DIFFRACTION BY A DEFECT IN AN OPEN WAVEGUIDE - CiteSeerX

c 2009 Society for Industrial and Applied Mathematics 

SIAM J. APPL. MATH. Vol. 70, No. 3, pp. 677–693

DIFFRACTION BY A DEFECT IN AN OPEN WAVEGUIDE: A MATHEMATICAL ANALYSIS BASED ON A MODAL RADIATION CONDITION∗ ANNE-SOPHIE BONNET-BEN DHIA† , GHANIA DAKHIA‡ , CHRISTOPHE HAZARD† , ` AND LAHCENE CHORFI§ Abstract. We consider the scattering of a time-harmonic acoustic wave by a defect in a twodimensional open waveguide. The scattered wave satisfies the Helmholtz equation in a perturbed layered half-plane. We introduce a modal radiation condition based on a generalized Fourier transform which diagonalizes the transverse contribution of the Helmholtz operator. The uniqueness of the solution is proved by an original technique which combines a property of the energy flux with an argument of analyticity with respect to the generalized Fourier variable. The existence is then deduced classically from Fredholm’s alternative by reformulating the scattering problem as a Lippmann–Schwinger equation by means of the Green’s function for the layered half-plane. Key words. transform

open waveguide, Helmholtz equation, radiation condition, generalized Fourier

AMS subject classifications. 35C15, 35J05, 78A45 DOI. 10.1137/080740155

1. Introduction. In this paper, we are concerned with the scattering of a timeharmonic wave by an obstacle located in an open waveguide. This problem occurs in many physical applications, such as geophysics (seismic waves are used to investigate the earth crust), ultrasonic nondestructive testing (detection of cracks in pipelines, in concrete reinforcing tendons, etc.), or integrated optics (e.g., junction of optical waveguides). We consider here a simple two-dimensional acoustic problem where the waveguide fills the half-plane R2+ := {(x, z) ∈ R2 ; z > 0}. In the absence of the defect, wave propagation is described by a real-valued wavenumber function which depends only on the z-variable, which defines the transverse direction of the waveguide:  k0 (z) :=

k1 k2

if 0 < z < h, if z > h,

with 0 < k2 < k1 ,

where the latter assumption ensures the existence of guided waves, which propagate in the longitudinal direction x. As shown in Figure 1.1, the presence of a defect is represented by a local perturbation of k0 (z) described by a bounded real-valued function k such that k(x, z) − k0 (z) has compact support. The scattering problem is then defined as follows. Considering a given incident ∗ Received by the editors November 7, 2008; accepted for publication (in revised form) April 3, 2009; published electronically July 22, 2009. http://www.siam.org/journals/siap/70-3/74015.html † ENSTA/POEMS, 32 Boulevard Victor, 75015 Paris, France (Anne-Sophie.Bonnet-Bendhia@ ensta.fr, [email protected]). ‡ Universit´ e Mohamed Khider, B.P. 145. RP., 07000 Biskra, Algeria (g [email protected]). § Universit´ e Badji Mokhtar, B.P. 12, 23000 Annaba, Algeria (l chorfi@hotmail.com).

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BONNET-BEN DHIA, DAKHIA, HAZARD, AND CHORFI

Fig. 1.1. The perturbed waveguide.

wave u0 solution to (1.1)

−Δu0 − k02 (z) u0 = 0

(1.2)

u0 (x, 0) = 0

in R2+ ,

for x ∈ R,

find the associated perturbed wave u solution to (1.3)

−Δu − k 2 (x, z) u = 0

(1.4)

u(x, 0) = 0 for x ∈ R,

in R2+ ,

and such that the scattered wave uS := u − u0 is outgoing. The way to characterize this outgoing nature of uS is the central issue of this paper. In the case of a homogeneous medium, the well-known Sommerfeld radiation condition [5] plays this role. For a layered medium, a natural idea is to use a generalized expression of this condition involving the different wavenumbers corresponding to the different layers. Such a generalization is possible in certain situations (see, e.g., [6]) but seems inadequate if the medium allows the presence of guided modes which propagate in the direction of the layers. Various solutions have been proposed. The condition introduced by Xu in [20, 21] for a three-dimensional layered medium consists in splitting the scattered field into a finite sum of guided waves and a “free” wave and imposing separately for each of them a usual Sommerfeld radiation condition with the appropriate wavenumber (two-dimensional conditions for the guided components, and a three-dimensional condition for the free one). More recently, Ciraolo and Magnanini [3] proposed a similar radiation condition based on the same splitting of the scattered field using a weaker form of Sommerfeld-type conditions for the various components (their study concerns only the uniform waveguide, but their condition probably applies for a locally perturbed waveguide). In a slightly different context of a medium which allows the propagation of surface waves similar to guided waves, Duran, Muga, and N´ed´elec [8] proposed an adaptation of the Sommerfeld condition by dividing the propagative medium into two regions and imposing in each of them a Sommerfeld-type condition. We can also mention the “upward propagating radiation condition” introduced by Chandler-Wilde and Zhang [2] which is based on an integral representation on an infinite surface outside the layer. But the conditions of application of the latter preclude the existence of guided waves: it actually controls the transverse behavior of the scattered field but not in the direction of the layers.

DIFFRACTION BY A DEFECT IN AN OPEN WAVEGUIDE

679

The modal radiation condition we propose here is inspired by the work of BonnetBen Dhia and Tillequin [1] about the junction of open waveguides. Its originality consists in using a generalized Fourier transform, which diagonalizes the transverse part (along z) of the unperturbed Helmholtz operator. The use of this transformation in the context of layered media is not new [7, 13, 18, 19]. It offers an efficient tool for studying the behavior of the acoustic field in the longitudinal direction x: outside the defect, the field appears as a superposition of guided modes and radiation modes. The modal radiation condition actually controls this longitudinal behavior: it ensures that only outgoing modes are involved in the latter superposition. From a physical point of view, this means that the energy of the field radiates towards infinity in the longitudinal direction. And such a condition is sufficient for characterizing the global outgoing behavior of the field: in the transverse direction, it is enough to impose a decay condition, which is contained implicitly in the modal radiation condition. Our main goal is to prove that the above scattering problem equipped with the modal radiation condition is well-posed. The general idea of the proof is rather classical. We reduce the scattering problem to a Lippmann–Schwinger integral equation for which Fredholm’s alternative applies. The existence of a solution then follows from uniqueness. These questions are dealt with in section 4. The two preceding sections are devoted to the uniform waveguide and provide us the main tools for establishing the solvability of the scattering problem. In section 2.1, we first recall the usual notion of guided and radiation modes. The decomposition on this family of modes yields the generalized Fourier transform associated with the guide, whose main properties are collected in section 2.2. This allows us to formulate the modal radiation condition in section 2.3. This condition is well adapted to describe the behavior of a wave generated by a localized excitation. We establish in section 3.1 the solvability of such an unperturbed radiation problem by introducing the associated Green’s function. We then show in section 3.2 a fundamental property of its solution by introducing the notion of longitudinal energy flux. Compared with other existing conditions, one may criticize the apparent complexity of our condition. Understanding only its formulation requires us to become familiar with an intricate tool: the generalized Fourier transform. But to a certain extent, this tool overcomes all the difficulties related to the layers. Once this tool and its application to the uniform waveguide have been understood, the treatment of the scattering problem becomes very simple, particularly the proof of uniqueness. This proof seems to be new. It combines two arguments. The first one is related to the longitudinal energy flux introduced in section 3.2. The idea is similar to that proposed in [2], where the energy flux is defined on an infinite boundary parallel to the layer. Here we use instead the energy flux on an infinite transverse section, orthogonal to the layer. The second argument is based on an analyticity property with respect to the generalized Fourier variable. It is inspired by the method developed in [18] for proving the absence of eigenvalues embedded in the essential spectrum of the acoustic propagator for a perturbed stratified medium, using the usual Fourier transform in the direction of the layers. Here we work instead with the generalized Fourier transform in the transverse direction, which simplifies considerably the proof. To our knowledge, the only other proof of uniqueness for the perturbed scattering problem is the one proposed by Xu [20, 21]. It is based on the study of the asymptotic expansion of the scattered field. However, as far as we understand it, this proof seems to be incomplete. Indeed Xu proves that without excitation, the asymptotic expansion vanishes [20, Lemma 3.2]. But in our opinion this does not imply that the field itself vanishes: the difference between the field and its asymptotic expansion may contain

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some evanescent component. The question remains open. We finally introduce some notations concerning certain usual functional spaces. If X is a subset of Rn and μ is a measure on X, we denote by L2 (X; dμ) the space of square integrable functions on X for the measure μ, equipped with the usual inner product and associated norm:  1/2 (ϕ, ψ)X;dμ := ϕ(x) ψ(x) dμ(x) and ϕX := (ϕ, ϕ)X;dμ . X

If μ is the Lebesgue measure on X, we shall omit dμ in these notations. For r ∈ R, r (X) is the set we denote by H r (X) the usual Sobolev spaces. If X is unbounded, Hloc r of functions ϕ such that ϕ|Y ∈ H (Y ) for all bounded Y ⊂ X. Throughout the paper, we shall use the following determination of the complex square root:  ζ := |ζ|1/2 ei(arg ζ)/2 with − 3π/2 < arg ζ ≤ π/2. √ In particular the square root of a negative real number ζ is given by ζ = −i|ζ|1/2 . 2. The modal radiation condition. 2.1. Modes of a uniform waveguide. In the case of the uniform waveguide, the modes are simply obtained by the method of separation of variables, which amounts to seeking solutions to (1.1)–(1.2) in the form u0 (x, z) = ϕ(z) e px for some p ∈ C. This leads to the following Sturm–Liouville eigenvalue problem: Find λ ∈ C and a nonzero function ϕ such that (2.1)

−ϕ − k02 ϕ = λ ϕ in R+ ,

(2.2)

ϕ(0) = 0,

where λ = p2 . For every λ ∈ C, consider the following basis of solutions to (2.1):     k02 (z) + λ (z − h) sin  k02 (z) + λ (z − h) and sλ (z) := cλ (z) := cos . k02 (z) + λ Note that these expressions do not depend on the choice  of a determination of the complex square root, since these are even functions of k02 (z) + λ. Hence, cλ (z) and sλ (z) are entire functions of λ ∈ C. The linear combination of these function given by (2.3)

Φλ (z) := cλ (0)sλ (z) − sλ (0)cλ (z)

satisfies the boundary condition Φλ (0) = 0. So it defines an entire family of solutions to (2.1)–(2.2). Noticing that for z > h, Φλ (z) can be decomposed as   √ √ 1 cλ (0) cλ (0) 1 +i k22 +λ (z−h) −i k22 +λ (z−h)   − s + s (0) e − (0) e , λ λ 2 i k22 + λ 2 i k22 + λ we see that Φλ (z) is in general unbounded as z → +∞, apart from two sets of values of λ. On one hand, if λ ∈ [−k22 , +∞), it is a bounded oscillating function. On the

DIFFRACTION BY A DEFECT IN AN OPEN WAVEGUIDE

other hand, if λ ∈ (−k12 , −k22 ) is such that cλ (0) = if λ is a solution of the dispersion equation (2.4)

 tan λ + k12 h = −



λ + k12 −λ − k22

681

 −k22 − λ sλ (0), or equivalently

with λ ∈ (−k12 , −k22 ),

then Φλ (z) is exponentially decreasing. These sets represent actually the two components of the spectrum of the operator associated with problem (2.1)–(2.2). Indeed, consider the unbounded self-adjoint operator A defined in L2 (R+ ) by (2.5)

Aϕ := −ϕ − k02 ϕ ∀ϕ ∈ D(A) := H 2 (R+ ) ∩ H01 (R+ ).

It is well known that its spectrum Λ is composed of two parts. Its point spectrum Λp ⊂ (−k12 , −k22 ) consists of the finite number of roots λj , j = 1, . . . , N , of (2.4): each Φλj belongs to L2 (R+ ) and is then an eigenfunction associated with the eigenvalue λj ∈ Λp . The continuous spectrum of A is Λc := [−k22 , +∞): for each λ ∈ Λc , Φλ ∈ / L2 (R+ ); it is a generalized eigenfunction. The modes of the uniform waveguide are then defined by (2.6)

Uλ± (x, z) := Φλ (z) e∓

√ λx

for λ ∈ Λ.

Assuming a time dependence of the form e−iωt , we can consider that Uλ+ (x, z) is a right-going mode, whereas Uλ− (x, z) is a left-going mode. Guided modes are associated with the eigenvalues λj ∈ Λp : they are confined near the core of the guide and propagate towards x → ±∞. Radiation modes correspond to the continuous spectrum Λc : they are either propagative towards x → ±∞ when λ ≤ 0 or evanescent in this direction when λ > 0 (and exponentially increasing in the opposite direction x → ∓∞). 2.2. The generalized Fourier transform. How can one obtain a diagonal representation of A? The generalized Fourier transform adapted to operator A yields the answer. The introduction of such a transformation for Sturm–Liouville operators with a continuous spectrum dates back to the first half of the last century, in the original works of Weyl and Titchmarsh (see [4, 17]). Its use in the context of wave propagation in stratified media is not new (see, e.g., [7, 13, 18, 19]). The generalized Fourier transform appears as the operator of “decomposition” on the family of eigenfunctions and generalized eigenfunctions {Φλ (z); λ ∈ Λ}:  (2.7)

(F ϕ)(λ) :=

R+

ϕ(z) Φλ (z) dz

∀λ ∈ Λ,

which makes sense, for instance, when ϕ ∈ L2 (R+ ) has compact support. As the usual Fourier transform, this expression leads to a unitary transformation from L2 (R+ ) to some spectral L2 -space. However the measure associated with this L2 -space contains not only a continuous part on Λc but also a pure point contribution corresponding to Λp . We denote dμ :=

 λ∈Λp

ρλ δλ + ρλ dλ|Λc ,

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where δλ is the Dirac measure at λ ∈ Λp , dλ|Λc is the Lebesgue measure restricted to Λc , and ⎧ 1 2(k12 + λ) ⎪ ⎪ = if λ ∈ Λp , ⎪ 2 ⎨ Φλ  + h + (−k22 − λ)−1/2 R ρλ :=  ⎪ ⎪ k22 + λ ⎪ ⎩ if λ ∈ Λc . π (c2λ (0) + (k22 + λ) s2λ (0)) The following classical properties of F derive from spectral theory of self-adjoint operators (see, e.g., [12]). Proposition 2.1. The transformation F defined by (2.7) for all ϕ ∈ L2 (R+ ) with compact support extends to a unitary operator from L2 (R+ ) to L2 (Λ; dμ) (still denoted by F ), which diagonalizes A in the sense that for every measurable function f : R → C, (2.8)

f (A)ϕ = F ∗ f (λ) F ϕ

∀ϕ ∈ D(f (A)) = F ∗ (L2 (Λ; (1 + |f (·)|2 )dμ)),

where f (λ) stands for the operator of multiplication by f (λ) in L2 (Λ; dμ). Its adjoint F ∗ = F −1 appears as the operator of “recomposition” on the family {Φλ ; λ ∈ Λ}:  ∗ (2.9) F ϕ = ϕ(λ)  Φλ dμ(λ) ∀ϕ  ∈ L2 (Λ; dμ). Λ

Remark 2.2. Formula (2.9) can be written more explicitly as follows, which makes  clear the discrete and continuous components of F ∗ ϕ:    = ρλ ϕ(λ)  Φλ (z) + ϕ(λ)  Φλ (z) ρλ dλ. (F ∗ ϕ)(z) λ∈Λp

Λc

Similarly to the usual Fourier transform, the integral on Λc has to be understood as an improper integral (which converges in L2 (R+ )), that is, the limit of integrals on bounded intervals of the form (−k22 + M −1 , M ) when M → ∞. Our aim is to use the generalized Fourier transform to solve (1.1)–(1.2). But Proposition 2.1 is not sufficient to do so, since for fixed x ∈ R, function u0 (x, ·) does not belong in general to L2 (R+ ). That is why we need to extend F to a larger space. As the usual Fourier transform, F can be interpreted in the sense of distributions, in a space similar to the Schwartz space S  (R) of tempered distributions [16]. The construction of such an extension is described in [12]. We recall here the main results. As for the usual Fourier transform, this extension is based on the fact that the action of F leads to an exchange of regularity and decay at infinity between the physical and spectral variables z and λ. But here, this exchange property holds under the assumption that  2 2 π 1 2 2 ∀n ∈ N, (2.10) k1 − k2 = n + 2 h2 which rules out the cutoff frequencies of the waveguide, that is, the cases when a guided mode becomes a radiation mode (the associated Φλ (z) becomes constant for z > h). We denote by SA (R+ ) the space of functions ϕ which are infinitely differentiable on both intervals (0, h) and (h, +∞), decay rapidly at infinity as well as their derivatives (in the sense that limz→+∞ dnz (z m ϕ(z)) = 0 for all n, m ∈ N, where dnz stands

DIFFRACTION BY A DEFECT IN AN OPEN WAVEGUIDE

683

for the nth derivative), and satisfy 2 2 n 2 2 n d2n z ϕ(0) = 0 and [(dz + k0 ) ϕ]h = [dz (dz + k0 ) ϕ]h = 0 ∀n ∈ N,

where [ψ]h denotes the gap of ψ at z = h, that is, [ψ]h := limε 0 {ψ(h+ ε)− ψ(h− ε)}. The latter conditions ensure that SA (R+ ) ⊂ D(An ) for all n ∈ N. Functions of SA (R+ ) play the role of test functions. This space is continuously embedded in L2 (R+ ) and dense. Moreover it is “adapted to A” in the sense that A appears as a continuous operator in SA (R+ ).  (R+ ) The corresponding space of distributions is then defined as the dual space SA + of SA (R ) (where the term “dual” means here the collection of antilinear continuous functionals). If we identify L2 (R+ ) with its dual space, it can be interpreted as a  (R+ ) in the functional scheme: subspace of SA (2.11)

 SA (R+ ) ⊂ L2 (R+ ) = L2 (R+ ) ⊂ SA (R+ ),

 where the duality product · , ·R+ between SA (R+ ) and SA (R+ ) appears as an exten2 + sion of the inner product of L (R ):

ϕ, ψR+ = (ϕ, ψ)R+

∀ϕ ∈ L2 (R+ ), ∀ψ ∈ SA (R+ ).

And A can be interpreted in the sense of distributions by setting

Aϕ, ψR+ := ϕ, AψR+

 ∀ϕ ∈ SA (R+ ), ∀ψ ∈ SA (R+ ).

The chain of spaces (2.11) is naturally converted into a spectral chain by the generalized Fourier transform. Indeed setting SA (Λ) := F (SA (R+ )), we can extend  F to distributions of SA (R+ ) by the formula  Λ := ϕ, F −1 ψ  R+

Fϕ, ψ

 ∀ϕ ∈ SA (R+ ), ∀ψ ∈ SA (Λ),

where · , ·Λ denotes the duality product between SA (Λ) and its dual space SA (Λ) which defines the space of spectral distributions. In other words, F defines an isomorphism between (2.11) and the spectral chain SA (Λ) ⊂ L2 (Λ; dμ) ⊂ SA (Λ) . The space of spectral test functions SA (Λ) is identified by the following lemma [12].  Λ ∈ CN and there Lemma 2.3. A function ψ belongs to SA (Λ) if and only if ψ| p exists ψ˜ ∈ S(R) such that   1  ψ(λ) =  2 k22 + λ ∀λ ∈ Λc . ψ˜ k2 + λ As a consequence, outside a vicinity of λ = −k22 , spectral distributions of SA (Λ) are similar to tempered distributions of S  (R). In particular the space of locally integrable tempered functions   : Λ → C; ϕ|  (−k22 ,M) ∈ L1 (−k22 , M ) ∀M > −k22 L1temp(Λ) := ϕ  (2.12) and limλ→+∞ λ−m ϕ(λ)  = 0 for some m ∈ N is contained in SA (Λ).

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BONNET-BEN DHIA, DAKHIA, HAZARD, AND CHORFI

2.3. Expression of the radiation condition. Consider a semi-infinite section of the uniform waveguide X ± × R+ , where X − := (−∞, −a) and X + := (+a, +∞) for some a > 0. Our aim is to exhibit a condition which selects outgoing solutions to (2.13)

−Δu − k02 u = 0

(2.14)

u(x, 0) = 0

in X ± × R+ ,

for x ∈ X ± .

Let us first solve formally this problem, which can be rewritten as −

(2.15)

∂2u + Au = 0 ∂x2

in X ± × R+ ,

thanks to the definition (2.5) of A. Applying the generalized Fourier transform then leads to −

(2.16)

∂2u λ + λu λ = 0 ∂x2

in X ± ,

where u λ (x) := (F u(x, ·))(λ). As a consequence, − u λ (x) = α + λ e

(2.17)

√

λx

+ +α − λ e

√ λx

,

where α ± λ does not depend on x. The inverse transform then yields    √ √ − + λx − λx (2.18) u(x, z) = α + Φλ (z) dμ(λ), e + α  e λ λ Λ

which means that u is a superposition of the right-going and left-going modes (2.6), more precisely, a finite superposition of guided modes together with a continuous superposition of radiation modes (see Remark 2.2). The radiation condition proposed in this paper consists in keeping only outgoing modes in (2.18), that is, imposing − + and α − α + λ = 0 in the left section X λ = 0 in the right section X . In order to state a proper expression of this condition, we first have to justify the above formal solution of (2.13)–(2.14). This problem has to be interpreted in a weak 1 (X ± × R+ ) ∩ L∞ (X ± × R+ ), (2.13) signifies that sense. Assuming that u ∈ Hloc    ∇u · ∇v − k02 u v dxdz = 0 X ± ×R+

for all v ∈ H 1 (X ± × R+ ) with compact support and which vanish on the boundary of X ± × R+ . If v is regular enough, the Green’s formula and condition (2.14) yield  u (−Δv − k02 v) dxdz = 0. X ± ×R+

In this equality, we can choose test functions of the form v(x, z) = η(x) ψ(z) with η ∈ D(X ± ) (that is, an infinitely differentiable function with compact support; see [16]) and ψ ∈ SA (R+ ) (which is rapidly decreasing as z → +∞, so the equality holds by a density argument, since u is assumed bounded). Therefore we have  u (−η  ψ + η Aψ) dxdz = 0 ∀η ∈ D(X ± ), ∀ψ ∈ SA (R+ ), X ± ×R+

DIFFRACTION BY A DEFECT IN AN OPEN WAVEGUIDE

685

 which means exactly that (2.15) is satisfied in the tensor product D (X ± ) ⊗ SA (R+ ).  + We can then apply the extension of the generalized Fourier transform to SA (R ), so  that (2.16) is satisfied in D (X ± ) ⊗ SA (Λ). This equation admits solutions which are not functions. For instance, the radiation modes (2.6) correspond to (2.17), where α ± λ ± are Dirac distributions. Here we shall consider only the case where α λ are functions of λ, which is enough to describe scattered waves but not incident waves such as radiation modes. Such an assumption actually imposes a certain decay of u in the transverse direction (z → +∞). We are now able to give a precise definition of our outgoing radiation condition. 1 (X ± × R+ ) ∩ L∞ (X ± × R+ ) to (2.13)– Definition 2.4. A weak solution u ∈ Hloc 1 (2.14) is said to satisfy the modal radiation condition if there exists α ± λ ∈ Ltemp (Λ) (see (2.12)) such that

(2.19)

− (F u(x, ·))(λ) = α ± λ e

√ λ |x|

for ± x > a and λ ∈ Λ.

3. Radiation in a uniform waveguide. 3.1. The radiation problem. In this section, we consider the following unperturbed radiation problem for a given excitation f ∈ L2 (R2+ ) which is assumed compactly supported, say, in [−a, +a] × R+ : ⎧ 1 Find u ∈ Hloc (R2+ ) ∩ L∞ (R2+ ) such that ⎪ ⎪ ⎪ ⎨ −Δu − k 2 u = f in R2 , 0 + (P0 ) ⎪ u(x, 0) = 0 for x ∈ R, ⎪ ⎪ ⎩ u satisfies the radiation condition (2.19). We want to verify that the only solution to this problem is given by the usual integral representation  G(M, M  ) f (M  ) dM  ∀M ∈ R2+ , (3.1) u(M ) = R2+

where G is the outgoing Green’s function of the uniform waveguide. Such a representation is well known (see, e.g., [10, 13, 21]). Our aim is to show that it is compatible with the modal radiation condition (2.19). We begin by a formal construction of (3.1) using the generalized Fourier transform. We proceed as in section 2.3 by rewriting the Helmholtz equation together with the Dirichlet condition in the form −

∂ 2u + Au = f. ∂x2

Setting u λ (x) := (F u(x, ·))(λ) and fλ (x) := (F f (x, ·))(λ), we thus have (3.2)

−

λ ∂ 2u + λu λ = fλ ∂x2

on R

for all λ ∈ Λ. By virtue of the radiation condition (2.19), we consider naturally the following Green’s function associated with this equation: γλ (x) :=

e−

√ λ |x|

√ 2 λ

for λ = 0.

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BONNET-BEN DHIA, DAKHIA, HAZARD, AND CHORFI

It is then easy to see that u λ (x) = (γλ ∗ fλ )(x) =

(3.3)

 (3.4)

=

R2+

 R

γλ (x − x ) fλ (x ) dx

γλ (x − x ) f (x , z  ) Φλ (z  ) dx dz 

is a solution to (3.2). Noticing that   √ +a ± λ x √ e   √ fλ (x ) dx e− λ |x| (3.5) u λ (x) = 2 λ −a

for ± x > a,

we see that this solution is outgoing in the sense of (2.19). We finally apply the inverse Fourier transform, which yields u(x, z) = (F −1 u λ (x))(z)   = Λ



 (3.6)

=

R2+

R2+











γλ (x − x ) f (x , z ) Φλ (z ) dx dz



Φλ (z) dμ(λ)

 γλ (x − x ) Φλ (z) Φλ (z ) dμ(λ) f (x , z  ) dx dz  . 



Λ

The last expression is nothing but the integral representation (3.1), where the Green’s function is given by  (3.7) G(M, M  ) := γλ (x − x ) Φλ (z) Φλ (z  ) dμ(λ) for M = (x, z), M  = (x , z  ). Λ

Of course the above inversion is formal and has to be justified, as does the uniqueness of the solution. Proposition 3.1. Under assumption (2.10), for every f ∈ L2 (R2+ ) with support in a given bounded domain Ω0 ⊂ R2+ , problem (P0 ) has a unique solution u given by the integral representation (3.1). And for every bounded domain Ω ⊂ R2+ , there exists C(Ω, Ω0 ) > 0 such that (3.8)

uH 1 (Ω) ≤ C(Ω, Ω0 ) f Ω0 .

Proof. Let us first prove the uniqueness of the solution. We proceed as in section 2.3 with X ± replaced by R. Suppose that u is a solution to the homogeneous  problem (i.e., f = 0). Then u λ (x) satisfies (2.16) in D (R) ⊗ SA (Λ). The radiation condition (2.19) imposes that u λ (x) is a function of λ for |x| > a but is not necessarily a function in (−a, +a). We thus have to solve (2.16) in the sense of distributions.   Considering the restriction of u λ to Λ \ {0}, that is, choosing √ test functions √ ψ ∈ SA (Λ) that vanish near λ = 0, we can define vλ± := (∂ uλ /∂x ± λ u λ ) exp(∓ λ x) (which does not near λ = 0 √ make sense when applied to test functions that do not vanish vλ± /∂x = 0, which since λ is not differentiable). We then deduce from (2.16) that ∂ shows that outside λ = 0, all the possible solutions are given by (2.17), where α + λ  and α − λ are arbitrary distributions of SA (Λ). But the radiation condition on the right + imposes that α − λ = 0 and on the left that α λ = 0. Thus we have proved that  Λ = 0 ∀x ∈ R and ψ ∈ SA (Λ) which vanish near λ = 0.

 uλ (x), ψ

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687

Since u λ (x) is a function for |x| > a, this implies that it vanishes there (in other words, u λ is a distribution with support in [−a, +a] × {0}). Hence u(x, z) = 0 for |x| > a. Owing to the principle of unique continuation (see, e.g., [5, Lemma 8.5], where the result is stated for C 2 functions, but remains valid in our situation), u vanishes everywhere. So (P0 ) has at most one solution. Consider now the question of existence. If Λ were bounded, (3.6) would simply follow from Fubini’s theorem. But here the integrals on Λ are understood as improper integrals, because of the possibly slow decay of the integrand when λ → ∞, which yields the singularity of the Green’s function at M = M  . It is thus necessary to use the interpretation of F in the sense of distributions, that is, u(x, ·), ψR+ =  uλ (x), F ψΛ for ψ ∈ SA (R+ ) and x ∈ R. The use of Fubini’s theorem is now justified by the decay of F ψ ∈ SA (Λ) (see Lemma 2.3), which yields   

u(x, ·), ψR+ = γλ (x − x ) Φλ (z  ) F ψ(λ) dμ(λ) f (x , z  ) dx dz  , R2+

Λ

or equivalently, 

u(x, ·), ψR+ =

R2+

 −1  F {γλ (x − x ) Φλ (z  )}, ψ R+ f (x , z  ) dx dz  .

Hence setting G(M, M  ) = F −1 {γλ (x − x ) Φλ (z  )}(z), which is nothing but (3.7), the integral representation (3.1) will derive from the latter equality if we can replace the duality product by an integral and apply again Fubini’s theorem. This can be justified by the following lines, where we study some properties of G in order to prove the stability property (3.8). Let Aλ (M, M  ) denote the integrand in the continuous component of the expression (3.7) of G, that is, Aλ (M, M  ) := γλ (x − x ) Φλ (z) Φλ (z  ) ρλ

for λ ∈ Λc ,

whose asymptotic behavior when λ → ∞ is easily worked out. We obtain Aλ =

(as) Aλ

+

(re) Aλ ,

where

(as) Aλ (M, M  )

:=

e−

√ λ |x−x |

2πλ

√ √ sin( λ z) sin( λ z  ),

(re) term Aλ (M, M  ) −3/2

and the remaining is a continuous function of (M, M  ) whose order ) uniformly with respect to M and M  in a bounded domain. of magnitude is O(λ (as) The integral of Aλ on (0, +∞) is well known (see, e.g., [11, page 491]): G(as) (M, M  ) :=

 0

+∞

Aλ (M, M  ) dλ = (as)

(x − x )2 + (z + z  )2 1 ln , 4π (x − x )2 + (z − z  )2

which is nothing but the Green’s function of the Laplace operator in the half-plane. We then have the decomposition G = G(as) +G(re) , where G(re) (M, M  ) is continuous with respect to (M, M  ). The properties of the volume potential associated with G(as) , that is, (3.1) with G replaced by G(as) , are well known [5]. On the other hand, the volume potential associated with G(re) clearly yields a continuous function. As a consequence, for f ∈ L2 (Ω0 ), (3.1) defines a function u ∈ L2loc (R2+ ) which is by construction a very  (R+ )). weak solution to −Δu − k02 (z) u = f (this equation is satisfied in S  (R) ⊗ SA The fact that the volume potential can be derived with respect to M cannot be

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deduced easily from the above decomposition of G (for the derivation of the integral which involves G(re) leads to an improper integral). We can use instead an argument of interior regularity for the Laplace operator (see, e.g., [14, Theorem 3.22]; note that the Dirichlet boundary condition does not raise any difficulty since our problem amounts to a problem set in R2 by symmetry), which ensures that this very weak 1 (R2+ )), so that the stability property (3.8) solution actually is a weak solution (in Hloc is satisfied. 1 It remains finally to verify that (3.5) defines two functions α ± λ ∈ Ltemp (Λ), so that the modal radiation condition is satisfied. Actually we shall prove in the next subsection a more precise result (see Proposition 3.3). 3.2. Longitudinal energy flux. The expression (3.7) of the Green’s function is based on the generalized Fourier transform along z. It is well adapted to describe the behavior of G, and thus of u, in the longitudinal direction x: our radiation condition (2.19) actually is a longitudinal radiation condition. Another useful and equivalent expression of G can be obtained using the usual Fourier transform along x (it can also be deduced from (3.7) by Cauchy’s theorem using a deformation of the integration path in the complex λ-plane). This expression is well adapted to study the asymptotic behavior of G in the transverse direction z (see, for instance, [10]). In particular, it is the basic tool for proving the following proposition. Proposition 3.2. The solution u(M ) = u(x, z) given by the integral representation (3.1) satisfies u(M ) = O(z −1/2 ), ∂u/∂x(M ) = O(z −3/2 ), and ∂u/∂z(M ) = O(z −1/2 ) as z → +∞ uniformly with respect to x in a bounded domain. This behavior allows us to define the notion of longitudinal energy flux. Consider the rectangle ΩR := (−a, +a) × (0, R), where R is chosen large enough so that ΩR contains the support of f. Using the Helmholtz equation and the Green’s formula gives      ∂u |∇u|2 − k02 |u|2 − u dγ = f u, ΩR ∂ΩR ∂n R2+ where ∂u/∂n denotes the exterior normal derivative of u on the boundary ∂ΩR of ΩR . Taking the imaginary part of this equality yields   ∂u Im u dγ = − Im f u, ∂ΩR ∂n R2+ where the left-hand side represents the energy flux going out of ΩR across ∂ΩR (in fact its mean during a period; see, e.g., [15]) and the right-hand side stands for the energy produced by the excitation f (again its mean during a period). In the boundary integral, we can distinguish three contributions: a transverse one, on (−a, +a) × {R} (note that the integral on (−a, +a) × {0} vanishes thanks to the Dirichlet boundary condition), and two longitudinal components, on {±a} × (0, R). By Proposition 3.2, the former vanishes when R → +∞, whereas the latter both have limits  ∂u (±a, z) u(±a, z) dz, (3.9) E ± (u) := Im R+ ∂|x| which define, respectively, the left-going (−) and right-going (+) energy fluxes. The energy conservation then writes as  (3.10) E + (u) + E − (u) = − Im f u. R2+

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DIFFRACTION BY A DEFECT IN AN OPEN WAVEGUIDE

The radiation condition (2.19) provides us precious information about these longitudinal energy fluxes. In particular it tells us that these fluxes are nonnegative, which justifies the word outgoing for the radiated field. Indeed we have the following property. Proposition 3.3. If u is solution to (P0 ), then the generalized Fourier coefficients α± λ involved in the radiation condition (2.19) belong to the space   V := ϕ  : Λ → C measurable; |λ|1/4 ϕ  ∈ L2 (Λ; dμ) ⊂ L1temp (Λ), and the longitudinal energy fluxes are given by    ± 2 (3.11) E ± (u) = |λ| αλ  dμ(λ). Λ∩R−

Remark 3.4. Note that the above integral is defined only on the negative part of the spectrum, which corresponds to propagative modes (guided and radiative; see section 2.1). Evanescent modes do not contribute to the energy flux. Proof. Formula (3.11) follows from the Parseval-like identity   ∂u ∂ uλ (±a, z) u(±a, z) dz = (±a) u λ (±a) dμ(λ), (3.12) ∂|x| ∂|x| + R Λ where u λ (±a) := (F u(±a, ·))(λ) and its derivative are given by the radiation condition (2.19): √ √ ∂ uλ − λa (±a) = − λ α± . λ e ∂|x| √  Taking the imaginary part of (3.12) yields the result, since λ = −i |λ| for λ < 0. Of course, we have to justify (3.12), which cannot be deduced directly from the unitary nature of F , for neither u(±a, ·) nor ∂u/∂|x|(±a, ·) belongs a priori to L2 (R+ ). To do so, we have to make precise the functional framework where this equality occurs. Consider the Hilbert space   V  := ϕ  : Λ → C measurable; |λ|−1/4 ϕ  ∈ L2 (Λ; dμ) , − u λ (±a) = α± λ e

√

λa

and

which is clearly dual to V when we consider the duality product     := (|λ|−1/4 ϕ  Λ;dμ

ϕ,  ψ  , |λ|+1/4 ψ) V ,V

∀ϕ  ∈ V  and ψ ∈ V .

It is readily seen that SA (Λ) is contained and dense in V and V  , so that both spaces  Λ appear as subspaces of SA (Λ), and the above duality product is nothing but ϕ,  ψ  1  if ϕ  or ψ belongs to SA (Λ). Actually both spaces V and V are contained in L (Λ) temp

(see (2.12)), since |λ|±1/4 is square integrable near λ = 0. Then using the generalized  Fourier transform in the sense of SA (Λ), we have     = F −1 ϕ,  R+

ϕ,  ψ  F −1 ψ V ,V

∀ϕ  ∈ V  and ψ ∈ SA (Λ),

and by density, this equality holds for all ψ ∈ V : this is the proper interpretation of (3.12). It remains to verify that our particular situation falls within this context.

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Let us prove that u λ (x) belongs to V for every fixed x ∈ R. From (3.3), we have    1/2   +a e−√λ |x−x | a     √ fλ (−a,+a) , | uλ (x)| =  fλ (x ) dx  ≤   −a 2|λ| 2 λ by the Cauchy–Schwarz inequality. Then   a 1/2 2 |λ| | uλ (x)| dμ(λ) ≤ |λ|−1/2 fλ 2(−a,+a) dμ(λ). 2 Λ Λ To see that the right-hand side is bounded, we have to split the integral into two parts, say, on Λ \ (−ε, +ε) for some positive ε < k22 . On one hand, we have   |λ|−1/2 fλ 2(−a,+a) dμ(λ) ≤ ε−1/2 fλ 2(−a,+a) dμ(λ) = ε−1/2 f 2R2 , Λ\(−ε,+ε)

+

Λ

since F is unitary. On the other hand, for λ ∈ (−ε, +ε), functions Φλ are uniformly bounded, so          |fλ (x )| =  f (x , z) Φλ (z) dz  ≤ C |f (x , z)| dz ≤ C  f (x , ·)R+ , R+

R+

where the last inequality now uses √ the fact that the support of f is bounded in the z-direction. Hence fλ (−a,+a) ≤ 2aC  f R2+ for all λ ∈ (−ε, +ε), and consequently 

+ε

−ε

|λ|−1/2 fλ 2(−a,+a) dμ(λ) ≤ 2aC 2



+ε

−ε

|λ|−1/2 ρλ dλ f 2R2 . +

To sum up, we have proved that there exists some C > 0 depending on the support of f such that  |λ|1/4 u λ (x)Λ;dμ ≤ Cf R2+

∀x ∈ R;

√ thus u λ (x) ∈ V . Notice finally that this implies that λ u λ (x), which is nothing but −∂ uλ (x)/∂|x|, belongs to V  . This completes the proof. 4. Scattering in a locally perturbed waveguide. We can now give a precise definition of the scattering problem introduced in section 1. Considering a given incident wave u0 solution to (1.1)–(1.2), which may be a superposition of guided and radiation modes (2.6), the question is to prove the well-posedness of the following problem:

(P)

⎧ 1 Find u ∈ Hloc (R2+ ) ∩ L∞ (R2+ ) such that ⎪ ⎪ ⎪ ⎨ −Δu − k 2 u = 0 in R2 , + ⎪ u(x, 0) = 0 for x ∈ R, ⎪ ⎪ ⎩ uS := u − u0 satisfies the radiation condition (2.19).

Remark 4.1. As for the unperturbed radiation problem (P0 ), we are interested in weak solutions. Note that as k 2 is bounded, classical regularity results [9] show that 2 u ∈ Hloc (R2+ ) and is continuously differentiable.

DIFFRACTION BY A DEFECT IN AN OPEN WAVEGUIDE

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Problem (P) can be classically written as a Lippmann–Schwinger equation, by noticing that the scattered wave uS satisfies −ΔuS − k02 uS = (k 2 − k02 ) u

in R2+

and the Dirichlet boundary condition on z = 0. Thus it is a solution to the unperturbed radiation problem (P0 ) for f = (k 2 − k02 ) u ∈ L2 (R2+ ) which is compactly supported. Proposition 3.1 then tells us that  G(M, M  ) (k 2 (M  ) − k02 (M  )) u(M  ) dM  ∀M ∈ R2+ . (4.1) uS (M ) = R2+

Let Ω0 be a bounded open subset of R2+ that contains the support of k 2 − k02 . Consider then the integral operator K defined in L2 (Ω0 ) by  G(M, M  ) (k 2 (M  ) − k02 (M  )) v(M  ) dM  ∀M ∈ Ω0 . (Kv)(M ) := Ω0

It is clear that if u is a solution to (P), its restriction to Ω0 (still denoted u for simplicity) is a solution to the following integral equation: (4.2)

(I − K)u = u0

in L2 (Ω0 ).

And conversely, if u ∈ L2 (Ω0 ) is a solution to the latter equation, it extends to the whole domain R2+ by setting u = u0 + uS , where uS is given by the integral representation (4.1), and this extension is a solution to (P). In other words, (P) is equivalent to the Lippmann–Schwinger equation (4.2). Lemma 4.2. K is a compact operator in L2 (Ω0 ). Proof. Proposition 3.1 shows that there exists C > 0 such that KvH 1 (Ω0 ) ≤ C (k 2 − k02 ) uΩ0 ≤ C  uΩ0

∀v ∈ L2 (Ω0 ).

Hence K appears as a continuous operator from L2 (Ω0 ) to H 1 (Ω0 ). As Ω0 is bounded, the compact embedding of H 1 (Ω0 ) into L2 (Ω0 ) yields the result. The above property shows that Fredholm’s alternative can be applied to (4.2): if its solution is unique, then it it well-posed. Theorem 4.3. Under the assumption (2.10), problem (P) has at most one solution. Proof. Suppose that u is a solution to (P) with u0 = 0, that is, u = uS . Following exactly the same lines as in section 3.2, with k0 replaced by k, the energy conservation (3.10) now writes as E + (u) + E − (u) = 0, where the energy fluxes E ± (u) are still defined by formula (3.9). Proposition 3.3 then tells us that both E + (u) and E − (u) vanish since these are nonnegative quantities. More precisely, we know that outside the defect, the generalized Fourier transform u λ (x) := (F u(x, ·))(λ) vanishes on the negative part of the spectrum: (4.3)

u λ (x) = 0

∀λ ∈ Λ ∩ R− and |x| > a.

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On the other hand, we have seen in section 3.1 that the integral representation (4.1) of uS = u is equivalent to the following integral representation of u λ (x) (see (3.4)):  u λ (x) =

e− R2+

√

λ |x−x |

√ 2 λ

Φλ (z  ) f (x , z  ) dx dz  ,

where f = (k 2 − k02 ) u. As mentioned √ in section 2.1, function Φλ (z) extends to an entire function of λ ∈ C. Moreover, λ is analytic in C \ iR+ (by virtue of our choice of a determination of the complex square root; see section 1). Therefore, as f is compactly supported, the above integral representation shows that u λ (x) extends to an analytic function of λ in C \ iR+ for every x ∈ R. From (4.3), we know that if |x| > a, this function vanishes on the segment (−k22 , 0). Analyticity implies that it vanishes in C \ iR+ , in particular for all λ ∈ Λ, which means that u(x, z) = 0 for |x| > a. Since k 2 is a bounded function, the principle of unique continuation [5, Lemma 8.5] can be applied, so u vanishes everywhere. Thanks to the equivalence between (P) and the Fredholm equation (4.2), we also know that (4.2) has at most one solution, and Fredholm’s alternative yields the main result of this paper, which can be formulated as follows. Corollary 4.4. Under assumption (2.10), the scattering problem (P) has a unique solution u which depends continuously on the incident wave u0 in the sense that for every bounded domain Ω ⊂ R2+ , there exists C(Ω) > 0 such that uH 1 (Ω) ≤ C(Ω) u0 Ω0 , where Ω0 contains the support of the defect. 5. Conclusion. Following the idea introduced in [1], we have proposed a new radiation condition for characterizing the behavior of a wave scattered by a defect in a uniform open waveguide, and we have proved the solvability of the scattering problem equipped with this condition, using an original proof for uniqueness. The already-existing conditions [3, 20, 21] could be viewed as mixed modal–Sommerfeld conditions: they isolate the guided components of the scattered wave and ensure that these components are outgoing, whereas the remaining component satisfies a usual Sommerfeld condition. Our condition is a full modal condition: thanks to the use of the generalized Fourier transform, the guided and radiative components of the scattered wave are dealt with in the same way. The question is not to decide whether one condition is better than another one, since they all play the same role. Our aim was rather to show that the generalized Fourier transform provides us a very powerful theoretical tool for studying scattering problems in a layered medium. Indeed, this transform is particularly well adapted for the description of the behavior of the acoustic field in the longitudinal direction, whereas the usual Fourier transform in the longitudinal direction is rather appropriate to its behavior in the transverse direction. The simplicity of the proof of uniqueness for the scattering problem brings to light the substantial gain offered by the use of the former transform. A similar approach can be developed in more complicated situations, for instance for the junction of two different open waveguides by a varying cross section part. Works on this subject are in progress.

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