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Une présentation mathématique de la méthode de Cagniard-de Hoop Partie I En dimension deux

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(1)Une présentation mathématique de la méthode de Cagniard-de Hoop Partie I En dimension deux Julien Diaz, Patrick Joly. To cite this version: Julien Diaz, Patrick Joly. Une présentation mathématique de la méthode de Cagniard-de Hoop Partie I En dimension deux. [Rapport de recherche] RR-5824, INRIA. 2006, pp.89. �inria-00070201�. HAL Id: inria-00070201 https://hal.inria.fr/inria-00070201 Submitted on 19 May 2006. 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..

(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Une présentation mathématique de la méthode de Cagniard-de Hoop Partie I En dimension deux Julien Diaz — Patrick Joly. N° 5824 Février 2006. N 0249-6399. ISRN INRIA/RR--5824--FR. Thème NUM. apport de recherche.

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(167) (x) eosuj@g

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(169) {XZ˜­+miwuj+|}UXZ¢ +. −. .

(170) . r c. . . . +. . +. .

(171) . . −. +. −. −. . . 1 2. p = ±i. Ξ. −1[ ∪ ]1; +∞[} . \^su9 k+‡ue^sunˆjXms‚A+giŒysukXj+moeGZ,{f7Z;§ x+œ£k+¼e ZLezDglq‡dwŸnq{f{fZ{ip mij@sugnœgop@ZkX|moZ lLZpˆœZ∈jXZZg]−∞; q  * w gom}ΩwK‡uZLmieox+Z^wuxImzgiwunœeVZ ˆ{fZLk¨e

(172) |xXsyqkXw‚jxXkX|misuZLY[eL§ xXˆZ¦fZ{flLœnˆY[n©gilZx+w‚m D Zg . Γ. Γ. _¨suj@gomisuj+e$Y%w‚nˆj@goZLj+w‚j@gp@kXZ Z Ξ(p) dp = − Z Ξ(p) dp. afsunœg ρ kXj¡molLZ x8syeon©gin©¹h™+{XZLezgonˆjXlª goZjI{fmoZ‡yZm}e +∞ ™fsujx8syeoZ2X Γ. D.

(173)

(174) Dρ = {p ∈ D | |p| < ρ} ,

(175)

(176)

(177)

(178) Γρ = {p ∈ Γ | |p| < ρ} ,

(179)

(180)

(181)

(182) Cρ = {p ∈ Ω | |p| = ρ} .. oe sunœg^ZLkXezg%j+Z|su|jIsykXehgimon©­8gi9kXZ/<l® su¹ºZLmi{fmonœZ¡ZLY[jyl{fg}ZywOZL§Vkfginœ¦ rFsyjkXw‚nqm}{feo|p@kžezkX¢¸|{XZ suZΞj@¢£|goZsyZm}kXeh|g˜mVˆZ[w‚ZLj+ezmog^wuZLœœ|}c@nqUXw‚goj@sunqp@gnqezkXnˆDZZ/{feokXZZm g goΩZLݜˆ™RZezsyeo{XsujžZŸmogonœgij@ZZgoˆpylLœZk+Œum}Z/ezw‚syœˆZmzZ/gieoZezZkXŒyp@mGYkXœZZ ZLj@|DgVsuj@milgi∪ZLsu kXCm

(183) ¹ºZL∪moezY[syΓn©lg D ∪C ∪Γ D xIw‚m}|suk+mok{Xw‚j+eVˆZ/ezZLj+e{fZeV‡Ow‚ˆZkXm}e

(184) |misunqeoeiw‚j@giZLeF` ‡usynœm A+ŒykXmoZ„<;$ZehgjdkXˆˆZ#X Z Z Z 9 :y§ ƒu<‰ ; Ξ(p)dp + Ξ(p)dp + Ξ(p)dp = 0. ¹¶– wy®|w‚nˆœxXZLmiY[WLe^Zj@ˆwgpy{fk+ l A+Z2j+X n©ginœsyj—{fZ ˆwŸmiwy|nˆjXZ|wumomilZp@kXZjXsyk+e^wK‡ysuj+eG|}UXsunqeznˆZZgGxXkXnqeipyk+Z x < 0 suj¨‡ylmiBn A+Z  ezn =m(p) ≥ 0 Zg <(p) 6= 0 <e(|y| 1 + p + ipx) > 0 Zg^{fsujI|˜p@kXZy™fxIsykXm

(185) gisukfg ψ ∈ [0 ; π/2[∪]π/2 ; π]. Cρ. ρ. ρ. ρ. ρ. ρ. ρ. ρ. ρ. Γρ. Dρ. 2. Cρ. 1 2. lim Reiψ Ξ(Reiψ ) = 0.. \^suk+eVx8suk+‡usujIe

(186) {fsuj+|wuxXxXˆnˆp@kXZLm

(187) œZˆZY[Y[Z{fZ usymi{+w‚jT= f™I€(@WX R7→+∞. _`_ba(ced3f3g.h. ρ.

(188) Kƒ :. D(. =m(p). i Γ−. Γ+ −i cos θ D. <e(p). −i. † nœŒykXmoZ,7eX:vZxXmilLeoZj@giw‚gonˆsuj¡{fZ Γ Zg Γ {Xwuj+eVˆZ/xXˆwuj|suY[xXˆZ¦fZ +. −. !_"$#.

(189)  

(190)    

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(192) 0(12

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(195)   . ( 15e<(!. .  

(196) !S.  .  . U. < . 

(197) e  . lim. –GsujI|. 5D/e(1 . 

(198) . C. 

(199) . ρ7→∞. lim. ρ7→∞. (z − z0 )f (z) = 0,. z0 Z. Z.  - . R > R0. 5D

(200) 

(201) 

(202) 5D. U. ψ2 ∈. 

(203) 3. |z−z0 |→+∞ z∈U. γ. z0. ψ1 < arg(z − z0 ) < ψ2 } (R0 > 0, ψ1. lim. (. e(5D, (5D ( #   .  

(204)  .   . :"7.  . ( 

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(206) { ® nˆj@golLŒum}wOgonˆsuj. \^suk+e

(207) {XlL{fkXnqeosuj+e

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(209) L„ :. D(. {Xsuj+| Z. e. – » 1 − sc |y|(1+p2 ) 2 +ipx. dp = −. Z. ". +∞. # 1 dγ + (t) dγ − (t) −st − e dt. 1 1 (1 + γ + (t)2 ) 2 dt (1 + γ − (t)2 ) 2 dt 1. £ j+Z/moZehgiZ˜x+œk+e^pyk3® ªl‡OwuœkXZLm

(210) ˆw[p@k+wujygin©gil . 1. Γ. (1 + p2 ) 2. r c. dγ ± (t) , dt (1 + γ ± (t)2 ) 1. x8suk+m|ZqwsujkfgonˆˆnˆeoZ{ ® w‚­8sum}{ 9 :u§^:LŠ;KX. 1 2. 1 + γ ± (t)2. eosunœgL™fZLj¡kfgonˆœnqeiw‚j@g

(211) >® Z¦fxXmiZLeieznˆsuj 9 :y§:‹;V{fZ 1 + γ ± (t). 1 2 2. =. r.  21. ct r. =. γ ± (t). X. u§ ƒuƒ. 9 : <;. − iγ ± (t) cos θ , | sin θ|.   ct | sin θ| c 2 t2 ∓ i cos θ . − 1  qr r2 c2 t2 − 1 r2. Gm™fZj¡{flLmonˆ‡Ow‚j@g 9 :u§^:L‹<;›™XsyjY[suj@gimoZ/p@kXZ2X Z. u§ ƒO„.   | sin θ| c  ct dγ ± (t) = ± qr − i cos θ . dt r c2 t2 −1. £ Zehg^wuœsymie ¹¶wy|nˆœZ/{fZ/Y[suj@gimoZLmVp@kXZ . 9 : Q;. r2. 1 dγ ± (t) = ±q dt (1 + γ ± (t)2 ) t2 −. u§ ƒu€. 9 : <;. 1. 1 2. x+kXnˆe

(212) p@kXZ2X eosunœg,X. Z. e Γ. u§ ƒ. 9 : 7 ;. » – 1 − cs |y|(1+p2 ) 2 +ipx. (1 + p2 ). 1 2. u ˜(x, y, s) =. Z. dp = 2. +∞ r c. 2π. Z. +∞ r c. e−st r. t2 −. 2π. r2. r2 c2. e−st r. t2 −. dt.. r2. dt,. u§ ƒ‚Š. 9 : <;. c2. u§ ƒ. 9 : <>;. † {XZ˜nˆj+qw‚w[ˆZY[Y[lZLgijyUXgsf™y{fZLZ"j*;™XkfjXginœsuˆkInˆeie

(213) w‚j@{fg$lL{f>®k+nœj‚nˆeothsuZL|›j+gieVnœ‡dˆZnœgol{fZ^qw/gimiwuj+ez¹ºsumiYlLZ{fZ G w‚xXqwu|Z 9 oe nœ¦dnˆWY[ZGZg${XZmijXnœWLmoZ^lgiwuxIZ c2. !_"$#.

(214) K€.  

(215)    

(216) !"#$%'&( )*,+-'./

(217) 0(12

(218) ! 3(4

(219) 56. d° d’ .  . y§.   . :. 2

(220) (?5D ( (5#O

(221)  2 9 : :<:';Y  

(222) 31 . u(x, y, t) = 0, u(x, y, t) = 2π. r. 1 t2 −. r2 c2. ,. t<. r c. t>. r c. SRº²"R¶°S N5D t `6Q[Y

(223) (B5D  u 3 3  &5D?

(224)  K! (5D   3 e .

(225) (5 `e(5DN e(1e . (

(226) 3  ( r(x) = ct    

(227) !

(228)  

(229) <3. .1Q&5D^(31 3 , u 5 

(230)   t

(231) 

(232)  2O<!

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(236)  

(237) *5  :

(238) ¡!

(239)  (

(240) 

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(242) |ZxXmosy­XˆWY[ZZLezg X h ≥ 0    9 ƒX§ :J; 1 ∂ U ∂ U 1 ∂U 1 ∂ = δ(x) δ(y − h) f (t) − + . . 2. µ(y). 2. ∂t2. wK‡uZ|. . ρ(y). ∂x2.   µ(y) = µ1 ,.  µ(y) = µ , 2. ∂y. ρ(y) ∂y. ρ(y) = ρ1 ,. y > 0,. ρ(y) = ρ2 ,. y<0. Zg c(y) = s µ(y) . ρ(y). X{  Z syY[YZV{Xwuj+e3ˆZLe:eoZL|gonˆsuj+e xXmilL|lL{fZLjygiZLe suje® nœj@gilmiZLeiezZ$w‚k |Lw‚q|kXd{fZVqw

(243) ¹ºsujI|›gonˆsuj{fZ»˜moZLZjezsyœkfginœsyj    9 ƒX§ <ƒ ; 1 ∂ 1 ∂ u ∂ u 1 ∂u − + = δ(x) δ(y − h) δ(t). 2.

(244). µ(y) ∂t2. 2. ρ(y) ∂x2. ∂y. ρ(y) ∂y.    (!     a% 2+,+,(! a .+ a)*+, (

(245) * )a (. _`_ba(ced3f3g.h. (0, 0). /*(+%3a. ct. 3.

(246) Š :. D(. .  

(247)  

(248) 'R S. ž°+² f² >°  \^suk+eVkXgonˆœnqezZLmosyj+eVx+wum

(249) ˆw[eokXn©giZ/œZe$¹ºsuj+|gonˆsuj+e

(250) {XZ/ˆw‡Ow‚minˆwu­XˆZ˜|suY[xXˆZ¦fZ z {fl A+j+nœZe

(251) x+w‚m,X  . ρ2 1 + z 2. R(z) =. Zg. T (z) =.  21. − ρ1. 1. ρ2 (1 + z 2 ) 2 + ρ1. ρ2. . c22 c21. . . c21 c22 c21 c22.  21 + z2  21 + z2. 1 2ρ2 1 + z 2 2 .  12 1 2 2 2 +z + ρ1 (1 + z ). + ˆ‘`S žL°8‘ K´ ¨‘ Ÿµ RSR²  +´ \^suk+e

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(254) ezZLmosyj@g {fl AIjXnœZ{fe

(255) Z xXˆk+‡ye

(256) lxXmiBn miA8lLw‚|j@nˆeog lY[ZLjyg

(257) w‚ki­¼§F\syk+eVkfgonˆˆnˆeoZmisuj+eVxIw‚mVˆw[eokXnœgoZ/ˆZLe

(258) |sdsymi{fsyjXjXlLZLe$xIsyˆwunœmiZLe (r, θ) x Zg y = r sin θ, x = r cos θ {mifwKZOcy° su>SR® j su² j+{XZ9{fµZLZœj¡° ‡uS ­XsuœˆZLkXk¨Y[ezZŸk+m

(259) {X8ˆwu#w°Ij+µ>A+e ‘ Œy’ œkXZmiYZ/nˆ<€ ˆ!;›nœZL§$S k \*:*suµ kIZe

(260) eh/’ g[w‚xXR¶kXµ x8Rj dZ‘ˆ{fsuZLj+Y[e "n©¢£|r:ZLmisu|kXœmZŸ>®k+{XnˆjyjžZgilnœ|j+miZnœezj@ZLgikXgowumim^j@Z g {fOZt {f|=ZsyjX{f(0,jXZlyY[™+0)nœˆ¢¸Z|ZZ¹ºm}g%mo|syˆ{fjyZuZg § c t Ω (t) x8ZkfglLŒyw‚ˆZY[ZLjyNg LgomiZ{X l A+jXn3x+w‚,m X Ω (t)   v  Z g r <t . Ω (t) = (x, y) ∈ |y > 0 c ‡uˆ,w syOœA+° kXŒuSRY[kX² Z^miZG{+€ w‚;Fj+µ Ze¬œg ° ˆZSj+suY[k+nˆe œnˆZw‚kxXx8ƒ/IZZLˆ°8suezµ¶j+g ‘ e kX’ j*|ZLmi|!œS >Z® nˆ {XjygiZ^µ lP|miZLnœZL/’ jykXgiR¶mmoµ Z R{fdOZ˜‘ |=Z^{X (0,ZY[–G0)n©Z*¢£|ZZLY g mi|L{fœY[ZZy§ZŸm}wKˆcuZ*syj¹ºmisucj@xItg ZL9kf{XZLg Z*j*lL>miŒy® susuw‚kXjIˆZŒy{fY[ZVZZLeo{fkXjyZmg Ω (t) Ω (t) LgimoZ{f

(261) l A+jXn¼xIw‚Y mX   v  Z g r Ω (t) = (x, y) ∈ |y < 0 <t .  . . . .

(262). . . 

(263). 1.

(264). .  . .  . . 2.  .  .  . 

(265). . . 1. R. R. . R. 2. 1. .  .  .  . . 

(266). . . 2. T. . T. T. 2. c2. !_"$#.

(267)  

(268)    

(269) !"#$%'&( )*,+-'./

(270) 0(12

(271) ! 3(4

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(273) 0 eosuª \nœgV{fsygimok+wusyejXn©giŒujXZZLsujyZgogg^syj+ªB(t) § j@adk syn©p@g k¼[C(t)D(t)] Ω (t) œkXZ%jXZ*eocdsyYj+l{fgoZŸmi[A(t)B(t)] nˆp@{XkXn©giZ%Z*{fsuZ j+{X[A(t)B(t)] + x u w  m i m u w X x I x y s z m / g  ª >  ® O w f ¦ Z § £ ¬  L Z z e / g X ­ ˆ n  Z j  | u s + j ® nœFeoZ%|milZ 9 nˆ|n ˆZ%Y[nœˆnœZLk :";{Xsuj@gˆZ[¹ºmosyjyg * Z f { % Z LgoZŸ{+w‚j+e/ˆZ*Y[nˆœnˆZk¥ˆZ*xXˆk+(Oy) g  e ˆ   Z @ j g { ® suj+{fZ/Zehg|suj+ezgonœgokXl{Xk¡eoZŒyY[Zj@g [A(t)B(t)] Zg^{fkžeoZŒuY[ZLjyg [C(t)D(t)]. ] d’“’     & 3 3   (Q&^?1#1 25D)* O 35Q (5

(274)

(275) 5e

(276) `1`  O6< 3  ?5Q -  ( 2-   -θN=B5Qarccos(c. 3 e(1/c  )5

(277)  &2   3   J

(278) 31) .  .  . .  . . 

(279). . . R. . . .  

(280) e   

(281) 5e . 1. c. . 2. [A(t)B(t)] . .  s  1 1 r cos θ    − 2+ , t = r sin θ  2  c1 c2 c2        . (.    . d’“°S ²

(282) X²'R¶°S. ¼Zgimonqw‚jXŒyœZ G. t = r sin θ. ”. OA(t)B(t). s. [C(t)D(t)]. ƒf§ uw. 9 7 ;. θ ∈ [0 ; θc ] ;. 1 1 r cos θ − 2− , c21 c2 c2. ƒf§ ‚­. 9 7 ;. θ ∈ [π − θc ; π]).. ZLezgmiZL|›g}w‚jXŒyœZ˜Zj 8 9 3g ;›™+{XZ/xXœkIe OA(t) = c t Zg OB(t) = c t X 2. cos θc =. 1. OB(t) c1 = . OA(t) c2. j 9 ƒX{X§ 7yZw;|sd9 sumiZLm}{feox¼su§j+jX9 ƒfl§Z7‚e ­; ;Vj¼® ZLezg^w‚mik¡nœZLx8j¡su{¼nˆj@® wug kfgomiZp@kX{fZ/Z|<® s@lsypymikI{XwOsugojXnˆsujXjžlLZL{fe k¨eoZŒyY[Zj@g¬thsunˆŒuj+wuj@g œGFZ/® lLx8p@suk+nˆj@w‚gog nˆsuA(t) (c t, 0) B(t) 2. 9. miZLeox¼§¬ˆZ˜x8sunˆj@g C(t) {fZ/|s@symi{XsujXjXlLZLe (−c t, 0) w‚k¡xIsynœj@g D(t) {fZ/|sdsum}{fsuj+jXlZe (c1 t cos θc ; c1 t sin θc ) 2. (−c1 t cos θc ; c1 t sin θc )). . f{\Z syk+ΩeFjX(t)sugo§ZLmoΩsyj+e (t)Ω x8(t)Zkfg>®lLZŒyjIwuezœZLZLYY­XZLˆj@Z^g|Lsygoj+miezZ/gonœ{fgol kXA+l^j+{Xn¼ZLx+e¬w‚gim,monqX w‚j+ŒuˆZLe OA(t)B(t) Zg OC(t)D(t) xXminœ‡yl s ) ( v c Zg r 1 1 r| cos θ| | y > 0, | cos θ| > <t< . Ω (t) = (x, y) ∈ r sin θ − + c c c c c † nœj+wuœZLY[Zj@gjXsukIeVjXs‚gisuj+e Ω(t) = Ω (t) ∪ Ω (t) ∪ Ω (t) § te. R. te. . te. 1. 2. 2 1. 2. R. _`_ba(ced3f3g.h. T. te. 2 2. 2. 2.

(283)  :. D(. – 9 3g ;. g. ΩR (t). 8 9 .;. Ωte (t). Ωte (t).   9 .g ;. g. θ. C 9 .;. Z. . ΩT (t). † nˆŒukXmiZ€DX¬v

(284) ZLxXmolezZLj@giwOginœsyj{Xk¹ºmisuj@g{ ® suj+{XZp@k+wuj+{  ¼°8µ¶‘ ²'R>°S  eS Xµ '² RP ‘ d  ZGjjXsugoZm}w

(285).

(286). . h=0. . r  c t c21 t2  1 +   γ1 (t) = −i cos θ + | sin θ| −1 ;   r r2       ! r  2 t2 c t c 1 υ1+ (t) = −i cos θ + | sin θ| 1 − 1 2 ;   r r      r   2 2    γ + (t) = −i c2 t cos θ + | sin θ| c2 t − 1. 2 r r2. \syk+e

(287) x8sukX‡ysuj+eVw‚ˆsum}e$ljXsyj+|ZLm

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(328) ’ 5e#(P  ‘ .  O  J

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(339) 5e

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(347)

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(368) x8syeosuj+e. p. i. i. 2π. §. x , r. t2 −. r∗ 2 c21. (y + h) r. 9 œZe

(369) |; sdZsug m}{fsuj+jXlZe x8suqw‚nˆmiZLe

(370) {fk¡x8sunˆjyg x {Xwuj+eVœZ/miZx8WmiZ/{fk¡xIsynœj@g^eosukXm}|ZnˆY%w‚ŒuZ ze rnœgok+Zl/g θZj eosu(0,j@g

(371) −h) r=. x2 + (y + h)2 ,. cos θ =. sin θ =.  r  c1 t c21 t2  +  γ (t) = −i cos θ + | sin θ| −1  1  r r2 ! r 2 t2  c c t  1 + 1  υ (t) = −i cos θ + | sin θ| 1 − 2 .   1 r r. _`_ba(ced3f3g.h.

(372) ƒ‚„. D(. r:sukXm{fl

(373) A+jXnˆm γ (t) jXsukIeVkfgonˆœnqeoZmisuj+eVqw ¹ºsuj+|gonˆsuj + 2. F(p, t) = −y 1 + p. 2.  21. +h. . c22 + p2 c21. œezZ¨k+nœgo‡OZw‚Y[j@gx+™XeŸ‡dmi{ wu® wun8mox8misunˆ‡uk+lm Zž{fZ >® suX j+{XZ¡gom}w‚j+eoY[nˆeoZžw‚kPxIsynœj@g.  21. + ipx − c2 t = 0,. ™. (x, y) t02 = t02 (x, y). Zg*œZ œZLYY[Z. x<0. d’“’. ]. . .  WNe(. . . . -

(374) ! 3  ( (e

(375) &    1 . m = min(1, c1 /c2 ). p ∈ [−im; im] 7→ g(p) = F(p, t) − c2 t = −y 1 + p. . 

(376) (1. (.. . h(q) = g(ip).  (  (  We(. q ∈ [−m; m] 7→ h(q) = −y 1 − q. 2.  12. 2.  12. +h. +h. . . c22 + p2 c21. c22 − q2 c21.  21. g.  12.  (

(377)  (. + ipx.. − qx.. 3 3 ,  5D

(378) 5e OO

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