## Abstract

The paper is concerned with the propagation of shear horizontal surface waves (SHSW) in semi-infinite elastic media with vertically periodic continuous and/or discrete variation of material properties. The existence and spectral properties of the SHSW are shown to be intimately related to the shape of the properties variation profile. Generally, the SHSW dispersion branches represent randomly broken spectral intervals on the (*ω*, *k*) plane. They may, however, display a particular regularity in being confined to certain distinct ranges of slowness *s*=*ω*/*k*, which can be predicted and estimated directly from the profile shape. The SHSW spectral regularity is especially prominent when the material properties at the opposite edge points of a period are different. In particular, a unit cell can be arranged so that the SHSW exists within a single slowness window, narrow in the measure of material contrast between the edges, and does not exist elsewhere or vice versa. Explicit analysis in the (*ω*, *k*) domain is complemented and verified through the numerical simulation of the SH wave field in the time–space domain. The results also apply to a longitudinally periodic semi-infinite strip with a homogeneous boundary condition at the faces.

## 1. Introduction

A surface acoustic wave travelling in a semi-infinite medium is understood to satisfy the boundary condition and the radiation condition, the latter demanding that the amplitude tends to zero at the infinite depth. Within this primary definition, the notion of surface waves diversifies if the medium is inhomogeneous. In a vertically inhomogeneous half-space, which is the case of broad significance in seismology and ultrasonics, a variation of material properties along the depth coordinate *y* implies that the surface wave is associated with a continuum of dispersive solutions in the Fourier domain of frequency *ω* and horizontal wavenumber *k*. This continuum is represented by, generally, a family of dispersion branches of, say, velocity *v*=*ω*/*k* versus *ω*. Local bulk wave velocity *c*(*y*) commonly assumes a constant limit *c*_{∞} at the infinite depth, in which case the surface-wave velocity spectrum contains continuous curve(s) *v*_{n}(*ω*) lying below *c*_{∞} (Aki & Richards 1980; ,Kennett 1983). An elementary example is a Love-type spectrum for a coated homogeneous substrate. However, any local minimum of *c*(*y*) as well as the Rayleigh velocity at the surface (for the P/SV modes), even if these are greater than *c*_{∞}, give rise to the localized wave channel in the sense of high-frequency asymptotics (so that possible leakage is negligible), see Alenitsyn (1964) and ,Alshits & Maugin (2005). The so-defined surface wave exists on the curves *v*_{n}(*ω*), which are continuous in (high) *ω* and are defined up to a certain velocity that is anyway less than the absolute maximum of *c*(*y*). As is well known, a remarkable exception, leading to a notably different spectral pattern, is specifically a periodic half-space, where the surface wave is a superposition of the Floquet harmonics with the Floquet wavenumbers lying in the stopbands. In consequence, the surface-wave velocity branches *v*_{n}(*ω*) may extend up to infinity and may contain randomly broken intervals, which terminate or originate at the edges of the stopbands. (For the former reason, it is suitable to move on to the slowness in the subsequent discussion.)

There is a further interesting particularity that pertains to the case of the shear horizontal surface wave (SHSW) in a vertically periodic half-space. This is that the dispersion curves for ‘physical’ and ‘non-physical’ SHSW solutions, attributed as such owing to their decrease or increase in the depth, are alternating intervals of the branches of another, relatively easier to handle, spectrum, namely, the spectrum for SH guided waves in a single unit cell (period) considered as a vertically inhomogeneous plate with traction-free faces. This dispersion spectrum may be said to be the reference one for the SHSW problem in hand (which is why the hat is added). Once this spectrum is obtained, the sought after SHSW curves *s*_{n}(*ω*) are identified as the spectral intervals on the branches where the surface wave is physical and hence does exist. Such a specific property of the SHSW has been noted and used in many papers dealing with the case of a unit cell composed of, as a rule, two or, rarer, of several homogeneous layers (Podlipenets 1982; ,Camley *et al*. 1983; El Boudouti *et al*. 1993, ,1996; ,Shul'ga 2003; ,Gatignol *et al*. 2007; ,Chen *et al*. 2008). Note also that the SHSW spectral density as a function of *ω* at fixed *k* was first calculated by Camley *et al*. (1983).

The present paper considers the SHSW in a vertically periodic half-space with an arbitrary unit cell profile, i.e. any continuous and/or discrete variation of material properties through a period. The aforementioned two-step procedure for defining the SHSW has been validated for this general case in Shuvalov *et al*. (2006). The SH spectrum for a vertically inhomogeneous plate, which in the present context represents a given unit cell and yields the reference branches , has been analysed in detail in Shuvalov *et al*. (2008). The thrust of this study will be the second step of dealing with the SHSW problem, i.e. partitioning the spectrum into the SHSW-‘physical’ and ‘non-physical’ parts. More precisely, the challenge is to predict, whenever possible, the spectral trends and bounds of the SHSW existence directly from the shape of a given unit cell profile.

Apart from the simple case of two homogeneous layers in a unit cell, the SHSW curves *s*_{n}(*ω*) may be expected to represent a fairly random distribution of spectral intervals on the reference branches so that they can be identified numerically only, by testing the radiation condition along each branch . At the same time, the main motivation for our study is to show that the existence of the SHSW can often be determined without any calculations, by merely inspecting the shape of the unit cell profile. As we will see, a direct link between this profile shape and the SHSW existence is especially compelling in a piecewise continuous periodic half-space whose material properties undergo a jump across the interface between periods, i.e. they are different at the opposite edges of a unit cell. In such a case, the SHSW dispersion spectrum may be remarkably regular in that it is confined within certain slowness values and these allow a simple analytical evaluation. Moreover, a unit cell profile can be arranged so that the SHSW exists within a single slowness window and does not exist elsewhere or vice versa. Such knowledge enables a judicious design of the unit cell profile for realizing the desirable spectral properties of the SHSW, which is useful for applications. It is also noteworthy that the results obtained for the SHSW in a half-space solve the problem of the SH waves localized at the free edge of a semi-infinite strip (see Adams *et al*. 2008), namely, the dispersion curves for the former case contain the set of natural frequencies for the latter case.

The paper is organized as follows. The background of the problem is described in §2 (more details may be found in Shuvalov *et al*. (2006, ,2008)). A simple case of a periodically bilayered half-space is explicitly resolved in ,§3. The general case of SHSW in an arbitrary vertically periodic half-space is covered in ,§4. Despite a fairly involved underlying formalism, an effort is made in this section to avoid ‘technicalities’ and hinge the exposition on plain asymptotic considerations and numerical examples, whereas some rigorous theorems are given in electronic supplementary material 1 (additional examples are provided in electronic supplementary material 2). ,Section 5 gives special attention to the case of SHSW existence in a single spectral range. This prediction is verified through simulating the wave field in the time–space domain. The conclusions are presented in ,§6.

## 2. Background

### (a) Equation of motion

Given is a unidirectionally inhomogeneous medium with density *ρ*=*ρ*(*y*) and (real) stiffness *c*_{ijkl}=*c*_{ijkl}(*y*), which are continuous or piecewise continuous (e.g. piecewise constant) along the axis *Y*. A piecewise continuous dependence implies welded-contact interfaces orthogonal to *Y*, which cause a jump of material properties but maintain continuity of elastic displacement and traction. The tensor *c*_{ijkl} is supposed to have a symmetry plane *XY*. Consider SH waves propagating in this plane and polarized along the axis *Z*. The wave displacement is sought in the form(2.1)where the frequency *ω* and the slowness *s*=*v*^{−1} along the axis *X* are assumed to be real. By (2.1),(2.2)where *k*=*ωs*; the prime denotes the differentiation with respect to *y*; and . The equation of motion written with respect to *A*(*y*) has the form(2.3)where the impact of *c*_{45}≠0 amounts to replacing *c*_{55} by the above-mentioned *C*_{55}.

Equation (2.3) may be recast as(2.4)with(2.5)where the second component of ** η**(

*y*) is the amplitude of

*σ*

_{yz}, and

*ω*,

*s*are chosen as a pair of dispersion parameters (

*k*is implicit). The solution to (2.4), related to an initial condition

**(**

*η**y*

_{0}) at any

*y*

_{0}, is(2.6)where

**(**

*M**y*,

*y*

_{0}) is the 2×2 matricant or propagator. According to (2.5), it is a unimodular matrix(2.7)The matricant expands into the Peano series of multiple integrals(2.8)where

**is the identity matrix (Pease 1965; ,Aki & Richards 1980; ,Kennett 1983). A matricant satisfies the product rule**

*I***(**

*M**y*,

*y*

_{0})=

**(**

*M**y*,

*y*

_{1})

**(**

*M**y*

_{1},

*y*

_{0}), which may be helpful when

**(**

*Q**y*)=const. in [

*y*

_{1},

*y*

_{2}]∈[

*y*

_{0},

*y*] and hence

**(**

*M**y*

_{2},

*y*

_{1})=exp[(

*y*

_{2}−

*y*

_{1})

**]; in particular,**

*Q***(**

*M**y*,

*y*

_{0}) is a product of exponentials when

**(**

*Q**y*) is piecewise constant in [

*y*

_{0},

*y*]. For real

*c*

_{ijkl}and

*ω*,

*s*(as is the case in the paper), it follows from (2.5) and ,(2.8) that(2.9)

### (b) Periodicity

We now specify the inhomogeneity along *Y* as periodic, so that ** Q**(

*y*)=

**(**

*Q**y*+

*T*), where

*T*denotes the least period. A unit cell, understood as

*y*∈[0,

*T*], may be either continuously inhomogeneous or stacked of different homogeneous and/or inhomogeneous layers in welded contact. Similarly, the aggregate medium may be continuously inhomogeneous throughout or may allow a jump of material properties across the welded-contact interfaces between unit cells (

*(*

**η***y*) is anyway continuous). The main interest lies with the primary types of periodic ordering, hereafter referred to as continuous (no distinct interfaces), piecewise constant (a periodic stack of homogeneous layers) and piecewise continuous (continuously inhomogeneous unit cells with a jump of properties across their interfaces). Regarding the case of material discontinuity between unit cells, which implies that , a careful remark on the notations is in order: in this case, the definition domain for any function

*f*(

*y*) over a unit cell [0,

*T*] is tacitly understood in the paper to be [0,

*T*), so that

*f*(

*T*) actually reads as

*f*(

*T*

^{−}).

A pair of the SH Floquet modes associated with the wave amplitude in the form *A*(*y*) (see (2.1)) is defined by the eigenvectors *w*_{α} and eigenvalues *q*_{α} of the propagator ** M**(

*T*, 0) through a period,(2.10)where

*K*is the Floquet wavenumber (Pease 1965). The eigenvalues

*q*

_{α}depend on

*ω*and

*s*. Hence, the plane {

*ω*,

*s*} is mapped out into the domains of two types: the passbands, where

*q*

_{1},

*q*

_{2}are complex conjugated numbers of a unit absolute value (

*K*is real), and the stopbands, where

*q*

_{1},

*q*

_{2}are real (

*K*is purely imaginary). They are separated by the lines

*q*

_{1}=

*q*

_{2}(=±1) of stopband edges.

Note a handy expression for the unimodular propagator ** M**(

*NT*, 0) through any number

*N*of unit cells (Born & Wolf 1980)(2.11)which by continuity extends to the degenerate case

*q*

_{1}=

*q*

_{2}.

### (c) Dispersion equation

Let the medium be bounded in the direction *Y* of periodic inhomogeneity. The boundary condition *σ*_{yz}=0 on a traction-free plane placed at some *y*=const. nullifies the second component of the state vector * η*(

*y*) at this

*y*, see (2.5)

_{2}. Hence, by ,(2.6) and ,(2.11), the dispersion equation for the SH guided waves in a finite periodic structure

*y*∈[0,

*NT*] with the traction-free faces at

*y*=0 and

*y*=

*NT*is(2.12)where

*M*

_{3}(

*T*, 0)=0 is in turn the dispersion equation for a single unit cell extracted from the periodic structure and subjected to the traction-free condition at

*y*=0 and

*y*=

*T*. It is noted that the eigenvalues

*q*

_{1}and

*q*

_{2}of

**(**

*M**T*, 0) with

*M*

_{3}(

*T*, 0)=0 are equal to

*M*

_{1}(

*T*, 0) and

*M*

_{4}(

*T*, 0), which are both real according to (2.9). Therefore, by ,(2.12), the dispersion spectrum for a finite structure of

*N*unit cells consists of the branches given by

*M*

_{3}(

*T*, 0)=0, which lie in the stopbands owing to

*q*

_{1},

*q*

_{2}being real (one branch per stopband, probably touching its edges), and of the branches given by sin

*NKT*=0 (sin

*KT*≠0), which lie within the passbands.

What happens when we consider SHSW in a semi-infinite periodic structure *y*≥0 with a free boundary? The answer readily follows from pure reasoning. By virtue of the radiation condition, the SHSW occurs strictly inside the stopbands and consists of a single Floquet mode (with |*q*_{α}|<1, see (2.10)), which must therefore fulfil the boundary condition *σ*_{yz}=0 at *y*=0 on its own. The mode, which does so, is specified thereafter by *α*=1. It must correspond to the eigenvector of ** M**(

*T*, 0) with a zero second (related to

*σ*

_{yz}) component, . This in turn necessitates(2.13)with real

*q*

_{1,2}. It remains now to come back to the radiation condition and to ensure that the surface-wave solution rendered by the Floquet mode

*α*=1 is ‘physical’ (vanishing at

*y*→∞), which implies that |

*q*

_{1}|=|

*M*

_{1}(

*T*, 0)|<1. Thus, invoking (2.6) with

*y*

_{0}=0 and

*(0)=i*

**η***ωA*(0)

*w*_{1}, the amplitude

*A*(

*y*) of the SHSW displacement taken at the depth equal to any number

*N*of periods

*T*is , where(2.14)

Recapping the above arguments, the SHSW dispersion spectrum is therefore defined by a pair of conditions(2.15)with the entries given by (2.8). Opting for the slowness versus frequency dependence, denote the SHSW branches by *s*_{n}(*ω*) *n*=0, 1, 2, … with *n*=0 indicating the fundamental (SH_{0}) branch. They can be found by the following two-step procedure. The first step is solving equation (2.15)_{1}, which describes the dispersion branches of SH guided waves in a free inhomogeneous layer *y*∈[0,*T*] consisting of one period of a given structure. The SHSW exists on these reference branches and nowhere else. The second step is identifying the intervals *s*_{n}(*ω*) of SHSW existence on by means of the inequality (2.15)_{2}. A complementary part of the branches , for which (2.15)_{2} has the opposite sense, is where the Floquet mode *α*=1 (satisfying the traction-free boundary condition) increases in depth, and is thereby a ‘non-physical’ solution for surface waves. A crossover between ‘physical’ and ‘non-physical’ solutions occurs when , i.e. when a curve comes into contact with a stopband edge (the inverse, though, may not be the case; see figure S1 in electronic supplementary material 2).

Similar considerations apply to the case of the clamped boundary condition *A*=0. For a finite periodic structure, setting its faces to be clamped implies replacing *M*_{3} in equation (2.12) by *M*_{2}. The first factor is not affected and hence neither is the family of branches lying in the passbands. Removing the second factor, *M*_{2}(*T*, 0)=0, gives the equation for a single clamped unit cell that defines the branches in the stopbands. Along these branches, the surface-wave solutions in a periodic half-space with the clamped surface *y*=0 are associated with the Floquet mode *α*=2, for which *q*_{2}=*M*_{4}(*T*, 0) and . They are ‘physical’ when |*M*_{4}(*T*, 0)|<1. Having noted this, we confine ourselves in the following to the case of a traction-free surface (see, however, §A4 in electronic supplementary material 1).

### (d) Provisions and notations

It is expedient to set forth two properties, which are instrumental for the present study. One of them (the reciprocity) was first established for the case of a bilayered unit cell in El Boudouti *et al*. (1993), and then both were proved for the general case of an arbitrary vertically periodic half-space in Shuvalov *et al*. (2006, ,2008). They can readily be observed from the identity , where *M*_{Q(−y)} is the matricant for equation (2.4) with ** Q**(−

*y*) in place of

**(**

*Q**y*), and

**is the matrix with zero diagonal and unit off-diagonal components. This identity shows that inverting an arbitrary profile of unit cell [0,**

*T**T*] about its middle point (in other words, turning the periodic structure ‘upside down’) amounts to interchanging the diagonal components

*M*

_{1}and

*M*

_{4}of

**(**

*M**T*, 0). Hence, this transformation keeps intact the reference dispersion branches defined by equation (2.15)

_{1}, but inverts the sense of their partitioning by ,(2.15)

_{2}into the intervals of SHSW existence and non-existence, i.e. formerly ‘physical’ solutions become ‘non-physical’ and vice versa. This first property is referred to below as the reciprocity. The second, closely related, property is that any symmetric (even) profile

**(**

*Q**y*)=

**(**

*Q**T*−

*y*) of a unit cell

*y*∈[0,

*T*] provides the curves , which fully merge with the stopband edges

*q*

_{1}=

*q*

_{2}(=±1). Hence, at no point of the spectrum for an even profile can the SH wave qualify for the ‘physical’ surface wave such that vanishes at the infinite depth. In this regard, a general, i.e. asymmetric, type of unit cell profiles is understood hereafter.

For future use, we introduce the auxiliary notations(2.16)where the subscripts allude to the directions parallel and normal to the boundary. The dependence on *y* is periodic, and thus may be considered within a single unit cell [0,*T*]. The absolute maximum, Max *s*_{∥}(*y*), is the upper bound for the branches , and the absolute minimum, Min *s*_{∥}(*y*), is a threshold separating the so-called supersonic and subsonic parts of the spectrum . (The meaning of Max and Min as the absolute extrema is tacit hereafter). The quantities *κ* and *Z* may be interpreted as vertical components of the local wavevector and impedance for an obliquely downgoing SH mode, whose horizontal slowness is taken on the branches . For supersonic , both *κ* and *Z* are real throughout the whole unit cell [0,*T*]; for subsonic , both *κ* and *Z* are imaginary where and real where (see also §4*b*). For brevity, we hereafter write *κ*(*y*) and *Z*(*y*) without the reference to , which is understood and suppressed. In the case of a piecewise homogeneous unit cell consisting of *J* welded layers *j*=1, …, *J* counted in the sense of increasing *y*, the above notations are specialized as follows:(2.17)where the reference of *κ*_{j} and *Z*_{j} to is omitted as agreed.

While the analytical considerations in this paper assume a single vertical symmetry plane *XY* only, the subsequent numerical examples use fictitious isotropic material data for the sake of transparency of modelling various possible options via a simple self-sufficient input. As a result the figure legends specify the input by two parameters only (instead of three for an anisotropic case), which are taken to be *z*_{⊥}=*ρc*_{⊥} and , with *c*_{⊥}=*c*_{∥} kept tacit. Using isotropic input data fixes mutual scaling of the cut-offs and plateaux (depending on *s*_{⊥} and *s*_{∥}, respectively) of the reference dispersion branches ; however, this leads to no loss of generality or completeness of pinpointing possible spectral features of the SHSW in question. This is confirmed by an anisotropic example with *c*_{⊥}≠*c*_{∥}, which is supplied in electronic supplementary material 2, figure S1.

## 3. SHSW in a periodically bilayered half-space

To get started, consider the simplest arrangement of a periodic half-space, which is when the unit cell is composed of two homogeneous layers. This case is fully tractable by elementary algebra and allows an exact evaluation of the slowness ranges of SHSW existence.

The propagator ** M**(

*T*, 0) through a bilayer of thickness

*T*=

*d*

_{1}+

*d*

_{2}may be found in many textbooks (e.g. Aki & Richards 1980). Its components involved in ,(2.15) are as follows:(3.1)where

*κ*

_{1,2}and

*Z*

_{1,2}are given in (2.17). In the subsonic domain , the equation

*M*

_{3}(

*T*, 0)=0 defining the reference spectrum is incompatible with the equality

*M*

_{1}(

*T*, 0)=

*M*

_{4}(

*T*, 0). Hence, the branches cannot contact the stopband edges and so the SHSW existence/non-existence (i.e. ‘physical’/‘non-physical’) crossover is ruled out throughout the subsonic domain. Precisely at the threshold point , such crossover occurs on each of the branches , which all touch the stopband edges. In the supersonic interval , the upper or lower inequality sign in entails the SHSW existence or non-existence, respectively. This inequality changes sign within the supersonic interval provided the root

*σ*

_{12}of equation

*Z*

_{1}=

*Z*

_{2}in is supersonic, i.e. its value(3.2)is real and less than . It is evidently the case iff the inequality between and is inverse to that between and .

On these grounds, the possible scenarios of the SHSW existence or non-existence on the reference spectrum amount to the following alternative options.

Let the top layer

*j*=1 be ‘faster’ than the bottom one in the sense that . Then, no SHSW exists in the subsonic domain . If , the SHSW exists along the supersonic extent of all the branches. If , the supersonic SHSW exists for and does not exist for .Let the top layer

*j*=1 be ‘slower’: . Then, the subsonic SHSW exists along the subsonic extent of all the branches. If , then no supersonic SHSW exists. If , the supersonic SHSW does not exist for and exists for .

The statements (i) and (ii) describe inverse layer arrangements and hence, in view of the reciprocity property (see §2*d*), their assertions are in fact the same up to swapping the attribution of SHSW as existing or not existing. In either case, the SHSW existence and non-existence slowness ranges have exactly evaluated constant bounds. It is noted that, in the special case and , the entire spectrum , except the fundamental SH_{0} branch (which is non-dispersive if *s*_{∥}=const.), corresponds to the SHSW existence if , and to its non-existence if .

Figure 1 exemplifies both types of the SHSW spectral pattern that are possible for a periodically bilayered half-space with .

## 4. SHSW in an arbitrary periodic half-space

### (a) Supersonic SHSW

Consider a periodic half-space whose unit cell [0,*T*] is continuously inhomogeneous or is a stack of *J* welded layers. In this section, we are concerned with the existence of SHSW on the supersonic part of reference dispersion branches , which is bounded from above by Min *s*_{∥}(*y*) or by , *j*=1, …, *J*. As proved in §A1 of electronic supplementary material 1 and recapped below, monotonicity of *z*_{⊥}(*y*) or over the period gives sufficient SHSW-existence criteria in an exact and explicit form. It will be shown that approximate predictions can also be made for much broader classes of continuous unit cell profiles.

Assume first the case of a piecewise homogeneous periodic half-space. The situation may be described as follows (see §A1 for details). Let both and be monotonically increasing over the unit cell, in the sense that and . Then the SHSW must exist on all the branches throughout the supersonic domain (figure 2*a*; the subsonic part of figures 2 and ,3 will be discussed in ,§4*b*). Owing to the reciprocity property, inverting the monotonicity to decreasing merely interchanges the SHSW existence and non-existence in this (and following) statements. Next, suppose that only is increasing in [0,*T*] while the variation of is arbitrary. Then, the SHSW also exists on all the branches at and above the cut-offs ; however, now the SHSW existence may switch to non-existence within the yet supersonic extent of different (probably not all) branches at correspondingly different slowness levels (figure 2*b*). The lowest value of *s*(*n*) is equal to the least of the roots *σ*_{j,j+1} given by (A11) in §A1. Thus, if this least value is real and less than (as it is in figure 2*b*), then a uniform existence of SHSW terminates yet within the supersonic domain. This is always so provided that an increasing is complemented by , for then the upper bound *σ*_{1J} of the least *σ*_{j,j+1} (equation (A12) in §A1) is definitely real and less than . This is the case in figure 2*b*, where . Finally, consider a general case of a non-monotonic profile of in [0,*T*]. Interestingly, then, the distribution of supersonic slowness ranges of SHSW existence on various dispersion branches is no longer regular (figure 2*c*).

In the remaining part of this subsection, we deal with the case of a continuously inhomogeneous unit cell. The results of §A1 for a monotonic unit cell profile are in essence the same regardless of whether it is continuous or piecewise constant. Similarly as above, they tell us that an increase or decrease of *z*_{⊥}(*y*) over [0,*T*] ensures a uniform existence or non-existence of the supersonic SHSW on all the branches at least within a certain vicinity of the cut-offs. At the same time, the particularity of the case of a continuous unit cell is that the regular spectral distribution of supersonic SHSW may actually come about under much looser restrictions than the monotonicity of *z*_{⊥}(*y*) in [0,*T*]. It basically suffices that the material properties are discontinuous across the (welded-contact) interfaces between unit cells. The ensuing predictions and estimates are, however, approximate in that they are based on the simplest form of the Wentzel–Kramers–Brillouin (WKB) asymptotical approach, which assumes frequency to be high enough with respect to the gradient of material properties, and which breaks near the supersonic/subsonic threshold Min *s*_{∥}(*y*).

Consider a periodic half-space, whose unit cell [0,*T*] is continuous and has a sufficiently pronounced difference of material properties at *y*=0 and *y*=*T* (see §2*b*). The leading-order WKB formula approximates the propagator through a unit cell as , where the columns of matrix are eigenvectors of ** Q**(

*y*), and

**is a diagonal matrix with components . Within such approximation, the supersonic onset of branches of the free unit cell is defined by the equation (**

*E**n*=1, 2 …), and

**(**

*M**T*, 0) on is estimated by a diagonal matrix with the components(4.1)On these grounds, the radiation condition (2.15)

_{2}deciding the SHSW existence or non-existence on reduces to the inequality between

*Z*(0) and

*Z*(

*T*) defined in (2.16). It is algebraically the same as in §3. The only difference is that now the root

*σ*

_{T}of equation

*Z*(0)=

*Z*(

*T*) in ,(4.2)given it is real, is to be compared with Min

*s*

_{∥}(

*y*), which certainly does not have to be equal to one of the edge values

*s*

_{∥}(0) or

*s*

_{∥}(T). Thus, equations (2.15)

_{2}, (4.1) and ,(4.2) lead to the following asymptotically valid predictions.

Let

*z*_{⊥}(0)<*z*_{⊥}(*T*). The SHSW existence dominates throughout the supersonic domain either if or if and (real) . Otherwise, the SHSW existence dominates within , being replaced by the non-existence within if and (real) . The latter is always the case when*z*_{⊥}(0)<*z*_{⊥}(*T*) is complemented by .Let

*z*_{⊥}(0)<*z*_{⊥}(*T*). The SHSW non-existence dominates throughout the supersonic domain either if or if and (real) . Otherwise, the SHSW non-existence dominates within , being replaced by the existence within if and (real) . The latter is always the case when*z*_{⊥}(0)>*z*_{⊥}(*T*) is complemented by .

Statement (ii) is an inverse of statement (i) by virtue of the reciprocity property. Carefully saying that the existence or non-existence ‘dominates’ accounts for the use of only a leading-order WKB term as a premise for the above statements, which may fail for relatively low-frequency branches and near the threshold Min *s*_{∥}(*y*).

The supersonic part for each of the cases displayed in figure 3*a–c* illustrates one of the three options established for *z*_{⊥}(0)<*z*_{⊥}(*T*) in statement (i). It can be seen that the slowness range of existence of the supersonic SHSW is asymptotically regular, despite the fact that the profiles *z*_{⊥}(*y*) in [0,*T*] are not monotonic. This is unlike the case of a piecewise unit cell (possibly including inhomogeneous layers as well). To this end, it is instructive to compare figures 2*c* and 3*b*, which are computed, respectively, for a discrete (non-monotonic) profile and for its ‘smoothed’ version slightly distorted by an appropriate half-sine function. Because the shape of the profiles is similar, so is the shape of the dispersion curves ; at the same time, there is a striking difference in spectral distribution of supersonic SHSW, which is irregular in figure 2*c* but is regular in figure 3*b* (with the exception of two branches at a relatively low frequency). Figure 3 also shows a limited accuracy of estimating the bounds of slowness ranges of supersonic SHSW by constant values *σ*_{T} and Min *s*_{∥}(*y*).

Another significant observation implied by equation (4.1) is that, in contrast to the above cases, the dispersion spectrum of supersonic SHSW is no longer regular if *Z*(0)=*Z*(*T*) for any , which is when the periodic half-space is ‘perfectly continuous’ in the sense that its material properties have no jump across interfaces between unit cells, and are therefore the same at the unit cell edges. This feature is exemplified in figure 3*d*(i) plotted for an asymmetric (see §2*d*) continuous unit cell profile with precisely the same trends at the opposite edges, whereby the supersonic ranges of SHSW existence on the reference branches are patchy. Interestingly, it is this spectral irregularity—not regularity—which is conditioned by an equality (of edge properties), and hence is perturbation-sensitive. Figure 3*d*(ii) shows that just a 1 per cent difference between the edge values (*z*_{⊥}(*T*)⪆*z*_{⊥}(0)) entails a noticeable arrival of a uniform existence of SHSW on the supersonic onset of the high-frequency branches . A larger difference extends the SHSW spectral regularity to lower frequency and to a greater supersonic slowness, which falls into better agreement with the aforementioned estimate *σ*_{T}.

### (b) Subsonic SHSW

We now proceed to the subsonic domain where the slowness branches of a free unit cell rise above Min *s*_{∥}(*y*) and ultimately tend towards Max *s*_{∥}(*y*). On doing so, they develop families of high-frequency collective asymptotes consisting of sequences of segments of successive branches, which themselves do not intersect. Each family is attracted by a local maximum of *s*_{∥}(*y*), close to which (from below) the corresponding asymptotes form plateaux with a certain fall-off rate. These dispersion trends reflect the local wave-field damping in the vertical direction wherever *κ*(*y*) is imaginary, i.e. exceeds *s*_{∥}(*y*) (see (2.16)). As a result, acoustic channels in the vicinity of different local maxima of *s*_{∥}(*y*) are uncoupled from each other at high enough *ω* by an intermediate ‘gap’ (or ‘barrier’ ). In this very basic context, there is no principal difference between continuous and piecewise constant inhomogeneity profiles. Their dissimilarity does reveal itself in a different dispersion rate near local slowness maxima or in other types of plateaux, caused by relatively broad slowness minima typical for a discrete profile. However, these aspects, which are thoroughly discussed elsewhere (see Shuvalov *et al*. 2008), concern the shape of the reference SH spectrum rather than its partitioning into ‘physical’ and ‘non-physical’ surface-wave solutions, which is the issue in hand.

Thus, with an exception of the onset of the fundamental SH_{0} branch (see §A2 in electronic supplementary material 1), predicting the high-frequency SHSW along the subsonic extent of the branches can be hinged on the simplified asymptotic view of the wave field in a unit cell [0,*T*] with a given profile *s*_{∥}(*y*) as locally damped where , and propagating where . Linking this perspective to the radiation criterion |*A*(*T*)|<|*A*(0)| (2.14) yields the following SHSW-existence considerations.

Suppose that everywhere in the unit cell [0,

*T*] except for an interval [*y*_{0},*T*] or [0,*y*_{0}] adjacent to the upper or lower unit-cell edge, in which (the damping factor Im*κ*∼*ω*and*y*_{0}/*T*are assumed not to be too small). The SHSW at such exists or does not exist, respectively. In consequence, a monotonic decrease or increase of*s*_{∥}(*y*) over [0,*T*] ensures, respectively, an asymptotically uniform existence or non-existence of the SHSW throughout the subsonic domain except for close to Min*s*_{∥}(*y*) (where Im*κ*is small).Suppose that within the intervals [0,

*y*_{1}] and [*y*_{2},*T*] adjacent to both unit-cell edges, while in between. Comparing decay of the amplitudes |*A*(0)| and |*A*(*T*)| relatively to |*A*(*y*_{1,2})|∼1 relates an asymptotic criterion for the SHSW existence or non-existence at such to, respectively, the upper or lower inequality sign in(4.3)Suppose that local maxima of

*s*_{∥}(*y*) are separated by a local minimum and are therefore asymptotically uncoupled. Then, the SHSW existence considerations similar to the above apply independently to different families of high-frequency asymptotes driven in the spectrum by these maxima of*s*_{∥}(*y*). If, for instance, the latter are located at the unit-cell edges, then SHSW exists on the asymptotes tending to*s*_{∥}(0) and does not exist on the asymptotes tending to*s*_{∥}(*T*).

Having been referred to a continuous profile, all the statements allow an evident re-wording for a piecewise constant or piecewise continuous profile. Note that they rely on the profiles of *s*_{∥} and possibly *κ* but do not explicitly involve the profile of *z*_{⊥}, which has in turn been most essential in the cut-off vicinity.

The above-formulated properties are illustrated through their explaining the subsonic part of spectra given in figures 2 and ,3 (remember that the branches, which at the scale of the plots may be seen as intersecting, in fact never do so). According to statement (i), a monotonic in [0,*T*] piecewise constant or continuous profile of *s*_{∥} leads to the asymptotically uniform non-existence of the subsonic SHSW in figure 2*a* and to its existence in figure 3*c*. Statement (i) also underlies the SHSW existence for in figure 2*b*,*c*, and for in figure 3*b*. According to statement (ii), the upper and the lower sign of the inequality (4.3) explains the SHSW existence for in figure 2*b* and its non-existence for in figure 3*d*(i),(ii) respectively. Figure 3*a* shows a slightly more elaborate case. First, note that the dip of *s*_{∥}(*y*) at *y*=0 is chosen to be rather sharp. In the spirit of statement (i), this causes the lower bound for the slowness range of SHSW non-existence to be visibly dispersive and approaching the threshold *s*_{∥}(0)=Min *s*_{∥}(*y*) only for markedly high *ω*. Second, profile *s*_{∥}(*y*) is constructed to demonstrate a crossover between the two options formulated by statement (ii). This crossover occurs when inequality (4.3) changes its sense from the lower to the upper sign with increasing above both *s*_{∥}(0) and *s*_{∥}(*T*). Hence, the upper bound of the SHSW non-existence range in figure 3*a* can be estimated by the value *σ*_{0}, at which the two sides of (4.3) are equal. Finally, statement (iii) accounts for a ‘separable’ existence and non-existence of the SHSW on the two families of subsonic asymptotes observed in ,figures 2*c* and 3*b*,*d*(i),(ii).

Thus, a simple ‘asymptotic’ methodology outlined in §4*b* proves to be efficient for analysing the subsonic SHSW in a periodic half-space with basically any type of unit cell profile. This conclusion is further upheld by additional examples for various unit cell profiles, which are provided in electronic supplementary material 2.

## 5. SHSW slowness window and its numerical simulation for a half-space and a strip

### (a) SHSW slowness window

One of the interesting aspects discussed above is that certain types of a unit cell profile lead to SHSW uniform existence (alternatively, to its non-existence) within a single slowness window, which can be squeezed by an appropriate choice of material parameters. This is always so in a periodically bilayered half-space with and (i.e. ), for which the SHSW exists on all the branches in the exactly defined range (figure 1*b*). By (3.2), this range is narrower, the greater the contrast is between the two layers. In loose words, the faster and stiffer a unit cell is at its top face relative to the bottom one, the more ‘adverse’ is the periodic structure towards the surface-wave existence. The physical sense is evident indeed. Interestingly, if a piecewise homogeneous unit cell consists of more than two layers, then a similar monotonic decrease of and over [0,*T*] also restricts the SHSW existence to a slowness range below , but the SHSW-existence intervals on different branches in this case are different and patchy (figure 2*b*).

For a continuous unit cell, a monotonic decrease of *z*_{⊥}(*y*) and *c*_{∥}(*y*) over [0,*T*] is a sufficient (though not necessary) condition for the asymptotically uniform SHSW existence on within a single slowness range. Its lower and upper bounds can be approximated by constant (disregarding possible low-frequency dispersion) values of *σ*_{T} and Min *s*_{∥}(*y*) based on the WKB approach (see §4*a*). ,Figure 4, which is computed for an isotropic unit cell with a constant density *ρ* and a linearly decreasing profile(5.1)for *c*_{∥}(=*c*_{⊥}), and hence also for *z*_{⊥}=*ρc*_{⊥}, demonstrates another specific aspect of the case of a continuous unit cell. This is that making the profile slope *C* steeper not only squeezes the SHSW slowness window but can also filter out its low-frequency content by shifting the SHSW to higher frequency such that validates the WKB assumption *κ*(*y*)*T*≫*C*. Note that taking a greater slope *C* has a twofold restrictive effect on this assumption owing to its ‘pinning’ the SHSW slowness window towards Min *s*_{∥}(*y*), and hence reducing the local vertical wavenumber *κ*(*y*) (see (2.16)).

### (b) Numerical simulation

To verify the above predictions made in the Fourier domain, a time–space SH wave field in a periodic half-space has been simulated by means of the second-order finite-difference time-domain (FDTD) method and then processed at the surface by two-dimensional fast Fourier transform (FFT). A point source generating a SH transient signal was taken in the *XY* plane on the traction-free boundary of periodically repeated domains (*T*=2 mm along the axis *Y*) surrounded on three sides by perfectly matched layers (PMLs). Each domain models a continuously inhomogeneous isotropic unit cell with a constant *ρ* and a linear profile of *c*_{∥}(*y*). Similar data for the case of a bilayered unit cell are presented in electronic supplementary material 2. More details of the numerical procedure may be found in Golkin (2009).

The results of the post-signal processing of the transient wave field at the surface are presented as a colour-scaled intensity over the {*ω*, *k*} plane in figure 5*a*,*b*, the former obtained for the decreasing unit cell profile of *c*_{∥}(*y*) (and hence of *z*_{⊥}(*y*)) in a unit cell and the latter for the inverse, increasing, profile. The intensity distribution is overlaid on the dispersion spectrum computed from (2.15) and ,(2.8). The dispersion branches, which are the same in ,figure 5*a*,*b* by virtue of the reciprocity property (§2*d*), contain a single narrow spectral sector of, respectively, the SHSW existence for the decreasing profile and its non-existence for the increasing profile. Accordingly, the highest wave-field intensity in ,figure 5*a* occurs on the dispersion branches within this sector, whereas a precisely inverse picture is observed in figure 5*b*. Thus, the simulation results are in good conformity with the theoretical prediction.

It is instructive to visualize an exponential envelope of the SHSW with a pure imaginary Floquet wavenumber *K*=i*K*″. Figure 6*a* shows an example of the distribution of the FDTD-simulated signal through 20 unit cells with the continuous profile used in figure 5*b*. The signal is taken at a fixed horizontal distance *x*_{0} (≈5.5 mm) from the source and at a fixed time *t*_{0} (≈81 μs), which is large enough for the initial wavefront emitted by the source to reach the PMLs and to disappear. In order to extract a single Floquet mode, the two-dimensional FFT in *x* and *t* is performed along each interface *y*=*nT* between the unit cells. Figure 6*b* shows the amplitude values *A*(*nT*) for a fixed point (*ω*_{0}, *k*_{0}), which lies on one of the reference dispersion branches and corresponds to the SHSW existence (see inset). Applying the exponential fit yields the dependence *A*(*nT*)∼exp(−*K*″*nT*) with *K*″≈0.349 mm^{−1}. A direct calculation of *K*″, which implies computing the matricant ** M**(

*T*, 0) for the given unit cell profile and the given values

*ω*

_{0},

*k*

_{0}by means of equation (2.8) and finding its eigenvalue

*q*

_{1}=exp(i

*KT*) (see (2.10)), yields

*K*″≈0.346 mm

^{−1}. A neat quantitative agreement confirms the accuracy of the numerical modelling. Note in passing that the probed point (

*ω*

_{0},

*k*

_{0}) is taken to be in the supersonic domain where the Floquet stopbands are narrower, and hence the damping coefficient

*K*″ is generally weaker than in the subsonic domain (see the inset to figure 6

*b*; also cf. figure S1 in electronic supplementary material 2).

Finally, a similar simulation of the transient SH wave field has been performed for a semi-infinite longitudinally periodic isotropic strip with the traction-free faces parallel to the axis *X* and a point source at the free edge. The strip thickness is *d*=7.5 mm and the period along *Y* is *T*=2 mm (the PML was added after repeating 10 periods). The unit cell profile is the same as used in figure 5*b*. A simple analytical consideration given in §A4 in electronic supplementary material 1 shows that cutting a vertical strip from a vertically inhomogeneous half-space and setting the strip faces free of traction amounts to replacing a continuous SH wavenumber *k* of the half-space dispersion spectrum by discrete values *k*_{m}=*πm*/*d*, *m*=0, 1, 2, … . In consequence, the spectral pattern for a strip shown in figure 7 is basically (with allowance for the numerical accuracy) a ‘*k*-discretized version’ of figure 5*b*. Numerical evaluation of the mean distance Δ*k* between several successive points of intensity maxima gives Δ*k*≈0.426 mm^{−1}, which is close enough to the theoretical value Δ*k*=*π*/*d*≈0.419 mm^{−1}.

## 6. Conclusions

The dispersion spectrum of the SHSW in a vertically periodic half-space is a locus of broken intervals of the dispersion branches of the free unit cell. These intervals may be either randomly patchy or more or less regular in terms of the horizontal slowness *s*=*ω*/*k*. The key factor is the shape of the unit cell profile. A profile with the same trends at both edges of the unit cell leads to a mostly random SHSW spectrum. A contrast of material properties at the opposite unit-cell edges (hence across the interfaces between unit cells) furthers the SHSW spectral regularity and may restrict its existence or non-existence to distinct slowness ranges. They can be predicted and estimated without any computation. The ‘control’ material parameters are the impedance *z*_{⊥} (most essential near the cut-offs), the horizontal slowness *s*_{∥} and the vertical wavenumber *κ* (possibly involved in the subsonic domain). A simple case of the unit cell composed of two homogeneous layers is fully describable by elementary algebra; the case of a monotonic unit cell profile for *z*_{⊥} and *s*_{∥} allows exact predictions; otherwise, the high-frequency asymptotical considerations prove efficient. It often suffices to know the values of *z*_{⊥} and *s*_{∥} at the unit-cell edges. Altogether, the present study yields simple guidelines to model unit cell profiles bringing about the desirable behaviour of SHSW.

One remarkable observation is the possibility of SHSW existence solely within a single slowness window. This is the case for a piecewise homogeneous bilayered unit cell with and (the first layer is the one adjacent to the surface), and also for a broad class of continuously inhomogeneous unit cells, e.g. those with monotonically decreasing profiles of *z*_{⊥} and *c*_{∥}. Increasing the contrast between their values at the unit-cell edges squeezes the slowness window of SHSW existence and, in the case of a continuous unit cell, may also eliminate the low-frequency content. For an inverse, increasing, profile, the same slowness window is where the SHSW does not exist while existing elsewhere (the reciprocity property). The prediction of such spectral selectivity has been verified through the numerical simulation of the SH wave field in the time–space domain and its post-signal processing. A similar simulation has also been performed for the case of SH waves in a longitudinally periodic semi-infinite strip, which is described by the same formalism as the case of a half-space.

Regarding future work, an interesting development would be ‘unfolding’ the events observed on the {*ω*, *k*} (or {*ω*, *s*}) plane into another perspective, related for a fixed *k* (or *s*) to the *ω*(*K*) dependence on the Floquet wavenumber *K* in the Brillouin zone. This would provide an insight into the resonant reflection/transmission and other such features. Note to this end that assuming monoclinic (rather than orthorhombic) anisotropy keeps intact the state of affairs projected on the {*ω*, *k*} plane (see §2*a*), but it can essentially distort curves *ω*(*K*) and create gaps within the Brillouin zone due to the absence of a horizontal symmetry plane (Kato 1997). Further research agenda also includes combining the present formalism, which allows an arbitrary but perfect periodicity in a certain direction, with other methods developed for tackling problems with a perturbed periodicity (e.g. ,Adams *et al*. 2008; ,Maurel & Pagneux 2008).

The considerations in this study are relevant to other cases of scalar waves dealt with in structural elasticity, fluid mechanics and optics. In particular, mathematically and physically similar issues arise in a cognate area of electromagnetic waves in dielectric waveguides (e.g. Zolla *et al*. (2008) and references therein). A cross-fertilization of ideas and results for phononic and photonic crystals offers a significant and as yet unexhausted potential for further progress in both these research fields.

## Footnotes

- Received November 11, 2008.
- Accepted January 20, 2009.

- © 2009 The Royal Society