Curved channels with constant cross sections may support trapped surface waves

Curved channels with constant cross sections are constructed which support a trapped surface wave. Since corresponding eigenvalues are embedded in the continuous spectrum of the water wave problem and therefore possess the natural instability, the construction procedure requires “fine-tuning” of several parameters in the (small) curvature of the channel as well as geometrical restrictions on the cross section. In particular, the mirror symmetry of the cross section with respect to the vertical axis disrupts the procedure, but examples of suitable non-symmetric cross sections are provided.


The formulation of the problem
Let Ω = R × ω be a straight cylinder in the Euclidean space R 3 with a cross section ω ⊂ R 2 bounded by the line segment γ = {x = (x 2 , x 3 ) : x 3 = 0, |x 2 | ≤ 1} and a piecewise smooth curve connecting the points (±1, 0) inside the lower half plane R 2 − = {x : x 3 < 0}, see Fig. 1. The rescaling has been performed in order to reduce the length of γ to 2. Considering Ω as the water domain, the free surface is denoted by Γ = R × γ and by Σ = ∂Ω\Γ the union of walls and bottom.
In the channel, we consider the linearized water-wave problem, see, for example, [14], consisting of the Laplace equation  on the free surface Γ = {x : z = 0, |n| < 1}. Here, u is the velocity potential and λ = 2 g −1 is the spectral parameter, where is the frequency of the time-harmonic oscillations and g > 0 the acceleration due to gravity.
In the sequel, the problem (1.3)-(1.5) is referred to as problem P , while at = 0 we obtain the problem P 0 in the straight channel Ω 0 = Ω.

Spectra of the problems
The continuous spectrum σ 0 co of the problem is the closed real semi-axis R + = [0, +∞) in the complex plane C. The threshold values 0 = Λ 0 < Λ 1 ≤ Λ 2 ≤ · · · ≤ Λ j ≤ · · · → +∞ (1.6) divide σ 0 co into the intervals of constant multiplicity. In what follows, we consider the first interval (0, Λ 1 ). The entries of the sequence (1.6) are the eigenvalues of the model problem on the cross section ∂ ν U (x ) = 0, x ∈ ∂ω \ γ, ∂ z U (x ) = ΛU (x ), x ∈ γ, (1.7) where Δ is the Laplacian in the coordinates x = (x 2 , x 3 ). The spectrum of the problem P 0 is absolutely continuous but, for > 0, the spectrum σ = R + of problem P may contain embedded eigenvalues forming the point spectrum σ p . Our main goal is to find out a domain ω and a curve Υ , that is, the curvature κ in (1.1), such that σ p includes at least one eigenvalue (1.8) Being embedded in the continuous spectrum, this eigenvalue possesses the intrinsic instability, i.e. a small "wrong" perturbation of the appropriate curvature removes the eigenvalue out of the spectrum and turns it into a point of complex resonance [1,15]. In other words, our choice of the desired channel must be very precise and requires for "fine-tuning" of several parameters in the curvature (1.1), although many geometrical characteristics of ω and Υ can be fixed rather arbitrarily so that we are able construct infinitely many channels with the desired property.

Description of the paper
In the literature, there are many examples of local perturbations of three-dimensional channels which support trapped surface water waves and the corresponding spectral parameter is an embedded eigenvalue, see, for example, [2,4,6,9,[14][15][16]18,20,22,23,[33][34][35][36][37]. The original paper [6] introduces an elegant trick which requires the mirror symmetry about the mid-plane {x : x 2 = 0} of a channel Ξ ⊂ R 3 . By imposing the Dirichlet condition on the surface {x ∈ Ξ : x 2 = 0}, one creates an artificially positive cut-off value λ + † . The obtained mixed spectral problem in the half-channel Ξ + = {x ∈ Ξ : x 2 > 0} may have the discrete spectrum σ + d ⊂ (0, λ + † ), while the odd extensions of the corresponding eigenfunctions in x 2 -variable become an eigenfunction of the original problem in the original waveguide Ξ. Hence, σ + d is a part of the point spectrum in Ξ. However, under various assumptions on the geometry, variational and asymptotic methods have been applied to detect eigenvalues in σ + d , see [2,6,23,39] and others. The above-mentioned approach does not apply here since the channel Ω , surely, does not possess the mirror symmetry. Instead, we employ an asymptotic method based on a sufficient condition for the existence of trapped modes involving an artificial object, the augmented scattering matrix S , see [11,24,26] and Theorem 1. The necessary definitions are provided in Sect. 2, while the asymptotics of S is constructed in Sect. 5. In Sect. 4, we implement the fine-tuning procedure and introduce the parametrization of the quantities in (1.1) and (1.8). Our asymptotic analysis allows us to reduce the sufficient condition to the abstract fixed point equation We will show that the operator T is a contraction in the ball In this way, we obtain the curved channel with a constant cross section which has a trapped mode and the embedded eigenvalue cf. (1.8) and (1.10). This result will be formulated in Theorem 2.
It is worth to mention that the components κ 0 0 and κ 0 ± in (1.9) must satisfy the normalization and orthogonality conditions (3.18), (4.5), (4.7) and (4.8) under the appropriate choice of the cross section ω, see (3.21), (3.19) and (3.14), (3.17), while afterwards the coefficients τ ± are defined uniquely from (1.11). These conditions can be verified with quite arbitrary ingredients in the definition of the channel Ω , so that infinitely many channels supporting trapped surface waves are constructed by our procedure.

Motivation
In the frequently quoted paper [5], it was proved that the Dirichlet problem which models the planar quantum waveguide of the unit width and a non-trivial curvature κ 1 ∈ C ∞ 0 (R) of the mid-line, has an eigenvalue λ 1 below the continuous spectrum σ 1 c = [π 2 , +∞). Notice that here we put the superscript = 1 in order to point out that this result holds true without the smallness assumption (1.1) on the curvature. The variational approach, proposed in [5], has been applied in many works, cf. [22,25,31,32], in particular, for three-dimensional waveguides which in the case of a non-trivial twisting of the waveguide axis may have an empty discrete spectrum; we refer to the monograph [7] for the detailed survey about spectra in quantum waveguides.
With the reasons mentioned above, the fine-tuning procedure, as in [24], becomes quite limited. Hence, to construct a trapped mode, we have to accept two restrictive conditions, see (3.21) and (3.17) in Sect. 3.3. The latter condition forbids a cross section which is symmetric about the x 3 -axis. For this reason, we are not able to prove the existence of a trapped mode in a curved channel with a rectangular cross section, i.e. with vertical walls and horizontal bottom. Notice that in this case, by factoring out the dependence on z = x 3 variable, the problem reduces to the Neumann problem for the Helmholtz operator on a curved strip of width 2, that is, on the free surface Γ . In this way, being an apparent modification of the quantum waveguide in [5], an example of a curved two-dimensional acoustic waveguide of constant width supporting a trapped mode is not known yet. In [28], several examples of curved acoustic waveguides in dimension d ≥ 3 with constant cross section have been given that support a trapped mode.
We also point out that the criteria for the existence of trapped modes in [21] and [29] are adopted mainly for computational simulations. Finally, we emphasize that Sect. 5.2 gives examples of asymmetric cross sections ω such that both the introduced conditions (3.17) and (3.21) are satisfied.

Asymptotic analysis of the model problem
In the sequel, we assume that the first positive eigenvalue in the sequence (1.6) is simple, that is By the max-min principle, cf. [3,Thm. 10.2.2] or [40,Ch. 22], the non-positive part of the M -spectrum of the problem consists of two eigenvalues M 0 < 0 and M 1 = 0. The corresponding eigenfunctions are denoted by V 0 and V 1 normalized in the Lebesgue space L 2 (ω): Let us construct the asymptotics of the eigenpairs {M , V } of the perturbed problem with the parameter λ in (1.8) 3) The correction terms in the asymptotic ansätze with q = 0, 1 Notice that this problem is obtained directly by inserting (2.4) into (2.3) and extracting terms of order 2 .
Since both eigenvalues M 0 and M 1 are simple, the only compatibility condition in the problem (2.5) reads as where (·, ·) ω is the natural scalar product in the Lebesgue space L 2 (ω). Thus, in view of the normalization of the eigenfunctions V q , we obtain (2.6)

Waves
In the straight channel, there are two oscillatory and two exponential waves and {M q , V q } is the eigenpair of the problem (2.3) satisfying (2.4) and (2.6). Furthermore, a q is a normalization factor: For the further use, we define the symplectic (sesquilinear and anti-Hermitian) form The form comes from Green's formula in problem P 0 . Therefore, it is independent on the parameter R ∈ R for the waves in (2.7), (2.8) and their linear combinations as well as for other solutions of the problem in the unit strip. A direct calculation demonstrates that, owing to the normalization coefficients (2.10)-(2.12), we have Furthermore, since the exponents im 0 and m 1 in the waves (2.7) and (2.8) are different, we conclude by the independence property of Q(·, ·) that (2.14)

The augmented scattering matrix
Following [11], see also [24], we introduce the exponential wave packets and readily observe that, according to (2.13) and (2.14), we have where δ j,k is the Kronecker symbol. It is known, see, for example, [11,24,26], that the orthogonality and normalization conditions (2.17) assure the existence of the following solutions to the problem P : Here, the remainders Z 0± (x) and Z 1 (x) get the exponential decay o(e −δ|ς| ), while the exponents δ > 0 are defined by the eigenvalues (2.1) and can be fixed independently on ∈ (0, 0 ], 0 > 0. The coefficients in (2.18) and which is called the augmented scattering matrix.
It should be mentioned that, following [26], we include the exponentially growing packets w 1± in the outlet Ω + only, while the decompositions (2.18) and (2.19) contain the decaying wave v 1+ , see (2.8) and (2.16). However, the proofs in [11,24] allow us to derive that the matrix S is unitary and symmetric due to the equalities (2.17) and w 1− = w 1+ .

The existence of a trapped mode
In contrast to the classical scattering matrix s arising from the solutions (2.15) to the standard diffraction problem, the augmented scattering matrix S is an artificial object, because (2.18) and (2.19) involve exponentially growing waves and lose physical sense. However, the very reason to introduce such a matrix can be explained by the following observation: in the case , the solution (2.19) takes the form and, therefore, becomes a trapped mode, since v 1± decay in Ω ∓ , see (2.8). To derive (2.21), we have used the relations (2.16) and the evident formula supported by the unitary property of S . In other words, (2.20) is a sufficient condition for the existence of a trapped mode. The next theorem assures that (2.20) also becomes necessary in our particular case and it will be proven at the end of Sect. 4.2.
Theorem 2.1. There exists a positive number q such that, in the case

Differential operators in curvilinear coordinates
Since the equation Ψ(n, z) = 0 describing the surface ∂Ω \Γ does not depend on the variable ς and the gradient operator takes the form the normal ν(ς 0 , n, z) to this surface coincides with the normal of the cross section {x ∈ Ω : ς = ς 0 }. Here, J (ς, n) = 1 + nκ (ς) is the Jacobian. Furthermore, the Laplace operator in the curvilinear coordinates reads as Taking (1.1) into account and extracting from (3.1) the terms of order 0 and 1 , we obtain where the ellipsis stands for higher-order terms, which are inessential in our formal asymptotic analysis.

Computation of the coefficients
Extracting from (3.8) and (3.9) the terms of order − 1 2 yields the conditions (3.6). Hence, since U 1 (x ) is the only bounded solution of problem P 0 in Ω 0 with the threshold spectral parameter λ 0 = Λ 1 , we conclude that Notice that the function (3.10) does not depend on ς and, therefore, two last terms on the right-hand side of the Poisson equation in (3.7) vanish.
Collecting the coefficients of √ in (3.8) and (3.9) provides the following behaviour of the solution Z 1 to the problem (3.7): Since the linear growth of this solution is allowed in (3.12) when ς → ±∞, the problem (3.7), (3.12) with compactly supported right-hand side F 0 (ς, n, z) = κ 0 (ς)∂ n Z 0 1 (n, z) always has a solution which is defined up to a function (c 0 + c 1 ς) × U 1 (n, z) linear in ς-variable. Hence, some of the coefficients in (3.12) cannot be determined. However, we can derive certain relations between them. To this end, we insert Z 1 149 Page 10 of 23 S. A. Nazarov and K. M. Ruotsalainen ZAMP and U 1 into Green's formula in the truncated straight cylinder Ω R = (−R, R) × ω and obtain in the limit In the computation, we have used the formulae (2.7) and (3.12). Moreover, according to (3.10), the left-hand side of (3.13) is equal to (3.14) Hence, applying (3.11), we conclude from (3.13) the relation Notice that the equality S 0 11 = −1 occurs at the point To assure μ 0 > 0, we must assume that and For the calculation of S 0 ±1 , we apply Green's formula to the functions Z 1 and w 0 ± as follows: In the rest of the paper, we will need the assumption Finally, based on (3.15), we conclude that In Appendix A, we will verify the following estimates for the remainders in the asymptotic expansions (3.4) and (3.5): where c 0 and 0 are positive constants depending on the cross section ω and C 2 -norm of the rescaled curvature: Analysing the solutions (2.18) in a similar manner yields the asymptotic formulae for the upper left corner of the augmented scattering matrix. Notice that the appearance of the involution J in (3.24) has an evident reason: the waves w 0 0± pass the straight channel without any distortion, so that the reflection and transition coefficients are equal to 0 and 1, respectively.
We postpone the derivation of the estimates (3.23) and (3.25) to Appendix. The procedure is quite standard and straightforward based on the technique of weighted spaces with detached asymptotics.
Using (1.10), (2.9) and (3.15) gives Observing that , we see that our one-term asymptotic formula (4.2) is not sufficient to obtain Re S 11 (τ ) = −1. For this reason, it is not possible to apply the sufficient condition (2.20) directly.

(4.10)
Since the coordinate change x → (ς, n, z) brings small perturbations into differential operators of the water wave problem in Ω 0 , which smoothly depend on the parameters τ = (τ o , τ + , τ − ), the solution Z 1 as well as coefficients in its representation (3.8) has a similar dependence on τ . (For our purpose, Lipschitz continuity would be enough.) Observing also that the first term on the right-hand side of (4.10) involves the factors τ 0 , τ ± , we conclude that the operator T in the equation (1.11), which is a short form of (4.6) and (4.10), satisfies the inequalities Hence, it is a contraction in the ball B ρ provided the radius ρ > 0 is small enough. Thus, the Banach fixed point theorem guarantees that there are positive and ρ, ρ 0 such that the equation (1.11) has a unique solution τ in the ball B ρ0 and the estimate (1.12) is valid.
Recall that the equation (1.11) is equivalent to the relations (4.3) and (4.4) (and also to the sufficient condition (2.20) for the existence of a trapped mode). Therefore, formulae (1.1), (1.8), (1.9) and (1.10) determine the curved channel which supports a trapped surface wave and the eigenvalue of the problem (1.3)-(1.5) embedded in its continuous spectrum.

Straight walls and bottom
Let the cross section be a rectangle In this case, the eigenvalues for the Steklov problem (1.7) can be computed explicitly by the separation of variables: Notice that all the eigenvalues in (5.1) are simple, fulfilling our assumption (2.1). Particularly, Λ 0 = 0 and with the eigenfunctions U 0 (x ) = 1 and where A 1 is the normalization constant. The negative eigenvalue M 0 = −m 2 0 of the problem (2.2) is defined by the root m 0 ∈ (0, π 2d ) of the transcendental equation Λ 1 = m 0 tan(m 0 d) , and the corresponding eigenfunction is Then, by direct integration, we have violating the condition (3.17).
The same phenomenon appears also in any cross section ω with the mirror symmetry about the x 3axis, because all the modes of oscillations are either symmetric or anti-symmetric about the ordinate axis x 2 = 0, see [8,19]. In particular, V 0 is symmetric and V 1 anti-symmetric, and hence, the relations (5.4) and (5.5) are valid, too. In view of (5.5), the result obtained in the previous section on the existence of a trapped mode does not apply. Nevertheless, it could be possible that constructing higher-order terms in the asymptotic expansion in Sect. 3 may help to satisfy the criterion (2.20). The authors do not know whether this would lead to a success.
Another way to fulfil both conditions (3.21) and (3.17) simultaneously is to give a small perturbation to the walls, see Fig. 3 and apply the asymptotic methods, as in [10,17], to compute J 1 (κ 0 ), K 0 and K 1 , cf. (3.14) and (3.19). The relation K 0 = 0 remains valid because the perturbation is small. Moreover, according to the general results in [12,Ch. 7] K 1 is a real analytic function with respect to the perturbation parameter. Hence, either K 1 is zero only at the isolated parameter values or vanishes identically. To prove that the last case is not possible, we have to solve explicitly at least two boundary value problems in the original domain, to find out the change of K 1 under a small regular or singular perturbation of the boundary. This calculation is quite long and cumbersome. Instead, in the next section, we consider the depth d as a small parameter and employ the traditional analysis of elliptic problems in thin domains (see [17,Ch. 15,16] and [4,27]) for which one needs to solve a system of ordinary differential equations and to compute an integral of only one solution in a semi-infinite strip.

An example of a feasible cross section
Let us assume that the cross section has the shape of a thin trapezoid, in Fig. 4 ω d θ = {(x, y) : y ∈ (−d, 0), cot(θ)y < x < 1} (5.6) where the depth d > 0 is a small parameter and θ ∈ (0, π/2) is the sharp angle of ω d θ . The skewed part of the boundary is denoted by [4,27], we know that the eigenvalues (1.6) of the problem (1.7) in ω d θ are small, of order d, and satisfy the estimate with some positive numbers c p (θ) and d p (θ) depending on the angle θ and the index k ∈ N. Clearly, as before Λ d 0 = 0 and the eigenfunction is a constant: Notice that the normalization of eigenfunctions is not needed in this section. Here, the fast variables are denoted by (ξ, η) = d −1 (x, y). The function Y (x) = cos(πx) is the solution of the limit spectral problem, when d → 0 + , and L = π 2 is the corresponding eigenvalue. The function is the solution of the problem Inserting (5.8) and (5.9) into the problem (1.7), we observe that the correction term W 1 satisfies on the semi-infinite pointed strip ∂ ν W 1 (ξ, η) = −π 2 sin(θ)ξ + cos(θ)(η + 1) =: G(ξ, η). (5.10) Here, we have denoted by Σ θ the sharp end of the strip P θ : Since the integral Σ θ G(ξ, η)ds = 0, the problem (5.10) has a solution that decays at the rate O(e −πξ ) as ξ → +∞, cf. [30,Ch. 2]. Since for small d, we have x = dξ, y = dη, the asymptotic representation of the function U d 1 (x, y) on Σ θ takes the form ). According to the asymptotic analysis, the remainder satisfies the estimate for some positive constants C θ and D θ . As was mentioned in Sect. 5.1, the requirement (3.21) is met for a small d > 0 because the main term cos(πx) in (5.8) is anti-symmetric with respect to the "middle line" {(x, y) : y ∈ (−d, 0), x = 1/2} of the trapezoid ω d . It remains to evaluate the integral K d 1 in (3.14). In view of the definition of the cross section (5.6), we have, using the Green formula, The first integral has the asymptotic representation Since the normal derivative on Σ d θ is ∂ ν = −sin(θ)∂ ξ + cos(θ)∂ η , the integral I 2 takes the form Adding the previous results together, we finally obtain that In other words, at least for small values of the depth of the channel the coefficient K 2 1 < 0 and the condition (3.17) are valid. Therefore, with a suitable local distortion of the straight channel we can create a trapped mode.
It is worth to repeat the fact that according to general results in [12,Ch.7] the function (0, +∞) d → K d 1 is real analytic and therefore it can vanish only for isolated sequence of values of the parameter d.

On multiple eigenvalues
In the previous sections, we have found the eigenvalue (1.8) near the first positive threshold Λ 1 . A similar procedure can be applied to higher thresholds. However, searching for an eigenvalue λ near the threshold Λ n in (1.6), our procedure requires more assumptions about the spectral problem (2.2) with the new Robin condition Namely, first the non-positive eigenvalues M 0 , M 1 , . . . , M n = 0 (5.14) must be simple and, secondly, the corresponding eigenfunctions must satisfy K k := ω V k (n, z)∂ n V k (n, z) dndz = 0, k = 0, . . . , n. (5.15) Indeed, the conditions (5.15) are absolutely necessary because the equations similar to (4.6) and (4.10) involve the multipliers K −1 k . Furthermore, if there is a multiple eigenvalue among (5.14), then it is not possible to fulfil requirements of type (4.8), because several integral operators of type (3.20) have the same kernels.
It may happen that our scheme works in the case of the eigenvalue M n = 0 of multiplicity J > 1. However, in this case the analysis becomes incomparably more complicated, since the sufficient condition [11] deals with the right bottom J × J-block of the augmented scattering matrix and requires that this block has the eigenvalue −1. Notice that also some results in Appendix A about the justification hold true only for a simple eigenvalue Λ 1 .
Finally, we mention that the question of the existence of a curved waveguide supporting two different trapped modes is fully open yet.