Generalized Kirchhoff approximation for Helmholtz equation

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Generalized Kirchhoff approximation for Helmholtz equation

François Cuvelier

To cite this version:

François Cuvelier. Generalized Kirchhoff approximation for Helmholtz equation. 2014. �hal-00946716�

HELMHOLTZ EQUATION

F. CUVELIER

Abstract. We give integral formulas to approximate solutions of Dirichlet and Neumann problems for Helmholtz equation at high frequencies. These approximations are valid in the complementary of a union of convex compact obstacles. The first step of the iterative procedure is the classical Kirchhoff approximation. Convergence is proved by comparison with the geometrical optics asymptotics. The method is shown to be numerically stable.

1. Introduction

Let Ω be an open set in R ³ and Ω ^′ = R ³ \ Ω. We study the high frequency diffraction problems of an incident plane wave in Ω for Helmholtz equation with respectively, Dirichlet and Neumann boundary conditions :

(D)







∆v(x) + k ² v(x) = 0 for x ∈ Ω, v(x) = 0 for x ∈ ∂Ω,

v(x) = e ^−ikhξ,xi + u(x) for x ∈ Ω,

(N )



 

 

∆v(x) + k ² v(x) = 0 for x ∈ Ω,

∂v

∂n (x) = 0 for x ∈ ∂Ω,

v(x) = e ^−ikhξ,xi + u(x) for x ∈ Ω.

Here u satisfies the Sommerfeld radiation condition : r ² ( ∂u

∂r + iku) bounded when r = | x |→ + ∞ .

The incident plane wave, e ^−ikhξ,xi , is given with the normalization | ξ | = 1. Here, high frequency means that the wave length is small with respect to ∂Ω curvatures.

So usual numerical methods, such as finite element method, boundary element method and so on, fall down.

A classical high frequency approximation is given by geometrical optics , which, for a point x ∈ Ω, allows us to compute, from optic rays going through x, an approximation of the diffracted wave by Ω ^′ . We obtain for Dirichlet problem (D) and Neumann problem (N ) respectively

(1.1) v _O.G. ^D (x) = X

e ^−ikϕ(x) a ^D ₀ (x) and

(1.2) v ^N _O.G. (x) = X

e ^−ikϕ(x) a ^N ₀ (x).

Here, the phase ϕ(x) is the length of optic rays going through x and computation of the amplitude a ^· ₀ (x) work out by propagation and reflection formulas along optic rays (see [Cuv13]). The main problem of this method is its numerical instability : in order to compute this approximation, it’s necessary to determine all the optic rays going through x. But, small errors in the numerical representation of ∂Ω can give large errors in the optic rays determination.

1

An other one is Kirchhoff approximation, based on integral representations. We give Kirchhoff approximation respectively for Dirichlet problem (D) and Neumann problem (N ) :

(1.3) v ^D Kir.( x) = e ^{−ikhξ ξξ,xi} + 1 4π

Z

∂Ω

ik ( h ξξξ, n n n(σ) i − |h ξξξ, n n n(σ) i| ) e ^{−ikhξ ξξ,σ σ σi} e ^{−ik|x−σ|}

| x − σ | dσ

(1.4)

v ^N Kir.( x)

= e ^{−ikhξ ξξ,xi}

+

1 4π

R

∂Ω ik

hξξ ξ,n n n(σ)i

|hξξ ξ,n n n(σ)i| − 1 D

x−σ

|x−σ| , n n n(σ) E

e ^{−ikhξ ξξ,σ σ σi e}

^{−ik|x−σ|}

_|x−σ| dσ

Here, n n n(σ) is the unit normal to Γ at point σ, exterior to Ω ^′ . But, the validity of this method is restricted to Ω ^′ beeing a strictly convex compact (see [MT85] for Dirichlet problem and [Yin79], [Yin83] for Neumann problem) and false otherwise.

This is due to the incapacity of this method to see multiple reflections.

The purpose of this paper is to determine an iterative integral method, numeri- cally stable, equivalent, at first order and high frequency, to the geometrical optic approximation sets Ω ^′ is a finite and disjointed union of regular and strictly convex compacts. In both case, the first step is given by Kirchhoff approximation.

It relies in ...(crire les principales tapes du papier) 2. Notations and definitions

2.1. Gradient and Hessian on Surfaces. Let K ⊂ R ³ be a compact and Γ it boundary. Let us suppose that Γ is a regular and orientable surface.

Definition 1. (Gradient on Surfaces). The gradient of a differentiable func- tion ϕ : Γ ⊂ R ³ → R is a differentiable map grad grad grad ϕ : Γ → R ³ which assigns to each point σ ∈ Γ a vector grad grad grad ϕ(σ) ∈ T σ (Γ) ⊂ R ³ such that

< grad grad grad ϕ(σ), v > σ = dϕ σ (v) ∀ v ∈ T σ (Γ).

Definition 2. (Hessian on Surfaces). The hessian of a twice differentiable function ϕ : Γ ⊂ R ³ → R is the function Hess ϕ : Γ → L (R ³ ) which assigns to each point σ ∈ Γ a matrix Hess ϕ(σ) ∈ L (T σ (Γ)) such that

h Hess ϕ(σ)v, v i = d ² ϕ σ (v).v ∀ v ∈ T σ (Γ).

Proposition 1. With previous definition, and by taylor’s expansion we obtain for t ∈ R and v ∈ T σ (Γ)

ϕ(σ + tv) − ϕ(σ) = t h grad grad grad ϕ(σ), v i + t ²

2 h Hess ϕ(σ)v, v i + o(t ² ) Definition 3. The gradient of a differentiable function ϕ :

N times z }| {

Γ × · · · × Γ ⊂ R ^3N → R is a differentiable map grad grad grad ϕ : Γ × · · · × Γ → R ^3N which assigns to each point ν = (σ 1 , . . . , σ N ) ∈ Γ ^N a vector grad grad grad ϕ(ν) ∈ (T σ

1

(Γ) × · · · × T σ

N

(Γ)) ⊂ R ^3N such that

h grad grad grad ϕ(ν ), ̟ i ν = dϕ ν (̟) ∀ ̟ ∈ (T σ

1

(Γ) × · · · × T σ

N

(Γ))

The function ∇ ∇ ∇ ^σ

i

ϕ : Γ ^N → R ³ which assigns to each point ν = (σ 1 , . . . , σ N ) ∈ Γ ^N a vector ∇ ∇ ∇ ^σ

i

ϕ(ν) ∈ T σ

i

(Γ) ⊂ R ³ is defined by

h∇ ∇ ∇ ^σ

ⁱ

ϕ(ν), ̟ i i = dϕ ν (̟) ∀ ̟ i ∈ T σ

i

(Γ)

with ̟ = (0, . . . , 0, ̟ i , 0, . . . , 0).

Definition 4. The hessian of a twice differentiable function ϕ : Γ ^N → R is the function Hess ϕ :

N times z }| {

Γ × · · · × Γ → L R ^3N

which assigns to each point ν = (σ 1 , . . . , σ N ) ∈ Γ ^N , a matrix Hess ϕ(ν ) ∈ L (T σ

1

(Γ) × · · · × T σ

N

(Γ)) such that

h Hess ϕ(ν)̟, ̟ i ν = d ² ϕ ν (̟).̟ ∀ ̟ ∈ T σ

1

(Γ) × · · · × T σ

N

(Γ).

The function H _i,j ϕ :

N times z }| {

Γ × · · · × Γ → L R ³

, 1 ≤ i 6 = j ≤ N, which assigns to each point ν = (σ 1 , . . . , σ N ) ∈ Γ ^N , a matrix H _i,j ϕ(ν) ∈ L T σ

i

(Γ), T σ

j

(Γ)

is defined by

h H _i,j ϕ(ν)ω i , ω j i = d ² ϕ ν (̟).̟ ∀ (ω i , ω j ) ∈ T σ

i

(Γ) × T σ

j

(Γ) with ̟ = (̟ 1 , . . . , ̟ N ), ̟ k = 0 for k 6 = i and k 6 = j, ̟ i = ω i and ̟ j = ω j .

The function H _i,i ϕ :

N times z }| {

Γ × · · · × Γ → L R ³

, 1 ≤ i ≤ N, which assigns to each point ν = (σ 1 , . . . , σ N ) ∈ Γ ^N , a matrix H _i,i ϕ(ν ) ∈ L (T σ

i

(Γ)) is defined by

h H _i,i ϕ(ν)ω i , ω i i = d ² ϕ ν (̟).̟ ∀ ω i ∈ T σ

i

(Γ)

with ̟ = (̟ 1 , . . . , ̟ N ) ∈ T σ

1

(Γ) × · · · × T σ

N

(Γ), ̟ k = 0 for k 6 = i, and ̟ i = ω i . 2.2. Geometrical notations.

• Let (K i ) i=1,··· ,N be a set of regular, disjoint and strictly convex compact in R ³ .

• We denote by Γ i , the boundary of K i , and Γ = S N i=1 Γ i .

• Thus Γ i is a regular and orientable surface. So, given a point σ of surface Γ i

we can choose the coordinate axis of R ³ so that origin O of the coordinates is at σ and the z axis is directed along the negative normal (i.e. the outer normal) n n n(σ) of Γ i in σ (thus, the xy plane agrees with T σ (Γ i ) : tangent plane of Γ i in σ). It follows that a neighborhooh of σ in Γ i can be represented in the form z = g i (u, v), (u, v) ∈ U ⊂ R ² , where U is an open set and g i is a differentiable function with g i (0, 0) = ^∂g _∂u

ⁱ

(0, 0) = ^∂g _∂v

ⁱ

(0, 0) = 0.

Let us assume further that the u and v axes are directed along the principal directions, with the axis u along the direction of maximum principal curvature. Thus

k ₁ ⁱ (σ) = ∂ ² g i

∂u ² (0, 0), k ₂ ⁱ (σ) = ∂ ² g i

∂v ² (0, 0), and ∂ ² g i

∂u∂v (0, 0) = 0

and, so we obtain by developing g i (u, v) into Taylor’s expansion about (0, 0) g i (u, v) = − 1

2 (k ⁱ ₁ u ² + k ⁱ ₂ v ² ) + o(u ² + v ² )

• We note R ⁱ _u (σ) = _k

i

¹

1

(σ) and R ⁱ _v (σ) = _k

i

¹

2

(σ) the principal radii of curva- ture.

• Let us denote ℜ ⁱ (σ) the orthonormal basis ℜ ⁱ (σ) = { u u u i , vvv i , n n n i } where n n n i is the negative normal of Γ i in σ, u u u i and vvv i are the principal directions of Γ i

in σ with u u u i the direction of maximum principal curvature.

• We set Ω ^′ = S N i=1

K i and we note Ω i = R ³ \ K i .

• Let M ^m,n (R) the set of real matrix of size m × n.

• We note Γ ^l _∗ =

(σ 1 , · · · , σ l ) ∈ Γ ^l with (σ j ∈ Γ p and σ j+1 ∈ Γ q ⇒ p 6 = q)

and the phase function ψ l : R ³ × Γ ^l _∗ → R defined, for all ν = σ 1 , · · · , σ l ∈ Γ ^l _∗ , by

ψ l (x; ν ) = h ξξξ, σ 1 i + X l−1 j=1

| σ j+1 − σ j | + | x − σ l |

• We call C ^l (x) the set of all l-uplet (σ 1 , · · · , σ l ) ∈ Γ ^l _∗ such that:

(1) h ξξξ, n n n(σ 1 ) i < 0 and h σ j+1 − σ j , n n n(σ j+1 ) i < 0 for all j ∈ { 1, · · · , l − 1 } , (2) The phase ψ l (x, • ) is stationary on Γ ^l _∗ at point ν = (σ 1 , · · · , σ l ) (i.e.

grad grad grad _ν ψ l (x; ν) = 0) We note C (x) = S

l C ^l (x).

• Let ν = (σ ₁ ^ν , · · · , σ ^ν _l ) ∈ C ^l (x). We note ℜ ^ν j = ℜ ^j (σ _j ^ν ) = { u u u ^ν _j , vvv ^ν _j , n n n ^ν _j } , j ∈ { 1, · · · , l } .

• Let ν = (σ ^ν ₁ , · · · , σ _l ^ν ) ∈ C ^l (x), we define ξξξ ^ν _j , for j ∈ { 1, · · · , l } by : σ ^ν _j+1 − σ ^ν _j = λ ^ν _j ξξξ ^ν _j with λ ^ν _j = | σ _j+1 ^ν − σ _j ^ν |

and note ξξξ ^ν _j = ξ _j,1 ^ν , ξ _j,2 ^ν , ξ _j,3 ^ν

ℜ

^ν_j

. We also note B(σ _j ^ν ) the curvature matrix of Γ j in σ ^ν _j : it’s the diagonal matrix with diagonal entries

1 U

_j^ν

, _V ¹

ν

j

, 0 in ℜ ^ν j where U _j ^ν = 1/k ^j ₁ (σ _j ^ν ) and V _j ^ν = 1/k ₂ ^j (σ _j ^ν ) are the principal radius of curvature. Due to strict convexity of compact we have U _j ^ν > 0 and V _j ^ν > 0.

• Let R ^ν _j defined by : R ^ν _j =





u u u ^ν _j+1 , u u u ^ν _j u u

u ^ν _j+1 , vvv ^ν _j

u u u ^ν _j+1 , n n n ^ν _j vvv ^ν _j+1 , u u u ^ν _j

vvv ^ν _j+1 , vvv ^ν _j

vvv ^ν _j+1 , n n n ^ν _j n n n ^ν _j+1 , u u u ^ν _j

n n n ^ν _j+1 , vvv ^ν _j

n n n ^ν _j+1 , n n n ^ν _j





• we call R ^l (x) the set of all l-uplet ρ = (σ 1 , · · · , σ l ) ∈ (∂Ω) ^l _∗ such that ρ is an optic ray going through x and σ i the point of i ^th reflection along this ray. We note R (x) = S

l R ^l (x).

• Let x ∈ Ω and ρ = (σ 1 , · · · , σ l ) ∈ C ^l (x). We say that ρ realize : (1) a transmission condition at point σ i (i = 1, · · · , l) if

( _σ

₂

_−σ

₁

|σ

2

−σ

1

| = ξξξ for i = 1

σ

i+1

−σ

i

|σ

i+1

−σ

i

| = _|σ ^σ

ⁱ_i

^−σ _−σ

ⁱ⁻¹_i−1

_| for i ∈ { 2, · · · , l } (2) a reflection condition at point σ i (i = 1, · · · , l) if ( σ

2

−σ

1

|σ

2

−σ

1

| = ξξξ − 2 h ξξξ, n n n(σ 1 ) i n n n(σ 1 ) for i = 1

σ

i+1

−σ

i

|σ

i+1

−σ

i

| = _|σ ^σ

ⁱ

^−σ

ⁱ⁻¹

i

−σ

i−1

| − 2 D _σ

i

−σ

i−1

|σ

i

−σ

i−1

| , n n n(σ i ) E n n

n(σ i ) for i ∈ { 2, · · · , l } Here σ l+1 = x.

• Let x ∈ Ω and ν = (σ ₁ ^ν , · · · , σ ^ν _l ) ∈ C ^l (x) we note δ ^ν (σ _j ^ν ) =

0 if σ _j ^ν is a transmission point 1 if σ _j ^ν is a reflexion point

• We note T = S

l T ^l with T ^l the set of points x ∈ Ω such that exists (σ 1 , · · · , σ l ) ∈ (∂Ω) ^l _∗ verifying

(1) h ξξξ, n n n(σ 1 ) i = 0 or ∃ j ∈ { 1, · · · , l − 1 } such that h σ j+1 − σ j , n n n(σ j+1 ) i = 0,

(2) The phase ψ l (x, • ) is stationary on (∂Ω) ^l _∗ in (σ 1 , · · · , σ l ).

2.3. Matrix applications.

• Let σ > 0, we note S _σ ⊂ M ^3,3 ( R ) the set of matrix A such that I + σ A is regular. We note S σ the following application :

S σ : S _σ −→ M ^3,3 (R) A 7−→ A(I + σA) ⁻¹

• Let B ∈ M ^3,3 (R) a symmetric matrix , ηηη ∈ R ³ , ζζζ ∈ R ³ . We suppose h ζζζ, η i 6 = 0. We note T _B ,η η η,ζ ζζ the application of M ^3,3 (R) given by :

∀ A ∈ M ^3,3 (R), ∀ x ∈ R ³

(T _B ,η η η,ζ ζζ (A))x = (A − 2 h ζζζ, ηηη i B)x − 2 h ηηη, x i (Aηηη + Bζζζ )

− 2 h A ηηη + B ζζζ, x i ηηη + 2 h

2 h A ηηη, ηηη i − ^hBζ hζ ζζ,η ^ζζ,ζζζi η ηi

i h ηηη, x i ηηη

• we define, for x ∈ Ω \ T and ν = (σ ₁ ^ν , · · · , σ ^ν _l ) ∈ C ^l (x), the following l matrices M ^ν _j in M ^2,2 ( R ):

M ^ν ₁ = H _1,1 ψ l (x, ν ) and, ∀ j ∈ { 2, · · · , l }

M ^ν _j = H _j,j ψ l (x; ν) − H _j−1,j ψ l (x; ν)

M ^ν _j−1 −1

H _j,j−1 ψ l (x; ν).

Remark 1. We shall see in Lemma 3 that M ^ν _j is regular.

• Let x ∈ Ω \ T and ν = (σ ^ν ₁ , · · · , σ _l ^ν ) ∈ C ^l (x). We define by recurrence the l symmetric matrices P ^ν _j in L ( R ³ ) such that

P ^ν ₁ = T _B(σ

₁^ν

),n(σ

₁^ν

),ξ (0) × δ ^ν (σ ^ν ₁ ) and, ∀ j ∈ { 2, · · · , l }

P ^ν _j = S λ

^ν_j−1

( P ^ν _j−1 ) × 1 − δ ^ν (σ _j ^ν )

+ T _B(σ

^ν_j

),n(σ

^ν_j

),ξ

^ν_j−1

S λ

^ν_j−1

( P ^ν _j−1 )

× δ ^ν (σ ^ν _j ).

• Let D ^l the function define on Γ ^l _∗ by D ^l (σ 1 , · · · , σ l )

= ( |h ξξξ, n n n(σ 1 ) i| − h ξξξ, n n n(σ 1 ) i )

l−1 Q

j=1 h

D

_σj+1−σj

|σj+1−σj|

,n n n(σ

j+1

) E

− D

_σj+1−σj

|σj+1−σj|

,n n n(σ

j+1

) Ei

|σ

j+1

−σ

j

|

• Let N ^l the function define on Γ ^l _∗ by N ^l (σ 1 , · · · , σ l )

= hξ ξξ,n n n(σ

1

)i

|hξ ξξ,n n n(σ

1

)i| − 1 l−1

Q

j=1

" D

_σj+1−σj

|σj+1−σj|

,n n n(σ

j+1

) E

D

_σj+1−σj

|σj+1−σj|

,n n n(σ

j+1

) E

− 1

# D

σ

j+1

−σ

j

|σ

j+1

−σ

j

| , n n n(σ j ) E 3. Geometrical optics approximation

We only give the main results. The geometrical optic approximation is given, for Dirichlet and Neumann problems, respectively by : ∀ x ∈ Ω \ T

(3.1)

v ^D _O.G. (x)

= e ^{−ikhξ ξξ,xi} + P

l≥1

( − 1) ^l P

ρ=(σ

^ρ₁

,··· ,σ

^ρ_l

)∈R

l

(x)

e

^{−ikψl(x;ρ)}

l

Q

j=1

√ _det(

I +|σ

^ρ_j+1

−σ

_j^ρ

| P

^ρj

)

and

(3.2)

v ^N _O.G. (x)

= e ^{−ikhξ ξξ,xi} + P

l≥1

P

ρ=(σ

₁^ρ

,···,σ

^ρ_l

)∈R

l

(x)

e

^{−ikψl(x;ρ)}

l

Q

j=1

√ _det(I+|σ

ρ

j+1

−σ

^ρ_j

|P

^ρj

)

Here σ l+1 = x, det I+ | σ _j+1 ^ρ − σ ^ρ _j | P _j ^ρ

is positive, and in corollary, the Maslov indice vanished. For more explanation, report to [eC89] or [Cuv13].

4. Iterative Kirchhoff approximation 4.1. Dirichlet problem. We introduce the following kernels series

p ^D ₁ (σ) = ik ( |h ξ, n(σ) i| − h ξ, n(σ) i ) e ^−ikhξ,σi ∀ σ ∈ ∂Ω and, for σ ∈ ∂K j

p ^D _l (σ) = ik 4π

Z

∂Ω\∂K

j

p ^D _l−1 (σ ^′ )

σ − σ ^′

| σ − σ ^′ | .n(σ)

− σ − σ ^′

| σ − σ ^′ | .n(σ)

e ^−ik|σ

^′

^−σ|

| σ ^′ − σ | dσ ^′ We set the iterative Kirchhoff approximation for Dirichlet problem (D) by (4.1)

( u ^D ₀ (x) = 0

u ^D _l (x) = u ^D _l−1 (x) + _4π ¹ R

∂Ω p ^D _l (σ) ^e

^{−ik|x−σ|}

_|x−σ| dσ That is to say with previous notations

(4.2) u ^D _l (x) = u ^D _l−1 (x) + ik

4π l Z

(∂Ω)

^l_∗

D ^l (σ 1 , · · · , σ l ) e ^−ikψ

^l

^(x;σ

¹

^{,··· ,σ}

^l

⁾

| x − σ l | dσ 1 · · · dσ l

We state the main result comparing the iterative method describe in (4.1) and the geometrical optic approximation given in (3.1) for problem (D) :

Theorem 1. Let Ω an open in R ³ , exterior of a regular domain Ω ^′ finite and disjointed reunion of strictly convex compacts. Let x ∈ Ω \ T . If C ^l (x) = ∅ for l > n then

(4.3) e ^−ikhξ,xi + u ^D n (x) − v 0.G. ^D (x) = O 1

k

locally uniformly in x.

4.2. Neumann problem. We introduce the following kernels series p ^N ₁ (σ) = ik

h ξ, n(σ) i

|h ξ, n(σ) i| − 1

e ^−ikhξ,σi ∀ σ ∈ ∂Ω and, for σ ∈ ∂K j

p ^N _l (σ) = ik 4π

Z

∂Ω\∂K

j

p ^N _l−1 (σ ^′ )



 D σ−σ

^′

|σ−σ

^′

| , n(σ) E

D σ−σ

^′

|σ−σ

^′

| , n(σ) E

− 1





×

σ − σ ^′

| σ − σ ^′ | , n(σ ^′ )

e ^−ik|σ

^′

^−σ|

| σ ^′ − σ | dσ ^′

We set the iterative Kirchhoff approximation for Neumann problem (N) by (4.4)

( u ^N ₀ (x) = 0

u ^N _l (x) = u ^N _l−1 (x) + _4π ¹ R

∂Ω p ^N _l (σ) D

x−σ

|x−σ| , n(σ) E

e

^{−ik|x−σ|}

|x−σ| dσ

That is to say with previous notations (4.5)

u ^N _l (x) − u ^N _l−1 (x)

=

ik 4π

^l R

(∂Ω)

^l_∗

N ^l (σ 1 , · · · , σ l ) D

x−σ

l

|x−σ

l

| , n(σ l ) E

e

^{−ikψl(x;σ1,···,σl)}

|x−σ

l

| dσ 1 · · · dσ l

We state the main result comparing the iterative method describe in (4.4) and the geometrical optic approximation given in (3.2) for problem (N ) :

Theorem 2. Let Ω an open in R ³ , exterior of a regular domain Ω ^′ finite and disjointed reunion of strictly convex compacts. Let x ∈ Ω \ T . If C ^l (x) = ∅ for l > n then

(4.6) e ^−ikhξ,xi + u ^N _n (x) − v ^N _0.G. (x) = O 1

k

locally uniformly in x.

5. Technical Lemmas and properties

To prove previous theorems we need some technical lemmas and properties.

5.1. Stationary phase points of ψ l (x; • ). We first remark that Remark 2. Let ν = (σ ^ν ₁ , · · · , σ _l ^ν ) ∈ (∂Ω) ^l _∗ , if

h ξ, n(σ ₁ ^ν ) i ≥ 0 or if exists j ∈ { 1, · · · , l − 1 } such that

* σ _j+1 ^ν − σ _j ^ν

| σ _j+1 ^ν − σ _j ^ν | , n(σ _j+1 ^ν ) +

≥ 0 then

D ^l (σ ₁ ^ν , · · · , σ ^ν _l ) = N ^l (σ ₁ ^ν , · · · , σ ^ν _l ) = 0

To find stationary phase points on (∂Ω) ^l _∗ , we have to compute, for all x ∈ R ³ , the set of points ν = (σ ₁ ^ν , · · · , σ _l ^ν ) ∈ (∂Ω) ^l _∗ which satisfies

(5.1) ∇ ^S ν ψ l (x; σ ₁ ^ν , · · · , σ _l ^ν ) = 0 We have

∇ ^ν ψ l (x; σ ^ν 1 , · · · , σ l ^ν ) =



 

 

 



ξ − |σ ^σ

^2ν2

^−σ −σ

^ν11

| σ

^ν₂

−σ

^ν₁

| ^σ

^ν2

^−σ

^ν1

| −

σ

₃^ν

−σ

₂^ν

| ^σ

^3ν

^−σ

^2ν

| .. .

σ

^ν_l

−σ

^ν_l−1

| ^σ

^νl

^−σ

^νl−1

| −

x−σ

_l^ν

| ^x−σ

^lν

|



 

 

 



The condition (5.1) is equivalent to the existence of (µ 1 , · · · , µ l ) ∈ R ^l such that

∇ ^ν ψ l (x; σ ₁ ^ν , · · · , σ ^ν _l ) =



 

µ 1 n(σ ^ν ₁ ) .. . µ l n(σ ^ν _l )



 

By hypothesis | ξ | = 1, so we obtain



 



 



µ 1 = 0 or µ 1 = 2 h ξ, n(σ ^ν ₁ ) i n(σ ₁ ^ν ) and ∀ j ∈ { 2, · · · , l }

µ j = 0 or µ j = 2

σ

_j^ν

−σ

^ν_j−1

| ^σ

^jν

^−σ

^νj−1

| , n(σ _j ^ν )

n(σ _j ^ν )

Then, using remark 2, we have the

Lemma 1. Let x ∈ Ω \ T and ν = (σ ₁ ^ν , · · · , σ _l ^ν ) ∈ C ^l (x). Then ν realize a trans- mission or a reflection condition on each points σ ^ν _j , j ∈ { 1, . . . , l } .

5.2. Relation between M ^ν _j and P ^ν _j . We have the fundamental Lemma

Lemma 2. Let x ∈ Ω \ T and ν = (σ ^ν ₁ , · · · , σ _l ^ν ) ∈ C l (x). Then, ∀ j ∈ { 1, · · · , l } we have :

(5.2) M ^ν _j = P ^ν _j

|T

∂Ω

(σ

^ν_j

) + 1 λ ^ν _j

I − (ξ _j ^{ν t} ξ _j ^ν ) |T

∂Ω

(σ

_j^ν

)

,

(5.3) det M ^ν _j =

ξ _j ^ν , n(σ _j ^ν ) λ ^ν _j

! 2

det(I + λ ^ν _j P ^ν _j ),

(5.4) sgn M ^ν _j = 2,

and

(5.5) M ^ν _j inversible.

Proof of lemma 2 :

The proof of the lemma worked out by recurrence.

step one of recurrence proof Using definition of P ^ν ₁ , we easily obtain

P ^ν _1|T

∂Ω

(σ

^ν₁

) = 2 h ξ ₁ ^ν , n(σ ₁ ^ν ) i δ ^ν (σ ₁ ^ν )B(σ ₁ ^ν ) _|T

_∂Ω

_(σ

₁^ν

₎ On the other hand, we have

M ^ν ₁ = H _σ ^S

₁

_,σ

₁

ψ l (x; ν)

= H σ ^S

1

,σ

1

( h ξ, σ ^ν 1 i + | σ ^ν 2 − σ 1 ^ν | )

= 2 h ξ ^ν ₁ , n(σ ^ν ₁ ) i δ ^ν (σ ₁ ^ν )B(σ ₁ ^ν ) |T

∂Ω

(σ

^ν₁

) + _λ ¹

ν

1

I − (ξ ₁ ^{ν t} ξ ₁ ^ν ) |T

∂Ω

(σ

₁^ν

)

and so we have proved formula (5.2) for j = 1.

To obtain formula (5.3), we first remark that P ^ν ₁ ξ ₁ ^ν = 0. We set H _j =





1 0 ξ _j,1 ^ν 0 1 ξ _j,2 ^ν 0 0 ξ _j,3 ^ν





Then, combining | ξ ₁ ^ν | = 1 and formula (5.2) gives det(I + λ ^ν ₁ P ^ν ₁ ) = _(det ¹ _H

1

)

²

det ( ^t H ₁ (I + λ ^ν ₁ P ^ν ₁ ) H ₁ )

=

λ

^ν₁

h ^ξ

^1ν

^,n(σ

^ν1

⁾ i 2

det M ^ν ₁ .

One finds easily that det(I + λ ^ν ₁ P ^ν ₁ ) > 0 and tr M ^ν ₁ > 0. So, we get sgn M ^ν ₁ = 2.

We deduce that det M ^ν ₁ > 0 and so M ^ν ₁ is regular.

Step j + 1 of recurrence proof

To prove formula (5.2) at step j + 1, we first have to compute (P ^ν _j+1 ) ℜ

j+1

and M ^ν _j+1 respectively in function of (P ^ν _j ) ℜ

^ν_j

and M ^ν _j . To simplify notations, we note (P ^ν _j ) ℜ

^ν_j

= [P r,s ] r,s∈{1,2,3} , and M ^ν _j = [M r,s ] _r,s∈{1,2} .

• Computation of (P ^ν _j+1 ) ℜ

j+1

By definition,

(P ^ν _j+1 ) ℜ

^ν_j+1

=

S λ

^ν_j

( P ^ν _j )

ℜ

^ν_j+1

× 1 − δ ^ν (σ _j+1 ^ν )

+ T _B(σ

^ν_j+1

),n(σ

^ν_j+1

),ξ

^ν_j

S λ

^ν_j

( P ^ν _j )

ℜ

^ν_j+1

× δ ^ν (σ _j+1 ^ν ).

We first evaluate S λ

^ν_j

( P ^ν _j ) ℜ

j

:

(S λ

^ν_j

(P ^ν _j )) ℜ

j

det(I + λ ^ν _j P ^ν _j ) = (P ^ν _j ) ℜ

j

+ λ ^ν _j Q ^ν _j + λ ^ν _j 2

I det P ^ν _j with Q ^ν _j = [Q r,s ] p,q∈{1,2,3} ∈ M ^3,3 (R) and

Q 11 = P 11 (P 33 + P 22 ) − P ₁₂ ² − P ₁₃ ² , Q 12 = P 12 P 33 − P 13 P 23 ,

Q 13 = P 13 P 22 − P 12 P 23 ,

Q 22 = P 22 (P 33 + P 11 ) − P ₁₂ ² − P ₂₃ ² , Q 23 = P 23 P 11 − P 12 P 13 ,

Q 33 = P 33 (P 22 + P 11 ) − P ₁₃ ² − P ₂₃ ² . As det P ^ν _j = 0, we obtain :

(S λ

^ν_j

(P ^ν _j )) ℜ

^ν_j

= 1 det(I + λ ^ν _j P ^ν _j )

(P ^ν _j ) ℜ

^ν_j

+ λ ^ν _j Q ^ν _j and

(S λ

^ν_j

(P ^ν _j )) ℜ

^ν_j+1

= _{det( I +λ} ¹

ν

j

P

^νj

) R ^ν _j (P ^ν _j ) ℜ

j

+ λ ^ν _j Q ^ν _{j t} R ^ν _j . Now, we have to compute T _B(σ

_j+1^ν

),n(σ

^ν_j+1

),ξ

^ν_j

S λ

^ν_j

( P ^ν _j )

ℜ

^ν_j+1

. In local coordinates, we have B(σ ^ν _j+1 ) =



 

1

U

_j+1^ν

0 0 0 _V ¹

ν

j+1

0

0 0 0



 

ℜ

^ν_j+1

, n(σ _j+1 ^ν ) ℜ

^ν_j+1

=



 0 0 1



 and ξ _j+1 ^ν

ℜ

^ν_j+1

=



 ξ _j+1,1 ^ν ξ _j+1,2 ^ν ξ _j+1,3 ^ν



 . So, if we note W =

(w pq ) p,q∈{1,2,3}

=

(S λ

^ν_j

(P ^ν _j )) ℜ

^ν_j+1

and T =

(t pq ) p,q∈{1,2,3}

=

T _B (σ

^ν_j+1

),n(σ

^ν_j+1

),ξ

^ν_j

S λ

^ν_j

(P ^ν _j )

ℜ

^ν_j+1

then, ∀ x ∈ R ³

T _B(σ

^ν_j+1

_),n(σ

^ν_j+1

_),ξ

^ν_j

S λ

^ν_j

(P ^ν _j )

ℜ

^ν_j+1

x

=



 



w 11 − 2 h ^ξ

^jν

^,n(σ

^νj+1

⁾ i

U

_j+1^ν

w 12 w 13

w 12 w 22 − 2 h ^ξ

^νj

^,n(σ

^νj+1

⁾ i

V

_j+1^ν

w 23

w 13 w 23 w 33



 





 x 1

x 2

x 3





− 2x 3



 



w 13 + h ^ξ

^jν

^,u

^νj+1

i

U

_j+1^ν

w 23 + h ^ξ

^jν

^{, v}

^νj+1

i

V

_j+1^ν

w 33



 



− 2



 





 



w 13 + h ^ξ

^νj

^,u

^νj+1

i

U

_j+1^ν

w 23 + ^ξ

ν j,2

V

_j+1^ν

w 33



 

 ,



 x 1

x 2

x 3







 





 0 0 1





+4







 w 13

w 23

w 33



 ,



 0 0 1







 x 3



 0 0 1





− h ^ξ

^jν

^{,n(σ 2}

^νj+1

⁾ i

h ^ξ

^νj

^{, u}

^νj+1

i

²

U

_j+1^ν

+ h ^ξ

^jν

^{, v}

^νj+1

i

²

V

_j+1^ν

x 3



 0 0 1





thus

t 11 = w 11 − 2 h ^ξ

^νj

^,n(σ

^j+1ν

⁾ i

U

_j+1^ν

, t 12 = w 12 ,

t 13 = − w 13 − 2 h ^ξ

^jν

^{, u}

^νj+1

i

U

_j+1^ν

, t 22 = w 22 − 2 h ^ξ

^νj

^,n(σ

^j+1ν

⁾ i

V

_j+1^ν

, t 23 = − w 23 − 2 h ^ξ

^jν

^{, v}

^νj+1

i

V

_j+1^ν

, t 33 = w 33 − h ^ξ

^jν

^{,n(σ 2}

^νj+1

⁾ i

h ^ξ

^νj

^{, u}

^νj+1

i

²

U

_j+1^ν

+ h ^ξ

^νj

^{, v}

^νj+1

i

²

V

_j+1^ν

. Using

ξ _j ^ν , n(σ _j+1 ^ν )

= 1 − 2δ ^ν (σ _j+1 ^ν ) ξ _j+1 ^ν , n(σ ^ν _j+1 )

, we obtain : T _B (σ

_j+1^ν

),n(σ

^ν_j+1

),ξ

^ν_j

S λ

^ν_j

(P ^ν _j )

| ^T

^∂Ω

^(σ

^j+1ν

⁾ =

(S λ

^ν_j

( P ^ν _j )) ℜ

j+1

| ^T

^∂Ω

^(σ

^νj+1

⁾ − 2 1 − 2δ ^ν (σ _j+1 ^ν ) ξ ^ν _j+1 , n(σ _j+1 ^ν )

B (σ _j+1 ^ν ) | ^T

^∂Ω

^(σ

^j+1ν

⁾ Finally, we have

(5.6)

(P ^ν _j+1 ) ℜ

j+1

| ^T

^∂Ω

^(σ

^ν1+1

⁾ =

(S λ

^ν_j

( P ^ν _j )) ℜ

j+1

| ^T

^∂Ω

^(σ

^νj+1

⁾ +2δ ^ν (σ ^ν _j+1 )

ξ _j+1 ^ν , n(σ _j+1 ^ν )

B (σ _j+1 ^ν ) | ^T

^∂Ω

^(σ

^νj+1

⁾

• Computation of M ^ν _j+1 By definition,

M ^ν _j+1 = H _σ ^S

^ν_j+1

_,σ

^ν_j+1

ψ l (x, ν ) − H _σ ^S

^ν_j

_,σ

_j+1^ν

ψ l (x, ν) M ^ν _j −1

H _σ ^S

_j+1^ν

_,σ

_j^ν

ψ l (x, ν ) We first evaluate H _σ ^S

^ν

j+1

,σ

_j^ν

ψ l (x; ν )[M ^ν _j ] ⁻¹ H _σ ^S

^ν

j

,σ

^ν_j+1

ψ l (x; ν ). By construction of ψ l , we have :

H _σ ^S

_j+1^ν

_,σ

^ν_j

ψ l (x; ν) = H _σ ^S

_j+1^ν

_,σ

_j^ν

| σ _j+1 ^ν − σ _j ^ν | . That’s give in ℜ ^j

H _σ ^S

^ν

j+1

,σ

_j^ν

ψ l (x; ν ) = 1 λ ^ν _j L ^ν _j where L ^ν _j = [L p,q ]

p,q∈{1,2}

with L 11 =

u ^ν _j , ξ _j ^ν u ^ν _j+1 , ξ ^ν _j

−

u ^ν _j , u ^ν _j+1 , L 12 =

v ^ν _j , ξ ^ν _j u ^ν _j+1 , ξ _j ^ν

−

v ^ν _j , u ^ν _j+1 , L 21 =

u ^ν _j , ξ _j ^ν v ^ν _j+1 , ξ _j ^ν

−

u ^ν _j , v _j+1 ^ν , L 22 =

v ^ν j , ξ ^ν j v j+1 ^ν , ξ j ^ν

−

v ^ν j , v ^ν j+1

. We have by hypothesis M ^ν _j = P ^ν _j

|T

∂Ω

(σ

_j^ν

) + _λ ¹

ν j

I − (ξ _j ^{ν t} ξ _j ^ν )

|T

∂Ω

(σ

^ν_j

) i.e. : M ^ν _j

ℜν j

= [M p,q ] _p,q∈{1,2}

=

P 11 P 12

P 12 P 22

+ 1

λ ^ν _j

"

1 − ξ _j,1 ^ν 2

− ξ ^ν _j,1 ξ _j,2 ^ν

− ξ _j,1 ^ν ξ ^ν _j,2 1 − ξ ^ν _j,2 2

#

By recurrence hypothesis M ^ν _j is regular, so we have : H _σ ^S

^ν

j+1

,σ

^ν_j

ψ l (x; ν)[M ^ν _j ] ⁻¹ H _σ ^S

^ν

j

,σ

^ν_j+1

ψ l (x; ν) = 1

λ ^ν _j 2 L ^ν _j [M ^ν _j ] ^{−1 t} L ^ν _j . Now, we evaluate H _σ ^S

_j+1^ν

_,σ

_j+1^ν

ψ l (x; ν) in ℜ ^j+1 . We find

H _σ ^S

^ν

j+1

,σ

^ν_j+1

ψ l (x; ν )

= H _σ ^S

^ν

j+1

,σ

^ν_j+1

| σ _j+1 ^ν − σ _j ^ν | + | σ _j+2 ^ν − σ ^ν _j+1 |

=

1

λ

^ν_j

I − ξ _j ^{ν t} ξ ^ν _j

| ^T

^∂Ω

^(σ

^νj+1

⁾ + _λ

ν

¹

j+1

I − ξ _j+1 ^{ν t} ξ _j+1 ^ν

| ^T

^∂Ω

^(σ

^νj+1

⁾

− 2δ ^ν (σ _j+1 ^ν )

n(σ _j+1 ^ν ), ξ _j ^ν

B (σ ^ν _j+1 ) | ^T

^∂Ω

^(σ

^νj+1

⁾ . Finally, we obtain

(5.7)

M ^ν _j+1

=

1

λ

^ν_j

I − ξ ^ν _j ^t ξ _j ^ν

| ^T

^∂Ω

^(σ

^νj+1

⁾ + _λ

ν

¹

j+1

I − ξ ^ν _j+1 ^t ξ _j+1 ^ν

| ^T

^∂Ω

^(σ

^νj+1

⁾

− 2δ ^ν (σ j+1 )

n(σ ^ν _j+1 ), ξ _j ^ν

B (σ j+1 ) | ^T

^∂Ω

^(σ

^j+1ν

⁾ − ( ^{λ 1}

^νj

)

²

L ^ν _j [ M ^ν _j ] ^{−1 t} L ^ν _j .

• Formula (5.2) at step j + 1 : We compute now D ^ν _j+1 = P ^ν _j+1

| ^T

^∂Ω

^(σ

^j+1ν

⁾ + _λ

ν

¹

j+1

I − (ξ j+1 t ξ j+1 ) | ^T

^∂Ω

^(σ

^νj+1

⁾ − M ^ν _j+1 . Combining formulas (5.6) et (5.7), we obtain

D ^ν _j+1 =

(S λ

^ν_j

(P ^ν _j )) ℜ

j+1

| ^T

^∂Ω

^(σ

^νj+1

⁾ + _λ ¹

ν

j

I − ξ _j ^{ν t} ξ ^ν _j

| ^T

^∂Ω

^(σ

^j+1ν

⁾ + ¹

( ^λ

^νj

)

²

L ^ν _j [M ^ν _j ] ^{−1 t} L ^ν _j . We prove that D ^ν _j+1 = 0 using relations P ^ν _j ξ _j ^ν = 0, | ξ _j ^ν | = 1 and ^t R ^ν _j R ^ν _j = I.

• Formula (5.3) at step j + 1 :

We remark that P ^ν _j+1 ξ ^ν _j+1 = 0. Then, we pose

H ^ν _j+1 =





1 0 ξ _j+1,1 ^ν 0 1 ξ _j+1,2 ^ν 0 0 ξ _j+1,3 ^ν





and we obtain det I + λ ^ν _j+1 P ^ν _j+1

= 1

(det H ^ν _j+1 ) ² det ^t H ^ν _j+1 I + λ ^ν _j+1 P ^ν _j+1 H ^ν _j+1

. So

ξ _j+1 ^ν , n(σ _j+1 ^ν ) 2

det I + λ ^ν _j+1 P ^ν _j+1

= det





1 + λ ^ν _j+1

P ^ν _j+1 u ^ν _j+1 , u ^ν _j+1

λ ^ν _j+1

P ^ν _j+1 u ^ν _j+1 , v ^ν _j+1

ξ ^ν _j+1 , u ^ν _j+1 λ ^ν _j+1

P ^ν _j+1 u ^ν _j+1 , v ^ν _j+1

1 + λ ^ν _j+1

P ^ν _j+1 v _j+1 ^ν , v _j+1 ^ν

ξ _j+1 ^ν , v ^ν _j+1 ξ _j+1 ^ν , u ^ν _j+1

ξ ^ν _j+1 , v ^ν _j+1

1



 .

As | ξ _j+1 ^ν | = 1, we get, with formula (5.2) at step j + 1:

det I + λ ^ν _j+1 P ^ν _j+1

= λ ^ν _j+1 ξ _j+1 ^ν , n(σ _j+1 ^ν )

! 2

det M ^ν _j+1 .

• Formulas (5.4) and (5.5) at step j + 1 : In fact, we proved that

∀ λ > 0 det(I + λP ^ρ _j+1 ) = λ ξ ^ρ _j+1 , n(σ _j+1 ^ρ )

! 2

det M ^ρ _j+1 where ρ = (σ ₁ ^ν , · · · , σ ^ν _j+1 ) ∈ C ^j+1 (λξ ^ν _j+1 ). Moreover, we have :

P ^ρ _j+1 = P ^ν _j+1 , for λ = λ ^ν _j+1 ,

M ^ρ _j+1 = M ^ν _j+1 and ∀ λ > 0

det(I + λP ^ρ _j+1 ) > 0 because P ^ρ _j+1 is a positive matrix. That’s give

∀ λ > 0, det M ^ρ _j+1 ≥ 0.

This quantity is positive for λ in a neighborhood of zero, then by continuity we have :

∀ λ > 0 tr M ^ρ _j+1 > 0 with

tr M ^ρ _j+1 = P 11 + P 22 + 1 +

ξ _j+1 ^ν , n(σ _j+1 ^ν ) 2

λ .

Thus, we obtain formula (5.4) to step j + 1.

We have also showed that M ^ν _j+1 is regular.

That’s close the proof of Lemma 22 5.3. Lemma of transmission.

Lemma 3 (of transmission). Let x ∈ Ω \ T and ν = (σ ^ν ₁ , · · · , σ ^ν _l ) ∈ C ^l (x) (1) if exists t ∈ R ⁺ _∗ such that

σ = σ ^ν ₁ − tξ ∈ ∂Ω and ξ.n(σ) < 0 then

µ = (σ, σ ₁ ^ν , · · · , σ ^ν _l ) ∈ C ^l+1 (x) (2) if exists j ∈ { 1, · · · , l − 1 } such that

σ ∈

]σ ^ν _j ; σ _j+1 ^ν [ ∩ ∂Ω and (σ − σ ^ν _j ).n(σ) < 0 then

µ = (σ ₁ ^ν , · · · , σ _j ^ν , σ, σ _j+1 ^ν , · · · , σ _l ^ν ) ∈ C ^l+1 (x) Noting µ = (σ ₁ ^µ , · · · , σ _l+1 ^µ ), we obtain in both cases

(5.8)

( − 1) ^{l e}

^{−ikψl(x;ν)}

l

Q

j=1

det ( I+|σ

j+1^ν

−σ

^ν_j

|P

^νj

)

1/2

+( − 1) ^{l+1 e}

^{−ikψl+1 (x;µ)}

l+1

Q

j=1

det ( I+|σ

^µj+1

−σ

_j^µ

|P

^µj

)

1/2

= 0 Proof of Lemma 3 :

In both case, using the strict convexity of compacts (K j ) j∈{1,··· ,N} , we obtain µ ∈ C ^l+1 (x).

Remark 3. In two cases, µ realize a transmission condition in σ, and thus

ψ l (x; ν ) = ψ l+1 (x; µ)

Now, we prove the formula (5.8), in both case. In fact, we only have to prove that

Y l

j=1

det I + | σ ^ν _j+1 − σ _j ^ν | P ^ν _j

1/2

=





l+1 Y

j=1

det I + | σ _j+1 ^µ − σ ^µ _j | P ^µ _j





1/2

Here σ _l+1 ^ν = σ _l+2 ^µ = x.

In the first case, the proof is immediate.

Under the hypothesis of the second case, we have

∀ j ∈ { 1, · · · , j } P ^ν _j = P ^µ _j and, µ realize a transmission condition in σ, hence

P ^µ _j+1 = S |σ

_j^ν

−σ| (P ^ν _j ) = P ^ν _j I+ | σ ^ν _j − σ | P ^ν _j −1

. Thus, we obtain

I+ | σ _j+1 ^ν − σ | P ^µ _j+1

I+ | σ ^ν _j − σ | P ^µ _j

= I+ | σ ^ν _j+1 − σ ^ν _j | P ^ν _j . Taking determinant of previous formula, we get

det I+ | σ _j+1 ^ν − σ | P ^µ _j+1

det I+ | σ _j ^ν − σ | P ^µ _j

= det I+ | σ _j+1 ^ν − σ _j ^ν | P ^ν _j . Moreover, we have

S |σ

_j+2^ν

−σ

^µ_j+1

| ( P ^µ _j+1 ) = S |σ

^ν_j+1

−σ|

S |σ

_j^ν

−σ| ( P ^ν _j ) As µ realize a transmission condition in σ we obtain

| σ _j+1 ^ν − σ | + | σ ^ν _j+1 − σ | = | σ _j+2 ^ν − σ ^µ _j+1 | thus

S |σ

^ν_j+2

−σ

_j+1^µ

| (P ^µ _j+1 ) = S |σ

^ν_j+1

−σ

_j^ν

| (P ^ν _j ).

Then, we have

∀ i ∈ { j + 1, · · · , l } P ^ν _i = P ^µ _i+1 That’s close proof of Lemma 3. 2

6. Proof of Theorem 1

To proof this theorem, we apply stationary phase technics to the formula (4.1) and compare the result to the geometrical optic approximation.

6.1. Stationary phase lemma. Due to (5.5) (M ^ν _j regular) we can apply the iter- ative stationary phase lemma to u ^D _l (x) (see [Cuv94]) : ∀ x ∈ Ω \ T

(6.1)

u ^D _l (x) − u ^D _l−1 (x)

=

ik 4π

l 2π k

l P

ν∈C

l

(x) e

i π4 l P j=1

sgnMν j

l

Q

j=1

det M

^νj

1/2

D ^l (ν) ^e

⁻

_|x−σ

^ikψl(x;ν)ν

l

| + O( _k ¹ ).

Thus, using definition of D ^l (ν) with ν ∈ C ^l (x), we obtain D ^l (ν) = 2 ^l

|h ξ, n(σ ^ν ₁ ) i|

l−1 Y

j=1

D _σ

ν j+1

−σ

^ν_j

|σ

^ν_j+1

−σ

^ν_j

| ,n(σ _j+1 ^ν ) E

| σ _j+1 ^ν − σ _j ^ν | Due to formula 5.3 and 5.4 (lemma 2), we have

X l j=1

sgn M ^ν _j = 2 ^l

and

Y l

j=1

det M ^ν _j

1/2

= Y l

j=1

ξ ^ν _j , n(σ ^ν _j ) λ ^ν _j

det(I + λ ^ν _j P ^ν _j ) ^1/2

= Y l

j=1

σ

^ν_j+1

−σ

^ν_j

| ^σ

^νj+1

^−σ

^νj

| , n(σ ^ν _j )

σ ^ν _j+1 − σ ^ν _j

det( I +

σ _j+1 ^ν − σ _j ^ν P ^ν _j )

^1/2

with σ _l+1 ^ν = x. Taking into account that, for ν ∈ C ^l (x), we have

σ ^ν ₂ − σ ₁ ^ν

| σ ^ν ₂ − σ ₁ ^ν | , n(σ ₁ ^ν )

= |h ξ, n(σ ₁ ^ν ) i| ,

* σ ^ν _j+1 − σ ^ν _j σ ^ν _j+1 − σ ^ν _j

, n(σ ^ν _j ) +

=

* σ _j ^ν − σ _j−1 ^ν σ _j ^ν − σ _j−1 ^ν

, n(σ _j ^ν ) +

, j = 2, . . . , l, we find that

(6.2)

u ^D _l (x) − u ^D _l−1 (x)

= ( − 1) ^l P

ν=(σ

^ν₁

,··· ,σ

_l^ν

)∈C

l

(x)

e

^{−ikψl(x;ν)}

l−1

Q

j=1

det ( I +|σ

^ν_j+1

−σ

_j^ν

| P

^νj

)

!

det ( I +|x−σ

^ν_l

| P

^νl

)

1/2

+ O( _k ¹ )

6.2. Comparison with geometrical optic approximation. To compare the previous formula with geometrical optic approximation given by formula (3.1), we have to study the contributions of the sets C ^l (x) and R ^l (x). We clearly have R ^l (x) ⊂ C ^l (x). Using lemma 3 we obtain

Remark 4. The contributions of points in C (x) \ R (x) cancel each other.

Let x ∈ Ω \ T . Suppose that C ^l (x) = ∅ for l > n, we conclude the proof of Theorem 1 using the following remark

Remark 5. The only components of C (x) having a real contribution are :

• all ν = (σ ^ν ₁ ) ∈ C ¹ (x) coming through x such that ν realize a transmission condition in σ ₁ ^ν and

∀ t > 0 σ ₁ ^ν − tξ ∈ Ω

• all ν = (σ ₁ ^ν , · · · , σ ^ν _j ) ∈ R ^j (x) coming through x (j ≤ n).

That’s close proof of Theorem 1.2

7. Proof of Theorem 2

To proof this theorem, we apply stationary phase technics to the formula (4.4) and compare the result to the geometrical optic approximation (3.2).

Due to (5.5) ( M ^ν _j regular) we can apply the iterative stationary phase lemma to u ^N _l (x)(see [Cuv94]),we obtain ∀ x ∈ Ω \ T

(7.1)

u ^N _l (x) − u ^N _l−1 (x)

=

ik 4π

^l _2π

k

l P

ν∈C

l

(x) e

i π4 l P j=1

sgnMν j

l

Q

j=1

det M

^νj

1/2

N ^l (ν ) D _x−σ

ν l

|x−σ

^ν_l

| , n(σ ^ν _l ) E

e

^{−ikψl(x;ν)}

|x−σ

^ν_l

| + O( ¹ _k )

Thus, we use (5.3), (5.4) and the definition of N ^l to get :

(7.2)

u ^N _l (x) − u ^N _l−1 (x)

= P

ν=(σ

₁^ν

,··· ,σ

_l^ν

)∈C

l

(x)

e

^{−ikψl(x;ν)}

l−1

Q

j=1

det ( I+|σ

j+1^ν

−σ

^ν_j

|P

^νj

)

!

det ( I+|x−σ

_l^ν

|P

^ν_l

)

1/2

+ O( ¹ _k )

To compare the previous formula with geometrical optic approximation given by formula (3.2), we use previous results from the proof of theorem 1.

8. Conclusion

We have proved the validity of the iterative Kirchhoff formulas (4.1) and (4.4) at high frequency. Numerical results for Dirichlet problem (D) can be founded in [Cuv94].

References

[Cuv94] F. Cuvelier. Th` ese: Etude th´ eorique de l’approximation de Kirchhoff pour l’´ equation de Maxwell, dans le compl´ ementaire d’une r´ eunion de convexes. Etude num´ erique. Univ.

Paris XIII, 1994.

[Cuv13] F Cuvelier. Geometrical optics formulaes for helmholtz equation.

http://hal.inria.fr/hal-00785774 , 2013.

[eC89] M.Balabane et C.Bardos. Equation des ondes : solutions asymptotiques et singularit´ es.

Compte Rendus INRIA de la session ´ electromagn´ etisme, 1989.

[MT85] R.B. Melrose and M.E. Taylor. Near peak scattering and the corrected kirchoff approxi- mation for a convex obstacle. Adv. in Math., 55(3):242, 1985.

[Yin79] D. Yingst. The Kirchhoff Approximation for the Neumann Problem. Thesis, 1979.

[Yin83] D. Yingst. The kirchhoff approximation for maxwell’s equation. Indiana Univ. Math.

Journal, 32(4):543, 1983.

F.Cuvelier : Universit´ e Paris XIII, Institut Galil´ ee, LAGA CNRS UMR 7539, 93430 Villetaneuse, France

E-mail address: cuvelier@math.univ-paris13.fr