## Abstract

We consider wave propagation along the surface of an elastic half-space, whose surface is flat except for a straight, infinite length, ridge or trench that does not vary in its cross-section. We seek to resolve the issue of whether such a perturbed surface can support a trapped wave, whose energy is localized to within some vicinity of the defect, and explain physically how this trapping occurs. First, the trapping is addressed by developing an asymptotic scheme, which exploits a small parameter associated with the surface variation, to perturb about the base state of a flat half-space (which supports a surface wave, as demonstrated by Lord Rayleigh in 1885). We then provide convincing numerical evidence to support the results obtained from the asymptotic scheme; however, no rigorous proof of existence is presented.

## 1. Introduction

Surface-guided waves are important in many applications, and since the work of Lord Rayleigh (1885), many articles and books have appeared with applications to, for instance, seismology (Chapman 2005), acoustic microscopy (Briggs 1992), surface acoustic waves (SAW) in ultrasonics (Cheeke 2002) that also play a role in microelectronic devices and acoustic sensors. Extensions to interfacial (Stoneley 1924) and fluid-coupled (Schölte 1947) waves and to dispersive plate (Rayleigh–Lamb) waves are all possible and provide yet more applications. It would not be far-fetched to say that Rayleigh's study of surface waves upon a semi-infinite elastic half-space has had fundamental and far-reaching effects upon modern life and many things that we take for granted today, stretching from mobile phones through to the study of earthquakes.

Among this huge literature, there are studies of topographically guided Rayleigh waves, and thereby transport of energy, in specific directions along defects in the surface. In early studies of SAW devices, there was considerable interest in this guiding of Rayleigh waves and Burridge & Sabina (1972*a*,*b*) and Lagasse *et al*. (1973) considered ridge-like (i.e. of a constant cross-section) defects upon the surface, but with a tall rectangular plate replacing the smooth (Gaussian) topography of figure 1, which were essentially thin rectangular plates attached to a semi-infinite half-space. A schematic of such a defect is provided in the inset in the upper right corner of figure 1.

Interestingly, numerical simulations indicated that a wave appeared to be trapped within the plate (topography) on the surface and propagated (unattenuated in the direction of propagation) along it, while being weakly coupled to the half-space. The wave speed of this trapped mode, limiting to that of a Rayleigh–Lamb plate wave for sufficiently tall and narrow plates, always remained below that of the free-surface Rayleigh wave. Heuristic arguments were also presented to motivate the existence of this trapped mode, and both seismic observations (Yomogida *et al*. 2002) and experiments with these SAW devices also apparently demonstrated the presence of this wave (Lagasse *et al*. 1973). Many years later, some existence proofs emerged (Bonnet-Ben Dhia *et al*. 1999; Duterte & Joly 1999), which strongly indicate that such trapped topographic Rayleigh waves should exist.

Our aim is to generate an asymptotic theory that captures this trapping effect, not for plates of high aspect ratio attached to a surface, but for smooth perturbations to the surface, thereby demonstrating conclusively that they are present and providing a convenient route to calculate the frequencies at which they are present.

Trapped elastic waves, either in plates or by edges of semi-infinite plates or else in elastic rods, have been of considerable recent theoretical interest (Gridin *et al*. 2005*a*,*b*; Kaplunov *et al*. 2005; Zernov *et al*. 2006), and it has been demonstrated that, in elastic plates, modes can be trapped in regions of either high local curvature (Gridin *et al*. 2005*a*) or thickness variation (Kaplunov *et al*. 2005; Postnova & Craster in press). These modes are eigensolutions of the problem, but have a finite energy, that are localized in space and exponentially decay with distance from the geometrical perturbation; this is exactly what one might envisage the localized Rayleigh wave to do. However, there is a fundamental physical difference: the elastic plate solutions consist of many Rayleigh–Lamb modes that can be cut-off (cease to propagate) if, say, the local thickness or curvature decreases. This cutting-off of modes is essential to the physical mechanism underlying the trapping for plates and rods; the trapping is directly associated with a mode that is cut-off away from the perturbation and cut-on in the vicinity of it, and it is also related to the group velocity of the modes. However, Rayleigh waves on a half-space are completely different: there is no obvious cut-off behaviour; the wave is non-dispersive with exponential decay in depth and is otherwise unattenuated; and there are no infinite sets of modes. Instead, there is merely a propagating surface wave with no characteristic length-scale present; for example, the thickness of an elastic plate, which would otherwise govern the propagation of the wave.

Nonetheless, a long-wave style approach proves successful in generating an asymptotic theory and the reasons for its success are summarized in §6. The paper is organized as follows. First, in §2, we present the geometry that we consider, together with the formulation of the problem that is convenient for our needs. In §3, the asymptotic scheme is then detailed, and it is shown that a relatively simple single ordinary differential equation (ODE) encapsulates the dominant physics. In §4, solutions from the full governing equations and the asymptotic scheme are then compared, using a numerical scheme; this confirms the existence of trapped modes. Finally, in §6, we offer some conclusions and a physical interpretation of the trapping phenomenon.

## 2. Formulation

We take the elastic material to be isotropic, with Lamé constants *μ* and *λ*, and of constant density *ρ*, and that all displacements are sufficiently small, such that the approximations of linear elasticity are applicable throughout. We assume that all displacements evolve in a time-harmonic manner and that there is propagation along the -axis, with dimensional wavenumber in arbitrary units. Limitations of the aspect ratio *H*_{0}/*L*_{0} are imposed in the asymptotic scheme, and throughout we assume that the surface profile flattens out at infinity in the cross-section variable , i.e. as .

We employ the Helmholtz representation to describe the displacement field in terms of a scalar potential and vector potential . Thus, summarizing the assumptions imposed on , together with the factor henceforth assumed understood and suppressed, we have the following form of the displacement field and potentials and :(2.1)(2.2)where a subscript variable after a comma denotes partial differentiation with respect to the variable in the subscript and the potential components of satisfy the wave equation. From this, it is immediate that the following Helmholtz equations hold:(2.3)(2.4)where wavenumbers and are defined as and , respectively. Longitudinal and transverse wave speeds, *c*_{L} and *c*_{T}, respectively, are defined with respect to the Lamé constants and the material density in the usual fashion, , . Further, we have introduced the wave-speed ratio *κ*=*c*_{L}/*c*_{T}.

It is assumed that the surface of the half-space is stress-free, namely , where is the usual isotropic stress tensor defined by the Lamé constants, and is normal to the surface at , given by . Hence, the stress-free conditions applied on are(2.5)We have to impose the gauge condition, , throughout the domain to remove the redundancy of a degree of freedom in the formulation. Before doing this, we note that on multiplying all equations of (2.4) by their respective unit vectors and summing, we obtain the Helmholtz equation, , after taking the divergence. For all materials, *c*_{R}<*c*_{T}<*c*_{L} (Achenbach 1984; eqn 5.11, p. 189), where *c*_{R} is the wave speed of the Rayleigh wave on a flat half-space, thus for oscillatory (non-trivial) solutions, . Should have a maximum within the domain, there. This is inconsistent with the Helmholtz equation derived for . It follows that cannot have a strict maximum within the domain, whence the maximum of must be achieved on the boundary. Applying the same argument to the function implies that imposing the condition at the boundary is sufficient for it to hold throughout the domain. This yields the additional boundary condition,(2.6)

We now suppose the defect to the surface has characteristic height *H*_{0}, and width *L*_{0}, as shown in figure 1, and proceed to non-dimensionalize all variables in the following way:and introduce the aspect ratio between height and width,For use in the numerical scheme, and for clarity, we apply the transformation *x*=*ξ* and *z*=*η*−*h*(*ξ*) to the non-dimensional analogues of the above equations (2.3), (2.4) and boundary conditions (2.5) and (2.6). This transforms the physical domain to a more convenient domain, the vertical coordinate is now 0<*η*<∞ and the stress-free surface is now at *η*=0. This convenience comes at a cost, which is that the governing equations and boundary conditions acquire considerable complication.

The following non-dimensional equations and boundary conditions of motion are obtained:(2.7)with(2.8)These are subject to(2.9)on the surface *η*=0; where a subscript variable after ∂ denotes partial differentiation with respect to the variable in the subscript. The entries of this matrix are rather lengthy and have been relegated to appendix A.

## 3. Asymptotic scheme

Our asymptotic scheme will rely on small perturbations to the surface and we consider deformations for which the aspect ratio *H*_{0}/*L*_{0} is small, i.e. *ϵ*≪1. We seek to solve the governing equations (2.7) subject to boundary conditions (2.9) via an asymptotic expansion, exploiting this small parameter *ϵ* and first suppose that the eigenvalue *ω* may be expressed as a regular expansion in *ϵ*. We also assume regular expansions for the unknown functions *ϕ*, *ψ*_{1}, *ψ*_{2}, *ψ*_{3},(3.1)(3.2)We then proceed in the conventional way by equating coefficients of powers of *ϵ* to form a hierarchy of equations, noting that it is to be expected that the leading order solution will be exactly the classical Rayleigh solution for a half-space.

### (a) Equations of motion and boundary conditions at *ϵ*^{0}

The *ϵ*^{0} terms in (2.7)–(2.9) produce equations(3.3)(3.4)Subject to boundary conditions applied on *η*=0,(3.5)(3.6)These equations and conditions decouple into pairs for *ϕ*^{(0)}, and , . Addressing the latter pair, namely , , we find that imposing these solutions to be non-trivial yields a determinant condition *ω*^{(0)2}=0, hence solutions are non-oscillatory. To avoid this triviality, solutions must be , . Now solving the former pair, we find(3.7)(3.8)where the functions *f*^{(0)}(*ξ*) and are as yet undetermined. Indeed, our task is to find these, as they contain the variation of the solution in the *ξ*-direction. Exponentially decaying solutions for finite eigenfrequencies will correspond to trapped modes.

Further, boundary conditions (3.5) show that *f*^{(0)} and are related via(3.9)Requiring that the boundary conditions are satisfied at the surface *η*=0 yields the following condition for non-trivial solutions, which is equivalent to the well-known equation for the phase speed of Rayleigh waves(3.10)It is well documented that this equation has precisely one positive real root for *ω*^{(0)}>0 (Achenbach 1984; eqn 5.11, p. 189).

### (b) Equations of motion and boundary conditions at *ϵ*^{1}

The *ϵ*^{1} terms in (2.7)–(2.9) produce equations(3.11)(3.12)These are subject to boundary conditions applied on *η*=0,(3.13)Again these equations and conditions decouple into pairs for *ϕ*^{(1)}, and , . Solving the equations yields(3.14)(3.15)(3.16)where functions *f*^{(1)}, , and are as yet undetermined. For the former pair, *ϕ*^{(1)}, , it has already been established that to avoid a triviality at the previous order, we needed to impose a determinant condition equivalent to the Rayleigh equation for the frequency (3.10). We may achieve consistency at this order with . For the pair, , , no determinant condition was imposed; this system may be inverted to obtain the following relationships between , in terms of :(3.17)It is necessary to proceed to the next order to establish an ODE for one of the unknown amplitude functions , which is related to *f*^{(0)} via (3.9).

### (c) Equations of motion and boundary conditions at *ϵ*^{2}

The *ϵ*^{2} terms in (2.7) produce equations(3.18)(3.19)These are solved subject to boundary conditions applied on *η*=0,(3.20)(3.21)(3.22)(3.23)The earlier conditions (3.13) have been used to simplify the right-hand side of (3.20). The solution of this system of equations is(3.24)(3.25)where the following notation has been employed for brevity:(3.26)(3.27)Substituting the expressions for *ϕ*^{(2)}, into the boundary conditions (3.21) and (3.22) and rearranging yields a system of equations for the amplitudes of the complementary function (solution of the analogous homogeneous system), *f*^{(2)}, , in terms of the unknown amplitudes *f*^{(0)}, . However, this system is, of course, a non-homogeneous version of that obtained when dealing with the equations of motion and boundary conditions at order *ϵ*^{0}, where it was imposed that the system must have zero determinant for non-triviality. Thus, we are required to invoke the Fredholm alternative; the system is consistent only if the right-hand side is orthogonal under the dot product to the solutions of the analogous homogeneous system. This yields a relationship between *f*^{(0)}, , independent of the amplitudes *f*^{(2)}, , taking the form of an ODE. We then proceed to form a single ODE by exploiting the derived relationship (3.9) between *f*^{(0)} and .

This is equivalent to substituting the found expressions for *ϕ*^{(2)} and into the boundary conditions (3.21) and (3.22) to form two simultaneous differential equations, and eliminating the unknown functions *f*^{(2)}, using the Rayleigh frequency equation derived at (3.10). Application of the previously derived relationship (3.9) between *f*^{(0)} and , yields a differential eigenvalue problem for the leading order amplitude, , and the frequency correction, ^{(2)},(3.28)which, for decaying eigensolutions, is subject to(3.29)The coefficients *A*, *B* and *C* are functions of *β* and *ω*^{(0)2},(3.30)(3.31)(3.32)

The choice to develop a single ODE for rather than *f*^{(0)}(*ξ*) is arbitrary, as the two are related by (3.9). It is perhaps remarkable that the asymptotic procedure has condensed the mechanics of the problem and the possibility of trapping, or otherwise, into a single ordinary differential eigenvalue equation (3.28), which must in general be solved numerically for . Nonetheless, one can use the aspects of Sturm–Liouville theory to gain some understanding of how we expect the solutions to behave.

We have no guarantee as yet that the differential equation found will have a solution at all; however, as a consequence of assumptions made in the problem formulation, *h*_{,ξξ}(*ξ*)→0 as *ξ*→±∞. Therefore, far from the defect , where . Further, the decay condition imposed on necessitates *A*/*C* to be of opposite sign to ^{(2)}. Therefore, we can infer that should a trapped wave exist, the sign of *A*/*C* will give the sign of the frequency perturbation about the Rayleigh-wave base state.

In a related work, for example for elastic plates (Gridin *et al*. 2005*a*), the dominant physics was reduced to a single ODE similar to that in (3.28),(3.33)and those authors were able to identify that under certain parameter constraints, the operator on the left-hand side was positive. Multiplying (3.28) throughout by and integrating on (−∞, ∞) (after using integration by parts) gives(3.34)If *h*_{,ξξ} were known to be positive throughout, it would become apparent that all integrals in (3.34) are of known sign. In particular, if *A*/*C*>0 and *B*/*C*<0, the operator on the left-hand side is positive, whence only positive eigenvalues may exist; thus, no trapping is possible under the above argument. However, necessarily *h*_{,ξξ} is not of definite sign. We will proceed to test this condition numerically for consistency in §4.

Moreover, we are able to use this argument to place restrictions on the function , which may satisfy the underlying equation; consistency may only be achieved provided , which gives us that the operator on the left-hand side will not be of definite sign, and the possibility of trapping remains open. For example, substituting the profile for a Gaussian hill *h*(*ξ*)=exp(−*ξ*^{2}) into the consistency condition implies that solutions must satisfy , or equivalently must have the same sign as *B*/*C*.

## 4. Results

The full elasticity problem (2.7) subject to boundary conditions (2.9) is solved numerically to determine the eigenfrequencies using a spectral Laguerre scheme, based upon the Newton–Kantorovich method, which automatically incorporates the exponential decay conditions. Moreover, this is free of the small parameter limitations of the above asymptotic scheme; thus, it can be used to assess the departure of the asymptotic scheme from the full numerical solution for various values of *ϵ*. It is also worth pointing out that the gauge condition, applied within the numerics as a boundary condition (owing to the minimum principle discussed earlier), is checked numerically to hold throughout the whole domain (which it does). The Laguerre scheme is for semi-infinite domains; hence, we split the problem into symmetric and anti-symmetric ones, in *u*_{1}, with *u*_{2}, *u*_{3} having opposite symmetry about *ξ*=0 and append a D or N superscript, respectively, when we refer to a particular case.

The numerical results presented in this section are used to illustrate the topographic trapping that has already been argued in §3 and compare the asymptotic scheme to the direct numerical scheme employed using spectral methods. Each of the plots included uses a value of *ϵ*=0.33 (*H*_{0}=1, *L*_{0}=3), in order to exaggerate the difference between the two.

Figure 2 shows the displacement profiles for a surface profile described by the ‘Gaussian bump’, . The dimensionless frequency eigenvalue produced from the numerical scheme, 9.260181, differs in square from the frequency eigenvalue produced by the asymptotic scheme, 9.244471, by 2.907×10^{−1}. This profile supports only a single surface wave using the -matrix ; the asymptotic scheme with the converse boundary conditions imposed by finds no physical eigenvalues to the problem. However, for sufficiently high , the surface is able to support a wave with those boundary conditions imposed by , a point we return to later.

Since we are solving an eigenvalue problem for *ω*^{2}, we present errors as a difference of squares. It is notable that computation in *ω*^{2} is advantageous for *ω*>1, since a disagreement in squares of (*ϑ*) leads to a disagreement of (*ϑ*/*ω*^{2}). This effect is magnified for larger *ω*. In particular, in the example presented in figure 2, the eigenvalues are found to have a difference of squares, 2.907×10^{−1}, but differ in value by only 1.571×10^{−2}.

Figure 3*a*,*b* shows the displacement profiles for a ‘Gaussian valley’ surface profile, . Two surface waves are given, as the asymptotic scheme generates one eigenvalue for each of the possible symmetric or anti-symmetric boundary conditions imposed at the boundary of symmetry *ξ*=0. The displacements are localized to a vicinity of either side of the perturbation to the surface; a physical interpretation of this is given in §5. The frequency eigenvalue produced from the numerical scheme in figure 3*a*, 9.254181, differs in square from the frequency eigenvalue produced by the asymptotic scheme, 9.250879, by 6.1106×10^{−2}. In figure 3*b*, the frequency from the numerical scheme, 9.254503, differs in square from the asymptotic value of 9.251344 by 5.847×10^{−2}.

Further simulations with smaller values of *ϵ* demonstrate that the errors in eigenvalue are significantly lower than expected; of order *ϵ*^{4}, instead of *ϵ*^{3}. This would suggest that if the asymptotic scheme were continued to higher orders, we would observe that ^{(3)}=0. This was perhaps to be expected; it is apparent that in a similar fashion to the equations of motion and boundary conditions at *ϵ*^{1}, the equations at *ϵ*^{3} would decouple into pairs for *ϕ*^{(3)}, and , . Since the left-hand side of the equations for *ϕ*^{(3)}, must be a non-homogeneous analogue of those obtained at order *ϵ*^{0}, we would be able to achieve consistency within the scheme with ^{(3)}=0.

Figure 4 shows the displacement profiles for a ‘Gaussian double-valley’ surface profile, when *d*=2.82, . As demonstrated in this figure, the surface supports four surface waves, two for each of the symmetric/anti-symmetric conditions. It is worth noting that *d*∼2.82 appears to be a critical value for supporting four surface waves. For *d*<2.82, the surface supports fewer than four waves and for *d*>2.82, the surface supports four or perhaps more; for sufficiently high *β*, the surface is able to support more waves, in a similar way as the ‘Gaussian hill’. One would expect, and indeed one finds, another such critical value, since the choice *d*=0, equivalent to the case shown in figure 3*a*,*b*, is demonstrated to support two waves.

As *β* decreases, numerical evidence suggests that this critical value of *d* increases, which may have been expected; as the wavenumber increases, the distance between the defects on a length-scale relative to the wave decreases, thus further separation is required to maintain these surface waves. Further, results suggest that the required value of *d* for trapping increases with an increase in *κ*.

As mentioned at the end of §3, we are able to check for consistency within the asymptotic scheme by evaluating the sign of . If this is positive, then the operator on the left-hand side is not of definite sign, whence negative eigenvalues (and hence trapping) are possible. In practice, checking this integral through the use of the trapezium rule on a non-constant grid demonstrates consistency, as the integral is seen to be positive for each of the examples undertaken in this section. Considering one example, shown in figure 2, and referring back to the condition derived at the end of §3 (that *B*/*C* must have the same sign as for solutions), we are able to interpret this condition physically as imposing that sufficient mass of the solution lies in a vicinity of *ξ*=0, since the integrand is only positive for , and it is found in practice that *B*/*C*<0.

We conclude this section by illustrating how the nature of the solution varies with *β*. Figure 5*a*–*d* shows contour plots for the *β*-values of 2, 10, 25, 50, respectively; as shown, as *β* increases, the trapping becomes more strongly localized to within a vicinity of the surface defect, a phenomena which holds true for the other components of displacement. It is also observed that the nature of the trapped wave is predominately transverse in the majority of the domain. Although the magnitude of the displacement field in the - and -directions on the surface of the space are comparable, as demonstrated in figures 2–4, the displacement in the -direction is very much more localized to within a vicinity of the surface than the displacement in the -direction, as shown in figure 6.

## 5. Physical interpretation

To understand the physics of the trapping, we turn our attention to the modes of a solid circular elastic bar; heuristically, one could imagine the smooth perturbation as being akin to a circular arc of material atop a half-space. Before pursuing this heuristic view, we note some useful characteristics of the mode structure within an elastic bar as being: increasing the ratio of bar radius (*a*) to wavelength , the curvature of the bar surface decreases relative to the wavelength-scale, as this ratio gets ever larger, it is envisaged that one of these solid-bar modes deforms continuously to become the Rayleigh surface wave; the limit corresponds to a half-space, and there is no physical mechanism to introduce a discontinuity in the limit.

It is therefore to be expected that a mode in the solid bar will deform continuously into the Rayleigh surface wave as ; we proceed to investigate this claim. If one constructs polar coordinates in a solid elastic bar of a constant circular cross-section in the natural fashion, the -axis lying along the axis of symmetry, radius being the perpendicular distance to the surface and the azimuthal coordinate *θ* taken to be increasing anti-clockwise round the bar, it is immediate that a wave with the character of the Rayleigh wave will have no azimuthal component of displacement and will travel in the -direction. Such a wave is referred to in the literature as a ‘longitudinal mode’.

As previously, we assume that the bar has the material properties of steel, *c*_{L}=5960 m s^{−1} and *c*_{T}=3260 m s^{−1}. Of the longitudinal modes in the solid circular bar, only the lowest mode travels at a speed slower than the Rayleigh speed *c*_{R}=3016 m s^{−1}, as demonstrated in figure 7. The point at which the speed of the lowest mode crosses the Rayleigh speed is shown in the top right inset to figure 7, at which point the bar radius is 1.4133 wavelengths. The lowest mode then achieves a minimum speed achieved at a bar radius of 2.425 wavelengths, indicated by a circle in the figure inset, before approaching the Rayleigh speed from below as .

Figure 8 compares displacement and stress profiles for the lowest longitudinal mode to the Rayleigh surface wave. For a guide of just three wavelengths (figure 8*a*,*b*), the Rayleigh-wave displacement profile is almost identical to that of the lowest solid-bar mode. For a guide of radius 10 wavelengths (figure 8*c*,*d*), it becomes difficult to distinguish any difference at all.

Given a fixed dimensional wavenumber , the solid circular bar supports a single mode with a lower frequency (hence a lower speed) than that of a Rayleigh wave upon a half-space. For a perturbed half-space with a Gaussian hill, as shown in figure 2, the surface near the perturbation is curved, and hence locally like that of a solid bar. Drawing on the analogy that a perturbed half-space is similar to a portion of a solid bar attached to a half-space, we argue heuristically that there exists a range of frequencies for which the curved section supports a wave, but the flat half-space does not. Therefore, a wave may propagate unattenuated in the curved section, yet may be ‘cut-off’ in the flat-space, and hence satisfies the decay conditions imposed upon the solution at . This coincides with the numerical results obtained.

For a Gaussian valley, as shown in figure 3*a*,*b*, we argue in a similar fashion. Conversely to a hill, the deepest part of the valley appears not to support a trapped wave, as the amplitude of all three displacement components decreases in modulus within the vicinity of the profile minimum. Locally, the surface has curvature similar to that of a hole in an infinite elastic medium. As previously, we suggest that given the absence of any physical grounds for introducing discontinuity, the displacement and stress profiles of this wave will be approached continuously by one of the longitudinal modes down the interior of a hole. If we consider the dispersion curves for a circular annulus, and let the radius tend to infinity, then it can be demonstrated that the second longitudinal mode in the circular annulus is localized to the inner radius and is the analogue of a Rayleigh surface wave.

The modal structure for longitudinal modes in the annulus is very similar to that shown in figure 7 and the second longitudinal mode behaves similarly, in that the speed of this mode does not fall below the Rayleigh surface wave speed. This suggests that there exists no range of frequencies, for which a wave can exist near the inner surface of a hole but not in the half-space; thus, trapping is not expected. This agrees with the observation of decreased amplitude near *ξ*=0 in figure 3*a*,*b*. The points of maximal amplitude occur on either side of the surface perturbation, which locally appear to be like sections of a solid bar, whence we appeal to the earlier argument to propose a physical explanation as to why trapping occurs there.

By providing a physical description of the above surface profile by likening it to that of two solid bars within the half-space (based on the local curvature), we further suggest that when the modes travelling down each bar are in-phase, we generate the symmetric profile (figure 3*a*), and when they are out-of-phase, the anti-symmetric profile of figure 3*b* is generated.

In understanding the double-valley scenario shown in figure 4 and the case of high mentioned in §4, we invoke a similar argument; when the two surface defects are sufficiently far apart, the surface locally appears like that of four distinct bars, as indicated in figure 9*a*. Indicating bar displacement profiles in *u*_{3} which are ‘in-phase’ by ‘+’, and ‘out-of-phase’ by ‘−’, we speculate that the profiles generated by imposing symmetric conditions (figure 4*a*,*b*) on *u*_{2}, *u*_{3} are analogues of waves we describe as (+−−+) and (++++), respectively. Those generated by imposing anti-symmetric conditions on *u*_{2}, *u*_{3} (figure 4*c*,*d*) are (−+−+) and (−−++).

As the distance between the profiles (measured by *d*) is reduced, we observe a change of regimes observed at *d*∼2.82, and we liken the physical situation to that of three bars within a half-space, indicated in figure 9*b*, whence only three surface profiles are available: (+++), (+−+) and (+0−), where 0 represents a bar to the left of which is an in-phase bar and to the right of which is an out-of-phase bar. This scenario is a degenerate case from both the (−−++) and (−+−+) cases: for sufficiently small *d*, the middle two bars +− and −+ are indistinguishable, at which point we draw on an analogy of a three-bar problem. As *d* gets smaller still (the critical value is numerically estimated at *d*∼1.05), we enter into a situation which we liken to that of only two bars (figure 3*a*,*b*), which occurs as the cases (+++) and (+−+) degenerate simply into (++), as indicated in figure 9*c*. Likewise, for a single bar, when is sufficiently large, the single hill resembles a profile consisting of two solid bars, which may either act in-phase or out-of-phase, thus generating two profiles.

## 6. Concluding remarks

In this paper, we have offered an asymptotic scheme to demonstrate that Rayleigh waves are trapped by a small topographic perturbation along the surface of an elastic half-space. In doing so, we have made a contribution to the existence of surface waves in topographic waveguides that are practically utilized in applications, such as microwave technology. We have provided no rigorous proof of their existence, but provide convincing evidence to support the asymptotic results using a direct numerical scheme based on spectral methods.

We have illuminated the results with a physical explanation of the trapping: topographic trapping is effectively caused by the local variation of the surface perturbation, which leads us to a convenient means to calculate the frequency at which trapping exists and the local extent of trapping. The nature of this localization on surface wave trapping is exploited in many of the SAW devices this paper supports. The trapping is also closely connected with the long-wave trapping of elastic waves in plates and rods in regions of local variations in curvature or thickness. However, it is important to note that the long-wave nature of the problem emerges as follows: if we fix the wavenumber in the -direction as , but vary *k*_{x}, the horizontal wavenumber, then there is a critical frequency , where *k*_{x}=0. At this frequency, the wavelength in the -direction is infinite and, in that sense, the wave is long; beneath this frequency the wave cannot exist and is cut-off, at least upon a half-space. The asymptotic method is a perturbation about this long-wave situation, and when viewed in this manner is akin to the long-wave waveguide trapping problems, which are governed by cut-offs.

Having shown the utility of the asymptotic methods developed here, it should be possible to extend the approach of this paper to look at topographically guided Rayleigh–Lamb waves along the surface of an elastic plate or else to trapping along the other surface defects that are no longer small in the asymptotic sense. Both of these problems are a significant challenge and will be the focus of future work.

## Acknowledgments

This work was supported by the EPSRC and Dstl through a CASE studentship for S.A.; we gratefully acknowledge this support. The authors wish to thank the Controller HMSO for granting permission to publish the Crown Copyright material contained in the text.

## Footnotes

- Received July 14, 2006.
- Accepted September 12, 2006.

- © 2006 The Royal Society