Scattering and correlations

Scattering and correlations Yv es Coli n de V erdi` ere ∗ No v em b er 19, 2018 In tro du ction Let us consider the propagation of scalar w a v es with the speed v > 0 giv en b y the w av e equation u tt − v 2 ∆ u = 0 outside a compact domain D in the Euclidean space R d . Let us put Ω = R d \ D . W e can assume fo r example Neumann b oundary conditions. W e will denote b y ∆ Ω the (self-adjoint) Laplace op era t o r with the b oundary conditions. So our stationar y w av e equation is the Helmholtz equation v 2 ∆ Ω f + ω 2 f = 0 (1) with the b oundary conditions. W e consider a b ounded interv al I = [ ω 2 − , ω 2 + ] ⊂ ]0 , + ∞ [ and the Hilb ert subspace H I of L 2 (Ω) whic h is the image of the sp ectral pro jector P I of our op erator − v 2 ∆ Ω . Let us compute the in tegr a l ke rnel Π I ( x, y ) of P I defined by : P I f ( x ) = Z Ω Π I ( x, y ) f ( y ) | dy | in to 2 different w a ys: 1. F rom general sp ectral theory 2. F rom scattering theory . T aking the deriv atives of Π I ( x, y ) w.r. to ω + , w e get a simple general and ex- act relation b et w een the correlation of scattered wa v es and the Green’s function confirming the calculations from [2] in the case where D is a disk. See also [1] for other relations b etw een correlations and Green’s functions used in passiv e imaging. ∗ Institut F ourier, Unit´ e mixte de re cherc he CNRS-UJF 5 5 82, BP 74 , 38402 -Saint Martin d’H ` eres Cedex (F ra nce); yves.colin-de-verdiere@ujf-grenoble.fr 1 1 Π I ( x, y ) from s p ectral theory Using the resolve n t k ernel (G reen’s function) G ( ω , x, y ) = [( ω 2 + v 2 ∆ Ω ) − 1 ]( x, y ) for Im ω > 0 and the Stone formula, w e ha v e: Π I ( x, y ) = − 2 π Im  Z ω + ω − G ( ω + i 0 , x, y ) ω dω  T aking the deriv ativ e w.r. to ω + of Π [ ω 2 − ,ω 2 + ] ( x, y ) , we get d dω Π [ ω − ,ω 2 ] ( x, y ) = − 2 ω π Im( G ( ω + i 0 , x, y )) . (2) 2 Short revi ew of scatte ring theory They are man y references for scattering theory: for example [3, 4]. Let us define for k ∈ R d the plane w av e e 0 ( x, k ) = e i< k | x> . W e are lo oking for solutions e ( x, k ) = e 0 ( x, k ) + e s ( x, k ) of the Helmholtz equation (1) in Ω where e s , the scattered wa v e satisfies the so-called Sommerfeld radiation condition 1 : e s ( x, k ) = e ik | x | | x | ( d − 1) / 2  e ∞ ( x | x | , k ) + O ( 1 | x | )  , x → ∞ . The complex function e ∞ ( ˆ x, k ) is usually called the sc attering amplitude . It is kno wn that the previous problem admits an unique solution. In more ph ysical terms, e ( x, k ) is the w a v e generated by the full scattering pro cess from the plane w av e e 0 ( x, k ). Moreo ver w e hav e a generalized F ourier transform: f ( x ) = (2 π ) − d Z R d ˆ f ( k ) e ( x, k ) | d k | with ˆ f ( k ) = Z R d e ( y , k ) f ( y ) | dy | . F rom the previous generalized F ourier transform, w e can get the k ernel of any function Φ( − v 2 ∆ Ω ) as fo llo ws: [Φ( − v 2 ∆ Ω )]( x, y ) = (2 π ) − d Z R d Φ( v 2 k 2 ) e ( x, k ) e ( y , k ) | d k | . (3) 1 As often, we deno te k := | k | and ˆ k := k / k 2 3 Π I ( x, y ) from s cattering the ory Using Equation (3) with Φ = 1 I the characteristic functions of some b ounded in terv a l I = [ ω 2 − , ω 2 ], w e get: Π I ( x, y ) = (2 π ) − d Z ω − ≤ vk ≤ ω e ( x, k ) e ( y , k ) | d k | . Using p ola r co ordinates and defining | dσ | as the usual measure on the unit ( d − 1) − dimensional sphere, w e get: Π I ( x, y ) = (2 π ) − d Z ω − ≤ vk ≤ ω k d − 1 dk Z k 2 = k 2 e ( x, k ) e ( y , k ) | dσ | . W e will denote b y σ d − 1 the total v o lume of the unit sphere in R d : σ 0 = 2 , σ 1 = 2 π , σ 2 = 4 π , · · · . T aking the same deriv ative as b efo r e, w e get: d dω Π [ ω 2 − ,ω 2 ] ( x, y ) = (2 π ) − d ω d − 1 v d Z vk = ω e ( x, k ) e ( y , k ) | dσ | . Let us lo ok at e ( x, k ) as a random wa v e with k = ω /v fixed. The p oin t-p oint correlation o f suc h a random wa v e C scatt ω ( x, y ) is giv en b y: C scatt ω ( x, y ) = 1 σ d − 1 Z vk = ω e ( x, k ) e ( y , k ) | dσ | . Then we hav e: d dω Π [ ω 2 − ,ω 2 ] ( x, y ) = (2 π ) − d ω d − 1 σ d − 1 v d C scatt ω ( x, y ) . (4) 4 Correlatio n of scattered p lane wa v es and Green ’s function: the sc alar case F rom Equations (2) and (4), w e get: (2 π ) − d ω d − 1 σ d − 1 v d C scatt ω ( x, y ) = − 2 ω π Im( G ( ω + i 0 , x, y )) . Hence, w e ha ve Theorem 1 F or the sc ala r w a ve e quation u tt − v 2 ∆ u = 0 outside a b ounde d domain in R d , w e have the fol lowin g ex p r ession of the c orr elation of sc a tter e d wave of fr e quency ω in terms of the Gr e e n ’s function: C scatt ω ( x, y ) = − 2 d +1 π d − 1 v d σ d − 1 ω d − 2 Im( G ( ω + i 0 , x, y )) . F or later use, w e put γ d = 2 d +1 π d − 1 σ d − 1 . (5) 3 5 The case of elast ic w a v es W e will consider the elastic w a v e equation in the do ma in Ω: ˆ H u − ω 2 u = 0 , with self-adjoint b oundary conditions. W e will assume that, at lar ge distanc es, w e hav e ˆ H u = − a ∆ u − b grad div u . where a and b are constan ts: a = µ ρ , b = λ + µ ρ with λ, µ the Lam´ e’s co efficien ts and ρ the densit y of the medium. W e will denote v P := √ a + b (resp. v S := √ a ) the sp eeds of the P − (resp. S − )w a v es near infinity . 5.1 The case Ω = R d W e w an t to deriv e the sp ectral decomp osition of ˆ H from the F ourier in v ersion form ula. Let us c ho ose, fo r k 6 = 0, by ˆ k , ˆ k 1 , · · · , ˆ k d − 1 an ortho no r mal basis of R d with ˆ k = k k suc h that these ve ctors dep ends in a measurable w ay of k . Let us intro duce P k P = ˆ k ˆ k ⋆ the orthogonal pro jector o n to ˆ k and P k S = P d − 1 j =1 ˆ k j ˆ k ⋆ j so that P P + P S = Id. Those pro jectors corresp ond resp ectiv ely to the p ola r izat io ns of P − and S − w av es. W e hav e Π I ( x, y ) = (2 π ) − d R ω 2 ∈ I ω d − 1 dω  v − d P R v P k = ω e i k ( x − y ) P k P dσ + v − d S R v S k = ω e i k ( x − y ) P k S dσ  . using the plane w a v es e O P ( x, k ) = e i k x ˆ k and e O S,j ( x, k ) = e i k x ˆ k j w e get the form ula 2 : Π I ( x, y ) = (2 π ) − d R ω 2 ∈ I ω d − 1 dω  v − d P R v P k = ω | e O P ( x, k ) i h e O P ( y , k ) | dσ + v − d S P d − 1 j =1 R v S k = ω | e O S,j ( x, k ) i h e O S,j ( y , k ) | dσ  . 2 W e use the “bra-ket” nota tio n of quantum mechanics: | e ih f | is the o pe rator x → h f | x i e where the br ack ets are linear w.r. to the second entry and anti-linear w.r. to the first one 4 5.2 Scattered plane w a v es There exists scattered plane w av es e P ( x, k ) = e O P ( x, k ) + e s P ( x, k ) e S,j ( x, k ) = e O S,j ( x, k ) + e s S,j ( x, k ) satisfying the Sommerfeld condition and f r o m whic h w e can deduce the sp ectral decomp osition of ˆ H . 5.3 Correlations of scat tered plane w a v es and Green’s function F ollow ing the same path as for scalar w a v es, w e get an iden tit y whic h holds no w for the full Green’s tensor Im G ( ω + iO , x, y ): Theorem 2 F or the elastic w ave e quation, we hav e the fol lowing ex p r ession of the imaginary p art of the Gr e en ’s f unc tion in terms of the c orr ela tion of the sc atter e d S- an d P- waves: Im G ( ω + iO , x, y ) = − γ − 1 d ω d − 2  1 σ d − 1 v d P R v P k = ω | e P ( x, k ) i h e P ( y , k ) | dσ + 1 σ d − 1 v d S P d − 1 j =1 R v S k = ω | e S,j ( x, k ) i h e S,j ( y , k ) | dσ  , with γ d define d by Equation (5). This form ula express es the fact that the corr elation of scattered plane w a v es randomized with the appropriate w eights ( v − d P v ersus v − d S ) is prop ort io nal t o the Green’s tensor. Let us insist on t he fact that this true ev erywhere in Ω ev en in the domain where a and b are not constan ts. References [1] Y. Colin de V erdi ` ere. Semi-classical analysis and passiv e imaging. Nonlin- e arity 22 :45–75 (2009). [2] F. Sanc hez-Sesma, J. P ´ erez-Ruiz, M. Campillo & F. Luz´ on. The elasto dy- namic 2D G reen’s f unction retriev al from cross-correlation: the canonical inclusion problem. Ge ophysic al R ese ar ch letters 33 :1 3 305 (2006). [3] A. R amm. Scattering b y obstacles, D. R eidel Publishing, D or dr e cht, Hol land, (1986). [4] M. Reed & B.Simon. Metho ds of Mo dern Mathematical Ph ysics I I I. Scat- tering Theory . A c ademic Pr ess, New Y ork, (1979). 5