Interfacial acoustic waves in one-dimensional anisotropic phononic bicrystals with a symmetric unit cell

Abstract

The paper is concerned with the interfacial acoustic waves localized at the internal boundary of two different perfectly bonded semi-infinite one-dimensional phononic crystals represented by periodically layered or functionally graded elastic structures. The unit cell is assumed symmetric relative to its midplane, whereas the constituent materials may be of arbitrary anisotropy. The issue of the maximum possible number of interfacial waves per full stop band of a phononic bicrystal is investigated. It is proved that, given a fixed tangential wavenumber, the lowest stop band admits at most one interfacial wave, while an upper stop band admits up to three interfacial waves. The results obtained for the case of generally anisotropic bicrystals are specialized for the case of a symmetric sagittal plane.

1. Introduction

The vast legacy of Professor Peter Chadwick includes many significant results in linear, nonlinear, uncon- strained and constrained elasticity and thermo- elasticity. In particular, his name is inseparable from the theory of surface and interfacial acoustic waves in anisotropic media (crystals). Inspired by the early work of Stroh [1], Chadwick laid, along with Lothe & Barnett [2], the foundation of what became known as the Stroh formalism of crystal acoustics in application to the theory of localized (Rayleigh and Stoneley) waves. A classical extensive article [3] by Chadwick & Smith largely established the rigorous basis of this formalism. The all-embracing reviews [4–7] provided comprehensive guides to bulk and surface waves in cubic and transversely isotropic media. Professor Chadwick's inherent approach to attacking problems at the deep and fundamental level has rendered his research papers as hugely influential in the development of anisotropic linear elastodynamics (as well documented in [8]). Some of his ideas underlie the origin of the present paper, which we dedicate to the memory of Peter Chadwick.

In recent decades, the centre of interest of theoretical and applied acoustics has moved to waves in metamaterials and periodic structures called phononic crystals [9–11]. Many studies have been performed on the spectra of surface acoustic waves (SAWs) in semi-infinite phononic crystals, where they become dispersive, in contrast to the case of homogeneous media. The impetus for these studies came from the pioneering works carried out in the 1970s and 1980s on SAWs in superlattices, i.e. periodically layered elastic media; see [12–14] and the bibliographies therein. Having intensified later on (see for example [15–23]), this research trend is nowadays associated with one-dimensional (1D) elastic phononic crystals, which remain highly topical side by side with piezoelectric and solid–fluid analogues and with two- and three-dimensional models. The combination of anisotropy and (periodic) spatial inhomogeneity essentially prevents explicit solutions to the boundary problem (which is already complicated enough in the case of homogeneous crystals) being reached, so SAW properties used to be investigated mainly by numerical means. However, it appears that the analytical approach underlying the Lothe–Barnett–Chadwick theory of SAWs in crystals can be developed and applied to the problem of the existence and number of SAWs in the spectra of 1D elastic and piezoelectric phononic crystals [24–26].

The present paper is concerned with the interfacial acoustic waves (IAWs) localized at the internal boundary of 1D phononic bicrystals, which are formed from two perfectly bonded periodically layered or functionally graded semi-infinite elastic media of arbitrary anisotropy. Such waves have been considered in elastic as well as in piezoelectric structures [27–32]. The question which we would like to answer is: how many IAWs can exist in a stop band of the Floquet spectrum of a phononic bicrystal? The motivation for our study comes from the results [33–35] on the existence of Stoneley waves at the interface between homogeneous anisotropic solids. Earlier papers [33,34] established that the Stoneley waves in bicrystals can exist within sectors of propagation directions rather than along discrete directions only. The paper [35] proved the uniqueness of the subsonic Stoneley wave and provided the criterion for its existence. The approach of [35] was based on the method of a surface impedance matrix [36,37] constructed from the eigenvectors of the Stroh matrix. Following [24], we are going to proceed from this method generalization, which invokes the eigenvectors of the transfer matrix through a unit cell of the periodic structure. In the present study, we consider phononic bicrystals with arbitrarily anisotropic but symmetrically assembled unit cells. The latter implies that the spatial variation of the material properties within the unit cell is symmetric relative to its midplane. According to [24], such an arrangement of the unit cell leads to more stringent conditions on the admissible number of SAWs by comparison with the case of the arbitrary (asymmetric) unit cell. Nevertheless, it will be shown that while the lowest stop band of the phononic bicrystal spectrum admits only one IAW at fixed wave number, which is similar to the case of a subsonic Stoneley wave in a homogeneous bicrystal [35], any upper stop band allows the existence of up to three IAWs. As we will see, this is because the surface impedance matrix in the upper stop bands may have pole(s) and is not constrained by the sign definiteness of its static limit.

The paper is organized as follows. The properties of the transfer and surface impedance matrices for a 1D periodic structure are described in §2. The results on the existence of IAWs are presented in §3, where we consider the cases of general anisotropy (§3b) and the sagittal plane being the plane of crystallographic symmetry (§3c). The summary is given in §4.

2. Transfer matrix and impedance for a phononic crystal with a symmetric unit cell

(a). Transfer matrix

Consider a 1D phononic crystal whose density ρ and stiffness tensor c_ijkl (i, j, k, l = 1, 2, 3) are periodic piecewise continuous functions of one coordinate y = n · r along the direction of unit vector n. This may be a functionally graded and/or multilayered elastic material with the perfectly bonded layer interfaces orthogonal to n. A harmonic wave field, which propagates with frequency ω and tangential wavenumber k along the direction of unit vector m orthogonal to n, can be sought in the form

$$\left(\begin{array}{c}\mathbf{u}(\mathbf{r},t)\\ \mathbf{f}(\mathbf{r},t)\end{array}\right)=\left(\begin{array}{c}\mathbf{a}(y)\\ -\mathrm{i}\mathbf{l}(y)\end{array}\right){\mathrm{e}}^{\mathrm{i}(kx-\omega t)},$$ 2.1

where u is the displacement, $\mathbf{f}=\hat{\mathit{\sigma }}\mathbf{n}$ is the normal traction, $\hat{\mathit{\sigma }}$ is the stress and x = m · r. According to [1–3,36], the equation of motion and the stress–strain law can be combined into the system of equations

$$\frac{\mathrm{d}\mathit{\xi }}{\mathrm{d}y}=\text{i}\hat{\mathbf{N}}\mathit{\xi }$$ 2.2

with respect to the vector of amplitudes

$$\mathit{\xi }(y)={(\mathbf{a}(y),\hspace{0.17em}\mathbf{l}(y))}^T,$$ 2.3

where^T means transposition. The 6 × 6 Stroh matrix of coefficients, presented here in the form of [36] (it allows for zero k, which is useful in the case of dispersion), is

$$\hat{\mathbf{N}}=-\left(\begin{array}{cc}k{(nn)}^{-1}(nm)& {(nn)}^{-1}\\ k^2[(mn){(nn)}^{-1}(nm)-(mm)]+\rho {\omega }^2\hat{\mathbf{I}}& k(mn){(nn)}^{-1}\end{array}\right),$$ 2.4

where (ab)_jk = c_ijkla_ib_l for a, b = n, m and $\hat{\mathbf{I}}$ is the 3 × 3 unit matrix. The Stroh matrix obeys the symmetry relation

$$\hat{\mathbf{N}}=\hat{\mathbf{T}}{\hat{\mathbf{N}}}^T\hat{\mathbf{T}},$$ 2.5

where the matrix $\hat{\mathbf{T}}$ has zero diagonal and unit off-diagonal blocks. We assume henceforth that ω and k are real, hence so is $\hat{\mathbf{N}}$.

The corresponding transfer, or propagator, matrix $\hat{\mathbf{M}}(T,0)\equiv \hat{\mathbf{M}}$ relating the state vector at the opposite edges y = 0 and y = T of the unit cell,

$$\mathit{\xi }(T)=\hat{\mathbf{M}}\mathit{\xi }(0),$$ 2.6

is either a product of propagators ${\hat{\mathbf{M}}}_i={\mathrm{e}}^{\mathrm{i}{h}_i{\hat{\mathbf{N}}}_i}$ through the constituent homogeneous layers, each with a constant matrix ${\hat{\mathbf{N}}}_i$ and thickness h_i, or a multiplicative integral of a continuously varying matrix $\hat{\mathbf{N}}(y)$ [38–40]. In either case, equation (2.5) provides the identity [38–40]

$${\hat{\mathbf{M}}}^{-1}=\hat{\mathbf{T}}{\hat{\mathbf{M}}}^{†}\hat{\mathbf{T}},$$ 2.7

where^† means Hermitian conjugation. By (2.7), the eigenvalues γ_α of $\hat{\mathbf{M}}=\hat{\mathbf{M}}(\omega ,k),$

$$\hat{\mathbf{M}}{\mathit{\zeta }}_{\alpha }={\gamma }_{\alpha }{\mathit{\zeta }}_{\alpha },\alpha =1,\dots ,6,$$ 2.8

are pairwise related; namely,

$$\mathrm{either}\hspace{0.17em}\left|{\gamma }_{\alpha }\right|=\left|{\gamma }_{\alpha +3}\right|=1\hspace{0.17em}\hspace{0.17em}\mathrm{or}\hspace{0.17em}\hspace{0.17em}{\gamma }_{\alpha }=1/{\gamma }_{\alpha +3}^{\ast }\hspace{0.17em}\mathrm{for}\hspace{0.17em}\left|{\gamma }_{\alpha }\right|\ne 1,\alpha =1,2,3,$$ 2.9

where* means complex conjugation. Each eigenvector

$${\mathit{\zeta }}_{\alpha }={({\mathbf{A}}_{\alpha },\hspace{0.17em}{\mathbf{L}}_{\alpha })}^T,$$ 2.10

being taken as the initial data for equation (2.2), may be said to generate an αth partial mode ξ_α(y), whose absolute value at the consecutive period edges stays constant in the case (2.9)₁ or increases or decreases in the case (2.9)₂. Correspondingly, the occurrences of the two options (2.9) map out the plane (ω, k) into the spectral zones called pass bands and stop bands. Aiming at the localized wave solutions, we will be interested in the full stop bands defined as the areas on the plane (ω, k) where all six eigenvalues obey (2.9)₂. The orthonormalization relation, fulfilled by the eigenvectors ζ_α owing to the identity (2.7), is in the full stop bands of the form

$${\mathit{\zeta }}_{\alpha }^{†}\hat{\mathbf{T}}{\mathit{\zeta }}_{\beta }={\delta }_{\alpha +3,\beta },\alpha =1,2,3;\hspace{0.17em}\beta =1,\dots ,6,$$ 2.11

where δ_αβ is the Kronecker symbol.

The present paper is concerned with the periodic structures whose unit cell is symmetric relative to its midplane. In the case of discretely inhomogeneous (layered) materials, a symmetric unit cell is composed of an odd number n = 2m + 1 of layers, the ith and (2m + 1 − i)th layers (i = 1, …, m) being identical. In particular, as noted in [41], it may be a structure of identical bilayers, where the period edge is referred to the middle of one of the layers. For functionally graded materials, this implies an even profile $\hat{\mathbf{N}}(y)=\hat{\mathbf{N}}(-y)$ of the Stroh matrix with respect to the middle of a unit cell. In the former case of a layered unit cell, the transfer matrix ${\hat{\mathbf{M}}}^{(S)}$ is

$${\hat{\mathbf{M}}}^{(S)}={\hat{\mathbf{M}}}_1{\hat{\mathbf{M}}}_2\cdots {\hat{\mathbf{M}}}_{m+1}\cdots {\hat{\mathbf{M}}}_2{\hat{\mathbf{M}}}_1,$$ 2.12

where we add the label (S) for ‘symmetric’. From equation (2.5), in view of (2.12) and real-valuedness of $\hat{\mathbf{N}},$ it follows that ${\hat{\mathbf{M}}}^{(S)}$ satisfies the identities

$${\hat{\mathbf{M}}}^{(S)}=\hat{\mathbf{T}}{\hat{\mathbf{M}}}^{(S)T}\hat{\mathbf{T}}\mathrm{and}{\hat{\mathbf{M}}}^{(S)-1}={\hat{\mathbf{M}}}^{(S)\ast }.$$ 2.13

The same relations hold for the case of an even functionally graded unit cell, as can be seen either from the equality ${\hat{\mathbf{M}}}_{\hat{\mathbf{N}}(y)}^{(S)}={\hat{\mathbf{M}}}_{{\hat{\mathbf{N}}}^T(-y)}^{(S)T}$ of the unit cell propagators for equation (2.2), with the matrices of coefficients indicated in the subscript, or from the limiting consideration of (2.12) with $h_i\to 0$ and $n\to \mathrm{\infty }.$ Note that a conjunction of the identities (2.13)₁ and (2.13)₂ is consistent with equation (2.7), but none of (2.13) is valid on its own for phononic crystals with a generic (asymmetric) unit cell.

According to (2.13)₁, the orthonormalization relation for the eigenvectors ζ_α of ${\hat{\mathbf{M}}}^{(S)}$ holds in the form

$${\mathit{\zeta }}_{\alpha }^T\hat{\mathbf{T}}{\mathit{\zeta }}_{\beta }={\delta }_{\alpha \beta },\alpha ,\hspace{0.17em}\beta =1,\dots ,6,$$ 2.14

which, owing to (2.13)₂, is complemented in the full stop bands by the relation

$${\mathit{\zeta }}_{\alpha }^{\ast }={\mathit{\zeta }}_{\alpha +3},\alpha =1,2,3.$$ 2.15

Equations (2.14) and (2.15) taken together are equivalent to two equalities

$${\mathit{\zeta }}_{\alpha }^{†}\hat{\mathbf{T}}{\mathit{\zeta }}_{\beta }=0,{\mathit{\zeta }}_{\alpha }^T\hat{\mathbf{T}}{\mathit{\zeta }}_{\beta }={\delta }_{\alpha \beta },\hspace{0.17em}\alpha ,\beta =1,2,3,$$ 2.16

where the former is consistent with (2.11) but the latter is essentially different. They may also be written in the matrix form,

$${\hat{\mathbf{A}}}^{†}\hat{\mathbf{L}}+{\hat{\mathbf{L}}}^{†}\hat{\mathbf{A}}=\hat{\mathbf{0}}\mathrm{and}{\hat{\mathbf{A}}}^T\hat{\mathbf{L}}+{\hat{\mathbf{L}}}^T\hat{\mathbf{A}}=\hat{\mathbf{I}},$$ 2.17

where $\hat{\mathbf{A}}$ and $\hat{\mathbf{L}}$ are the 3 × 3 matrices whose columns are the vectors A_α and L_α, α = 1, 2, 3, respectively. Note that equations (2.14)–(2.17) are valid for any α under the assumption of a semisimple (diagonalizable) transfer matrix, adhered to below. The contrary case takes place at the pass band/stop band borders; it may exceptionally also occur within the bands, but this option does not affect the subsequent results.

It can be seen that the symmetric arrangement of the unit cell leading to the identities (2.13) in place of (2.7) has no effect on the properties of the transfer matrix eigenvalues (2.9) and hence on the band structure. On the other hand, this symmetry provides additional eigenvector relations (2.14) and (2.15) instead of (2.11). It is this particularity which underlies an essential difference in the considerations of the existence of localized waves developed for the cases of symmetric and asymmetric unit cells. As a consequence, as demonstrated for SAWs in [24,26], the results for the former case do not follow as a simple corollary of the results for the latter.

In the following, we address the full stop bands (2.9)₂ as frequency intervals ω_l < ω < ω_u at fixed k, with ω_l and ω_u implying the lower and upper band edges. For future use, let us specialize the eigenvalue numbering in full stop bands as

$$\left|{\gamma }_{\alpha }\right|<1,\left|{\gamma }_{\alpha +3}\right|>1,\hspace{0.17em}\alpha =1,2,3.$$ 2.18

This means that the vector of amplitudes (2.3) for a localized (evanescent) wave in a half-space y≥0 is a superposition of partial modes α = 1, 2, 3, generated by the initial value taken at the surface y = 0 in the form

$$\mathit{\xi }{|}_{y=0}=\sum _{\alpha =1}^3{b}_{\alpha }{\mathit{\zeta }}_{\alpha }\equiv {(\mathbf{A},\mathbf{L})}^T,$$ 2.19

where b_α are disposable constants.

(b). Impedance matrix

Consider a semi-infinite phononic crystal y≥0. Assuming ω and k in the full stop band and bearing in mind (2.19), introduce the surface impedance $\hat{\mathbf{Z}}$ by analogy with the case of a homogeneous half-space [2,3,36,37] via any one of the equivalent formulae

$${\mathbf{L}}_{\alpha }=-\text{i}\hat{\mathbf{Z}}{\mathbf{A}}_{\alpha }\hspace{0.17em}(\alpha =1,2,3)\mathrm{and}\hat{\mathbf{L}}=-\text{i}\hat{\mathbf{Z}}\hat{\mathbf{A}},\hspace{0.17em}\hat{\mathbf{Z}}=\text{i}\hat{\mathbf{L}}{\hat{\mathbf{A}}}^{-1},$$ 2.20

where $\hat{\mathbf{A}}$ and $\hat{\mathbf{L}}$ are defined below equation (2.17). In view of equation (2.17)₁, the matrix $\hat{\mathbf{Z}}$ is Hermitian.

The impedance possesses certain sign-definiteness properties, which may be proved similarly to [2,3]. However, differently from the above-cited works, we will not invoke the Lagrangian function and appeal directly to kinetic and strain energy of the wave (2.1). In particular, the time average local kinetic energy per unit surface can be written in the form [36]

$$E(y)=\frac{\rho {\omega }^2}{4}|\mathbf{u}{|}^2=-\frac{\omega }{8}{\mathit{\xi }}^{†}\left(\frac{\mathrm{\partial }}{\mathrm{\partial }\omega }\hat{\mathbf{T}}\hat{\mathbf{N}}\right)\mathit{\xi }=\frac{\text{i}\omega }{8}\frac{\mathrm{d}}{\mathrm{d}y}({\mathit{\xi }}^{†}\hat{\mathbf{T}}\frac{\mathrm{\partial }\mathit{\xi }}{\mathrm{\partial }\omega }),$$ 2.21

where ξ(y) is the vector of amplitudes (2.3). Apply (2.21) to the evanescent wave field generated in a half-space y≥0 by the initial data (2.19) and tending to zero as $y\to \mathrm{\infty }$. Then, in view of the continuity of ξ(y) at the layer interfaces, the total kinetic energy per unit surface $\overline{E}={\int }_0^{\mathrm{\infty }}E(y)\hspace{0.17em}\mathrm{d}y$ is

$$\overline{E}=-\frac{\text{i}\omega }{8}{\left({\mathit{\xi }}^{†}\hat{\mathbf{T}}\frac{\mathrm{\partial }\mathit{\xi }}{\mathrm{\partial }\omega }\right)}_{y=0}=-\frac{\text{i}\omega }{8}({\mathbf{A}}^{†}\frac{\mathrm{\partial }\mathbf{L}}{\mathrm{\partial }\omega }+{\mathbf{L}}^{†}\frac{\mathrm{\partial }\mathbf{A}}{\mathrm{\partial }\omega })=-\frac{\omega }{8}{\mathbf{A}}^{†}\frac{\mathrm{\partial }\hat{\mathbf{Z}}}{\mathrm{\partial }\omega }\mathbf{A},$$ 2.22

where the last equality assumes that the vector A does not depend on frequency and it takes into account definition (2.20) of $\hat{\mathbf{Z}}$ along with its Hermiticity. Thus, since the kinetic energy is a positive quantity and A is arbitrary, it follows that $\mathrm{\partial }\hat{\mathbf{Z}}/\mathrm{\partial }\omega \hspace{0.17em}$at fixed k is a negative-definite matrix in a full stop band. As a consequence,

$$\begin{array}{rl}& \mathrm{the}\hspace{0.17em}\mathrm{eigenvalues}\hspace{0.17em}\mathrm{of}\hspace{0.17em}\hat{\mathbf{Z}}\hspace{0.17em}\mathrm{decrease}\\ & \mathrm{with}\hspace{0.17em}\mathrm{increasing}\hspace{0.17em}\mathrm{frequency}\hspace{0.17em}\mathrm{in}\hspace{0.17em}\mathrm{a}\hspace{0.17em}\mathrm{full}\hspace{0.17em}\mathrm{stop}\hspace{0.17em}\mathrm{band}.\end{array}$$ 2.23

Next consider the time-averaged local elastic energy per unit surface

$$W=\frac{1}{8}({\sigma }_{ij}\frac{\mathrm{\partial }{u}_i^{\ast }}{\mathrm{\partial }{x}_j}+{\sigma }_{ij}^{\ast }\frac{\mathrm{\partial }{u}_i}{\mathrm{\partial }{x}_j}).$$ 2.24

Referring (2.24) to the wave (2.1), using the equation of motion and taking the static limit ω = 0 yields

$$W(y){|}_{\omega =0}=\frac{1}{8}\frac{\mathrm{d}}{\mathrm{d}y}({\mathbf{u}}^{†}\mathbf{f}+{\mathbf{f}}^{†}\mathbf{u}).$$ 2.25

As above, consider the evanescent wave field evolving from (2.19) in a half-space y≥0. Then the total static elastic energy per unit surface $\overline{W}={\int }_0^{\mathrm{\infty }}W(y)\hspace{0.17em}\mathrm{d}y$ reduces to

$$\overline{W}{|}_{\omega =0}=-\frac{1}{8}({\mathbf{u}}^{†}\mathbf{f}+{\mathbf{f}}^{†}\mathbf{u}){|}_{y=0}=\frac{\text{i}}{8}({\mathbf{A}}^{†}\mathbf{L}-{\mathbf{L}}^{†}\mathbf{A})=\frac{1}{4}{\mathbf{A}}^{†}\hat{\mathbf{Z}}\mathbf{A},$$ 2.26

where $\hat{\mathbf{Z}}={\hat{\mathbf{Z}}}^{†}$ was used. Since the strain energy is positive and the vector A is arbitrary, it follows that $\hat{\mathbf{Z}}$ is a positive-definite matrix at ω = 0, and so

$$\mathrm{the}\hspace{0.17em}\mathrm{eigenvalues}\hspace{0.17em}\mathrm{of}\hspace{0.17em}\hat{\mathbf{Z}}\hspace{0.17em}\mathrm{are}\hspace{0.17em}\mathrm{positive}\hspace{0.17em}\mathrm{at}\hspace{0.17em}\omega =0.$$ 2.27

Note that equation (2.27) takes into account the fact that the line ω = 0 gives rise to a full stop band once k≠0. This can be proved by invoking the static limit of the local strain energy W_α(y) of the αth partial mode generated at y = 0 by the eigenvector ζ_α (α = 1, …, 6). Integrating it over the unit cell and using (2.8) yields

$${⟨W_{\alpha }⟩}_{\omega =0}={\int }_0^T{W}_{\alpha }(y){|}_{\omega =0}\hspace{0.17em}\mathrm{d}y=-\frac{\text{i}}{8}(|{\gamma }_{\alpha }{|}^2-1)({\mathbf{A}}_{\alpha }^{†}{\mathbf{L}}_{\alpha }-{\mathbf{L}}_{\alpha }^{†}{\mathbf{A}}_{\alpha }).$$ 2.28

Since 〈W_α〉_ω=0 cannot turn to zero at k≠0, it follows that |γ_α|≠1 at ω = 0 and hence, by continuity, at some non-zero ω.

A central role in the subsequent analysis is played by the real part of the impedance $\hat{\mathbf{Z}}$, which we denote by $\hat{\mathbf{G}}$. Since $\hat{\mathbf{Z}}$ considered in the full stop band is Hermitian, its real part is a symmetric matrix, so that

$$\hat{\mathbf{G}}\equiv \text{Re}\hat{\mathbf{Z}}={\hat{\mathbf{G}}}^T.$$ 2.29

By this definition, the properties (2.23) and (2.27) of $\hat{\mathbf{Z}}$ carry over to the matrix $\hat{\mathbf{G}}$, so that

$$\begin{array}{rl}& \mathrm{the}\hspace{0.17em}\mathrm{eigenvalues}\hspace{0.17em}\mathrm{of}\hspace{0.17em}\hat{\mathbf{G}}\hspace{0.17em}\mathrm{are}\hspace{0.17em}\text{positive}\hspace{0.17em}\mathrm{at}\hspace{0.17em}\omega =0\hspace{0.17em}\mathrm{and}\\ & \mathrm{they}\hspace{0.17em}\mathrm{decrease}\hspace{0.17em}\mathrm{with}\hspace{0.17em}\mathrm{increasing}\hspace{0.17em}\mathrm{frequency}\hspace{0.17em}\mathrm{in}\hspace{0.17em}\mathrm{a}\hspace{0.17em}\mathrm{full}\hspace{0.17em}\mathrm{stop}\hspace{0.17em}\mathrm{band}.\end{array}$$ 2.30

The above-outlined features characterize the impedance of a periodic structure with any unit cell. The next following relation holds specifically in the case of a symmetric unit cell [24]. It can be obtained by combining identity (2.17)₂ with definition (2.20) of $\hat{\mathbf{Z}}$ and taking into account the Hermiticity $\hat{\mathbf{Z}}={\hat{\mathbf{Z}}}^{†}$, so that

$${\hat{\mathbf{A}}}^T\hat{\mathbf{L}}+{\hat{\mathbf{L}}}^T\hat{\mathbf{A}}={\hat{\mathbf{A}}}^T[\hat{\mathbf{L}}{\hat{\mathbf{A}}}^{-1}+{({\hat{\mathbf{A}}}^T)}^{-1}{\hat{\mathbf{L}}}^T]\hat{\mathbf{A}}=-i{\hat{\mathbf{A}}}^T(\hat{\mathbf{Z}}+{\hat{\mathbf{Z}}}^{\ast })\hat{\mathbf{A}}=\mathbf{I}.$$ 2.31

Hence, with regard to (2.29),

$$\hat{\mathbf{G}}=-{\hat{\mathbf{Q}}}^{-1},\mathrm{where}\hspace{0.17em}\hat{\mathbf{Q}}=2\text{i}\hat{\mathbf{A}}{\hat{\mathbf{A}}}^T.$$ 2.32

This relation is similar to that for the matrices built from the eigenvectors of the Stroh matrix of a homogeneous material [2,3,38].

The matrix $\hat{\mathbf{Q}}$ can be shown to be finite inside any full stop band, including possible exceptional points where the unit cell transfer matrix is non-semisimple. Hence $det\hat{\mathbf{G}}\ne 0$ and so

$$\mathrm{none}\hspace{0.17em}\mathrm{of}\hspace{0.17em}\mathrm{the}\hspace{0.17em}\mathrm{eigenvalues}\hspace{0.17em}\mathrm{of}\hspace{0.17em}\hat{\mathbf{G}}\hspace{0.17em}\mathrm{vanishes}\hspace{0.17em}\mathrm{in}\hspace{0.17em}\mathrm{a}\hspace{0.17em}\mathrm{full}\hspace{0.17em}\mathrm{stop}\hspace{0.17em}\mathrm{band}.$$ 2.33

According to [24], the dispersion equation $det\hat{\mathbf{L}}=0$ for SAW on a traction-free surface with $\hat{\mathit{\sigma }}\mathbf{n}=0$ can have at most one solution per full stop band at fixed k in a semi-infinite phononic crystal with a symmetric unit cell. Hence at most one eigenvalue of $\hat{\mathbf{Z}}$ can vanish in a full stop band at fixed k. In addition, contrary to the case of a homogeneous elastic half-space, a semi-infinite phononic crystal admits the existence of SAWs on a clamped surface with the boundary condition of zero displacement u = 0, which provides the dispersion equation $det\hat{\mathbf{A}}=0$. Specifically, it was shown in [24] that a phononic crystal with a symmetric unit cell subjected to a clamped boundary condition cannot support SAWs in the lowest full stop band 0 < ω < ω_u and admits at most one SAW at fixed k per upper full stop band ω_l < ω < ω_u. Denote the frequency of SAW on a clamped surface by ω_cl [ =ω_cl(k)]. We also note from the definition of $\hat{\mathbf{Q}}$ in (2.32) that $det\hat{\mathbf{Q}}=-8i{(det\hat{\mathbf{A}})}^2$ and that the eigenvalues of $\hat{\mathbf{Q}}$ and hence of $\hat{\mathbf{G}}$ can vanish only pairwise in a full stop band [24]. Thus, taking into account (2.23), it follows that

$$\begin{array}{rl}& \mathrm{the}\hspace{0.17em}\mathrm{eigenvalues}\hspace{0.17em}\mathrm{of}\hspace{0.17em}\hat{\mathbf{Z}}\hspace{0.17em}\mathrm{and}\hspace{0.17em}\mathrm{of}\hspace{0.17em}\hat{\mathbf{G}}\hspace{0.17em}\mathrm{are}\hspace{0.17em}\mathrm{continuous}\hspace{0.17em}\hspace{0.17em}\\ & \mathrm{monotonically}\hspace{0.17em}\mathrm{decreasi}\mathrm{n}\mathrm{g}\hspace{0.17em}\mathrm{functions}\hspace{0.17em}\mathrm{of}\hspace{0.17em}\mathrm{frequency}\hspace{0.17em}\\ & \mathrm{throughout}\hspace{0.17em}\mathrm{the}\hspace{0.17em}\mathrm{lowest}\hspace{0.17em}\mathrm{stop}\hspace{0.17em}\mathrm{band};\end{array}$$ 2.34

and

$$\begin{array}{rl}& \mathrm{one}\hspace{0.17em}\mathrm{eigenvalue}\hspace{0.17em}\mathrm{of}\hspace{0.17em}\hat{\mathbf{Z}}\hspace{0.17em}\mathrm{and}\hspace{0.17em}\mathrm{simultaneously}\hspace{0.17em}\mathrm{two}\hspace{0.17em}\mathrm{eigenvalues}\hspace{0.17em}\mathrm{of}\hspace{0.17em}\hat{\mathbf{G}}\\ & \mathrm{can}\hspace{0.17em}\mathrm{have}\hspace{0.17em}\mathrm{one}\hspace{0.17em}\mathrm{simple}\hspace{0.17em}\mathrm{pole}\hspace{0.17em}{\omega }_{cl}\hspace{0.17em}\mathrm{in}\hspace{0.17em}\mathrm{an}\hspace{0.17em}\mathrm{upper}\hspace{0.17em}\mathrm{stop}\hspace{0.17em}\mathrm{band}\hspace{0.17em}\mathrm{at}\hspace{0.17em}\mathrm{fixed}\hspace{0.17em}k.\end{array}$$ 2.35

As a consequence of (2.30), (2.33) and (2.35),

$$\mathrm{the}\hspace{0.17em}\mathrm{matrix}\hspace{0.17em}\hat{\mathbf{G}}\hspace{0.17em}\mathrm{is}\hspace{0.17em}\text{positive}\hspace{0.17em}\mathrm{definite}\hspace{0.17em}\mathrm{in}\hspace{0.17em}\mathrm{the}\hspace{0.17em}\mathrm{lowest}\hspace{0.17em}\mathrm{stop}\hspace{0.17em}\mathrm{band}.$$ 2.36

In the upper stop bands, two eigenvalues of $\hat{\mathbf{Z}}$ and one eigenvalue of $\hat{\mathbf{G}}$ are continuous functions of ω, the latter having a permanent sign due to (2.33). Note in conclusion that, except ω = 0, the eigenvalues of $\hat{\mathbf{Z}}$ and of $\hat{\mathbf{G}}$ at the stop band edges may generally have arbitrary signs (or turn to zero).

3. Interface waves in phononic bicrystals with symmetric unit cells

(a). Dispersion equation

Consider a phononic bicrystal formed of two perfectly bonded periodic half-spaces with symmetric unit cells. We will be interested in IAWs localized at the bicrystal interface and decaying away from it. In this regard, we assume that ω and k lie within a full stop band, which is common for both phononic crystals, that is, we consider a frequency interval ω_l < ω < ω_u, where ω_l = max(ω⁽¹⁾_l, ω⁽²⁾_l) and ω_u = min(ω⁽¹⁾_u, ω⁽²⁾_u) with ω⁽¹⁾_l,u and ω⁽²⁾_l,u being the edges of two overlapping full stop bands for crystals 1 and 2.

Let crystals 1 and 2 occupy the half-spaces y≥0 and y ≤ 0, respectively (figure 1). The evanescent wave field decaying with depth $y\to \mathrm{\infty }$ in crystal 1 is produced by the initial value (2.19) at y = 0. The initial value, which gives rise to the evanescent wave field decaying with depth $y\to -\mathrm{\infty }$ in crystal 2, is

$${\mathit{\xi }}^{(2)}{|}_{y=0}=\sum _{\alpha =1}^3{b}_{\alpha }^{\left(2\right)}{\mathit{\zeta }}_{\alpha }^{\left(2\right)\ast }\equiv {({\mathbf{A}}^{\left(2\right)},{\mathbf{L}}^{\left(2\right)})}^T,$$ 3.1

where ζ^(2)*_α = ζ⁽²⁾_α+3 (see (2.15)) are the eigenvectors of the transfer matrix M₂(T, 0)≡M₂ corresponding to its eigenvalues |γ⁽²⁾_α| < 1, and we have taken into account the identities M^(S)₂( − T, 0) = M^(S)−1₂(T, 0) = M^(S)*₂(T, 0) (see (2.13)₂ for the latter).

Figure 1.

Example of a phononic bicrystal with symmetric unit cells. Crystal 1 is made of materials 1a and 1b. Crystal 2 is made of materials 2c and 2b. The dashed line y = 0 is the boundary between crystals. The thickness of extreme layers 1a_e and 2c_e is half the thickness of layers 1a and 2c, respectively. The line y = 0 and the dotted midlines of layers 1a and 2c are the edges of symmetric unit cells. Unit vectors m and n specify the direction of propagation and the normal to the layer surfaces, respectively.

The boundary condition at the interface y = 0 implies continuity of displacement and normal traction. Hence, by (2.19) (with the superscript⁽¹⁾ referring to crystal 1) and (3.1), the sought IAW must satisfy equality

$$\sum _{\alpha =1}^3{b}_{\alpha }^{(1)}{\mathit{\zeta }}_{\alpha }^{(1)}=\sum _{\alpha =1}^3{b}_{\alpha }^{\left(2\right)}{\mathit{\zeta }}_{\alpha }^{(2)\ast }.$$ 3.2

Extending definition (2.20) to the surface impedance of crystal 2, so that ${\mathbf{L}}^{(2)}=i{\hat{\mathbf{Z}}}_2^{\ast }{\mathbf{A}}^{(2)},$ and using its Hermiticity ${\hat{\mathbf{Z}}}_2^{\ast }={\hat{\mathbf{Z}}}_2^T$ yields the dispersion equation of a form similar to the case of homogeneous media [35], namely

$$det{\hat{\mathbf{Z}}}_{\mathrm{I}}=0,$$ 3.3

where

$${\hat{\mathbf{Z}}}_{\mathrm{I}}={\hat{\mathbf{Z}}}_1+{\hat{\mathbf{Z}}}_2^T$$ 3.4

and the subscript I implies ‘interfacial’. Our further objective is to analyse the existence and maximum possible number of solutions of equation (3.3). With this in mind, we introduce the notation for the real symmetric part of ${\hat{\mathbf{Z}}}_{\mathrm{I}}={\hat{\mathbf{Z}}}_{\mathrm{I}}^{†},$

$$\text{Re}\hspace{0.17em}{\hat{\mathbf{Z}}}_I={\hat{\mathbf{G}}}_I={\hat{\mathbf{G}}}_1+{\hat{\mathbf{G}}}_2={\hat{\mathbf{G}}}_I^T,$$ 3.5

where ${\hat{\mathbf{G}}}_1=\text{Re}{\hat{\mathbf{Z}}}_1$ and ${\hat{\mathbf{G}}}_2=\text{Re}{\hat{\mathbf{Z}}}_2$ have the properties listed in §2b. We will also denote the eigenvalues and eigenvectors of ${\hat{\mathbf{Z}}}_I$ by λ_i and e_i, and the eigenvalues and eigenvectors of ${\hat{\mathbf{G}}}_{1,2}$ by g^(1,2)_i and t^(1,2)_i, i = 1, 2, 3.

(b). Unrestricted anisotropy

Consider the lowest stop band 0 < ω < ω_u of the bicrystal. It is the case that

$$\begin{array}{rl}& \mathrm{at}\hspace{0.17em}\mathrm{most}\hspace{0.17em}\mathrm{one}\hspace{0.17em}\mathrm{IAW}\hspace{0.17em}\mathrm{can}\hspace{0.17em}\mathrm{exist}\\ & \mathrm{in}\hspace{0.17em}\mathrm{the}\hspace{0.17em}\mathrm{lowest}\hspace{0.17em}\mathrm{stop}\hspace{0.17em}\mathrm{band}\hspace{0.17em}\mathrm{at}\hspace{0.17em}\mathrm{fixed}\hspace{0.17em}k.\end{array}$$ 3.6

The proof of this fact is the same as that given by [35] for the case of homogeneous anisotropic media. In view of the properties (2.27), (2.34) and equation (3.3), the statement (3.6) implies that not more than one eigenvalue of ${\hat{\mathbf{Z}}}_I$ can vanish in the lowest stop band at fixed k. Assume on the contrary that two eigenvalues λ₁ and λ₂ vanish at frequencies ω_I1 and ω_I2 > ω_I1, so that λ₁, λ₂ < 0 above ω_I2. Multiplying the spectral decomposition $Z_I=\sum _{i=1}^3{\lambda }_i{\mathbf{e}}_i\otimes {\mathbf{e}}_i^{†}$ from both sides by the (real) vector product q = Re(e₃) × Im(e₃) yields

$${\mathbf{q}}^T{\hat{\mathbf{Z}}}_I\mathbf{q}={\mathbf{q}}^T{\hat{\mathbf{G}}}_I\mathbf{q}=\sum _{i=1}^2{\lambda }_i|{\mathbf{q}}^T{\mathbf{e}}_i{|}^2<0.$$ 3.7

However, the inequality in (3.7) violates (2.36), and this contradiction proves (3.6). Note also two other observations of [35] which remain valid within the lowest stop band of a phononic bicrystal. Given a fixed k, they read that if IAW exists in this band then it must have a frequency greater than or equal to that of the minimal frequency of SAW in one of the crystals, and if none of the crystals admits a SAW then the IAW does not in the lowest stop band.

There is a different situation in the upper full stop bands. This is due to two factors. First, the impedance ${\hat{\mathbf{Z}}}_I$ is not sign definite at the lower edges of upper stop bands ω_l≠0. Second, by (2.35), the eigenvalues of ${\hat{\mathbf{Z}}}_1$ and/or of ${\hat{\mathbf{Z}}}_2$ can have a pole associated with SAW on a clamped surface of crystal 1 and/or crystal 2, hence the eigenvalues of ${\hat{\mathbf{Z}}}_I$ can have one or two poles in an upper stop band at fixed k. In the following, the reference to a fixed k is understood and kept tacit unless otherwise specified.

To begin with, assume that the eigenvalues of ${\hat{\mathbf{Z}}}_I$ have no poles and are therefore continuous monotonically decreasing functions λ_i(ω) in a given upper stop band. Since ${\hat{\mathbf{G}}}_I$ no longer has to be positive definite, each eigenvalue λ_i(ω) at fixed k can vanish once per stop band. Hence, three IAWs are possible in this case.

Now consider an upper full stop band where one of the eigenvalues of ${\hat{\mathbf{Z}}}_I$ has a pole at frequency ω = ω_cl, which corresponds to the pole of the eigenvalue of, say, ${\hat{\mathbf{Z}}}_1$ and hence of two eigenvalues g⁽¹⁾₁ and g⁽¹⁾₂ of the matrix ${\hat{\mathbf{G}}}_1$. In view of (2.33), g⁽¹⁾₁, g⁽¹⁾₂ < 0 at ω < ω_cl and g⁽¹⁾₁, g⁽¹⁾₂ > 0 at ω > ω_cl. The third eigenvalue g⁽¹⁾₃ of ${\hat{\mathbf{G}}}_1$ and the eigenvalues g⁽²⁾_i (i = 1, 2, 3) of ${\hat{\mathbf{G}}}_2$ are continuous functions of a permanent sign within a stop band. Let us successively examine different options regarding their signs.

First, suppose that g⁽¹⁾₃ < 0 and g⁽²⁾_i < 0 (i = 1, 2, 3), so that the matrix ${\hat{\mathbf{G}}}_I$ is negative definite at ω < ω_cl. In this case, the contraction ${\mathbf{q}}^T{\hat{\mathbf{Z}}}_I\mathbf{q}$ appearing in (3.7) taken at ω < ω_cl is negative, which rules out the occurrence of two positive eigenvalues of ${\hat{\mathbf{Z}}}_I$; hence at least two of them must be negative at ω < ω_cl. Similarly, if g⁽¹⁾₃ > 0 and g⁽²⁾_i > 0 (i = 1, 2, 3), then the matrix ${\hat{\mathbf{G}}}_I$ is positive definite at ω > ω_cl and therefore at least two eigenvalues of ${\hat{\mathbf{Z}}}_I$ must be positive at ω > ω_cl. In either of these two cases, not more than two eigenvalues of ${\hat{\mathbf{Z}}}_I$ can vanish; see figure 2. Hence at most two IAWs can exist.

Figure 2.

Possible frequency dependence of the eigenvalues λ_i of matrix ${\hat{\mathbf{Z}}}_I$ (curves 1,2,3) in an upper full stop band, where one eigenvalue has a pole and either at least two eigenvalues are negative at ω < ω_cl (a) or at least two eigenvalues are positive at ω > ω_cl (b). In this and subsequent figures, the origin point of the frequency axis implies the lower stop band edge ω = ω_l, not ω = 0.

Next, let g⁽¹⁾₃ < 0 and g⁽²⁾_i > 0 (i = 1, 2, 3). Multiplying ${\hat{\mathbf{Z}}}_I$ from both sides by the vector product q₁ = t⁽¹⁾₃ × t⁽²⁾₃ of two (real) eigenvectors of ${\hat{\mathbf{G}}}_1$ and ${\hat{\mathbf{G}}}_2$ yields

$${\mathbf{q}}_1^T{\hat{\mathbf{Z}}}_I{\mathbf{q}}_1={\mathbf{q}}_1^T{\hat{\mathbf{G}}}_I{\mathbf{q}}_1=\sum _{i=1}^3{\lambda }_i|{\mathbf{q}}_1^T{\mathbf{e}}_i{|}^2=\sum _{j=1}^2\sum _{i=1}^2{g}_i^{\left(j\right)}{({\mathbf{q}}_1^T{\mathbf{t}}_i^{(j)})}^2.$$ 3.8

The right-most sum is positive at ω > ω_cl, meaning that ${\hat{\mathbf{Z}}}_I$ cannot be negative definite, hence at least one eigenvalue of ${\hat{\mathbf{Z}}}_I$ is positive. If g⁽¹⁾₃ < 0 and g⁽²⁾₁, g⁽²⁾₂ < 0, while g⁽²⁾₃ > 0, then the right-most sum in (3.8) is negative at ω < ω_cl and hence so is at least one eigenvalue of ${\hat{\mathbf{Z}}}_I.$ In either of these two cases, not more than three eigenvalues of ${\hat{\mathbf{Z}}}_I$ can turn to zero (figure 3), so at most three IAWs can occur.

Figure 3.

Possible frequency dependence of the eigenvalues λ_i of matrix ${\hat{\mathbf{Z}}}_I$ (curves 1,2,3) in an upper full stop band, where one eigenvalue has a pole and either at least one eigenvalue is negative at ω < ω_cl (a) or at least one eigenvalue is positive at ω > ω_cl (b).

Given four continuous eigenvalues of ${\hat{\mathbf{G}}}_1$ and ${\hat{\mathbf{G}}}_2$, if two are positive and two are negative, e.g. g⁽¹⁾₃, g⁽²⁾₃ < 0 and g⁽²⁾₁, g⁽²⁾₂ > 0, then the contraction of ${\hat{\mathbf{Z}}}_I$ with q₂ = t⁽²⁾₁ × t⁽²⁾₂,

$${\mathbf{q}}_2^T{\hat{\mathbf{Z}}}_I{\mathbf{q}}_2={\mathbf{q}}_2^T{\hat{\mathbf{G}}}_I{\mathbf{q}}_2=\sum _{i=1}^3{\lambda }_i|{\mathbf{q}}_2^T{\mathbf{e}}_i{|}^2=\sum _{i=1}^3{g}_i^{\left(1\right)}{({\mathbf{q}}_2^T{\mathbf{t}}_i^{(1)})}^2+g_3^{\left(2\right)}{({\mathbf{q}}_2^T{\mathbf{t}}_3^{(2)})}^2,$$ 3.9

is negative at ω < ω_cl and the contraction (3.8) is positive at ω > ω_cl. Therefore, at least one eigenvalue of ${\hat{\mathbf{Z}}}_I$ must be negative at ω < ω_cl and simultaneously at least one of them must be positive at ω > ω_cl. Under these conditions, the eigenvalues of ${\hat{\mathbf{Z}}}_I$ vanish at most twice (see examples in figure 4), so at most two IAWs can exist.

Figure 4.

Possible frequency dependence (a) and (b) of the eigenvalues λ_i of matrix ${\hat{\mathbf{Z}}}_I$ (curves 1,2,3) in an upper full stop band, where one eigenvalue has a pole and at least one eigenvalue is negative at ω < ω_cl and at least one is positive at ω > ω_cl.

Now we proceed to the case where the eigenvalues of ${\hat{\mathbf{Z}}}_I$ have two poles corresponding to the pole ω = ω_cl1 of the pair of eigenvalues g⁽¹⁾_i of ${\hat{\mathbf{G}}}_1$ and to the pole ω = ω_cl2 of the pair of eigenvalues g⁽²⁾_i of ${\hat{\mathbf{G}}}_2$ (i = 1, 2) (figure 5). The remaining eigenvalues g⁽¹⁾₃ of ${\hat{\mathbf{G}}}_1$ and g⁽²⁾₃ of ${\hat{\mathbf{G}}}_2$ are either positive or negative throughout the given stop band. Consider the possible options of their signs. Let g⁽¹⁾₃ and g⁽²⁾₃ be of the same sign, e.g. g⁽¹⁾₃, g⁽²⁾₃ > 0. Then ${\hat{\mathbf{G}}}_1$ and ${\hat{\mathbf{G}}}_2$ and hence ${\hat{\mathbf{G}}}_I$ are positive-definite matrices at ω > ω_cl2 (where g^(j)_i > 0, i, j = 1, 2; figure 5). Hence the contraction (3.7) is positive and so at least two eigenvalues of ${\hat{\mathbf{Z}}}_I$ must be positive at ω > ω_cl2. At the same time, the contraction (3.8) taken at ω < ω_cl2 (where g^(j)_i < 0, i, j = 1, 2) is negative, hence at least one eigenvalue of ${\hat{\mathbf{Z}}}_I$ is negative at ω < ω_cl1. By drawing possible frequency dependences of eigenvalues λ_i(ω) of ${\hat{\mathbf{Z}}}_I$ with the above properties, we observe that λ_i(ω) can cross the zero at most twice (as exemplified in figure 6), that is, at most two IAWs can exist.

Figure 5.

Frequency dependence of the discontinuous pairs of eigenvalues g⁽¹⁾₁, g⁽¹⁾₂ (curves 1 and 2) and g⁽²⁾₁, g⁽²⁾₂ (curves 3 and 4) of the matrices ${\hat{\mathbf{G}}}_1$ and ${\hat{\mathbf{G}}}_2,$ respectively, in an upper full stop band.

Figure 6.

Possible frequency dependence of the eigenvalues λ_i of matrix ${\hat{\mathbf{Z}}}_I$ (curves 1,2,3) in an upper full stop band, where one eigenvalue has two poles and at least one eigenvalue is negative at ω < ω_cl1 and at least two are positive at ω > ω_cl2.

Still in the framework of the case depicted in figure 5, assume that g⁽¹⁾₃ and g⁽²⁾₃ have opposite signs, e.g. g⁽¹⁾₃ < 0 and g⁽²⁾₃ > 0. Then the contraction (3.8) is negative at ω < ω_cl1 and positive at ω > ω_cl2. Hence at least one eigenvalue of ${\hat{\mathbf{Z}}}_I$ is negative at ω < ω_cl1 and at least one of them is positive at ω > ω_cl2. Under this condition, it follows that the eigenvalues of ${\hat{\mathbf{Z}}}_I$ can vanish not more than three times (figure 7), so at most three IAWs can exist. Last, let g⁽¹⁾₃ < 0 and g⁽²⁾₃ < 0, so that ${\hat{\mathbf{G}}}_1$ and ${\hat{\mathbf{G}}}_2$ and hence ${\hat{\mathbf{G}}}_I$ are negative-definite matrices at ω < ω_cl1. Now the contraction (3.7) is negative at ω < ω_cl1 and hence so are at least two eigenvalues of ${\hat{\mathbf{Z}}}_I$, whereas the contraction (3.8) is positive at ω > ω_cl2 and hence so is at least one eigenvalue of ${\hat{\mathbf{Z}}}_I.$ Such an option admits at most two zero eigenvalues of ${\hat{\mathbf{Z}}}_I$ (figure 8) and therefore at most two IAWs within a given stop band. Note that the poles, assumed in figures 6–8 to occur on the same eigenbranch λ₁(ω) of ${\hat{\mathbf{Z}}}_I$, may certainly come about on different eigenbranches, but this does not affect the overall conclusions regarding the maximum possible number of zero eigenvalues.

Figure 7.

Possible frequency dependence of the eigenvalues λ_i of matrix ${\hat{\mathbf{Z}}}_I$ (curves 1,2,3) in an upper full stop band, where one eigenvalue has two poles and at least one eigenvalue is negative at ω < ω_cl and at least one is positive at ω > ω_c2.

Figure 8.

Possible frequency dependence of the eigenvalues λ_i of the matrix ${\hat{\mathbf{Z}}}_I$ (curves 1,2,3) in an upper full stop band, where one eigenvalue has two poles and at least two eigenvalues are negative at ω < ω_cl1 and at least one is positive at ω > ω_cl2.

Summing up the above considerations, we conclude that

$$\begin{array}{rl}& \mathrm{at}\hspace{0.17em}\mathrm{most}\hspace{0.17em}\mathrm{three}\hspace{0.17em}\mathrm{IAWs}\hspace{0.17em}\mathrm{can}\hspace{0.17em}\mathrm{exist}\\ & \mathrm{in}\hspace{0.17em}\mathrm{an}\hspace{0.17em}\mathrm{upper}\hspace{0.17em}\mathrm{full}\hspace{0.17em}\mathrm{stop}\hspace{0.17em}\mathrm{band}\hspace{0.17em}\mathrm{at}\hspace{0.17em}\mathrm{fixed}\hspace{0.17em}k.\end{array}$$ 3.10

(c). Symmetric sagittal plane

Suppose that the sagittal plane spanned by the vectors m and n is the plane of crystallographic symmetry for both upper and lower phononic crystals. Then the components (ab)_i3 and (ab)_3i, i = 1, 2, of the matrices (ab) in (2.4) are identically zero. As a result, the Stroh matrix along with the transfer matrix split into 4 × 4 and 2 × 2 submatrices, which describe respectively, four partial modes of sagittal polarization (S modes) and two partial modes of shear horizontal polarization (SH modes), the two mode families being fully decoupled from each other. Correspondingly, the elements Z_j,13 and Z_j,23 of the impedance matrices ${\hat{\mathbf{Z}}}_j$ of the media j = 1, 2 vanish, so there remain upper 2 × 2 diagonal blocks ${\hat{\mathbf{Z}}}_j^{2\times 2}$ and the 33rd diagonal elements Z_j,33≡G_j,33. Thus, on denoting ${\hat{\mathbf{Z}}}_I^{2\times 2}={\hat{\mathbf{Z}}}_1^{2\times 2}+{({\hat{\mathbf{Z}}}_2^{2\times 2})}^T$ and G_I,33 = G_1,33 + G_2,33, the dispersion equation (3.3) splits into two equations

$$det{\hat{\mathbf{Z}}}_I^{2\times 2}=0,G_{I,33}=0,$$ 3.11

defining the S-IAWs and the SH-IAWs waves, respectively. Note that the full stop bands for decoupled S and SH modes, in which the solutions of, respectively, (3.11)₁ and (3.11)₂ are sought, are different and independent of each other.

Consider the SH-IAWs. Such waves are known to be ruled out in the case of homogeneous half-spaces. Similarly, since G_I,33 is strictly positive in the lowest stop band owing to (2.36), it follows that

$$\mathrm{no}\hspace{0.17em}\mathrm{SH}-\mathrm{IAWs}\hspace{0.17em}\mathrm{exist}\hspace{0.17em}\mathrm{in}\hspace{0.17em}\mathrm{the}\hspace{0.17em}\mathrm{lowest}\hspace{0.17em}\mathrm{full}\hspace{0.17em}\mathrm{stop}\hspace{0.17em}\mathrm{band}.$$ 3.12

However, the SH-IAWs can arise in the upper full stop bands of a phononic bicrystal where G_j,33, j = 1, 2, may be of different signs. By virtue of (2.34) and (2.35), G_I,33 is a continuous monotonic function of frequency, hence

$$\begin{array}{rl}& \mathrm{at}\hspace{0.17em}\mathrm{most}\hspace{0.17em}\mathrm{one}\hspace{0.17em}\mathrm{SH}-\mathrm{IAW}\hspace{0.17em}\mathrm{can}\hspace{0.17em}\mathrm{exist}\\ & \mathrm{in}\hspace{0.17em}\mathrm{an}\hspace{0.17em}\mathrm{upper}\hspace{0.17em}\mathrm{full}\hspace{0.17em}\mathrm{stop}\hspace{0.17em}\mathrm{band}\hspace{0.17em}\mathrm{at}\hspace{0.17em}\mathrm{fixed}\hspace{0.17em}k.\end{array}$$ 3.13

Let us address the S-IAWs. Analysis of the existence and maximum possible number of solutions of the dispersion equation (3.11) can be carried out along similar lines to that done for the general equation (3.6) in the case of unrestricted anisotropy. Consistent with (3.12), statement (3.6) remains intact with respect to S waves; namely,

$$\begin{array}{rl}& \mathrm{at}\hspace{0.17em}\mathrm{most}\hspace{0.17em}\mathrm{one}\hspace{0.17em}\mathrm{S}-\mathrm{IAW}\hspace{0.17em}\mathrm{can}\hspace{0.17em}\mathrm{exist}\\ & \mathrm{in}\hspace{0.17em}\mathrm{the}\hspace{0.17em}\mathrm{lowest}\hspace{0.17em}\mathrm{stop}\hspace{0.17em}\mathrm{band}\hspace{0.17em}\mathrm{at}\hspace{0.17em}\mathrm{fixed}\hspace{0.17em}k.\end{array}$$ 3.14

Consider an upper full stop band. Suppose first that none of the eigenvalues λ₁ and λ₂ of ${\hat{\mathbf{Z}}}_I^{2\times 2}$ have poles, i.e. they are continuous monotonic functions of ω. Then each of them can vanish only once within the stop band (fixed k being understood) and so at most two S-IAWs are possible. Suppose now that one eigenvalue of ${\hat{\mathbf{Z}}}_I^{2\times 2}$ has a pole at ω = ω_cl corresponding to the pole of two eigenvalues of, say, ${\hat{\mathbf{G}}}_1^{2\times 2}$. Let the eigenvalues g⁽²⁾₁ and g⁽²⁾₂ of ${\hat{\mathbf{G}}}_2^{2\times 2}$ be either both positive or both negative in the given stop band. This means that the matrix ${\hat{\mathbf{G}}}_I^{2\times 2}$ is positive definite at ω > ω_cl in the former case and it is negative definite at ω < ω_cl in the latter case. Hence either at least one eigenvalue of ${\hat{\mathbf{Z}}}_I^{2\times 2}$ is positive at ω > ω_cl or at least one eigenvalue of ${\hat{\mathbf{Z}}}_I^{2\times 2}$ is negative at ω < ω_cl, respectively. In both cases, it follows that the eigenvalues of ${\hat{\mathbf{Z}}}_I^{2\times 2}$ can vanish not more than twice (for example, see figure 4a without curve 2 and figure 4b without curve 3), and so at most two S-IAWs can occur. Next, let the eigenvalues of ${\hat{\mathbf{G}}}_2^{2\times 2}$ be of opposite signs, e.g. g⁽²⁾₁ > 0 and g⁽²⁾₂ < 0 throughout the stop band. Multiplying ${\hat{\mathbf{Z}}}_I^{2\times 2}$ from both sides by the real vector p_i = (t⁽²⁾_i,y, − t⁽²⁾_i,x)^T, where t⁽²⁾_i,x and t⁽²⁾_i,y are components of the eigenvectors t⁽²⁾_i = (t⁽²⁾_i,x, t⁽²⁾_i,y)^T, i = 1, 2, of the matrix ${\hat{\mathbf{G}}}_2^{2\times 2}$, we observe that

$$\begin{array}{rl}& {\mathbf{p}}_1^T{\hat{\mathbf{Z}}}_I^{2\times 2}{\mathbf{p}}_1=\sum _{i=1}^2{g}_{1,i}{({\mathbf{p}}_1^T{\mathbf{t}}_i^{(1)})}^2+g_{2,2}{({\mathbf{p}}_1^T{\mathbf{t}}_2^{(2)})}^2<0\mathrm{at}\hspace{0.17em}\omega <{\omega }_{cl},\\ & {\mathbf{p}}_2^T{\hat{\mathbf{Z}}}_I^{2\times 2}{\mathbf{p}}_2=\sum _{i=1}^2{g}_{1,i}{({\mathbf{p}}_2^T{\mathbf{t}}_i^{(1)})}^2+g_{2,1}{({\mathbf{p}}_2^T{\mathbf{t}}_1^{(2)})}^2>0\mathrm{at}\hspace{0.17em}\omega >{\omega }_{cl}.\end{array}\}$$ 3.15

Hence at least one eigenvalue of ${\hat{\mathbf{Z}}}_I^{2\times 2}$ is negative at ω < ω_cl and at least one eigenvalue is positive at ω > ω_cl (see figure 4b with curve 2 removed). In this case, only one eigenvalue zero of ${\hat{\mathbf{Z}}}_I^{2\times 2}$ occurs and so only one S-IAW can exist. Last, suppose that the pair of eigenvalues of ${\hat{\mathbf{G}}}_1^{2\times 2}$ and that of ${\hat{\mathbf{G}}}_2^{2\times 2}$ each has a pole in a given full stop band, as shown in figure 5, so that ${\hat{\mathbf{G}}}_I^{2\times 2}$ is negative definite at ω < ω_cl1 and positive definite at ω > ω_cl2. Correspondingly, at least one eigenvalue of ${\hat{\mathbf{Z}}}_I^{2\times 2}$ is negative at ω < ω_cl1 and one is positive at ω > ω_cl2. Examining possible shapes of frequency dependence of eigenvalues of ${\hat{\mathbf{Z}}}_I^{2\times 2}$ consistent with the above condition, we conclude that they can vanish at most twice (as it is for example in figure 7 without curve 2), and hence at most two S-IAWs can exist. Thus, as expected,

$$\begin{array}{rl}& \mathrm{at}\hspace{0.17em}\mathrm{most}\hspace{0.17em}\mathrm{two}\hspace{0.17em}\mathrm{S}-\mathrm{IAW}\mathrm{s}\hspace{0.17em}\mathrm{can}\hspace{0.17em}\mathrm{exist}\\ & \mathrm{in}\hspace{0.17em}\mathrm{an}\hspace{0.17em}\mathrm{upper}\hspace{0.17em}\mathrm{full}\hspace{0.17em}\mathrm{stop}\hspace{0.17em}\mathrm{band}\hspace{0.17em}\mathrm{at}\hspace{0.17em}\mathrm{fixed}\hspace{0.17em}k.\end{array}$$ 3.16

4. Conclusion

The paper has considered the problem of the existence of IAWs in 1D phononic bicrystals with a symmetrically arranged unit cell. The main objective was to deduce how many IAWs with a fixed tangential wavenumber k can exist per full stop band of the Floquet frequency spectrum. The analysis was based on the properties of the surface impedance matrix, which is a generalization of a similar concept pivotal in the theory of surface and interfacial waves in homogeneous anisotropic media. It was found that the maximum possible number of IAWs in the lowest stop band is different from that in the upper ones. The lowest stop band admits at most one IAW at fixed k. This observation is similar to the theorem of uniqueness of the Stoneley wave at the interface between homogeneous half-spaces [35]. In contrast to that, an upper stop band can admit up to three IAWs with a fixed k. The difference comes from the facts that the impedance matrix, in contrast to being positive definite at zero frequency, is not sign definite at the lower frequency edge of an upper stop band and that, specifically, the upper stop bands admit the existence of SAW solutions on a mechanically clamped boundary, which causes the poles of the impedance. The admissible number of IAWs is split between the sagittally and shear horizontally polarized IAWs in the case of a symmetric sagittal plane.

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Authors' contributions

Both authors contributed equally to the problem statement, analytical derivations and writing of the manuscript.

Competing interests

We declare we have no competing interests.

Funding

The work by A.N.D. was supported by the Ministry of Science and Higher Education of the Russian Federation.