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

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) [26,29,42,43,46]). 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(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(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 ≠ 2 k 2 or equivalently, λ ≠ 2 k tanh ( 2 k d ) .

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Fig. 1.

(a) Gently sloping bottom. (b) Bottom with wells.

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(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 ≠ 2 k 2 which appears when considering smooth perturbations (Fig. 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(b) and Section 3.3.

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. 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 (1.2) 0 = λ 1 ε < λ 2 ε ⩽ λ 3 ε ⩽ ⋯ ⩽ λ n ε ⩽ ⋯ → + ∞ has the following distinguishing feature: the limit set (1.3) { λ n 0 = lim ε → 0 + λ n ε } is the union of three spectra, namely the spectra of the Neumann Laplacian in Ω ± and the Dirichlet problem for the operator − ∂ z 2 on the axis of Q ε . After the pioneering works [1,4], this problem has been investigated in many papers in the original, Fig. 2(a), and modified, Fig. 2(b), formulations with different methods and goals, see e.g. [3,11,12,16–18,21,30,31,34,40]. 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 [21,30,31] by the method of compound asymptotic expansions in dimensions d ⩾ 3 and d = 2 .

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Fig. 2.

(a) Dumbbell. (b) Junction of a massive body and thin rods.

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 [3,40] 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(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. [21,30,31] 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 [22,23,34,36] where reactions of thin fragments on longitudinal and transversal loadings are very discrepant.

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 [5–7] which allows us to construct non-reflecting 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.

2.1. 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 (2.1) P j − = ( y j − , − d ) and P j + = ( y j + , − d ) with y j − 1 + < y j − < y j + < y j + 1 − , see Fig. 3(a). The length of L j is 2 ℓ j > 0 . In a neighbourhood of L j we introduce the local coordinates ( n j , s j ) where s j ∈ ( − ℓ j ; ℓ 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 ˆ j a smooth extension of L j inside Π for the values s ∈ [ − ℓ j − δ j ; ℓ j + δ j ] for some small δ j > 0 . We define the domain (2.2) T j ε : = { x : s j ∈ ( − ℓ j − δ j ; ℓ j + δ j ) , − ε H j − ( s j ) < n j < ε H j + ( s j ) } entering the channel Π (see Fig. 3(b)) and set the thin curved strip (tunnel) ϖ j ε : = T j ε ∖ Π . Here H j ± ∈ C ∞ [ − ℓ j − δ j ; ℓ j + δ j ] are smooth profile functions such that H j : = H j + + H j − > 0 and ε > 0 is a small parameter.

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Fig. 3.

(a) Straight channel Π and defining curves L j . (b) Tunnel defined by a curve.

We consider the channel (2.3) Π ε : = Π ∪ ϖ 1 ε ∪ ⋯ ∪ ϖ J ε with several thin curved tunnels under the bottom, see Fig. 4(a). We denote Γ 0 : = R × { 0 } the free surface of Π ε . We study the water-wave problem consisting of the Helmholtz equation (2.4) − Δ u ε + k 2 u ε = 0 in Π ε , the Neumann (no penetration) boundary condition (2.5) ∂ ν u ε = 0 on ∂ Π ε ∖ Γ 0 , and the (kinematic) Steklov condition on the free surface (2.6) ∂ ν u ε = λ u ε on Γ 0 . Here, u ε is the velocity potential, Δ 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 Section 3.3, we will consider wells as drawn in Fig. 4(b). In this case, we impose the Neumann condition (2.5) at the ends of the wells.

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Fig. 4.

(a) Tunnels and (b) wells in the water domain Π ε .

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 λ † = λ † ( k ) : = k tanh ( k d ) = k e k d − e − k d e k d + e − k d . Note in particular that for k = 0 , we have λ † = 0 . For any λ > λ † , introduce the functions w ± such that (2.7) w ± ( y , z ) = e ± i y a 2 − k 2 ( e a ( z + d ) + e − a ( z + d ) ) , where the exponent a = a ( λ ) > 0 solves the equation a tanh ( a d ) = 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 − i ω t ).

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 , (2.8) χ ± = 1 for ± y ⩾ 2 ℓ 0 and χ ± = 0 for ± y ⩽ ℓ 0 . The parameter ℓ 0 > 0 is chosen such that the tunnels (2.2) are all contained in the region { x : | y | < ℓ 0 } (in particular, we have − ℓ 0 < y 1 − < y 1 + < ⋯ < y J − < y J + < ℓ 0 ). In the following, we shall say that a function u satisfying (2.4)–(2.6) is outgoing if it admits the expansion (2.9) u = χ − A w − + χ + B w + + u ˜ for some coefficients A , B ∈ C and some remainder u ˜ which decays exponentially for | y | ⩾ ℓ 0 . More precisely, one can prove that necessarily u ˜ decays with the rate O ( e − b 2 + k 2 | y | ) , where b is the root of the transcendental equation (2.10) − b tan ( b d ) = λ in ( π / ( 2 d ) , 3 π / ( 2 d ) ) . 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 (2.11) u − ε = χ − ( w + + R − ε w − ) + χ + T − ε w + + u ˜ − ε u + ε = χ + ( w − + R + ε w + ) + χ − T + ε w − + u ˜ + ε , where R ∓ ε , T ∓ ε ∈ C and u ˜ ∓ ε decay exponentially for | x | ⩾ ℓ 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 | ⩾ ℓ 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 T ε = T − ε = T + ε . The scattering matrix (2.12) S ε : = R − ε T ε T ε R + ε is uniquely defined, symmetric and unitary. In particular, we have (2.13) | R − ε | 2 + | T ε | 2 = 1 and | R + ε | 2 + | T ε | 2 = 1 (conservation of energy). 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 ϖ 1 ε , … , ϖ J ε is non-reflecting.

3.1. Asymptotics of the scattering coefficients

In what follows, for the reflection and transmission coefficients we shall consider the expansions: (3.1) R − ε = R − 0 + ε R − ′ + ε R ˜ − ε ; T ε = T 0 + ε T ′ + ε T ˜ ε . 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 (3.2) R − ′ = − i 2 N a 2 − k 2 ∑ j = 1 J ∫ − ℓ j ℓ j H j ( s ) ( ( d v j 0 d s ( s ) ) 2 + k 2 ( v j 0 ( s ) ) 2 ) d s and (3.3) T ′ = − i 2 N a 2 − k 2 ∑ j = 1 J ∫ − ℓ j ℓ j H j ( s ) ( | d v j 0 d s ( s ) | 2 + k 2 | v j 0 ( s ) | 2 ) d s . Here N : = ( 2 a ) − 1 ( e 2 a d − e − 2 a d ) + 2 d is a normalisation factor and H j ( s ) = H j + ( s ) + H j − ( s ) > 0 corresponds to the rescaled thickness of the curved strip ϖ j ε . Moreover, the functions v j 0 are defined as the solutions to the problems (3.4) − d d s ( H j ( s ) d v j 0 d s ( s ) ) + k 2 H j ( s ) v j 0 ( s ) = 0 for s ∈ ( − ℓ j ; ℓ j ) v j 0 ( ± ℓ j ) = w + ( y j ± , − d ) . In Section 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 (3.5) y j ± , L j , ℓ j , H j ± , j = 1 , … , J , 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 ϖ 1 ε , ϖ 2 ε (reindex the tunnels if necessary) are changed into ϖ 1 ε ( h ) , ϖ 2 ε ( h ) with (3.6) ϖ j ε ( h ) : = { x = ( y , z ) : ( y − h j , z ) ∈ ϖ j ε } , j = 1 , 2 . 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 (3.7) Π ε ( h ) : = Π ∪ ϖ 1 ε ( h ) ∪ ϖ 2 ε ( h ) ∪ ϖ 3 ε ∪ ⋯ ∪ ϖ J ε . Similarly to (3.1), we have the expansion (3.8) R − ε ( h ) = 0 + ε R − ′ ( h ) + ε R ˜ − ε ( h ) . Here R − ′ ( h ) is given by (3.2) with, for j = 1 , 2 , v j 0 defined as the solution to (3.4) with a data w + ( y j ± , − d ) replaced by w + ( y j ± + h j , − d ) . One observes that the map h ↦ R − ′ ( h ) is analytic. Therefore, we have the expansion (3.9) R − ′ ( h ) = R − ′ ( 0 , 0 ) + h 1 ∂ R − ′ ∂ h 1 ( 0 , 0 ) + h 2 ∂ R − ′ ∂ h 2 ( 0 , 0 ) + R ˜ − ′ ( h ) 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) 0 = ε − 1 ℜ e R − ε ( h ) = h 1 ℜ e ∂ R − ′ ∂ h 1 ( 0 , 0 ) + h 2 ℜ e ∂ R − ′ ∂ h 2 ( 0 , 0 ) + ℜ e R ˜ − ′ ( h ) + ℜ e R ˜ − ε ( h ) , 0 = ε − 1 ℑ m R − ε ( h ) = h 1 ℑ m ∂ R − ′ ∂ h 1 ( 0 , 0 ) + h 2 ℑ m ∂ R − ′ ∂ h 2 ( 0 , 0 ) + ℑ m R ˜ − ′ ( h ) + ℑ m R ˜ − ε ( h ) . This is equivalent to have (3.11) M h = G ε ( h ) with G ε ( h ) : = − ℜ e R ˜ − ′ ( h ) − ℜ e R ˜ − ε ( h ) − ℑ m R ˜ − ′ ( h ) − ℑ m R ˜ − ε ( h ) (3.12) and M : = ℜ e ∂ R − ′ ∂ h 1 ( 0 , 0 ) ℜ e ∂ R − ′ ∂ h 2 ( 0 , 0 ) ℑ m ∂ R − ′ ∂ h 1 ( 0 , 0 ) ℑ m ∂ R − ′ ∂ h 2 ( 0 , 0 ) ∈ R 2 × 2 . To continue the procedure, we have to assume that the geometrical parameters y j ± , L j , ℓ j , H j ± are also such that the matrix M is invertible (again, see Section 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 (3.13) Find h ∈ R 2 such that h = G ε ( h ) with G ε ( h ) : = M − 1 G ε ( h ) . Thus we obtain a fixed point equation. Let us prove that for a given ϱ > 0 small enough, G ε ( · ) is a contraction in the closed ball B ϱ ‾ : = { h ∈ R 2 : | h | ⩽ ϱ } for all ε ∈ ( 0 ; ε 0 ( ϱ ) ] . First, since h ↦ R − ′ ( h ) is analytic, we have (3.14) | R ˜ − ′ ( h ) | ⩽ c | h | 2 and | R ˜ − ′ ( h ( 1 ) ) − R ˜ − ′ ( h ( 2 ) ) | ⩽ c | h ( 1 ) + h ( 2 ) | | h ( 1 ) − h ( 2 ) | , ∀ h , h ( 1 ) , h ( 2 ) ∈ B ϱ ‾ . We emphasize that c in (3.14) and in the estimates below is independent of ϱ 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 Section 6.6) the estimates, for all δ ∈ ( 0 ; 1 / 2 ) , (3.15) | R ˜ − ε ( h ) | ⩽ c ε 1 / 2 − δ and | R ˜ − ε ( h ( 1 ) ) − R ˜ − ε ( h ( 2 ) ) | ⩽ c ε 1 / 2 − δ | h ( 1 ) − h ( 2 ) | , ∀ h , h ( 1 ) , h ( 2 ) ∈ B ϱ ‾ . This allows us to write, using also the invertibility of M , | G ε ( h ) | ⩽ c | G ε ( h ) | ⩽ c ( | h | 2 + ε 1 / 2 − δ ) and guarantees that G ε ( · ) maps B ϱ ‾ to B ϱ ‾ for ϱ small enough and all ε ∈ ( 0 ; ε 0 ( ϱ ) ] . On the other hand, for h ( 1 ) , h ( 2 ) ∈ B ϱ ‾ , we have | G ε ( h ( 1 ) ) − G ε ( h ( 2 ) ) | ⩽ c ( | R ˜ − ′ ( h ( 1 ) ) − R ˜ − ′ ( h ( 2 ) ) | + | R ˜ − ε ( h ( 1 ) ) − R ˜ − ε ( h ( 2 ) ) | ) (3.16) ⩽ c ( | h ( 1 ) | + | h ( 2 ) | + ε 1 / 2 − δ ) | h ( 1 ) − h ( 2 ) | , which ensures that G ε ( · ) is a contraction for ϱ small enough and all ε ∈ ( 0 ; ε 0 ( ϱ ) ] . Now the Banach contraction mapping principle proves the existence of a unique solution h sol to (3.13) in B ϱ ‾ . Moreover, we have the estimate (3.17) | h 1 sol | + | h 2 sol | ⩽ c 0 ε 1 / 2 − δ . This leads us to the announced assertion on non-reflectability.

4.1. Preliminaries calculus

The first step in the procedure presented in the previous section consists in finding geometrical parameters y j ± , L j , ℓ 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 .

⋆ Case k = 0 . When k = 0 , the solution of problem (3.4) satisfies H j ( s ) d s v j 0 ( s ) = constant for s ∈ ( − ℓ j ; ℓ j ) . Using the boundary conditions, we deduce that v j 0 ( s ) = w + ( y j − , − d ) + ( w + ( y j + , − d ) − w + ( y j − , − d ) ) ( ∫ − ℓ j ℓ j 1 H j ( s ) d s ) − 1 ( ∫ − ℓ j s 1 H j ( s ) d s ) . Set H j : = ( ∫ − ℓ j ℓ j 1 H j ( s ) d s ) − 1 . We have d s v j 0 ( s ) = ( w + ( y j + , − d ) − w + ( y j − , − d ) ) H j / H j ( s ) . Inserting the latter relation in (3.2) and using that w + ( y j ± , − d ) = 2 e i y j ± a 2 − k 2 , we obtain (4.1) R − ′ = − i 2 N a ∑ j = 1 J H j ( w + ( y j + , − d ) − w + ( y j − , − d ) ) 2 = − 2 i N a ∑ j = 1 J H j ( e i y j + a − e i y j − a ) 2 . Take J = 3 , H j = H , ℓ j = ℓ , 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 (4.2) y 1 − = − π 3 a ; y 2 − = 0 ; y 3 − = π 3 a ; and y j + = y j − + η a with η ∈ ( 0 ; π 3 ) . In this case, according to (4.1), we have R − ′ = − 2 i N a H ∑ j = 1 3 ( e i η − 1 ) 2 e 2 i y j − a = − 2 i N a H ( e i η − 1 ) 2 ( e − 2 i π / 3 + 1 + e 2 i π / 3 ) = 0 . Then, when we translate the position of the tunnels 1 and 2 respectively by h 1 and h 2 , according to (4.1), we find R − ′ ( h 1 , h 2 ) = − 2 i H ( N a ) − 1 ( e i η − 1 ) 2 ( e 2 i ( y 1 − + h 1 ) a + e 2 i ( y 2 − + h 2 ) a + e 2 i y 3 − a ) . We deduce ∂ R − ′ ∂ h 1 ( 0 , 0 ) = 4 N H ( e i η − 1 ) 2 e 2 i y 1 − a and ∂ R − ′ ∂ h 2 ( 0 , 0 ) = 4 N H ( e i η − 1 ) 2 e 2 i y 2 − a . Thus the matrix M in (3.12) is invertible if and only if the ratio e 2 i y 2 − a / e 2 i y 1 − a = e 2 i π / 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 j 0 does not depend on s (the width of the tunnels is constant). Then the function v j 0 satisfies − d 2 v j 0 d s 2 ( s ) + k 2 v j 0 ( s ) = 0 for s ∈ ( − ℓ j ; ℓ j ) v j 0 ( ± ℓ j ) = w + ( y j ± , − d ) . We deduce that v j 0 is given by v j 0 ( s ) = w + ( y j − , − d ) cosh ( k ( s + ℓ j ) ) + ( w + ( y j + , − d ) − w + ( y j − , − d ) cosh ( 2 k ℓ j ) ) sinh ( k ( s + ℓ j ) ) sinh ( 2 k ℓ j ) . We can also write v j 0 ( s ) = A j e k s + B j e − k s with A j = e k ℓ j e i y j + a 2 − k 2 − e − k ℓ j e i y j − a 2 − k 2 sinh ( 2 k ℓ j ) and B j = − e − k ℓ j e i y j + a 2 − k 2 + e k ℓ j e i y j − a 2 − k 2 sinh ( 2 k ℓ j ) . Then we have ∫ − ℓ j ℓ j ( d v j 0 d s ( s ) ) 2 + k 2 ( v j 0 ( s ) ) 2 d s = 2 k ( A j 2 + B j 2 ) sinh ( 2 k ℓ j ) . Using in particular that H j = 2 ℓ j H j , we deduce R − ′ = − i 2 N a 2 − k 2 ∑ j = 1 J ∫ − ℓ j ℓ j H j ( ( d v j 0 d s ( s ) ) 2 + k 2 ( v j 0 ( s ) ) 2 ) d s = − i k N a 2 − k 2 ∑ j = 1 J H j ( A j 2 + B j 2 ) sinh ( 2 k ℓ j ) (4.3) = − 2 i N a 2 − k 2 ∑ j = 1 J H j ( A j ( e 2 i y j − a 2 − k 2 + e 2 i y j + a 2 − k 2 ) − 2 B j e i ( y j − + y j + ) a 2 − k 2 ) with A j = 2 k ℓ j e 2 k ℓ j + e − 2 k ℓ j e 2 k ℓ j − e − 2 k ℓ j and B j = 4 k ℓ j e 2 k ℓ j − e − 2 k ℓ j . Note that one can verify that taking the limit k → 0 in (4.3), we get back the relation (4.1) obtained for k = 0 . In order to have R − ′ = 0 , we take J = 3 , H j = H , ℓ j = ℓ , 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 (4.4) y 1 − = − 19 12 π a 2 − k 2 , y 2 − = 3 4 π a 2 − k 2 , y 3 − = 25 12 π a 2 − k 2 , y 1 + = − 13 12 π a 2 − k 2 , y 2 + = 5 4 π a 2 − k 2 , y 3 + = 31 12 π a 2 − k 2 . Using that y j + − y j − = π / ( 2 a 2 − k 2 ) , we find R − ′ = − 2 i N a 2 − k 2 H ( − 2 B ( e − 2 i π / 3 + 0 + e 2 i π / 3 ) + ∑ j = 1 3 A e 2 i y j − a 2 − k 2 ( 1 − 1 ) ) = 0 . Now translating the position of the tunnels 1 and 2 respectively by h 1 and h 2 , according to (4.1), we find R − ′ ( h 1 , h 2 ) = − 2 i N a 2 − k 2 H ∑ j = 1 3 ( A ( e 2 i ( y j − + h j ) a 2 − k 2 + e 2 i ( y j + + h j ) a 2 − k 2 ) − 2 B e i ( ( y j − + h j ) + ( y j + + h j ) ) a 2 − k 2 ) with h 3 = 0 . Then for j = 1 , 2 , we obtain ∂ R − ′ ∂ h j ( h 1 , h 2 ) = 4 N H ( A ( e 2 i ( y j − + h j ) a 2 − k 2 + e 2 i ( y j + + h j ) a 2 − k 2 ) − 2 B e i ( ( y j − + h j ) + ( y j + + h j ) ) a 2 − k 2 ) and so ∂ R − ′ ∂ h j ( 0 , 0 ) = 4 N H ( A ( e 2 i y j − a 2 − k 2 + e 2 i y j + a 2 − k 2 ) − 2 B e i ( y j − + y j + ) a 2 − k 2 ) . Thus for our choice (4.4) of the positions of the tunnels, we obtain ∂ R − ′ ∂ h 1 ( 0 , 0 ) = − 8 N H B e − 2 i π / 3 and ∂ R − ′ ∂ h 2 ( 0 , 0 ) = − 8 N H B . As a consequence, the matrix M is indeed invertible.