On the existence of high-frequency boundary resonances in layered elastic media

We analyse the asymptotic behaviour of high-frequency vibrations of a three-dimensional layered elastic medium occupying the domain Ω=(−a,a)3, a>0. We show that in both cases of stress-free and zero-displacement boundary conditions on the boundary of Ω a version of the boundary spectrum, introduced in Allaire and Conca (1998 J. Math. Pures. Appl. 77, 153–208. (doi:10.1016/S0021-7824(98)80068-8)), is non-empty and part of it is located below the Bloch spectrum. For zero-displacement boundary conditions, this yields a new type of surface wave, which is absent in the case of a homogeneous medium.


Problem formulation and background
There is a growing interest in transport properties of elastic waves in periodic structures. On smaller length scales, new research opportunities emerge in audio filters, nanoscopic phononic lasers, perfect reflectors, phononic integrated circuits. On larger length scales, one may wish to control surface seismic waves in structured soils for civil engineering applications. It may therefore be of importance to consider the asymptotic analysis of spectral problems for the Navier operator of linearized elasticity, with rapidly oscillating coefficients. As we discuss below, the related spectra are shown to accumulate around two sets, the 'Bloch spectrum' and the 'boundary spectrum', whose eigenfunction sequences behave in essentially different ways that correspond to 'bulk' wave propagation and 'surface' wave propagation, respectively. The objective of this work is to demonstrate mathematically rigorously that the boundary spectrum  can be non-empty, and that it can in fact be located outside the Bloch spectrum. One notable feature of linearized elastic systems in homogeneous media with boundaries is a non-trivial coupling between pressure and shear waves at the boundary, which results in, e.g., the existence of surface waves, known as 'Rayleigh waves', even in the case of constant coefficients. In what follows we exploit some of the extra freedom provided by the general periodic setting in order to give an explicit example of elements of the boundary spectrum. As a by-product, we show the existence of a new kind of surface waves in layered periodic media with a clamped boundary: indeed, it is well known that a clamped boundary does not support surface acoustic waves on any homogeneous half space [1,2].
Consider an ε-periodic linearized elastic medium occupying the domain Ω = (−a, a) 3 ⊂ R 3 , where ε ∈ Ξ := {ε > 0 : ε −1 a ∈ N}. We study the following eigenvalue problem, understood in the weak sense: (1.1) Here, and throughout this article, we sum over repeated indices and use a comma for partial derivatives, for example u ε k,j := ∂u ε k /∂x j . The functions ρ ε (x) = ρ(x/ε), C ε = (C ijkl (x/ε)) 3 i,j,k,l=1 are the mass density and the elastic tensor of the medium, respectively. We assume that ρ, ρ −1 ∈ L ∞ (Y), Y := [0, 1) 3 , and that the elastic tensor C ∈ [L ∞ (Y)] 81 has the 'major' symmetries C ijkl = C klij , i, j, k, l = 1, 2, 3, and is uniformly elliptic, i.e. there exists ν > 0 such that It is well known (e.g. [3]) that, for each ε, the spectrum σ ε of (1.1) consists of the set of all first elements λ ε of pairs (λ ε , u ε ), u ε = 0 that satisfy (1.1), with λ ε m → ∞ as m → ∞. We study the high-frequency spectrum of (1.1); more precisely, we describe the asymptotic behaviour of the set ε 2 σ ε when the period ε ∈ Ξ goes to zero. Without loss of generality, we shall study the set of pairs (μ ε , Indeed, if μ ε and u ε satisfy (1.3) then (1.1) holds for u ε and λ ε = ε −2 (μ ε − 1). In other words, one has lim ε→0 ε 2 σ ε = lim ε→0σ ε , whereσ ε + 1 is the spectrum of ( Note that while R ε are compact they converge in the 'strong two-scale sense' [4][5][6] to a non-compact operator. In fact, as we recall next, the limit set lim ε→0σε contains a union of intervals. Indeed, for any m ∈ N 3 a subsequence of R ε strongly two-scale converges (e.g. [6]) to an operator R (m) defined on the space (1.5) (Henceforth, we use the subscripts ,y l , ,y j , . . . rather than ,l , ,j , . . . to refer to the derivatives with respect to the 'cell variable' y ∈ mY of a function of the pair (x, y) ∈ Ω × mY.) In particular, the spectrum of R (m) is contained in the set lim ε→0 (σ ε + 1) −1 . The arbitrary choice of m implies that σ Bloch ⊂ lim ε→0σ ε , where (σ Bloch + 1) −1 is the union of the spectra of R (m) over all m ∈ N 3 . The description of the 'Bloch spectrum' σ Bloch corresponding to the differential expression in (1.5) can be found in a number of standard references, such as e.g. [7]. In the study of lim ε→0 σ ε , the question that remains is if this limit set is fully described by the Bloch spectrum, i.e. whether lim ε→0σ ε = σ Bloch hold. For problems in the whole space and for problems on a 'supercell' torus (cf. [6]), it can be shown that this is indeed the case. However, introducing a boundary may result in the existence of sequences of eigenvalues of (1. as suggested by the numerical analysis of waves propagating in layered elastic media with a stress-free boundary [8][9][10]. We begin our discussion of boundary phenomena for the equation (1.1) with reference to the work [11] (see also [12,13]), where it is noted that the Bloch spectrum may not be sufficient for the description of lim ε→0σ ε . The paper [11] discusses, among other things, implications of a possible discrepancy between these two sets in the case of scalar elliptic partial differential equations (PDEs). Therein, a subset σ boundary of lim ε→0σ ε is introduced that was shown to contain any such 'leftover' spectrum. However, it remained to be seen whether such a discrepancy does occur. This leads one to natural questions about technical aspects of σ boundary , namely is it non-empty and if, in general, it is not contained in σ Bloch . The main objective of our present work is to elucidate this matter by providing an example of a PDE system based on the family (1.1) that possesses eigenvalue sequences with accumulation points that belong to σ boundary and lie outside σ Bloch . In our example, we consider a layered medium, where the elastic parameters C ijkl depend on a single spatial variable. We investigate both cases of the 'stress-free' (i.e. Neumann-type) and 'zero-displacement' (i.e. Dirichlet-type) boundary conditions on ∂Ω. We show that in each case a new surface wave is present, in addition to the classical Rayleigh wave in a homogeneous medium. In particular, for a zero-displacement boundary, this demonstrates a surface effect that has not been previously addressed. The layout of the article is as follows. In §2, we prove that the set lim ε→0σ ε is the union of σ Bloch and σ boundary for the PDE system (1.1) with Neumann boundary conditions. In §3, we provide a characterization for the set σ boundary in terms of a family of canonical half-space problems. In §4, by analysing the canonical family introduced in §3, we show for a specific example of (1.1) that σ boundary is neither empty nor a strict subset of σ Bloch . In the same section, we also give a new secular equation for the speed of a surface wave that occurs in problems of type (1.1) subject to a zero-displacement condition on part of the boundary of the domain Ω. Appendix A contains the technical details concerning the proof of the main characterization result, presented in §2, of the set lim ε→0σ ε for the PDE family (1.3) with Neumann boundary conditions.

2.
Homogenization and the characterization of the limit spectrum (a) The inclusion σ Bloch ⊂ lim ε→0σ ε The result of this section is demonstrated by a series of standard results that we shall introduce, without proof, and cite where appropriate. To begin with, we shall review the notion of two-scale convergence [4,14]. (i) We say that u ε two-scale converge to u 0 (x, y) ∈ L 2 (Ω; [L 2 (mY)] 3 ), and write u ε (ii) We say that u ε strongly two-scale converge to u 0 , and write u ε 2 → u 0 , if for all v ε 2 v 0 one has 3 and on a closed linear subspace H of [L 2 (Ω × mY)] 3 , respectively. We say that A ε strongly two-scale resolvent converge to A 0 and denote it by where P : The main result of this section, corollary 2.4, follows directly from the following standard result (e.g. [6]).

Then the inclusion
Note that the bilinear form is closed and non-negative on H ε , hence it generates a self-adjoint operator A ε whose domain Similarly, for each m ∈ N 3 we denote by H (m) the closure of and note that the bilinear form defines a self-adjoint operator A (m) whose domain is a dense linear subset of [L 2 ρ (Ω × mY)] 3 . Clearly, the spectrum of the operator A (m) coincides with the set σ (m) defined above. Further, lemma 2.3 implies that A ε 2 → A (m) , and since the spectrum of A ε is given byσ ε , the inclusion σ (m) ⊂ lim ε→0σ ε follows. Applying lemma 2.3 for all m ∈ N 3 yields The above inclusion, along with the fact that lim ε→0σ ε is closed, provides the inclusion (2.2).
The inclusion σ Bloch ⊂ lim ε→0σ ε is now demonstrated by the following proposition.
Proof. By the min-max variational characterization of eigenvalues, κ can readily be shown to be continuous with respect to θ . The continuity of κ(θ) and the fact that the set of rational numbers is dense in R imply that Therefore, as clos( m∈N 3 σ (m) ) is closed, it is sufficient to prove the claim of the corollary for the case when (2π For any κ, u as above one has 3 . Note that since u, C are ρ are Y-periodic, and therefore mY-periodic, one has by direct calculation that where w := e θ u. In view of the freedom in the choice of Ψ , we infer that κ is an eigenvalue of A (m) with eigenfunction w.

(b) Completeness of spectrum
We shall now introduce, in analogy with [11], the set We now present, and prove in this section, the following result.
Theorem 2.6. Under the above definitions ofσ ε , σ Bloch and σ boundary , one has Denote by H the space with inner product (u, v) = R 3 ρu · v. To prove theorem 2.6, it suffices to show that all κ ∈ lim ε→0σ ε \ σ boundary belong to the spectrum of the operator A defined on a dense linear subset of H by the formula This is sufficient since, by the Bloch decomposition, one has σ (A) ⊂ σ Bloch . We shall prove an equivalent claim, namely that (κ + 1) −1 is an element of the spectrum of the operator i.e. (κ + 1) −1 is an element of the spectrum of B = (A + 1) −1 . Suppose that κ = lim ε→0 κ ε , where κ ε ∈σ ε for all ε. Our construction of a suitable Weyl sequence for the operator B is based on the following lemma proved in appendix A.
Proof of theorem 2.6. Let U ε be one of the sequences provided by lemma 2.7. We will show that U ε is a Weyl sequence for (κ + 1) −1 and B, i.e. that where I is the identity operator. First, we note that Note, that for A given by (2.6), the function p : 3 . Therefore, to prove the theorem it is sufficient to show the convergence To this end, note that by (2.7), we have and, by (2.6), we find that This goes to zero uniformly in ϕ as ε → 0 by (2.8), the assumption that κ ε → κ, and the fact that for some C > 0.

A half-space limit problem
Throughout this section, we use the notation Π m 1 ,m 2 := (0, m 1 ) × (0, m 2 ) × (0, +∞) for all (m 1 , m 2 ) ∈ N 2 . We show that the 'boundary' part of the spectrum of the operator generated by the differential expression −ρ −1 (C ijkl (· k ) ,l ) ,j in the half-space R 2 × (0, +∞) is contained in the limit spectrum lim ε→0σ ε . More precisely, for all values of the 'in-plane quasi-momentum' (θ 1 , θ 2 ) ∈ [0, 2π ) 2 consider the operator A θ 1 ,θ 2 in [L 2 ρ (Π 1,1 )] 3 defined by the bilinear form where ∂ l u k denotes the partial derivative of the field component u k with respect to the lth independent variable. In the above formula, we denote by [L 2 ρ (Π m 1 ,m 2 )] 3 the space with inner product  3 as the space of 'H 1 -functions periodic in the first two variables', or equivalently the closure in H 1 (Π 1,1 ) of the space S(Π 1,1 ) defined below.

Remark 3.3.
Note that the continuous part of the spectrum of A θ 1 ,θ 2 is also contained in lim ε→0σ ε , although we do not use this fact in what follows.

Examples with 'boundary' eigenvalue limits outside the 'bulk' spectrum
Here we analyse the spectrum of a reduced, two-dimensional, version of the half-space set-up discussed in the previous section.
(a) Two-dimensional elasticity for a layered half-space We consider the problem (1.1) for the case when the coefficients C ε ijkl and ρ ε are independent of the variable x 2 and restrict ourselves to the solutions (eigenfunctions) that are also x 2 -independent. In what follows we consider a half-space filled with layers of homogeneous and isotropic materials so that the coefficients C ijkl are given by where λ, μ are the constant 'Lamé coefficients'. In this case, system (1.1), which is understood in the weak sense, decouples into two systems, one for the component u ε 2 and the other for the components u ε 1 , u ε 3 . Following the argument of §3, we are led to the study of the boundary spectrum of the differential operator −ρ −1 (C ijkl (· k ) ,l ) ,j in the half-space R 2 × (0, +∞). The limit spectral problem is obtained by performing a re-scaling of the independent variable in (1.1) with the factor ε −1 and setting lim ε→0 ε 2 λ ε =: ω 2 .
In what follows, we consider a particular case when the unit normal (n j ) 3 j=1 to the layers is given by (0, 0, 1), in other words the layers are parallel to the (x 1 , x 2 )-plane (and hence to the boundary of the half-space in question). The decomposition of the problem (1.1) in two independent systems induces a decomposition of the limit boundary spectrum into the 'out-of-plane' part, obtained by solving a problem for the second component u 2 = u 2 (x 3 ) of the displacement: − ρ −1 (μu 2 ) = ω 2 u 2 , (4.1) and the 'in-plane' part, obtained by solving a system of equations for the components u 1 = u 1 (x 3 ), Each of systems (4.1) and (4.2) leads to a classical system in each individual layer subject to the conditions of the continuity across interfaces of the unknown function and of the normal component 3 k=1 C i3k3 u k,3 , i = 1, 2, 3, of the stress tensor: where the square brackets denote the jump across the interface.
In what follows we focus on system (4.2) subject to the interface conditions (4.3).
(b) Rayleigh waves in a layered elastic half space: the case of two homogeneous isotropic layers per period of the half-space as plane waves with wavenumber k and decaying exponentially away from the surface, i.e. solutions to (4.2) and (4.3) of the form Introducing the unknowns it is a straightforward calculation to check that (4.2) is equivalent to the system where λ and μ take indices 'A' and 'B', depending on the layer in which the solution is considered.
The above system is considered subject to the condition of continuity across the AB interfaces, which is equivalent to the interface conditions (4.3). In what follows we look for solutions that decay exponentially as x 3 → ∞, which thereby belong to the set σ point θ 1 ,θ 2 . We use the notation (4.5) where c := ω/|k| is the absolute value of the phase velocity, which we refer to as the 'wave speed'.
In what follows we generally treat the quantities 1,2 ,˜ 1,2 as functions of c 2 .

(i) Zero normal stress boundary conditions
Consider the case when the equations (4.2) and (4.3) are subject to the condition of zero normal stress on the boundary {x 3 = 0} of the half-space. The well-known formula (e.g. [17]) for the Rayleigh wave speed c R,N in a homogeneous half-space filled with material A is where we set c = c R,N in the formulae for 1 , 2 . We consider those values of c 2 for which the functions 1,2 ,˜ 1,2 above take positive real values. This obviously gives one constraint on the values of c = ω/k : Under the condition a surface wave with the Rayleigh wave speed c R,N given by (4.6) persists, thanks to the decay of the wave amplitude over the period cell implied by (4.7). In the case of the opposite inequality an analogue of the Rayleigh wave exists, whose wave speedĉ R is found from the equation where we set c =ĉ R,N in the formulae for 1,2 ,˜ 1,2 and denotek = d|k|. Note that for fixed values of the elastic parameters of the materials A and B, and provided the inequality (4.8) is satisfied, the wave speedĉ R,N depends on the product hk. In the limit hk → 0, the secular equation (4.9) takes the form 4˜ 1˜ 2 = (˜ 2 1 + 1) 2 , i.e. one hasĉ R,N →c R,N as expected, wherec R,N is the Rayleigh wave speed for material B. This observation also serves as a proof of the existence of the surface wave in question, at least for sufficiently small values of the parameters h,k (note that condition (4.8) is satisfied for small values of h.) For the speed of this wave as a function of the non-dimensional wavenumberk, see figure 1.

(ii) Zero-displacement boundary condition
It is well known (following an argument of [17]) that when (4.2) and (4.3) are subject to the condition of zero displacement on the boundary {x 3 = 0}, no surface waves are found in the case of a homogeneous half-space, i.e. when the materials A and B are identical. Suppose now that the materials A and B are different, in other words the half-space is filled with a medium that is 'genuinely layered', in a periodic manner. Under the same condition (cf. (4.8)) as above, a surface wave of a new kind with wave speedĉ R,D is found. The corresponding analogue of the above secular equation (4.9) for this case has the form  hasĉ R,D → 0, which is consistent with the fact that in the homogeneous half-space this wave is absent. The analysis of the eigenvalue problem (1.1) subject to the Dirichlet condition on part of the boundary of Ω is similar to that given in § §2 and 3. In particular, appropriate versions of corollaries 2.4 and 3.2 hold. We omit the details of this analysis.
(c) The Bloch spectrum of the layered elastic space The Bloch spectrum (or 'bulk' spectrum, as it is often referred to) of the two-dimensional problem described in §4 is the union of the two sets: (i) the Bloch spectrum of the 'in-plane displacement' problem (4.2) (subject to boundary conditions at x 3 = 0), for which the classical Rayleigh waves are eigenfunctions of the homogeneous half-space problem with stress-free boundary; (ii) the Bloch spectrum of the 'out-of plane displacement' problem (4.1) (subject to corresponding conditions at x 3 = 0), for which the Love waves are the eigenfunctions of the homogeneous half-space problem. As in the previous section, we aim at the spectrum of the first of these, using the fact that the two spectra can be analysed independently.
In what follows θ = θ 3 is the third component quasi-momentum (see §3), θ ∈ [0, 2π ). We use the notation 1,2 ,˜ 1,2 as above, see (4.4) and (4.5). We also denote and Then, for a given value of k 2 (equivalently,k), the set of wave speeds c such that ω 2 = k 2 c 2 is in the Bloch spectrum for the value θ of the quasi-momentum is given by solutions to the equation where as beforek = d|k|. Typical plots of the corresponding dispersion diagrams are shown in figures 3 and 4. For the in-plane problem in the half-space, this is the continuous part of the spectrum while the point part is given by the Rayleigh eigenvalues. Next we show that the corresponding set of values of ω is situated above the 'boundary frequencies' found in §4.   Here S, T, L, M are given by formulae (4.11) and (4.12). Solving the above equation for ( 2 1 −˜ 2 1 ) 2 yields (4.14) It is verified by a direct calculation of the leading-order behaviour of the right-hand side of (4.14) that for small values ofk the equation (4.14) implies which is impossible unless μ A /ρ A = μ B /ρ B and θ = 0. This shows that the wave speedsĉ R,N , c R,D , which were discussed in §4b(i),(ii), are situated below Bloch wave speeds. The same relation therefore holds between the corresponding spectra, due to the formula ω 2 = c 2 k 2 , where k is the wavenumber (see the beginning of §4b) and c is the corresponding wave speed.

Conclusion
We have analysed the location of the Bloch spectrum and of part of a boundary spectrum for highfrequency vibrations of an ε-periodic layered elastic medium. We consider the 'scale-interaction' regime, when the frequencies are of the order ε −2 as ε → 0. Formulae (4.9) and (4.10) are the secular equations of two kinds of surface waves that are shown to exist in such a medium: a version of the usual Rayleigh wave in a homogeneous half-space with stress-free boundary (equation (4.9)), and a new kind of wave for the zero-displacement problem (equation (4.10)). The latter does not have an analogue in the case of a homogeneous medium. We prove that the frequencies of the waves of these two kinds are situated below the Bloch frequencies of the fullspace formulation. In particular, the 'boundary' part of the original spectra (equation (1.1)) is non-empty for sufficiently small values of ε.
The existence of Rayleigh-type surface waves in a layered elastic half-space with zero displacement on the boundary may have consequences in applications where structures could be affected at frequencies below the Bloch spectrum, e.g. in seismology.
Data accessibility. Data used to generate the graphs in figures 1-4 were automatically deleted by the Matlab software during the analysis. No other data were created during this study.