Spectral gaps for the linear water‐wave problem in a channel with thin structures

We consider the linear water‐wave problem in a periodic channel Πh⊂R2$\Pi ^h \subset {\mathbb {R}}^2$ , which consists of infinitely many identical containers and connecting thin structures. The connecting canals are assumed to be of constant, positive length, but their depth is proportional to a small parameter h. Motivated by applications to surface wave propagation phenomena, we study the band‐gap structure of the essential spectrum in the linear water‐wave system, which forms a spectral problem where the spectral parameter appears in the Steklov boundary condition posed on the free water surface. We show that for small h there exists a large number of spectral gaps and also find asymptotic formulas for the position of the gaps as h→0${h} \rightarrow 0$ : the endpoints are determined within corrections of order h3/2${h}^{3/2}$ . The width of the first spectral band is shown to be O(h)$O({h})$ .

F I G U R E 1 Periodic channel with thin connecting canals on the free surface of the water filled domain, see for example the monograph [11]. The bands and gaps of the continuous spectrum are called passing and stopping zones for waves.
The special feature of the linear water-wave equation is the appearance of the spectral parameter in the Steklov boundary condition on the free water surface. This makes a direct application of the classical Sobolev-space methods difficult, see for example [11], especially for an approach based on the application of the Dirichlet-to-Neumann-(or Steklov-Poincaré-) operator, which is a non-local operator and thus complicated from the point of view of applying the methods of the asymptotic analysis; also, see the review paper [12]. We follow here the modified techniques used for example in [19,22] which are based, among other things, on an unconventional definition of the problem operator with mixed types of inner products containing both volume and surface integrals; see (1.14). This method has been used for example for proving or disproving the existence of eigenvalues in some frequency interval.
For a fixed ℎ, the essential spectrum ess of the original problem is non-empty due to the unboundedness of the domain. More precisely, due to the periodicity of the domain, it follows from the Floquet-Bloch-Gelfand(FBG) -theory (see for example the books [10,23,24] and papers [15]; [18], Theorem 2.1; [21], Theorem 3.4.6, for a presentation relevant to the case of this paper) that ess has the band-gap structure where the spectral bands ℎ are compact subintervals of the positive real axis. By we denote here the Floquet parameter is the sequence of the eigenvalues of a "model problem" obtained from (1.5)-(1.7) by using the FBGtransform. In general, the spectral bands may and often do overlap, in which case the essential spectrum is connected. However, in between the bands there may also appear gaps which are free of the essential spectrum and which describe "forbidden" frequencies with no wave propagation (or, stopping zones between passing ones). The position of such gaps is in general of interest in physical applications, since they may be wanted for example for the design of wave filters and dampers.
In this paper we study the asymptotic position of the bands ℎ as ℎ → +0 and apply the results to detect gaps in the essential spectrum (1.1). In the main result, Theorem 5.1, we show that for all ∈ ℕ (with constants > 0 not depending on ℎ or ), where the sequence Λ 0 ( ) consists of the eigenvalues of a "limit problem" corresponding to the case ℎ = 0, or vanishing canals: it is a system of finitely many boundary value problems for certain simple ordinary differential equations with certain peculiar boundary conditions connecting them. The numbers Λ 0 are solutions of an explicit transcendental equation (3.6), and as we will show in Section 3, it is possible to get a lot of information on them by using computational arguments and in particular to show that infinitely many spectral gaps indeed exist in the case of the limit problem. Then, the estimate (1.2) implies that the spectrum ess also has at least any given number of gaps, if ℎ > 0 is sufficiently small; see Theorem 5.2. However, since the constants in (1.2) also depend on , we can only assure that finitely many gaps exist for a fixed ℎ. The proofs of Theorem 5.1 and Theorem 5.2 will be presented in Sections 5 and 6, and they consist of the justification of the formal asymptotic analysis of the model problem in Sections 2 and 4; the latter section contains the construction of the approximate eigenfunctions of the model problem. One more main tool is the so-called convergence lemma, which is Lemma 6.1 in Section 6.
Besides the Steklov spectral condition and periodic structure, one more characteristic feature of the problem under consideration is described by junctions of massive bodies with thin ligaments. For example, the dumbbell, which is a union of two massive domains connected by a thin cylinder, is a classical object in asymptotic analysis. The spectrum of the Laplace-Neumann problem in such a domain has been studied in many papers starting from the pioneering works [1,2]. Aymptotic expansions for eigenvalues and eigenfunctions have been constructed and applied in many other works including [9,16,17]; recently, for example in [4]. Concerning asymptotic methods, we will here partly follow the approach in [20].

Formulation of the problem, operator theoretic tools
Let us proceed with the exact formulation of the problem. We consider an infinite two-dimensional periodic channel Π ℎ ⊂ ℝ 2 consisting of water containers connected by narrow canals of diameter (ℎ). The coordinates of the points in the channel are denoted by = ( 1 , 2 ) = ( , ). We choose the coordinate system in such a way that the axis of the channel is in 1 -direction and the free surface is on the line { ∶ 2 = 0}. In more detail, the periodic channel Π ℎ is defined as the interior of the set where the domains Ω ℎ are translates of the periodicity cell Ω ℎ , The periodicity cell (see Fig. 2.a)) Ω ℎ ⊂ (−1∕2, 1∕2) × (− , 0), where > 0 is the depth of the channel, consists of two main parts, the container Ω and the connecting canals ℎ 0 , … , ℎ , more precisely, . Here, the following notation and conventions are used. The container Ω ⊂ ℝ 2 is a domain with a Lipschitz boundary and compact closure, and it is contained in the rectangle (− , ) × (− , 0), where 0 < < 1∕2. Moreover, we assume that for some 0 < ′ < the line segments {± } × (− ′ , 0) are part of the boundary Ω. The boundary of Ω consists of the free water surface Γ 0 = Ω ∩ { = 0} and the wall and bottom Σ = Ω ∩ { < 0}. The connecting canals ℎ ∶= ℎ are determined by the depth positions and relative widths We also denote ℎ ∶= ⋃ =0 ℎ and ± = ( ± , ) for all and signs. The parameter ℎ > 0 is assumed so small that the canals ℎ do not touch each other. We also write .
The free surface of the periodicity cell Ω ℎ (independent of ℎ) is denoted by Γ = Ω ℎ ∩ { 2 = 0}, and the wall and bottom part of the boundary is i.e. we leave out the lateral ends of the canals from Σ ℎ . Finally, the free water surface of the entire channel Π ℎ and its wall/bottom are defined, respectively, as Remark 1.1. We will use the following general notation. We write ℝ + 0 for the set of non-negative real numbers. Given a domain Ξ ⊂ ℝ , the symbol |Ξ| stands for its volume in ℝ and (⋅, ⋅) Ξ stands for the natural scalar product in 2 (Ξ), and (Ξ), ∈ ℕ, for the standard Sobolev space of order on Ξ. The norm of a function belonging to a Banach function space is denoted by ‖ ; ‖. For > 0 and ∈ ℝ , ( , ) (respectively, ( , ) ) stand for the Euclidean ball (resp. ball surface) with centre and radius . By , (respectively, , , ( ) etc.) we mean positive constants (resp. constants depending on a parameter ) which do not depend on functions or variables appearing in the inequalities, but which may still vary from place to place. The gradient and Laplace operators ∇ and Δ act in the variable , unless otherwise indicated. We write = ∕ etc. , and for the outward normal derivative on the boundary of a given planar domain.
In the framework of the linear water-wave theory we consider the spectral Steklov-Neumann problem in the channel Π ℎ , Here = ( ) = ( ; ℎ) is the velocity potential, = (ℎ) = −1 2 is a spectral parameter related to the frequency of harmonic oscillations = (ℎ) > 0 and the acceleration of gravity > 0 (the dependence of on ℎ will usually not be displayed). By the geometric assumptions made above, the derivative is defined almost everywhere on Σ ℎ . It coincides with on the free surface Γ tot .
(1. 16) Clearly, according to [3, Thm. 10.1.5, 10.2.2] the spectrum of  ℎ ( ) consist of null, which is a point in the essential spectrum, and a positive sequence of eigenvalues (counting multiplicities) convergent to 0; these can be calculated from the usual min-max principle where the minimum is taken over all subspaces ⊂  ℎ ( ) of co-dimension − 1. Using (1.14) and (1.15), we can write a max-min principle for the eigenvalues of the problem (1.13): On the other hand, formula (1.16) and the properties of the sequence ( ℎ ( ) )∞ =1 mean that the eigenvalues (1.17) form an unbounded sequence where multiplicities have been taken into account. We denote by ℎ, ∈  ℎ ( ) the eigenfunction corresponding to Λ ℎ ( ) and assume that these eigenfunctions are normalized so as to form, for fixed ℎ and , an orthonormal sequence in the space 2 (Γ). The functions ↦ Λ ℎ ( ) are continuous and 2 -periodic (see for example [7,Ch. 9], [10, Sec. 3.1]). Hence, the spectral bands ℎ = { Λ ℎ ( ) ∶ ∈ [0, 2 ) } of (1.1) indeed are compact intervals.

Equations for the terms of the ansatz
In Sections 2-4 we apply the method of matched asymptotic expansions, see [6,13,25] and others, to construct approximating near-eigenfunctions for the model problem (1.8)-(1.12). We fix and for this section and usually suppress them from the notation, denoting for example Λ ℎ ∶= Λ ℎ ( ), and ℎ ∶= ℎ, and similarly for the other quantities. We will next adapt to our problem the asymptotic ansätze used in [20] for a parameter independent Steklov problem. Inside the container Ω we write the outer expansion where the constant = ( ) and the function ′ = ′ are to be determined, and the dots indicate higher order terms in ℎ which are inessential for our formal analysis. The outer expansion in the canals ℎ ± , = 0, … , , reads as where the functions = , and = , are to be determined and the variable is stretched in -direction, For Λ ℎ = Λ ℎ ( ), we use the expansion where Λ 0 = Λ 0 ( ) is to be found.

Outer expansions in the canals
Let us derive equations for the terms introduced above. Putting the expansion (2.2) into (1.8) yields Collecting terms of the lowest order ℎ 0 in (2.4) leads to the equation so we obtain from the terms of order ℎ the boundary condition for equation (2.5) 0 ( , 0) = 0 ( ) ∶= Λ 0 0 ( ) , ∈ Υ (2.6) (see (1.4) for the notation). Since 0 = 0 everywhere on the bottom of the canal, Equation (1.9) leads to the boundary condition We consider the problem (2.5), (2.6), (2.7) as a one-dimensional Neumann problem for the function 0 in the variable so that is regarded as a parameter. The compatibility condition in this problem reads as On the other hand, by (2.5), for ∈ Υ ± , hence, using (2.6), (2.7) we get the following differential equation for 0 : If = 1, … , , we derive an equation for in the same way, except that the homogeneous Neumann condition (1.9) is used instead of (1.10) so that 0 is omitted. As a result we get the equations − 2 ( ) = 0 for = 1, … , , ∈ Υ. (2.9) As for the boundary conditions associated with (2.8), (2.9), the outer expansion does not contribute to the behaviour of the ansatz near = ±1∕2 hence, all must satisfy the quasiperiodic boundary conditions In addition, due to the leading term of the outer expansion (2.1), we require that for all = 0, … , there holds This condition connects all Equations (2.8) and (2.9).

Boundary layer phenomenon
Near the points ± = ( ± , ) the narrow canals ℎ ± , = 0, … , , are joined with the large container Ω. The geometry is thus crucially different from that of the isolated container, and there arises boundary layer effects, which influence the solutions of (1.8)-(1.12). In the framework of the method of matched asymptotic expansions, these effects are described by the inner expansions where 0 ± = 0, ±, and ′ ± = ′ ±, and we use the stretched coordinates .
(2.13) For = 0, … , , the coordinate dilation ↦ ± and formal substitution ℎ = 0 transform the singularly perturbed domain Ω ℎ into the unbounded one Ξ = ∪ ℙ , where the quadrants, half-spaces and strips are denoted by so that the term ℎ 2 Λ 0 0 disappears in the limit after multiplication by ℎ and the homogeneous Neumann condition occurs again.
We obtain from the expansion (2.2) in the canal ℎ ± that Looking at (2.15) we observe that the terms of the inner expansion (2.12) must behave linearly in the strip outlets ℙ . Let us list such solutions of (2.14). The first one is evident: the constant function, denoted by 0 ± = 1. The second solution 1 ± must grow as ± −1 ± in ℙ . Since the flux in the strip outlet does not vanish, the flux in the angular outlet is nonzero, too. This leads to the decompositions 1 0± where ± = | | ± | | for all . Notice that the numbers ∕2 and in (2.16) and (2.17) are nothing but the angles of opening of 0 and , respectively. A basic result in harmonic analysis assures the existence of the solutions 1 ± = − 1 ∓ , and their uniqueness follows from the requirement since the constant is annulled here.
Remark 2.1. We have ln ± = ln ± − ln ℎ, where ± is the distance between the points and ( ± , − ) . The factor ln ℎ in (2.20) can be hidden into the term ℎ ± so that the constant becomes independent of the large parameter | ln ℎ| and can be fixed at the next step of the asymptotic procedure. □ Comparing (2.20) and the outer expansion (2.1) in Ω, we conclude that matching at the level 1 = ℎ 0 yields the relation (2.11), where = 0, … , and is a constant, while matching at the level ℎ requires the following behavior of the correction term, when | ( ± , − ) | approaches 0:

Outer expansion in the container
The detached logarithmic terms in (2.21) can be interpreted as Poisson kernels and in this way the correction term ′ in the ansatz (2.1) for ℎ is obtained as the solution of the following Neumann problem in Ω, which is to be understood in the sense of distribution theory: Here, denotes the Dirac mass at the point of a one-dimensional manifold, namely the boundary curve Σ. The compatibility condition ∫ Ω ′ = 0 in the Neumann problem (2.22)-(2.24) leads to the relation To describe the solution of (2.22)-(2.24), we note that the boundary of Ω coincides with a line segment in a neighbourhood of every ± with ≥ 1, but the two points 0± are corners of Ω. This leads to the observation that, given a constant ∈ ℂ such that (2.25) holds, the problem (2.22)-(2.24) has a (harmonic) solution ′ in Ω, where ± = ± 1 + ,0 (± ) for = 0, … , (2.27) and the harmonic functioñin Ω satisfies The form of the singularities (2.26) at ± , ≥ 1, can be deduced from basic distribution theory, namely, in the space of distributions on the one-dimensional space of -axis there holds where 0 is the Dirac measure at 0. However, the corner points 0± are more complicated, and the asymptotic form of the solution in their vicinity as well as the estimates (2.28) are determined by the theory of elliptic boundary problems in corner and conical domains (see [8,14] and, e.g., [21, Ch. 2, Sect. 3.6]).

LIMIT PROBLEM AND ITS EIGENVALUES
Equations ( The problem (2.8)-(2.11) can be solved explicitly for a fixed in the sense that the eigenvalues Λ 0 are determined as solutions of a transcendental equation. We need to derive this equation and also prove results on the possible values of the solutions in order to establish the existence of spectral gaps. Due to the linearity of Equations (2.8)-(2.10) we see that of (2.11) can be considered as a normalization constant; it becomes fixed via the normalization just after (3.9). The solution 0 of (2.8) has the expression where , are obtained imposing condition (2. . The solutions , ≥ 1, of (2.9), (2.11) with = 1 are given by Note that does not depend on and, in particular, Replacing the solutions (3.3)-(3.5) into (2.25), we obtain the following transcendental equation which implicitly expresses the dependence of Λ 0 = Λ 0 ( ) on : Notice that according to the above derivation of the equation (3.6), we only take into account its positive solutions Λ 0 . The following observation can be proven by an elementary argument. Proof. We first remark that the set of solutions of (3.6) does not have finite accumulation points. Indeed, it suffices to consider points Λ 0 = ∈ (0, +∞) such that −1 0 Λ 0 (1 − 2 ) ≠ for any ∈ ℕ 0 (so that the denominator in (3.6) is nonzero). Then, for a fixed , the expression on the left-hand side of (3.6) is a real analytic function of Λ 0 in a neighborhood of , and does not vanish identically in , which can be seen directly from its expression in (3.6). By well known properties of analytic functions or power series, the point cannot be an accumulation point of zeros of , which proves the claim.
In order to show that there exist infinitely many solutions, we first fix ∈ (0, ∕2], hence, cos ∈ [0, 1), and let Λ 0 ≥ 1. Then, (3.6) is equivalent to where 1 and 2 are nonzero constants and 3 ( ) is a number uniformly bounded with respect to . Let us consider any ∈ ℕ so large that on the interval the value of the left-hand side of (3.7) is at least 1. Then, the value of the function Λ 0 ↦ cos ( 2 Λ 0 ) − cos decreases monotonely on from some number ( ) > 0 to 0, and on the same interval, the value of the function Λ 0 ↦ sin ( 2 Λ 0 ) increases monotonely from 0 to some number ′ ( ) > 0. This shows the right-hand side of (3.7) is a continuous function of Λ 0 in the interior of , the values of which decrease monotonely from +∞ to 0. The left-hand side of (3.7) defines a continuous function on , the values of which stay on some compact interval contained in [1, +∞), by assumption. By Rolle's theorem, the equation must have a solution in the interval (3.8).
Although we have not exactly defined the essential spectrum 0 ess of the limit problem, let us denote By Lemma 3.1, the set 0 ess ⊂ ℝ + 0 is unbounded. Thus, 0 ess contains infinitely many "gaps" as explained in the following result, which is a direct consequence of Lemmas 3.3 and 3.4.

MAIN RESULT ON THE ASYMPTOTIC POSITION OF SPECTRAL BANDS
We now state our main result concerning the asymptotic position of the spectral bands. It also justifies the formal asymptotic analysis of the previous sections and motivates the use of the approximate eigenfunctions (4.3). The result yields the existence of spectral gaps in the essential spectrum (1.1). We recall that the eigenvalues Λ ℎ ( ) of the model problem were defined in (1.18), and those of the limit problem Λ 0 ( ) in Lemma 3.1 and (3.10). The final step of the proof will be completed only in the last section. We start the proof of Theorem 5.1 by stating a classical lemma on near eigenvalues and eigenvectors (see [26] and, e.g., [3,Ch. 6]).
where the sum is taken over all eigenvalues of the operator  contained in the interval [ˆ− ,ˆ+ ] with multiplicities taken into account, and  are the corresponding eigenvectors orthonormalized with respect to each other in , while the coefficients are normalized by ∑ | | 2 = 1.
It will also be useful to formulate an intermediate step in the proof of the main result. Note that if Λ 0 ( ) is a simple eigenvalue in the following lemma, then = holds for these indices.  (3.9), of the eigenvalue Λ 0 ( ) in (3.10). Then, for some ℎ ′ > 0 there holds where ℎ = ℎ ( ) = 1∕ ( 1 + ℎΛ 0 ( ) ) .
Proof of Theorem 5.1. Here, given we will find an eigenvalue Λ ℎ ( ) with an unspecified index , such that (5.1) holds with this eigenvalue in the place of Λ ℎ ( ). We will show only in Section 6 that = , because this conclusion will require some additional arguments. So, let now and and thus also the number Λ 0 ( ) be fixed. We aim to apply Lemma 5.3 to the operator ( =)  ℎ ( ) ∶  ℎ →  ℎ (= ) of (1.15). Let us define an approximate eigenvalue and eigenvector of  ℎ ( ) by (ˆ=) ℎ ( ) = 1∕ ( 1 + ℎΛ 0 ( ) ) , (5.4) where  ℎ ( ) is defined as in (4.3) by using any of the eigenvectors ⃖⃗( ), (3.9), of the eigenvalue Λ 0 ( ). Definition (5.4) is motivated by the relation (1.16). From now on, we mostly suppress the indices and from the notation and denote ℎ ( ) =∶ ℎ ,  ℎ ( ) =∶  ℎ , Λ 0 ( ) =∶ Λ 0 and so on. Our aim is to show that of Lemma 5.3 can be chosen as small as in other words, we prove Lemma 5.4. Then, Lemma 5.3 gives an eigenvalue ℎ of  ℎ with the estimate Since the modulus of the coefficients of the operator [Δ, ℎ ] have the upper bound where also ‖ ‖ ;  ℎ ‖ ‖ ≤ 1 was taken into account. In the subdomains ℎ , where = 0, … , , we write using the equalities (2.9) and the Kronecker delta Then, (4.4) and the harmonicity of 1 ± imply ( ) Let us next consider the term ℎΛ 0 (  ℎ , ) Γ in (5.10). Here, we remind that the definition of the norm of  ℎ implies and then we apply (2.26), (2.28) to get We also have ‖ ‖˜; 2 ( Γ ∩ Ω ) ‖ ‖ ≤ | log ℎ|, so that the Cauchy-Schwartz-inequality and (5.19), (5.20) yield The contribution of the term with˜is of order ( ℎ 2 | log ℎ| ) also in the subdomain ℎ 0 . Moreover, (5.19) and the size of the support of the cut-off-function. We thus obtain since the subdomains ℎ , ≥ 1, do not contribute to this term. ( ) We consider the term (  ℎ , ) Ω ℎ . We have, by (2.23), (2.24), where the contribution of (2.24) vanishes, since the Dirac masses are concentrated outside the support of ℎ , and we used To estimate the contribution of the term with˜we remark that by definition, satisfies the homogeneous Neumann conditions at least everywhere in Ω ℎ ∩ ( ± , ) for some positive constant independent of ℎ. Evidently, the same holds also for the functions log |( ± , ± )| in the sets Ω ∩ ( ± ,  . Summarizing these observations we get This is at most ( ℎ 2 ) by (4.5), (4.6). These estimates yield ( ) Let us turn to the final estimate. According to (5.18), (5.21) and (5.22), the expression on the right of (5.10) is bounded by We have Here, the integrand of the first term is constant in the -variable so that the integral equals, by (2.8), Γ∩ ℎ 0 so that the second term in (5.23) is cancelled and we are left with the second term on the right of (5.24). This is small, as seen by the estimate where the Cauchy-Schwartz-Bunyakowski inequality was used to obtain the factor ℎ 1∕2 . Here we used that the function ′′ 0 is uniformly bounded, as a consequence of (2.8), (2.9). This completes the proof of the bound (5.6). □

CONVERGENCE THEOREM AND THE END OF THE PROOF OF THEOREM 5.1
To finish the proof of Theorem 5.1, i.e., to show that the indices and are the same in the proof of the previous section, we need the following result. Namely, fixing and , we found in the proof of Theorem 5.1 for all ∈ ℕ an index ∈ ℕ such that | | ℎΛ 0 ( ) − Λ ℎ ( ) | | ≤ ℎ 3∕2 . This estimate cannot hold for the same and two different numbers Λ 0 ( ) and Λ 0 ′ ( ) (just by the triangle inequality), hence for a large enough we must have ≥ . This yields Λ ℎ ( ) ≤ Λ ℎ ( ) ≤ ℎΛ 0 ( ).
By Lemma 3.1, the function ↦ Λ 0 ( ) is continuous and thus bounded on the compact interval [0, 2 ] so that the bound on the right-hand side can be made independent of . This proves (6.1).
Using (6.1) and compactness we find a sequence {ℎ } ∞ =1 converging to 0 such that for some number Λ ∶=Λ ( ) ∈ [0, +∞), ℎ −1 Λ ℎ →Λ as → +∞; (6.2) here, the same sequence {ℎ } ∞ =1 can be taken in the case several are considered simultaneously, or the sequence could be defined as a subsequence of any given sequence converging to 0 (see the last statements of the theorem). The same is true also in the later steps of the proof so that we do not comment these aspects any more.