Asymptotic Models for Scattering from Strongly Absorbing Obstacles: the Scalar Case
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(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Asymptotic Models for Scattering from Strongly Absorbing Obstacles: the Scalar Case H. Haddar — P. Joly — H.M. Nguyen. N° 5199 Mai 2004. N 0249-6399. ISRN INRIA/RR--5199--FR+ENG. THÈME 4. apport de recherche.
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(212) $. / n0jk[]dRsst_'nst_q© jt[_ st_yzlon v lq} v _y} j GIBC 2 ` 4 dfs±n]luj}tlo¡]gsxjy© §[]dmhf_jk[]_ st_yz'lqn v lo} v _'} ±j 2 * dfsy¥ / n¬lqjt[]_y}±§5lq} v s2 * dfsi}klq¡]gsxj±o_'}sxdflqn1lu!2 ` ¥¦"lqnz'_'}kn]dfn]jt[_jt[dm} v lq} v _'} GIBC zjk'[]lq_ n v d vdjk_'dmlonYnjtdms'jt©Ydf4 _yn]s_yR djk[]_'}5jt[_ j GIBC4 2 q 4 n]lq}"jt[_ ±j GIBC 2 4 *S* 4 dRsJ}klq¡]g sj¥ / n v _'_ v ©elqn]_[iqs Z. Z. Γ. Dε,3 ϕ · ϕ ds =. α ε3 2. ϕ · N ε,3 ϕ ds =. αε 2. Z. Z. Γ. |∇Γ ϕ|2 ds + ε¯ α. |∇Γ ϕ|2 ds +. α ¯ ε. Z. Z. [1+ Γ. [1+. εH ε2 −i (3H2 − G + ω 2 ) ] |ϕ|2 ds, α 2. ε2 εH +i (H2 − G + ω 2 ) ] |ϕ|2 ds. α ¯ 2. £}klq^ §[]dfz[1lqn_ _yiqstdfhmwz'lq^:p]grjk_ys$jk[ij O2 }t_y^-_y^,¡D_'} α = Γ. Γ. Γ. √ 2 2. +i. √ 2 2. 4R. √ 3Z √ Z Z 2ε ε 2 2 ε,3 |∇Γ ϕ| ds − ρε |ϕ|2 ds, D ϕ · ϕ ds = Im 4 2 Γ 1 Γ Γ √ Z √ Z Z 2ε 2 2 ε,3 Im |∇Γ ϕ| ds − ρε |ϕ|2 ds. ϕ · N ϕ ds = 4 2ε Γ 2 Γ Γ. §[_'}k_jk[]_#£g]nzjtdflqns zloneq_'}kq_72£g]nd£lo}t^:hfw lqn jtl §[]_'n jk_'n v jkl:M2 iun v iq}t_jt[egsp lYsº dmjtdfq_£lq} ε st^iuhfh_'nlqg]ρo[ 4 ¥±Z$[_p]}klq¡]hf_'^jt[_'n¬dRsΓjk[4 ijjt1[]_ dmnYjt_yqε}iuhRsdfn |∇ ϕ| zlq^:_§dmjt[Pjt[]_ §}klqn-stdmon4¥ F s-§"_1sx[iqhmh5n]l§_'{ephfiqdmn4©Edmj:dfs-pDlosksxdf¡]hf_jtl¯zlonsxjt}kgz®j-}tlo¡]gsxj s lq±lq} v _y} * ¥¢Z$[]_d v _yi dRsEjkl,gst_±stlq^:_±iqp]p]}klqp]}kdfiujt_|Ei v iup]p}tl*{rdf^:iujtdflqn-luªjk[]_±df^iuqdfniqGIBC }tw piu}tjJluMjk[]_±¡Dlqg]n v iu}kw lop _y}xº iujtlo}ks5vjk[ij£lo}t^iuhfhfwqdfq_s5jt []_,skiu^:_ lo} v _y}$luAv iup]p* }tl*{r/ df^:v iujtdfv lqn¡]grj±}k_ysxjtlo}t_#iu¡stlq}kprjkdmlonp]}klqpD_'}tjwq¥ ¦"lonstd _y}5£lq}dfnsjiunz'_#jt[]_ j GIBC lu%lq} _'} ¥ n _'_ ε j. Γ. ImD. ε,3. 2. √ √ 2 ε2 2 2 = −ε 1 − ε 2H + (3H − G + ω + ∆Γ ) . 2 2. n_#zyiunjt[]_y}t_'£lq}k_£lq}k^:iqhmhfw§}kdjk_ ImD. ε,3. √ √ 2 ε2 ε2 1 − ε 2H + ∆Γ + O(ε4 ). = −ε 1 + (3H2 − G + ω 2 ) 2 2 2. lujk_±jk[ij©]iqs H − G = (c − c ) ©r§[]_'}k_ c iqn v c iq}t_±jt[]_j§5l p}tdfnzdfpiqhMz'g]}k*iujtg]}k_ys5iuhflqn 2 sx_y_sx_z®jtdflqn1`¥f 4 ©]§"_#[i*o_ Γ . 2. 1 4. 1. 2. 2. 1. (1 +. 2. ε2 (3H2 − G + ω 2 )) > 0. 2. √ / j dRsjt[]_yn stg zdf_'nYj#jtl´st_'_y¨Pi pDlostdmjtdfq_iup]p}tl*{rdf^:iujtdflqn¬lq lo¡rjkiqdmn_ v ¡Ywzlqn sxd v _'}kdfn],jk[]_#£lq}k^iuhªdfneq_'}st_q©eniu^:_yhmw (1 − ε 2H + √ ε2 1 − ε 2H + ∆Γ = 2. ½
(213) ½ÆT"U9V@WW. . √ ε2 1 + ε 2H + (4H2 − ∆Γ ) 2. −1. ε2 2 ∆Γ ). + O(ε3 ).. §[]dRz[ zyiun ¡ _.
(214) 9A. S S) S(. Z$[_'}k_£lq}k_q© ImD. √ √ ε2 ε2 2 = −ε (1 + (3H2 − G + ω 2 )) (1 + ε 2H + (4H2 − ∆Γ ))−1 + O(ε4 ). 2 2 2. ε,3. lu%lq} v _'} * Rd s$lq¡rjiudfn]_ v ¡ew }k_'p]hRiqz'dmn. }klq¡gsj j ±º hfdm¨o_ F. ¡ew. GIBC Dε,3 √ 2 ε2 2 2 1 − (3H − G + ω + ∆Γ ) := ε 2 2 √ √ ε2 ε2 2 −iε (1 + (3H2 − G + ω 2 )) (1 + ε 2H + (4H2 − ∆Γ ))−1 . 2 2 2. 2* Q 4. √ √ ε2 ε2 2 2 2 2 1 − (3H − G + ω + ∆Γ ) ϕ − iε (1 + (3H2 − G + ω 2 ))ψ := ε 2 2 2 2. 2* D 4. √ ε2 ∆Γ ψ + (1 + ε 2H + 2ε2 H2 )ψ = ϕ. 2. 2* A 4. Drε,3. Z$[dfs$_'{ep}t_ststdmlon §dfhmh v _ gst_ v dmn0p]}iqzjtdRz_#dmn1jk[]_#£lqhfhml§dfn]:sx_ynst_SR Drε,3 ϕ. §[_'}k_ ψ dRsstlqhfgrjtdflqnjtlER. n_#zyiun1_yiosxdfhmw q_'}kdm£wjt[iuj Z. −. √ Z √ ε2 2 ε2 2 2 (1 + (3H − G + ω )) (1 + ε 2H + ε2 2H2 )|ψ|2 + |∇Γ ψ|2 ds. ds = −ε 2 2 2 Γ 2*. Z$[_}kdfq[Yj[iun v std v _dRsn]lonpDlostdmjtdfq_£lq}iqhmh ε §[]_'nz'_jt[]_iq¡sxlo}tp]jtdflqn p}tlop _y}xjw£lo} D ¥ 4 ±Ezlog]}sx_lqn]_zyiun£lqhfhml§ isxdf^:dmhRiu}p}tlrz_ v g]}k_±jkl v _'}kdfq_#}tlo¡]gsxj j jt[]df} v lo} v _'} GIBC ¥5Z$[]_ _'{rp]}t_ststdflqn lqjk[]dfs±zlqn v djkdmlondRs¡iosx_ v lon jk[]_iupp]}tl*{rdf^ijtdflqn R √ 2O`o 4 ε ε 2 (1 + (H − G + ω )) {1 − ∆ } + O(ε ). ImN = 2ε 2 2 L±_'nz'_q©]}k_'phfiozdfn] N ¡Yw Γ. Drε,3 ϕ · ϕ. ε,3 r. 2. ε,3. 2. 2. 2. −1. Γ. 2. ε,3. Nrε,3. :=. √ √ 2 ε2 2 2 1 + ε 2H − (H − G + ω + ∆Γ ) 2ε 2 √ 2 ε2 ε2 +i (1 + (H2 − G + ω 2 )) {1 − ∆Γ }−1 , 2ε 2 2. 2O` 4. dfn>2 ** 4 odmo_ys4iqn]lujk[]_'}4jt[]df} v lq} v _'} ±j GIBC ¥%Z$[]dRszlon v dmjtdflqn dRs4}klq¡]g sjdfn edm_y§¯lurjt[_E£lqhfhml§dfn] d v _ynojkdjwo©]§[]_y}t_jt[_ }tdfq[Yj$[iun v std v _dRsn]lonpDlostdmjtdfq_£lq}iqhmh ε © √ Z Z 2£`e 4 ε ε 2 ϕ · N ϕ ds = − (1 + (H − G + ω )) |ψ| + |∇ ψ| ds Im 2. ε,3. Γ. 2ε. Γ. 2. 2. 2. 2. 2. 2. 2. Γ. »£¼%½4»¿¾.
(215) .
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