YZQSQSB9AMh'RHG2[SGwPbY6?'YZ<XPXP>Gw[SGc=<XLSRHACIKACjTJLHTJ<X?9Tc=w[aEkIV<XPX<>GJE8£oU jw\[email protected]?<>hJ<]ESLHG%[[email protected]<[email protected]<>ACL[aE[email protected]<XtL.YZP DMES<[email protected][email protected]Pd\^YZEStIKGJLM?'[email protected]<>ACL[SG6PbYfB9TcDFEHGJLHhcG[SGQ0ACIVQ.YZjGtN][email protected]'YZ<XB9GJIKGJLM³? sfPd\mGo§]GJ?2Q~At=9=9<XeSP>G6[SG PbY[a<¢§[email protected]<>ACL[SGc=ACLH[SGc=:[SG2Q0ACIVQ.YZjG[email protected] P>G[email protected]<XtL.YZP[STJIKA*[aESP>Tt YZEB9TJt<XIKGl[SG G|=9GchcACLH[hOYZLH[a<>[[email protected]'YZ<X?4[email protected]>GVQ.YC=9=YZjG|[aEB9TJt<XIKGl[SG|[email protected]*?'Yc# s [email protected]? s f = 160 kHz j GJQ0GJLH[HYZLF?ONCYcijGchvP>[email protected]<XI|[email protected]<>ACLH= [SG¾PbY P>ACLStEHGJESB [SfGv[a=<¢§~60B'[email protected] <>ACL s f = 60 kHz GJ? [SG¦PbYP>ACLStEHGJESB[]\^[email protected][email protected]<>ACL[SG¦Pd\mACLH[SG¾[SG¾Q~ACIVQ.YZjGtN ` = πf /c(ω )a ' 25 cm GJ? 20 cm . ` . 50 cm N − g m s a s m m 0 a a d p p p d p p 2 p a ¹ #º¨ªµ ¦§¢<¢©w²«&¦±!¦¦«%¬¨ ¥ §]GJ?:QSRF[email protected]<>DMEHG j ACLH[a<[email protected]<>ACLH= ¥ §]GJ?[email protected] P>[email protected]<XtL.YZP Sg B9TcDFEHGJLHhcG4[SG%Q~ACIVQ.YZjG [email protected]<>[STJB9T [SG%I|YZLS<¢Gc=9?'[email protected]<>ACL [STJKI AM[aESP>Tl[email protected] Pd\^Y®nSG{£ ¨ [email protected]ª ¯w<>[email protected][email protected]<>ACL[SG2i*<X?9Gc=9=9G τ τ X< LM?9TJtB'[email protected]<>ACL f ¨ R.YZES?9Gc=ª ¯w<¢§[email protected]<>ACL[SGc= X< LM?9TJtB'[email protected]<>ACL f ACLH[SGc=[SG%Q0ACIVQ.YZjG τ τ τ ¨ [email protected]ª «TJt<XIKG[SG2[email protected]*?'Yc (Ω/ω)` ` X< LM?9TJtB'[email protected]<>ACL f ¨ [email protected]ª jvR.YZIVQQSB9AMh'RHG2Q~ACESB X< LM?9TJtB'[email protected]<>ACL ` ∼d f P>Gw°YZ<>=9hcGOYZE g ¨ R.YZES?9Gc=ª j ACLH[a<[email protected]<>ACLPX<XIV<X?9G S[ [email protected]<[email protected]<>ACL λ ≥ ` ,` f PX<XeSB9G}GJL z = 0 − s m p m a a d p p a p Ω a p f )f ' º¢§©¦£® ¦p«°±¢¦ ¦±´¦§¢:£ ©%« µ ¢p¦Á² · «¨.±¢¨p¢¤uµn¢ 9§¬¨§©%«¬¨¨ ¢¯x¢¨U©¸²´¢µ2· ª¨U©:¢¨¨ ¢£ª¤Zª¯x¥§©%¤p«°±¢ ¹ 8 ¢p¦¢ º¢§©¦l¦p± ¤lµ ¢ ¦« ;¨ªµ>²´¥¯x¬ ²±!µ ¥P¦¬¨©5²¬¨U¨ ¥p¦l¤B¢µ ª©%«/.¢¯x¢¨U© ª± ¤B¥ ;«¯x¢(²!¢ 7 ¢p¦©:¢¤+.¢µ © µnª¦p¦p«°±¢ 5¦ª¨¦ 7¹8 ¢¸¦p« ;¨ªµ.²!¢»£¤2¢¦¦p«¬¨"¢p¦©hªµ ¬¤p¦ ²«&¦£¢¤p¦p«¬¨% ²« b±!¦p«¬¨'¢©R£_¬± ¤¸± ¨ ¢ B¬¨²«%©%«%¬¨Sµ «&¯³«%©:¢)¤« ;«²´¢ ¢¨ z =0 £¤2¬p£_¬¤©%«%¬¨¨ ¢G µ 3¸µnª4²!¥¤p« .¥B¢¶¦ ¢2B¬¨ ²´¢³²´¢¶µ ª 9§¬¨§©%«¬¨(²´¢¶¯ ¬*²±!µnª©%«¬¨ ¹ PbY|hcACLH[a<[email protected]<>ACLWLHTchcGc=9=YZ<XB9G (Ω/ω)` ` L\[email protected]?wQ.YC=wB9GJIVQSPX<>GV[HYZLH=,P>Gc=%[email protected]<>GJLHhcGc={8¦\^YZLM?9GJLSLHGVQ.YZB'[email protected]@<>DMEHG ACQ~UJB9G2[SACLHh?9ACECACESB9=,[HYZLH= P>G2B9TJt<XIKG[SG%[email protected][email protected]?w[HYZLH= P>Gc=:[email protected]<>GJLHhcGc=QSB9Tc=9GJLF?9TcGc={ £ G[email protected]<XtL.YZP[STJIKA*[aESP>TtN]P>ACB9=9DMEHG4PbY|B9TcDFEHGJLHhcG[SG6Q0ACIVQ.YZ[email protected]?w°YZ<XeSP>G¤?`*QS<>DMEHGJIKGJLF?4YZES?9ACESB[SG Q~GJES?4LHG|Q.YC=[email protected]?9Tf[HYZLH==9ACL´h9R.YZIVQ´P>AC<XLM?'YZ<XLNht\mGc=9³? s[a<XB9G|Q0GJES?6LHG|Q.YC=6[email protected][a<¢§0B'YChJ?9Tt¥L´60Go§0kHz GJ?ON PbY P>ACLStEHGJESB2[]\^[email protected][email protected]<>ACLN][email protected]<XLHhJ<XQ.YZP>GJIKGJLM?}YC=9=9A*hJ<>Tc<G s|Pd\^[email protected][email protected]<>ACL"[HYZLH=%hcG6hOYC={NHL\[email protected]?%Q.YC=,[email protected]=,<XLa°[email protected]<>GJESB9G s PbY}[a<>=9?'YZLHhcGw[]\mACeH='[email protected][email protected]<>ACL d ' 20 cm£o j GJQ~GJLH[HYZLM?ONSPbY[email protected]'YZLH=9<[email protected]<>ACL[email protected]=vPbY6[[email protected]<XijTcG%=9GchcACLH[SG%[]\ ESLHG A [email protected]<>GJLSLHGtN0hOYZB'[email protected]<>[email protected]?@<>DMEHGK[]\ [email protected][email protected]<>ACL[HYZLH=%P>GVh9R.YZIVQ"P>AC<XLM?'YZ<XL[aE[email protected]<XtL.YZP[STJIKA*[aESP>T¤B9TJt<XIKGV[SG [email protected][email protected]?£oN.Gc=9?:QSB9AC[email protected]<XijGEH=9DME\ s H¯%Gc=[email protected][email protected]<XI|[email protected]<>ACLH=,Gon*[email protected]<XIKGJLF?'YZP>Gc=%Go§][email protected]=:Q0ACESB[SG °YZ<XeSP>Gc=[a<>[email protected]?'YZLHhcGc=:[SG2QSB9ACQ.YZ[email protected]<>ACL´¤?`*QSf<>DFEH=GJIK160GJLM? kHz £ ACLF?[[email protected]}DMEHG%PbYV[a<>[email protected]?'YZLHhcGhOYZB'[email protected]<>[email protected][email protected]<>DMEHG 3 − 5 cm []\^[email protected][email protected]<>ACLQ~ACESBESLHG6B9TcDMEHGJLHhcGK[SG Gc=9?[SGK=9GJESP>GJIKGJLF?4DFEHGJP>DMEHGc=[email protected]<[email protected]=kGJLFi*<XB9ACL N I|Y®n*<XI4ESI 10 cm =9GJP>ACL´PbYghcACLa¡HtESB'[email protected]<>AC100L8£o¦kHz `P [email protected]?[SACLHhk[a< r|hJ<XP>G[SG4 [email protected][email protected]<¢¡.GJBVDMEHGlPd\[email protected][email protected]<>ACL²[aE±[email protected]<XtcmL.YZP [STJIKAM[aESP>Tk=9G|dYZ<X?6[HYZLH==9ACL´h9R.YZIVQ±QSB9A*h9RHG shcGc=}B9TcDMEHGJLHhcGc=6[SGlQ0ACIVQ.YZjG" s f = 100 kHz Pd\mACB9[aB9Gk[SG [[email protected]<[email protected]<>ACL p ' 1.45£oNDMES<[email protected]?9TJLMEHTcGc=keS<>GJLYcitYZLF?kPbY[a<>[email protected]?'YZLHhcGW[]\[email protected][email protected]<>ACL[email protected]<XtL.YZPw[STJIKA*[aESP>T [a<>[email protected]=9GtNS[HYZLH=:hcGhOYC=ONS[]\ ESLHG}[a<>[email protected]?'YZLHhcG[SG2QSB9ACQ.YZ[email protected]<>ACLPX<XeSB9G[email protected]_E r|='YZLM?9G2Q~ACESB:[email protected]}[a<¢§~B'YChJ?9Tt G[[email protected]<>GJBlhOYZLH[a<>[[email protected]='[email protected]<XeSP>G[]\mGon*QSPX<>DMEHGJBKPd\ <XLM?9TJtB'[email protected]<>ACLºDMES< <[email protected]*<>GJLF?lQ0ACESBKP>Gc=VQSPXEH=VdYZ<XeSP>Gc=B9To_ DFEHGJLHhcGc=[SG2Q~ACIVQ.YZjG}[email protected]? PbYV[a<>[email protected][email protected]<>ACL[SG2i*<X?9Gc=9=9GtS¥L[email protected]<XPX<>='YZLF?:P>GIKAM[SUJP>G[SG2[a<>[email protected]<>ACL[SG2iM<X?9Gc='=9G%[]\ ESLHG h9R.Y >LHGw[SGeS<XPXP>Gc=vESLS<>[a<[email protected]<>ACLSLHGJPXP>GK¤QSB9Tc=9GJLM?9T%[HYZLH=¾P>Gwh9R.YZQS<[email protected]³G m{£oNj<[email protected]?vQ~At=9=9<XeSP>G[]\[email protected][email protected]<XIKGJB P>Gc=¾itYZP>GJESB9= [SGc= Q.YZB'[email protected]=¾[aEKIKA*[SUJP>G[STJijGJP>ACQSQ0T QSB9TchcTc[SGJIVIKGJLM?[HYZLH=¾hcG:h9R.YZQS<[email protected] τ (ω) YOijGch c (ω) N c GJ? τ t¥¦L|QSB9GJL.YZLF? ESLHG P>ACLStEHGJESB[]\^[email protected][email protected]<>ACL ` [email protected]<XIKTcG[SG Pd\mACB9[aB9G,[SG 30 cm NtP>G ?9GJIVQH= hOYZB'[email protected]<>[email protected][email protected]<>DMEHG[]\^[email protected][email protected]<>ACL¤[email protected]<XLa_ hJ<XQ.YZP>GJIKGJLF?:Q.YZB:[email protected][email protected]<>ACL s4hcGc= B9TcDMEHGJLHhcGc=£¾iCYZES?:YZP>ACB9= τ ' ` /c (ω) ' 0.3/300 = 10 s N*hcG%DMES<][email protected]? 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