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Surface waves in a channel with thin tunnels and wells at the bottom: non-reflecting underwater tomography

We consider the propagation of surface water waves in a straight planar channel perturbed at the bottom by several thin curved tunnels and wells. We propose a method to construct non reflecting underwater topographies of this type at an arbitrary prescribed wave number. To proceed, we compute asymptotic expansions of the diffraction solutions with respect to the small parameter of the geometry taking into account the existence of boundary layer phenomena. We establish error estimates to validate the expansions using advances techniques of weighted spaces with detached asymptotics. In the process, we show the absence of trapped surface waves for perturbations small enough. This analysis furnishes asymptotic formulas for the scattering matrix and we use them to determine underwater topographies which are non-reflecting. Theoretical and numerical examples are given.


Introduction.
1.1. Non-reflecting and invisible obstacles in waveguides. We investigate the propagation of surface water-waves in a planar channel in time-harmonic regime. We assume that the channel coincides, outside a region where the bottom is geometrically perturbed, with the reference straight channel. We consider a situation where an incident wave propagates through the channel, hits the geometrical defect and gives birth to a scattered field. One commonly denotes by R the reflection coefficient, which corresponds to the amplitude of the backscattered farfield, and by T the transmission coefficient, which corresponds to the amplitude of the transmitted farfield. Due to conservation of energy, these two complex numbers satisfy the relation (1.1) |R| 2 + |T | 2 = 1.
The scattering coefficients R, T depend on the geometry and satisfy R = 0, T = 1 in the reference straight channel. In this context, a question of growing interest is to find situations where one has good transmission properties (see in particular the literature concerning so-called Perfect Transmission Resonances (PTRs) [43,42,26,46,29]). In particular, one can wish to have perfect transmission in energy, that is |T | = 1. Due to (1.1), this is equivalent to have R = 0. In this case, following the terminology introduced in [6], we shall say that the perturbation of the bottom is non reflecting. Note that in this situation, the transmitted wave in general exhibits a phase shift with respect to the incident field. One can be more demanding and look for channels where R = 0 and T = 1. In this case, we shall speak of perfect invisibility.
Usually R and T depend analytically on the geometric parameters defining the channel/waveguide. As a consequence, non reflecting and invisibility situations are unstable: a small change of the setting may ruin them. Eigenvalues embedded into the continuous spectrum behave in a similar way, see for example [2]. In [35,37], the notion of enforced stability of eigenvalues was introduced and the method of fine-tuning the geometric parameters to maintain the eigenvalues in the continuous spectrum while perturbing the problem was developed and rigorously substantiated. To summarize, this method boils down to mimic the proof of the implicit functions theorem considering a certain indicator of existence of eigenvalues as a function of the parameters of the problem (geometry of the waveguide, physical coefficient in the equation, ...). It was adapted in [6] to the problem of invisibility. More precisely, in [6] the authors study an acoustic problem in a waveguide with locally gently sloping walls ( Fig. 1.1, a)) and propose a method to construct non reflecting perturbations of the reference straight geometry. We emphasize that the approach of [6] does not allow one to control the phase shift between the incident field and the transmitted field to impose T = 1. As a consequence, in general the distortion of the wall is not perfectly invisible.
The method of [6] was then adapted in [7] to the water-wave problem in a planar channel with a gently sloping bottom (see again Fig. 1.1, a)). In [7], it is shown that for the water-wave problem, the above technique does not only allow one to find non straight channels where R = 0, |T | = 1 but also geometries where R = 0, T = 1 (without phase shift between the incident and the transmitted fields). In other words, it allows one to construct perfectly invisible perturbations of the bottom. To be exhaustive, we must mention however that the technique does not apply for the particular case where, with the notation below, a 2 = 2k 2 or equivalently, λ = √ 2k tanh( √ 2kd). In order to obtain perfect invisibility (R = 0, T = 1) for the acoustic problem considered in [6], another way to perturb the reference straight geometry was studied in [5]. It consists in working with waveguides with several thin rectangles perpendicular to the wall, see Fig. 1.1, b). This singular perturbation gives a different differential of the scattering coefficients with respect to the geometry (have in mind the proof of the implicit functions theorem mentioned above) and in [5], it is proved that it allows one to achieve perfect invisibility in monomode regime by choosing the dispositions and lengths of the rectangles properly.
In the present work, we adapt the approach of [5] to the water-wave problem (2.4)-(2.6) in the much more tangled geometry with curved thin tunnels. We will show that the approach of [5] can be used to get R = 0 for all k ≥ 0 and λ > λ † (k) (again see the notation below) without the above restriction a 2 = 2k 2 which appears when considering smooth perturbations ( Fig. 1.1, a)) of the geometry. We will also prove that this manner of deforming the bottom does not allow one to get T = 1. In most of the article, we will perturb the reference straight channel by digging thin tunnels. We emphasize that the results of this article cover the more simple case of rectangular well shaped perturbations, see Fig. 1.1, b) and §3.3.

1.2.
Junctions of domains with different limit dimensions. Invisibility questions are one motivation of the present article. The technique that we will propose to get R = 0 relies heavily on asymptotic analysis in junctions of massive domains with thin ligaments and developing new results in this field is another goal of the paper. The dumbbell (see Fig. 1.2, a)), namely two massive domains Ω ± connected by a thin cylinder Q ε with a cross-section of diameter ε, is a classical object in asymptotic analysis. There are many studies of the Neumann Laplacian showing that its spectrum 0 = λ ε 1 < λ ε 2 ≤ λ ε 3 ≤ . . . ≤ λ ε n ≤ . . . → +∞ (1.2) has the following distinguishing feature: the limit set is the union of three spectra, namely the spectra of the Neumann Laplacian in Ω ± and the Dirichlet problem for the operator −∂ 2 z on the axis of Q ε . After the pioneering works [4,1], this problem has been investigated in many papers in the original, Fig. 1.2, a), and modified, Fig. 1.2, b), formulations with different methods and goals, see e.g. [11,22,30,31,40,12,34,16,17,18,3]. In particular, complete asymptotic expansions of eigenvalues and eigenfunctions were constructed in [12] by the methods of matched asymptotic expansions. Expansions for the solutions to stationary mixed boundary-value problems were found in [22,30,31]  It is remarkable that convergence theorems for the spectra (1.2) and (1.3) have been obtained for domains Ω ± with arbitrary shapes (see e.g. [4]), but the applications of asymptotic analysis have only been made under additional simplifying assumptions that the boundaries ∂Ω ± are flat near the junctions and that the ligaments Q ε are perpendicular to them. The main asymptotic terms are the same for curved surfaces, but the higher-order terms are influenced by the curvature (cf. [30,31]). Besides, the derivation of asymptotically sharp estimates becomes much more complicated in the curved case, see [40,3] for details. However, the ligaments connecting massive domains are always assumed to be straight.
In this work, we diverge from the traditional formulation for junction problems by, first, considering curved tunnels (2.2) with variable widths (treating the straight wells Fig. 1.1, b) as a special case in Section 3.3) and, second, allowing the mid-line of the tunnels to meet the bottom non perpendicularly.
On the other hand, to simplify the treatment of the boundary layers, we make the assumption that the boundary of the geometry consists of straight segments near the junction zones.
The first asymptotic terms, which are needed in the fine-tuning procedure to find geometries where R = 0, are constructed by means of the method of matched asymptotic expansions. However, we emphasize that if one wishes to construct infinite asymptotic series, it is better to apply the method of compound expansions which crucially simplifies the iterative process (cf. [22,30,31] and others). The reason is that the limit problems are solved in the same function spaces, whereas the method of matched asymptotic expansions requires for the solutions with singularities of ever growing orders. In order to shorten the article, we decide not to present the construction of infinite asymptotic series. But this can be done.
The very novelty of the asymptotic analysis in this paper lies in the justification scheme in Section 6. First, we will work with the traditional weighted spaces with detached asymptotics, the norms of which contain the moduli of the scattering coefficients. Consequently, proving error estimates with these norms directly implies the justification of the asymptotics of the scattering matrix. Second, we will consider a Sobolev space endowed with a rather exotic norm, which is defined as the infimum of the norms of several components related to the structure of the junctions (see Section 6). On this occasion, it is worth to mention that the Sobolev and Hölder norms, even their weighted variants with diversified weights on functions themselves and their derivatives in different directions, cannot properly reflect entangled composite asymptotic structures of solutions in junctions of thin and massive domains, especially in the vectorial case, like in elasticity. In this way, our innovative trick allows us to take into account miscellaneous contributions of all geometric parts of the junctions in the norm. This approach is entirely new and is certainly expected to be helpful in examining other boundary value problems with singular perturbations for example in hybrid domains [27,30,33,38] and for elastic junctions [23,21,34,36] where reactions of thin fragments on longitudinal and transversal loadings are very discrepant.
1.3. Outline of the paper. We start in Section 2 by presenting the setting of the problem. In Section 3, first we give the main terms appearing in the asymptotic expansions of the scattering coefficients R, T . Then we present the fine-tuning procedure introduced in [6,7,5] which allows us to construct nonreflecting underwater topographies for surface waves at any prescribed k ≥ 0 and λ > λ † (k). In Section 4, we explain how to implement the method numerically and give several examples of non reflecting channels. Section 5 is dedicated to the formal asymptotic expansion of the scattering solutions. This provides us in a rather direct way the main asymptotic terms in the decomposition of the scattering coefficients R ε − , T ε (with the notation below). The most technical part, Section 6, contains the operator formulation of the problem (2.4)-(2.6) and the derivation of asymptotically sharp estimates for the solutions with respect to the norms of weighted spaces with detached asymptotics. We emphasize that these norms are closely connected with the asymptotic structures derived in Section 5, which makes it quite easy to obtain error estimates. In this part, we also prove the absence of trapped modes in the channel Π ε for ε > 0 small enough. We end the article with some concluding remarks. The main results of this article are Theorem 3.1 (non-reflecting geometries) and the approach of Section 6 to prove error estimates for asymptotic expansions using well-chosen norms.

2.
Setting of the water-wave problem.
2.1. Notation. Let Π := {x = (y, z) ∈ R × (−d; 0)} be a straight two-dimensional channel. Let L j , j = 1, . . . , J with J ∈ N := {0, 1, . . . }, be a simple smooth curve inside the lower half-plane {x : z < −d}, connecting the points Fig. 2.1, a). The length of L j is 2 j > 0. In a neighbourhood of L j we introduce the local coordinates (n j , s j ) where s j ∈ (− j ; j ) is the arc length and n j is the oriented distance to L j . We assume that L j intersects the bottom Γ d := R × {−d} of the channel at the angles α ± j ∈ (0; π) (with the line (y ± j ; +∞)×{−d}) and denote by L j a smooth extension of L j inside Π for the values s ∈ [− j −δ j ; j +δ j ] for some small δ j > 0. We define the domain entering the channel Π (see Fig. 2.1, b)) and set the thin curved strip (tunnel) ε j := T ε j \ Π. Here H ± j ∈ C ∞ [− j − δ j ; j + δ j ] are smooth profile functions such that H j := H + j + H − j > 0 and ε > 0 is a small parameter.  We consider the channel with several thin curved tunnels under the bottom, see Fig. 2.2, a). We denote Γ 0 := R × {0} the free surface of Π ε . We study the water-wave problem consisting of the Helmholtz equation the Neumann (no penetration) boundary condition ∂ ν u ε = 0 on ∂Π ε \ Γ 0 , (2.5) and the (kinematic) Steklov condition on the free surface Here, u ε is the velocity potential, ∆ is the Laplace operator and ∂ ν stands for the outward normal derivative so that ∂ ν = ∂ z on Γ 0 . Moreover k ≥ 0 is the wave number in the direction perpendicular to the plane R 2 x while λ = g −1 ω 2 is the spectral parameter, ω > 0 being the frequency of time-harmonic oscillations and g > 0 the acceleration of gravity. Note that in §3.3, we will consider wells as drawn in Fig. 2.2 , b). In this case, we impose the Neumann condition (2.5) at the ends of the wells.

2.2.
Surface waves and scattering matrix. It is known, see e.g. [13,24], that the continuous spectrum of the problem (2.4)-(2.6) coincides with the closed semi-axis [λ † ; +∞), where the cut-off value is given by Note in particular that for k = 0, we have λ † = 0. For any λ > λ † , introduce the functions w ± such that where the exponent a = a(λ) > 0 solves the equation a tanh(ad) = λ.
Observe that w ± satisfy the Helmholtz equation (2.4), the Neumann condition (2.5) on the bottom Γ d of the straight channel and the Steklov condition on the free surface Γ 0 . They correspond to surface waves propagating along the channel Π from ∓∞ to ±∞ (with a convention of a time-harmonic regime in e −iωt ). Now we focus our attention on the scattering of the waves w ± in the singularly perturbed channel Π ε . To describe separately the behaviours at +∞ and −∞, first we introduce two smooth cut-off functions χ ± ∈ C ∞ (R) such that 0 ≤ χ ± ≤ 1, The parameter 0 > 0 is chosen such that the tunnels (2.2) are all contained in the region {x : |y| < 0 } (in particular, we have − 0 < y − 1 < y + 1 < . . . < y − J < y + J < 0 ). In the following, we shall say that a function u satisfying (2.4)-(2.6) is outgoing if it admits the expansion for some coefficients A, B ∈ C and some remainderũ which decays exponentially for |y| ≥ 0 . More precisely, one can prove that necessarilyũ decays with the rate O(e − √ b 2 +k 2 |y| ), where b is the root of the transcendental equation in (π/(2d), 3π/(2d)). One can show that the problem (2.4)-(2.6) in Π ε admits a solution u ε − (resp. u ε + ) such that u ε − − w + (resp. u ε + − w − ) is outgoing. Due to (2.9), we have the representations where R ε ∓ , T ε ∓ ∈ C andũ ε ∓ decay exponentially for |x| ≥ 0 . The function u ε ∓ represents the total field associated with the incident field w ± incoming from ∓∞. Uniqueness for u ε ∓ occurs if and only if trapped modes are absent. We remind the reader that trapped modes are solutions to (2.4)-(2.6) (without source term nor incident field) which decay exponentially for |x| ≥ 0 . They can appear only for a discrete set of wavenumbers k (see e.g. [10]). In what follows (see Theorem 6.6), we shall prove that trapped modes do not exist for ε small enough so that u ε ∓ are well-defined. In (2.11), R ε ∓ and T ε ∓ are usually called reflection and transmission coefficients. It is known that T ε − = T ε + and to simplify we shall denote is uniquely defined, symmetric and unitary. In particular, we have When R ε − = 0, the backscattered field associated with u ε − (see (2.11)) is evanescent. Note that in this situation, due to (2.13), we also have R ε + = 0, |T ε | = 1 and as mentioned in the introduction, we say that the family of tunnels ε 1 , . . . , ε J is non-reflecting.

Non-reflecting topographies
Below we will compute asymptotic expansions of the functions u ε ± defined in (2.11) with respect to ε. Then we will derive expansions of the scattering coefficients of the matrix S ε . Since the procedure is a bit long, we first give the main results and explain how to use them to construct non-reflecting topographies, that is to obtain R ε − = 0. For the construction of the asymptotics and the proof of error estimates, we refer the reader to Sections 5 and 6 respectively.

Asymptotics of the scattering coefficients.
In what follows, for the reflection and transmission coefficients we shall consider the expansions: Here R 0 − , R − , T 0 , T are complex constants which are independent of ε and R ε − , T ε correspond to some abstract remainders. First we will show that R 0 − = 0 and T 0 = 1. This is natural because when ε → 0, the thin tunnels disappear and the incident wave w + propagates in Π without being perturbed. Next for the correction terms in (3.1), we will establish the following important formulas Here N := (2a) −1 (e 2ad − e −2ad ) + 2d is a normalisation factor and H j (s) = H + j (s) + H − j (s) > 0 corresponds to the rescaled thickness of the curved strip ε j . Moreover, the functions v 0 j are defined as the solutions to the problems In §4.1, we will give explicit examples of setting where the geometrical parameters are such that R − = 0. In this case, the family of tunnels is almost non-reflecting. More precisely, in this situation a perturbation of the straight channel Π of order ε produces a reflection of order ε 2 only. However, due to the presence of the remainder R ε − in the representation (3.1), the identity R − = 0 does not yet suffice to guarantee the non-reflectability. Therefore, we have to refine our strategy.
3.2. The fine-tuning procedure. Let us fix once for all a family of geometrical parameters such that R − = 0. Now we shift two tunnels along the abscissa axis. More precisely, for h = (h 1 , h 2 ) ∈ R 2 we assume that ε 1 , ε 2 (reindex the tunnels if necessary) are changed into ε We regard h = (h 1 , h 2 ) ∈ R 2 as small parameters which are independent of ε. We denote respectively R ε − (h) and T ε (h) the reflection and transmission coefficients in the geometry Here where R − (h) is an abstract remainder. Using that R − (0, 0) = 0 (this results from the particular choice of the geometrical parameters (3.5)), we find that there holds R ε − (h) = 0 if and only h ∈ R 2 solves the following system of transcendental equations (3.10) This is equivalent to have To continue the procedure, we have to assume that the geometrical parameters y ± j , L j , j , H ± j are also such that the matrix M is invertible (again, see §4.1 for examples where this assumption is satisfied). Then h solves (3.11) if and only if it is a solution to the problem Thus we obtain a fixed point equation. Let us prove that for a given > 0 small enough, G ε (·) is a contraction in the closed ball B := {h ∈ R 2 : |h| ≤ } for all ε ∈ (0; ε 0 ( )].
We emphasize that c in (3.14) and in the estimates below is independent of small enough. Moreover, we will prove that h → R ε − (h) is smooth. We will deduce that the remainder h → R ε − (h) in (3.8) is also smooth. More precisely, we will show (see Theorem 5.2 and §6.6) the estimates, for all δ ∈ (0; 1/2), This allows us to write, using also the invertibility of M, and guarantees that G ε (·) maps B to B for small enough and all ε ∈ (0; ε 0 ( )]. On the other hand, which ensures that G ε (·) is a contraction for small enough and all ε ∈ (0; ε 0 ( )]. Now the Banach contraction mapping principle proves the existence of a unique solution h sol to (3.13) in B . Moreover, we have the estimate This leads us to the announced assertion on non-reflectability.
in the channel Π ε (h sol ), the wave w + passes the family of tunnels without reflection.
Note that according to formula (3.3), we have m T (h sol ) ≤ 0. And there holds m T (h sol ) < 0 as soon as the parameters are not such that [k = 0 and w + (y − j , −d) = w + (y + j , −d), ∀j = 1, . . . , J]. When m T (h sol ) < 0, at least for ε small enough, the transmission coefficient T ε (h sol ) has a negative imaginary part: m T ε (h sol ) < 0. This means that after passing the tunnels the wave w + certainly gets a phase shift, although it is of the order ε only.
Remark 3.2. To achieve non-reflectability we have only varied the positions of the tunnels, although the other geometric parameters (3.5) could evidently be varied as well. However, doing so, the smooth dependence of the reflection coefficient R ε − (h) on the perturbation parameter h would require other arguments, which would possibly be more involved than the simple change of coordinates in (6.68).
3.3. The case of wells. We mainly keep the notation of Section 2.1 but replaced the tunnels ε j by the wells Fig. 2.2, b)) and consider the original water-wave problem (2.4)-(2.6) in the new channel Π ε . Then adapting the approach below, we find that the reflection and transmission coefficients for the scattering solution u ε − (see (2.11)) admits the asymptotic expansions Here again N = (2a) −1 (e 2ad − e −2ad ) + 2d and H j (s) = H + j (s) + H − j (s) > 0 corresponds to the rescaled thickness of the strip ε j− . Moreover, the functions v 0 j are defined as the solutions to the problems Note that the last Neumann condition originates from the no-penetration condition ∂ ν u ε (x) = 0 on the end face {x ∈ ε j : s j = 0} of the curved well ε j− . The mixed boundary-value problem (3.19) is still uniquely solvable. The justification of asymptotics in this case is completely similar to the case of tunnels below. As a consequence, the fine-tuning scheme of §3.2 still works and provides examples of families of non-reflecting wells.

Numerical experiments.
In this section, we implement numerically the approach leading to the Theorem 3.1 above.
4.1. Preliminaries calculus. The first step in the procedure presented in the previous section consists in finding geometrical parameters y ± j , L j , j , H ± j , j = 1, . . . , J, such that the coefficient R − in (3.2) satisfies R − = 0 and the matrix M in (3.12) is invertible. To proceed, we divide the analysis according to the value of k.
Inserting the latter relation in (3.2) and using that w Take J = 3, H j = H, j = , j = 1, 2, 3 so that H j = H, j = 1, 2, 3. In other words, we consider a situation with three similar tunnels. Then set In this case, according to (4.1), we have Then, when we translate the position of the tunnels 1 and 2 respectively by h 1 and h 2 , according to Thus the matrix M in (3.12) is invertible if and only if the ratio e 2iy − 2 a /e 2iy − 1 a = e 2iπ/3 is not a real number, which is indeed the case.
Case k > 0. When k > 0, to simplify the presentation, we assume that H j (s) = H 0 j does not depend on s (the width of the tunnels is constant). Then the function v 0 j satisfies We can also write v 0 Then we have Note that one can verify that taking the limit k → 0 in (4.5), we get back the relation (4.1) obtained for k = 0. In order to have R − = 0, we take J = 3, H j = H, j = , j = 1, 2, 3 so that A j = A, B j = B, H j = H, j = 1, 2, 3. In other words, again we consider three similar tunnels. Then we set Now translating the position of the tunnels 1 and 2 respectively by h 1 and h 2 , according to (4.1), we find with h 3 = 0. Then for j = 1, 2, we obtain Thus for our choice (4.4) of the positions of the tunnels, we obtain As a consequence, the matrix M is indeed invertible.
Remark 4.1. If J = 2, which corresponds to the case of only two tunnels, the matrix M is always non invertible because |R ε − | is invariant with respect to the coordinate change (y, z) → (y − h 0 , z). In this case we would only have the real parameter h 2 − h 1 instead of the couple (h 1 , h 2 ) which is not enough to cancel one complex coefficient.
Above we presented two examples. Of course, the list can be enlarged readily.

4.2.
Numerical procedure. Numerically, we solve the fixed point problem (3.13) using an iterative procedure. More precisely, we start from some arbitrary h 0 := (h 0 1 , h 0 2 ) ∈ R 2 whose norm is not "too large" and then, for a given h n ∈ R 2 , we set .
Let us explain how to compute the right hand side in (4.6). Using (3.9) in (3.8), we get Extracting the real and imaginary parts, we deduce from the definition of G ε (h n ) (see (4.6)) that there holds .
Using the latter relation in (4.6), we see that h n+1 is related to h n by the simple formula .
is computed at each step n ≥ 0 solving the scattering problem Note that the geometry of the channel Π ε (h n ) depends on the step n ≥ 0. Then according to the representation (2.11), the coefficient R ε − (h n ) is given by where N = (2a) −1 (e 2ad − e −2ad ) + 2d (see after (3.3)). At each step n ≥ 0, we approximate the solution of problem (4.8) with a P2 finite element method in Π ε 11 (h n ) := {x = (y, z) ∈ Π ε (h n ) | |y| < 11}. We emphasize in particular that at each step, it is necessary to mesh the domain. At y = ±11, a truncated Dirichlet-to-Neumann map with 10 terms serves as a transparent boundary condition. Computations are implemented with FreeFem++ 1 while results are displayed with Paraview 2 .

4.3.
Results. The results we obtain are displayed in Figures 4.1-4.3. We start with h 0 = (0, 0) and set ε = 0.2. For the Figure 4.1, we take k = 0, λ = 1 and we fix the geometrical parameters as in (4.2) with y − 1 = −π/(3a) − 2π/a, y − 2 = 0, y − 3 = π/(3a) + 2π/a and y + j = y − j + π/2. For the Figure 4.2, we set the parameters as in Figure 4.2 except for the definition of the third tunnel. Here ε 3 is constructed from the half circle passing through P ± 3 = (y ± 3 , −d) with y − 3 = π/(3a) + π/a and y + 3 = y − 3 + 3π/4 (we enlarge the radius of the third tunnel to break the symmetry). We emphasize that in this setting we do not have R − (0, 0) = 0. However, this is not a problem and one can prove that the sequence (h n ) constructed via the recursive relation (4.7) converges to some h sol ∈ R 2 such that R ε − (h sol ) = 0. Finally, for the Figure 4.3, we take k = 0.5, λ = 1 and the geometrical parameters as in (4.4). For each setting, we represent the real parts of u ε − (h n ) (top), w + (middle) and u ε − (h n ) − w + (bottom) after 15 iterations (n = 15). As expected, we observe that the amplitude of the scattered field u ε − (h n ) − w + is very small in the incident direction (no reflection). Interestingly, the fixed point procedure converges though the parameter ε (the width of the tunnels) is not very small (here ε = 0.2). As predicted by the theory, one can also observe a small phase shift between the incident and the transmitted fields.

Derivation of the asymptotic expansions
In this section, we compute an asymptotic expansion of the function u ε − defined in (2.11) with respect to ε. In the straight channel Π, at least "far" from the junctions with the tunnels, we make the ansatz Here and in what follows, the dots stand for inessential higher-order terms. Our goal is to identify the functions u 0 , u . This will allow us to compute the terms in the ansatz for the reflection and transmission coefficients The justification of the expansions with the proof of error estimates will be given in Section 6.   5.1. Main asymptotic terms. In Π, as a first approximation of u ε − , it is natural to take w + because when ε tends to zero, the tunnels disappear and then the incident wave does not suffer from scattering. Therefore we set u 0 = w + . Now let us focus our attention on the approximation of u ε − in the thin tunnels ε j . To simplify the notation, we omit the index j. As usual, we stretch the transversal section of the tunnel considering the change of variables In (5.3), ∂ n = ∂/∂n and so on, A(n, s) = 1 + κ(s)n and κ(s) is the curvature of L at the point s ∈ (− ; ). The unit outward normal vector ν ε ± on the lateral sides of the tunnel ε is as follows: Notice that this vector is written in the local coordinate system (n, s) and therefore Following the standard dimension reduction procedure in thin domains, see e.g. [28,Chap. 15], in ε , we consider the ansatz where the functions v 0 , v and v have to be determined. Inserting (5.6) into the initial problem (2.4)-(2.6) restricted to ε , using (5.4), (5.5) and collecting the terms of order ε −1 , we arrive at Since the problem is homogeneous, we have to set v (ζ, s) = v (s). Now collecting the terms at order ε 0 = 1 when inserting (5.6) in (2.4)-(2.6) restricted to ε , we obtain The compatibility condition for (5.7) writes Since the length of Υ(s) is equal to H(s) = H + (s) + H − (s) > 0 (the rescaled thickness of the curved strip ε ), (5.8) turns into the ordinary differential equation In order to close problem (5.9), we have to impose boundary conditions. To proceed, we match the value of v 0 with the one of u ε − | Π at order ε 0 = 1 at the junction points P ± . This gives us v 0 (± ) = w + (y ± , −d).

Correction terms.
For the moment, we have formally derived an expansion at order ε 0 = 1 of u ε − in Π ε . Now, we wish to get a better approximation at order ε. In particular, we want to identify the term u in the ansatz u ε − = w + + εu + . . . in Π (see (5.1)). Inserting this expansion in (2.4)-(2.6) and making ε → 0, we find that u must satisfy In order to define completely (uniquely) u , we have to impose conditions at the junction points P ± j . To proceed, we will employ the method of matched asymptotic expansions, cf. [44,15], which consists in matching the behaviour of u ε − in a neighbourhood of P ± j with the behaviour of a inner expansion of u ε − at infinity. Note that to obtain a non zero corrector u , we have to allow singular behaviours at the P ± j . In the following, to simplify notation, again we omit the index j. In the junction zone near the endpoints P ± of the curve L, we anticipate the existence of a boundary layer phenomenon. To capture it, in this region the asymptotic behaviour of u ε − will be described in the stretched coordinates To simplify the proofs, especially the justification procedure in Section 6, cf. Remark 5.1, we assume that the profile functions H ± in (2.2) are constants and that the curve L is straight for s ∈ (− −δ; − +ρ) and s ∈ ( − ρ; + δ) for some ρ > 0. Thus, we see that changing the coordinate x → ξ ± and setting formally ε = 0 transform the domain Π ε into the union of the upper half-plane R 2 + := {ξ = (ξ 1 , ξ 2 ) : ξ 2 > 0} and the rotated strip θ ± := {ξ : (η ± , ζ ± ) := (ξ 1 cos α ± − ξ 2 sin α ± , ξ 1 sin α ± + ξ 2 cos α ± ) ∈ R × Υ(± )} (5.13) (see Figure 5.1). In the vicinity of the point P ± , we introduce the inner expansion of u ε − : Since ∆ x − k 2 = ε −2 (∆ ξ ± − ε 2 k), the Laplacian is the main asymptotic part of the Helmholtz operator in the variables ξ ± . Furthermore, on the boundary ∂Π ε near P ± , the normal derivative ∂ ν(x) is nothing but ε −1 ∂ ν(ξ ± ) , where ν(ξ ± ) is the outward normal on the boundary of the junction Θ ± := R 2 + ∪ θ ± . Therefore, we deduce that the functions U 0± , U ± must satisfy According to the conditions one imposes at infinity in Θ ± , Problems (5.15) can admit non zero solutions. Clearly Y 0 (ξ) = 1 is one of them. One can also show (see Proposition 6.7 below) that there are some functions Y 1± , Y 2± ∈ H 1 loc (Θ ± ) satisfying (5.15) such that when |ξ| → +∞, Here, K 1± , K 2± are constants depending on the width and the tilt angle of the strip θ ± . Moreover H 1 loc (Θ ± ) denotes the set of functions whose H 1 norm is finite in any bounded subdomain of Θ ± . We will look for U 0± , U 1± as combinations of the Y 0 , Y 1± , Y 2± . One could add other solutions of (5.15) in the expansions but this is not needed for the calculus of the first two terms. Now we match the behaviours of the two outer expansions of u ε − in Π and in ε with the behaviour of the inner expansion of u ε − in an intermediate region where |x − P ± | → 0 and |ξ ± | → +∞. Using the stretched coordinates (5.12), a Taylor expansion at the point P ± yields Here, we took into account that ∂ z w + (y ± , −d) = 0 and set r ± := |x − P ± | = (|y − y ± | 2 + |z + d| 2 ) 1/2 . On the other hand, in the vicinity of P ± we also have Identifying the powers in ε 0 = 1, first, we obtain In the inner expansion u ε − (x) = U 0± (ξ ± ) + εU ± (ξ ± ) + . . ., let us look for U ± as When |ξ ± | → +∞, we have Comparing (5.18), (5.19) and (5.22), we deduce that we ought to take c 1± = ∓H(± )∂ s v 0 (± ) and c 2± = ∂ y w + (y ± , −d).

(5.23)
Note that the constants c 0± in (5.21) still remain unfixed. Moreover, according to (5.22), the representation of u ε − (x) for ξ ± ∈ R 2 + such that |ξ ± | → +∞ includes the logarithmic term Therefore, we will impose that the correction term in the outer expansion u ε − = w + + εu − + . . . in Π solves (5.11) and behaves as as r ± → 0 + . One can prove the existence of a unique outgoing solution of this problem. We denotẽ u (P ± ) the corresponding constant in (5.25). According to the definition of outgoing solutions (see (2.9)), we have the decomposition u = χ − R − w − + χ + T w + + . . . (5.26) where the dots correspond to some term which is exponentially decaying at infinity in Π. Once u has been fixed, one obtains that the constants c 0± in (5.21) are given by Note that the term ln ε appears because of the change of variable ξ ± → x in (5.24). Continuing the matching procedure, one would write a problem similar to (5.9)-(5.10) for the correction term εv (s, ln ε) in the outer expansion in the strip ε and regard (5.27) as the data in this problem. This term depends linearly on ln ε, but it is not important in the sequel.
Remark 5.1. Note that considering the stretching (5.12) in the strip ε leads for the variables (s, n) to the transformation (s, n) → ε −1 (s ∓ l, n). Omitting the above assumption that L is straight in a neighbourhood of P ± would lead to a much more complicated change of variables The Laplacian is still the main asymptotic part of the Helmholtz operator in the new variables (5.28), but the remainder is a second order differential operator with degenerating coefficients at the point P ± . This would make the estimation of the discrepancies in Section 6 much more cumbersome, cf. [3].

5.3.
Computing the scattering coefficients. Once the singular solution u − has been found, it is straightforward to compute the coefficients R − , T in (5.26). To this end, first we observe that the quantities are independent of D > 0 (to show this property, integrate by parts and use that u − , w ± satisfy the homogeneous Helmholtz equation in Π). Inserting the representation (5.26) in (5.29) and taking the limit as D → +∞, we obtain (5.30) Here N is the same as in (3.3), such that with Π(D) := {(y, z) ∈ Π : |y| < D}. Integrating by parts in Π δ (D) for a given D > 0 and taking the limit δ → 0 + , we also find (5.31) Here, (r ± j , ϕ ± j ) ∈ R + × (0; π) is the polar coordinate system centered at P ± j . Note that the logarithmic singularities in (5.25) have been taken into account to obtain the second line of (5.31). Now using that v 0 j (± j ) = w + (y ± j , −d) (see (5.10)), we get which are nothing but formulas (3.2) and (3.3). As for the asymptotic expansions the following estimate will be proven in Section 6.5.

5.4.
The global asymptotic approximation. The matching method we have used up to now yields asymptotic expansions of u ε − in different zones. One may be interested in computing a global approximation of u ε − in the whole domain Π ε . To obtain such an approximation, and more generally, to model boundary-value problems of type (2.4)-(2.6) on junctions of domains with different limit dimensions, an approach based on self-adjoint extensions of differential operators has been proposed in [30,38]. Using this approach one can indeed model the initial problem by the system consisting of the equations (5.9)-(5.10), (5.11)-(5.26) and derive error estimates. Unfortunately we obtain limited accuracy and these estimate are not sufficient for the main goal of this paper.
Instead of using this model, we will work on the different expansions of u ε − and glue them to obtain a global approximation. The traditional approach [15] to do that, based on the use of partitions of unity, does not provide sufficient accuracy for the purpose of the paper. Instead, we will employ a trick with cut-off functions with overlapping supports, as introduced in [28,Chap. 2]. Note that the asymptotic structures we get with this technique have been shown to be equivalent with the method of compound asymptotic expansions (see [45,28]).
The tunnel T ε j enters the strip Π through the two junction segments (see the bold line in Fig. 2.1, b)) Introduce smooth cut-off functions such that for j = 1, . . . , J, we have Then we define X ε as In (5.37), the radii 0 < R ± j < R ± j are chosen such that X ε± j = 1 in a neighbourhood of Ξ ε± j . Finally we set X ε 0 := X ε | Π and X ε j := X ε | ε j , j = 1, . . . , J. Note that with a slight abuse of notation, we make no distinction between these functions and their extensions by zero to Π ε . Observe that X ε 0 is supported in Π while X ε j is supported in ε j , j = 1, . . . , J.
Now we have everything to define our global approximation. In the straight part Π, set In the curved thin strips ε j , we set These formulas need explanations. In Π, the right hand side of (5.39) first contains the outer expansion (5.1) multiplied by the cut-off function X ε 0 which annuls it in a neighbourhood of the junction segments (5.36). It also contains (second line) the inner expansions (5.14) multiplied by cut-off functions which annul them far from the junctions. Note that the terms matched in §5.2 are present in both expansions. However, their duplications are cancelled by the last subtrahend in (5.39) (which coincides exactly with these matched terms). A similar structure is found in (5.40). However, the corrector εv j (ln ε, s j ) of the outer expansion in ε j is not included to the first line on the right hand side of (5.40). This fact simplifies the subtrahend in (5.40), although it will lead to additional difficulties in the estimation of the discrepancies in Section 6.5. The authors do not know a simpler asymptotic structure which still provides asymptotically sharp error estimates.

Justification of asymptotics.
In this section, we prove error estimates to justify the asymptotic expansions derived formally in §5.
The next problem we consider is the mixed boundary value problem in the tunnel ε j : Here, H 1 0 ( ε j ; Ξ ε± j ) is the Sobolev space of functions satisfying the Dirichlet conditions on the junction segments Ξ ε± j defined in (5.36). Noticing that d t ((h + t) −1 ) = −(h + t) −2 and using an integration by parts, one can establish the following classical one-dimensional Hardy inequality for all > 0, h > 0. This allows one to show that the standard Sobolev norm in H 1 0 ( ε j ; Ξ ε± j ) is equivalent to the weighted norm where ρ j (s) = min(|s − j |, |s + j |). This observation and the Riesz representation theorem imply the following simple assertion. Lemma 6.3. If F ε j ∈ H 1 0 ( ε j ; Ξ ε± j ) * , then problem (6.18) has a unique solution V ε j ∈ H 1 0 ( ε j ; Ξ ε± j ) and there holds the estimate where, for some ε j > 0, the number C j is independent of F ε j and ε ∈ (0; ε j ]. Finally, we consider the problem (see a picture of the unbounded domain Θ ± j in Fig. 5.1) in the weighted space W 1 τ,σ (Θ ± j ) endowed with the norm Lemma 6.4. Let τ ∈ (−1; 0), σ ∈ (0; π/H j (± )) and F ± j ∈ W 1 −τ,−σ (Θ ± j ) * . Then, the problem has a unique solution U ± j ∈ W τ,σ (Θ ± j ) and the estimate 6.4. Construction of an approximate inverse operator. We will find the operator (6.36) as the sum (6.39) where the terms will be defined below. We fix a functional F ε ∈ W 1,ε −β,−γ (Π ε ) * and denote its norm by N ε .
Step 3. Let us calculate the discrepancy (mismatch with respect to the data F ε ) left in the problem (6.3) by the sum (6.46) u ε = u ε 0 + u ε 1 + . . . + u ε J . We denote u εj 0 = u ε 0 ε j . For every ψ ∈ W 1 −β,−γ (Π ε ) with compact support (we can work by density), we find In the following, we shall say that a functional 0 is a sum of terms located in ε-neighbourhoods of the points P ± 1 , . . . , P ± J . One can prove that it is O(ε 0 ). We will compensate it by boundary layers in the next step. In contrast, we prove now that the f εj 0 are small as ε tends to zero.
Observe that the L 2 -norm of ψ ∈ W 1,ε −β,−γ (Π ε ) has the weights of order ε −γ−1 in the vicinity of the points P ± j and of order ε −γ in ε j , see (6.24) and (6.25). Note also that u εj 0 is constant in B R ± j (P ± j ). Recalling the formulas (5.3)-(5.5) and the relation |A(n, s) − 1| ≤ c ε in ε j , we find (6.48) Now using (6.41), in ε j we obtain one can write On the other hand, one can show the estimate Inserting (6.49) and (6.50) in (6.48), finally we obtain (6.51) |f εj 0 (ψ)| ≤ cN ε ε γ+1/2 ψ; W 1,ε −β,−γ (Π ε ) . Now let us turn to the discrepancy left by the terms u ε j in the sum (6.46). These functions vanish at the segments Ξ ε± j and we have Finally, since F ε is antilinear, we get (6.53) Note that to obtain the identity, we used in particular the sequence of equalities The second element of the right hand side of (6.53) is supported in ε-neighbourhoods of the points P ± 1 , . . . , P ± J . Now we compensate it by boundary layers terms.
Here, we have taken into account the width of the tunnel O(ε) and the evident relation ρ t e −2δρ/ε dρ ≤ c t,δ ε t+1 .
For the remaining term I ε j , proceeding as in Section 6.4, Step 3, we obtain (6.66) ε γ (ε + ρ j )I ε j ; L 2 ( ε j ) ≤ cε γ+3/2 (1 + | ln ε|). Comparing with the other previous estimates, we find that this is the largest term. This concludes the proof of Theorem 5.2. Indeed, the positive number δ = −γ in (5.35), cf. (6.15), can be chosen as small as one wishes. However, it is not possible to set δ = 0 because R − (ln ε) is a linear function of ln ε.
6.6. Fine-tuning. Now we explain how to prove that there is > 0 small enough such that the constant c δ appearing in the estimate (5.35) of Theorem 5.2 can be chosen independent of h ∈ B = {h ∈ R 2 : |h| ≤ } for all ε ∈ (0; ε 0 ( )]. Let be the channel (2.3) with two tunnels shifted according to (3.6). We set x h = (y h , z) with y h = 1 − χ 1 (y) − χ 2 (y) y + χ 1 (y)(y − h 1 ) + χ 2 (y)(y − h 2 ) (6.68) and χ j is a smooth cut-off function such that The change of coordinates x → x h is non-singular for small h ∈ R 2 and transforms Π ε (h) into Π ε . Moreover, it is independent of ε. The corresponding change affects the differential operators of the problem (2.4)-(2.6) only in the Laplacian ∆ which turns into where D is a first order differential operator with smooth and compactly supported coefficients, depending smoothly also on the parameter h = (h 1 , h 2 ). Then working as in the classical proofs of perturbations theory for linear operators (see [19,Chapter 7,§6.5], [14,Chap. 4]), we can use the above change of variables to compare the solutions of the problem (2.4)-(2.6) in the same geometry Π ε and show that they have smooth dependence with respect to h. We refer the reader to [5, §6.3] for more details. Proof. First, for a source term F satisfying the conditions (6.71) below, we consider the problem The natural variational formulation of the Neumann problem (6.69), cf. [25], writes ∇U, ∇V Θ ± = (F, V) Θ ± , ∀V ∈ H. (6.70) Here (·, ·) Θ ± is the scalar product of L 2 (Θ ± ) and H is the completion of C ∞ c (Θ ± ) (infinitely differentiable and compactly supported functions) with respect to the norm V; H := ∇V; L 2 (Θ ± ) 2 + V; L 2 (Θ ± ∩ B R (0)) 2 1/2 .

Conclusion
In this article, we presented a method to construct non-reflecting perturbations of the bottom of a channel for a water-wave problem. To proceed, we considered singular perturbations of the geometry with thin curved channels. With this approach, we showed how to get R = 0 and we proved that we cannot get T = 1. As a consequence, the transmitted field exhibits a phase shift with respect to the incident field. Let us mention that acoustic waveguides and water channels are somehow opposite of each other in the following sense: for the former, regular perturbations of the boundary, Fig. 1.1, a), may at most be non-reflecting (R ε = 0) while wells, Fig. 1.1, b), may be perfectly invisible (T ε = 1). For the latter these properties are reversed. To obtain the main result of this article (construction of non reflecting perturbations of the bottom), we extend the results known for asymptotic expansion in junctions of massive bodies and thin ligaments. More precisely, we considered situations where the ligaments have a non constant width and where the ligaments do not arrive perpendicularly to the massive body. These studies were not performed in literature. Finally, let us mention that another approach to achieve invisibility has been proposed in [8,9] for acoustics problems. It would be relevant to study if we can adapt it to deal with water-wave problems.