## Abstract

Just as uni-directional Rayleigh waves at the traction-free surface of a transversely isotropic elastic half-space and Stoneley waves at the interface between two such media may have arbitrary waveform and may be represented in terms of a single function harmonic in a half-plane, it is shown that surface-guided waves travelling simultaneously in all directions parallel to the surface may be represented, at each instant, in terms of a single function satisfying Laplace’s equation in a three-dimensional half-space. That harmonic function is determined so that its normal derivative at the surface equals the normal displacement of the surface (or interface). It is shown, moreover, that the time evolution of that normal displacement may be *any* solution to the membrane equation with wave speed being equal to that of classical, uni-directional, time-harmonic Rayleigh or Stoneley waves. A similar representation is also shown to exist for Schölte waves at a fluid–solid interface, in the non-evanescent case. Thus, every surface- or interface-guided disturbance in media having rotational symmetry about the surface normal is governed by the membrane equation with appropriate wave speed, provided that the combination of materials allows uni-directional, time-harmonic waves that are non-evanescent. Conversely, each solution to the membrane equation may be used to construct a representation of either a Rayleigh wave, a Stoneley wave or a (non-evanescent) Schölte wave. In each case, the disturbance at all depths may be represented at each instant in terms of a single function harmonic in a half-space.

## 1. Introduction

In linear elasticity, time-harmonic waves guided by a traction-free surface (Rayleigh waves), waves travelling along the interface between two elastic media (Stoneley waves) and waves at a fluid–solid interface (Schölte–Gogoladze waves) are non-dispersive, since each is a solution to a boundary-value problem that contains no natural scale of length or time. Chadwick (1976) derived, in the first two cases, more general waves travelling in one direction, in which the profile of the normal displacement at the free surface or interface is arbitrary, yet the waves travel without distortion. Within these waves, the displacements at all depths are described in terms of a single function harmonic in a half-plane. In this paper, we show that, whenever all material behaviour is invariant under rotations around the surface normal and the material combinations allow uni-directional time-harmonic waves guided by a surface or interface to propagate without radiation and consequent attenuation, this representation may be generalized to describe surface waves or interfacial waves travelling simultaneously *in all directions*. Within these disturbances, all components of material displacement may be written in terms of a single function *Φ*(*x*,*y*,*Z*;*t*), which, at each instant *t*, is a harmonic function of *x*, *y* and *Z* in the half-space *Z*>0 (the variable *Z* differs from the physical coordinate *z* by a scaling factor). Moreover, the normal derivative of *Φ* at *Z*=0 equals the normal displacement at the surface and is a solution of the membrane equation (the two-dimensional wave equation) with wave speed being equal to that of the appropriate time-harmonic uni-directional wave (surface or interface). In fact, to each solution of the membrane equation, there correspond solutions describing both Rayleigh waves and interfacial waves.

In isotropic elasticity, the speed *c*_{R} of time-harmonic Rayleigh waves is independent both of wavelength and of direction of travel across the traction-free surface of a half-space *z*>0, where (*x*,*y*,*z*) are Cartesian coordinates and *t* is time. Thus, displacements of the form
1.1
which decay with depth ( as ), have dispersion relation *c*_{R}^{2}**k**⋅**k**=*ω*^{2}, which is the same as that of the two-dimensional wave equation (the *membrane equation*).

We first show that, for *any* superposition of waves of the form (1.1), *each* displacement component *u*_{i} (*i*=1,2,3) at *any* fixed depth *z* satisfies the membrane equation
1.2
where is the two-dimensional D’Alembertian associated with speed *c*_{R}. We then obtain a representation of those displacements in terms of the single function *Φ*(*x*,*y*,*Z*;*t*) harmonic in an abstract half-space *Z*>0 (corresponding to various scalings of *z*>0) and such that ∂*Φ*(*x*,*y*,0;*t*)/∂*Z* equals the normal displacement at the free surface. We also show that this free surface displacement may be *any solution* to the membrane equation (1.2), thus generalizing the representation of uni-directional Rayleigh waves of arbitrary form in terms of a pair of conjugate harmonic functions (Chadwick 1976) originally proposed by Friedlander (1948). Moreover, we show that these results concerning the membrane equation and a related *harmonic function* defined in a half-space apply also to omni-directional waves at the plane interface between two isotropic media (Stoneley waves) and to waves at a fluid–solid interface (non-evanescent Schölte waves).

This work extends a number of recent investigations for elastic surface and interfacial waves. For time-harmonic disturbances, it was previously shown (Achenbach 1998; Deutsch *et al.* 1999) that surface waves on a transversely isotropic half-space and waves in isotropic plates may have (complex) amplitude being any solution to the reduced-membrane (two-dimensional Helmholtz) equation (the disturbances at all depths are described as being ‘carried by’ the solution of the Helmholtz equation). In Kiselev (2004), elastic surface waves of Chadwick type yet having displacements varying linearly with the transverse coordinate have been constructed. While generalizing these results to time-harmonic surface waves on a half-space of arbitrary anisotropy and having dependence on the lateral coordinates of arbitrary algebraic degree, the present authors (Parker & Kiselev 2009) highlighted why *rotational invariance* of material properties about O*z* yields dramatic simplification, leading to Achenbach’s representation in terms of solutions to the Helmholtz equation—a result recently generalized (Kiselev & Rogerson 2009; Parker 2009) to arbitrarily graded media. Parker and Kiselev showed, also, how waves that are not time-harmonic have a compact representation provided that they are surface waves, even if they have arbitrary directional dependence. In fact, such waves may be ‘carried by’ any solution to the membrane equation (the two-dimensional wave equation). This property arises because surface waves are both non-dispersive—time-harmonic waves of all frequencies travelling in a chosen direction have the same wave speed—and rotationally invariant—the wave speed is independent of propagation direction (the first result itself follows from the scale invariance of the boundary-value problem defining the depth dependence of displacements within surface waves on *any* homogeneous elastic half-space). Since waves at the interface between any two uniform, isotropic half-spaces (elastic, or fluid) also are governed by equations and boundary conditions that are invariant both to rotation about O*z* and to rescaling (the moduli are constant in each of *z*<0 and *z*>0), if non-evanescent interfacial waves (Stoneley, or Schölte–Gogoladze) exist, they likewise correspond to any solution to the membrane equation. Details of the disturbances at any depth carried by that solution to the membrane equation are straightforward extensions to those of the surface wave (described in Achenbach (1998) and Parker & Kiselev (2009)).

The paper is organized as follows. It is first observed that, for a material with rotational symmetry about O*z*, each component of any surface-guided disturbance at any chosen depth satisfies the two-dimensional wave equation (the membrane equation) with appropriate wave speed. Moreover, for that membrane equation the initial conditions may be chosen arbitrarily. Then, in §3, disturbances are analysed that simultaneously satisfy this equation while also satisfying the Navier equations, boundary and decay conditions of isotropic elasticity. They are found to correspond to a normal displacement at the surface, which is *any* solution to the membrane equation, with wave speed being equal to the Rayleigh speed. Moreover, they give a compact representation for the displacements *at any depth* within an isotropic material in terms of a bounded solution to Laplace’s equation in a half-space, with normal derivative at the boundary being equal to the surface displacement, which itself evolves as a solution to the membrane equation. In §4, for Stoneley waves at the interface between two isotropic elastic media, a similar representation is shown to apply in each elastic half-space. Since, for interfacial waves, an inviscid fluid is effectively an elastic medium with vanishing shear modulus (Craster 1996), a representation for non-evanescent Schölte waves travelling simultaneously in all directions exists and is given in §5. In §6, the significance of the membrane equation is emphasized. It is the scalar equation having dispersion relation *c*^{2}**k**⋅**k**=*ω*^{2} identical to that of non-evanescent surface or interfacial waves when elastic half-spaces and fluid each have rotational symmetry around the surface normal. The membrane equation governs the evolution of normal displacements at the surface *z*=0; moreover, it actually governs *each* displacement component at *each* depth *z*. At each instant, the displacements at all depths are related to the normal displacement at the surface in terms of a *single* solution to Laplace’s equation in a half-space, with boundary data being equal to the solution of the membrane equation at that instant. While details are given only for isotropic materials, it is clear that the essential ingredients of rotational invariance about O*z* and of scale invariance allow similar results for combinations of transversely isotropic materials.

## 2. Surface and interface waves governed by the membrane equation

In this section, we present a general derivation of the membrane equation applicable to each of the three physical situations considered in the paper. We show, also, how initial conditions for solutions to the membrane equation relate to functions appearing within the Fourier representation. For any wave vector **k**=*k*_{1}**e**_{1}+*k*_{2}**e**_{2}, standard time-harmonic surface or interface waves have the form
2.1
with *A*(**k**) an arbitrary complex amplitude and with **e**_{1}, **e**_{2} and **e**_{3} being unit vectors parallel to Cartesian axes O*x*, O*y* and O*z*. The region *z*>0 contains a homogeneous elastic material, while *z*<0 is either free space (Rayleigh waves), is occupied by a homogeneous elastic material (Stoneley waves) or an inviscid fluid (Schölte–Gogoladze waves—non-evanescent case). The complex depth dependence **V**(*z*,**k**) is composed of contributions that are exponentials in |**k**|*z* (decaying away from *z*=0 wherever material is present) and constructed as a (suitably normalized) solution to a boundary-value problem for ordinary differential equations, which defines the wave speed *c*=*c*(**k**) as an eigenvalue. Whenever the materials are either isotropic or transversely isotropic with O*z* as the axis of rotational symmetry, *c* is entirely independent of **k** and it is possible to *choose* **V**(*z*,**k**) so that the normal component *W*≡**V**⋅**e**_{3} is real, with tangential component **V**−*W***e**_{3} imaginary and parallel to **k**. Additionally, **V** may be normalized so that *W*(0,*k*)=1, while rotational invariance about O*z* implies that displacements of the form (2.1) are confined to the sagittal plane, so that if **V**(*z*,*k***e**_{1})=*U*(*z*,*k*)**e**_{1}+*W*(*z*,*k*)**e**_{3} for *k*>0, then *U*(*z*,*k*) is purely imaginary, *W*(*z*,*k*) is real and, for any **k**,
Since 180^{°} rotation about O*z* replaces by , then
2.2
with the bar denoting complex conjugation. Moreover, *scale invariance* (Hunter 1989; Parker 2005) of the defining boundary-value problem shows that rescaling of wavelength causes only a similar rescaling of depth, so that **V**(*z*,*Λ***k**)=**V**(*Λz*,**k**) for each *Λ*>0, so allowing the representation
2.3
These properties may be confirmed in the classical case in which the material in *z*>0 is isotropic with Lamé constants *λ* and *μ*, with density *ρ* and with *z*=0 traction-free. In this case, explicit expressions for Rayleigh wave displacements are
2.4
where *β*_{1}≡(1−*ρc*_{R}^{2}/*μ*)^{1/2}, *β*_{2}≡[1−*ρc*_{R}^{2}/(*λ*+2*μ*)]^{1/2} satisfy 4*β*_{1}*β*_{2}=(1+*β*_{1}^{2})^{2}, which is the *secular equation* defining the unique (positive) speed *c*≡*c*_{R} of Rayleigh waves.

Now, integrating expression (2.1) over all **k** gives the general surface or interface wave as a Fourier superposition
2.5
where **x**≡*x***e**_{1}+*y***e**_{2}+*z***e**_{3}. By exploiting the property , this may be converted to polar form as
2.6
where , , and , with *r*≥0 and *k*≥0.

As a special case of equation (2.5), the case *A*(**k**)=*δ*(*k*_{2})*a*(*k*_{1}) gives **k**⋅**x**=*k*_{1}*x* so that
2.7
where, after exploiting the identity , it is found that
2.8
with
so that for all *k*. This ensures that each of the right- and left-travelling displacement fields **u**^{+}(*x*−*ct*,*z*) and **u**^{−}(*x*+*ct*,*z*) is real. This describes plane-strain disturbances (*u*_{2}≡0) as a superposition of two waves travelling without distortion along O*x* (cf. Chadwick 1976) in the positive and negative directions, respectively. (It may be noted that the right-travelling disturbance involves *a*(*k*) only for *k*>0, while the left-travelling disturbance is defined by *a*(*k*) for *k*<0. This is an example of ‘splitting into positive and negative frequency components’, e.g. Perel & Sidorenko (2009) and references therein.)

At *z*=0, the normal displacement takes the form , where (since *W*(0,*k*)=1)
so that
2.9
Since is the general solution to the one-dimensional wave equation , it is relevant to note that it may be specified in terms of *initial conditions* and through
2.10
where use of *x*,*y*,*z*,*t*,*Z* or *θ* as subscripts denotes partial differentiation. Thus, plane-strain surface or interfacial waves have normal displacement at *z*=0, which is the general solution to . In the corresponding displacements (2.7), each of **u**^{+}(*x*−*ct*,*z*) and **u**^{−}(*x*+*ct*,*z*) may be expressed in terms of *A*^{±}(*k*) through equation (2.8) and alternatively may be written in terms of a single pair of conjugate harmonic functions, as shown for Rayleigh and Stoneley waves travelling along O*x* by Chadwick (1976). For Rayleigh waves, inserting the specific forms (2.4) for and into expression (2.7) for **u**^{+}(*θ*,*z*), where *θ*≡*x*−*ct*, gives the right-travelling wave as **u**^{+}=*u*^{+}**e**_{1}+*w*^{+}**e**_{3}, where
2.11
in terms of two conjugate harmonic functions *ϕ*(*θ*,*Z*),*ψ*(*θ*,*Z*) that satisfy *ϕ*_{θ}(*θ*,*Z*)=*ψ*_{Z}(*θ*,*Z*), *ϕ*_{Z}(*θ*,*Z*)+*ψ*_{θ}(*θ*,*Z*)=0, which decay as and are related to the *arbitrary* normal displacement at the surface of right-travelling waves through and . A similar representation exists for the left-travelling wave, involving a second pair of harmonic functions.

Since the boundary-value problem defining **u**^{+}(*θ*,*z*) is invariant under rotations about the axis O*z*, non-distorting waveforms of arbitrary shape may also travel in the direction of any unit vector . Within these, displacements depend upon only and *z*. Indeed, by defining
expression (2.6) reduces to
2.12

Equation (2.12) gives a set of displacements decaying with depth and satisfying the traction-free boundary condition at *z*=0. Then, by inspection, each contribution **u**^{(γ)} to the integral in equation (2.12) is seen to satisfy , so that each Cartesian component of **u**^{(γ)} satisfies the membrane equation with appropriate wave speed *c*. Furthermore, the same holds for each component *u*, *v* or *w* of the integral (2.12).

As a special case, the normal displacement at *z*=0 satisfies
2.13
Since may be any (continuous) function of its first argument, expression (2.12) suggests that surface or interface waves exist such that the normal displacement of the surface
2.14
is *any* solution to the membrane equation (2.13) (just as plane-strain disturbances are linked to *any* solution to the one-dimensional wave equation). In fact, a compact representation in terms of harmonic functions is readily obtained, as in §3 for Rayleigh waves on an isotropic half-space. This generalizes to propagation in all directions both the result (2.7) and the representation of each of **u**^{±} in terms of harmonic functions in a half-plane. Moreover, the representation (2.5) shows that may be a solution to equation (2.13) with *arbitrary initial conditions* and .

Rearrangement of equation (2.5), using , gives
where, within the second contribution, the substitution −**k**↦**k** has been made. Then, using the normalization condition shows that
and
Introducing complex functions *B*(**k**) and *C*(**k**) through
shows that
2.15
so that 2*πB*(**k**) and 2*πC*(**k**) are the Fourier transforms of the *real* functions and and satisfy and . The corresponding rearrangement of equation (2.5) is
2.16
Hence, displacements (2.5) may indeed correspond to solutions of equation (2.13) with arbitrary initial conditions and , since inversion of formulae (2.15) determines *B*(**k**) and *C*(**k**) in the representation (2.16), which is equivalent to equation (2.5).

## 3. Displacements within a general Rayleigh wave

It is widely known (e.g. Love 1942; Hunter 1976; Brekhovskikh & Goncharov 1985) that the Navier equations for linear isotropic elasticity
3.1
(a dot denoting time differentiation) imply that **∇**×**u** and **∇**⋅**u** each satisfy the wave equation with *c*^{2}=*μ*/*ρ*≡*c*_{1}^{2} and *c*^{2}=(*λ*+2*μ*)/*ρ*≡*c*_{2}^{2}, respectively. Then, for *any* disturbance travelling parallel to O*x* without change of form at a speed (so that , ), it is clear that
and
These *elliptic* partial differential equations (pdes), coupled with traction-free conditions at *z*=0 and the decay condition as , yield one procedure for analysing displacements within Rayleigh waves that are independent of *y* and for constructing displacement fields in terms of solutions to [*β*_{α}^{2}∂^{2}/∂*x*^{2}+∂^{2}/∂*z*^{2}]*h*=0 when (a condition selected by the boundary conditions). They also provide a clue crucial in constructing a representation for displacements within a *general* three-dimensional elastic surface wave.

Observe how, for displacements **u**=**u**(*x*,*z*,*t*)=*u***e**_{1}+*w***e**_{3}, inserting the travelling wave conditions
3.2
into the Navier equations leads to a coupled pair of elliptic pdes for *u*(*x*,*z*,*t*) and *w*(*x*,*z*,*t*) in which *t* plays the role of a parameter. Since the natural generalization of the travelling wave condition *h*_{t}±*c*_{R}*h*_{x}=0, which is invariant under rotations about O*z* is *c*_{R}^{2}(*h*_{xx}+*h*_{yy})−*h*_{tt}=0 (cf. equation (1.2)), we seek solutions of equation (3.1) that *also* satisfy the equation
3.3
Then, writing **u**=*u*(*x*,*y*,*z*,*t*)**e**_{1}+*v*(*x*,*y*,*z*,*t*)**e**_{2}+*w*(*x*,*y*,*z*,*t*)**e**_{3} and splitting into components converts equation (3.1) to
3.4
The boundary conditions at *z*=0 become
3.5
while decay requires that as .

Use of the identities *μ*/*ρc*_{R}^{2}=(1−*β*_{1}^{2})^{−1} and (*λ*+2*μ*)/*ρc*_{R}^{2}=(1−*β*_{2}^{2})^{−1} converts the elliptic system (3.4) to
3.6
The treatment of three-dimensional disturbances is motivated by first considering solutions of equations (3.6) that are independent of *y*. In these, equation (3.6)_{2} becomes *β*_{1}^{2}*v*_{xx}+*v*_{zz}=0, for which the *only* solution *v*(*x*,*z*,*t*) satisfying *v*_{z}(*x*,0,*t*)=0 and as is *v*≡0. Thus, the solutions describe plane-strain disturbances. Within these, the isochoric (volume-preserving) contribution to *u*,*w* has *u*_{x}+*w*_{z}=0. This condition reduces equations (3.6)_{1} and (3.6)_{3} to *β*_{1}^{2}*u*_{xx}+*u*_{zz}=0 and *β*_{1}^{2}*w*_{xx}+*w*_{zz}=0, so giving , where the pair of functions *ϕ*(*x*,*Z*;*t*), *ψ*(*x*,*Z*;*t*) satisfy the Cauchy–Riemann equations *ϕ*_{x}=*ψ*_{Z}, *ϕ*_{Z}+*ψ*_{x}=0 and as with *Z*≡*β*_{1}*z* (observe that the dependence upon *t* is purely parametric). Analogously, the irrotational contribution satisfies *u*_{z}=*w*_{x}, for which equations (3.6)_{1} and (3.6)_{3} reduce to *β*_{2}^{2}*u*_{xx}+*u*_{zz}=0 and *β*_{2}^{2}*w*_{xx}+*w*_{zz}=0, so giving for a second conjugate harmonic pair of functions and satisfying , with as and *Z*≡*β*_{2}*z*.

Forming the linear combination in which and then applying the boundary conditions (3.5)_{1} and (3.5)_{3} yield
These are consistent conditions on the boundary values of two functions harmonic in *Z*>0 only for with *b*/*a*=−(1+*β*_{1}^{2})/2*β*_{2} and 4*β*_{1}*β*_{2}=(1+*β*_{1}^{2})^{2} (the secular equation). The travelling wave condition then gives *ϕ*_{t}+*c*_{R}*ϕ*_{x}=0, so that this construction recovers the representation (2.11) for sagittally polarized Rayleigh waves.

To extend this description of surface waves to three-dimensional disturbances, observe that the isochoric contribution satisfying *u*_{x}+*v*_{y}+*w*_{z}=0 allows solutions to equations (3.6) in which *u*_{z}=*β*_{1}^{2}*w*_{x} and *v*_{z}=*β*_{1}^{2}*w*_{y}, so that *v*_{x}−*u*_{y}=0. Moreover, these solutions allow the representation of displacements
3.7
in terms of some function *Φ*(*x*,*y*,*Z*;*t*) satisfying
3.8
namely a harmonic function in the abstract half-space *Z*>0 at *each* instant *t*.

Somewhat analogously, the irrotational contribution for which *w*_{x}=*u*_{z}, *w*_{y}=*v*_{z} and *u*_{y}=*v*_{x} leads to
3.9
These then describe displacements that have the representation
3.10
in terms of a second function that is harmonic in *Z*>0 and decays as , but here with *Z*=*β*_{2}*z*.

Forming the linear combination , which is the direct generalization of that used in the plane-strain case, then applying all three conditions in equation (3.5) thus yields
As in the plane-strain case, these are consistent if 4*β*_{1}*β*_{2}=(1+*β*_{1}^{2})^{2} and the two functions *Φ* and are related by
Making the substitution then gives the displacements within the *general Rayleigh wave* on an isotropic half-space as
3.11
where *Φ*(*x*,*y*,*Z*;*t*) is the solution of Laplace’s equation (3.8) in the half-space *Z*>0, with as at each instant *t* and with boundary condition . Then, making the choice *a*=2/(1−*β*_{1}^{2}) gives a representation for *all displacements* at each instant *t*, in terms of the single function *Φ*(*x*,*y*,*Z*;*t*) that is the solution to Laplace’s equation
3.12
with as for each *t*, and with normal derivative at the boundary *Z*=0 equal to the shape
3.13
of the free surface. Moreover, itself evolves as a solution to the membrane equation (2.13) with *c*=*c*_{R}.

The representation (3.11)–(3.13) is surprisingly compact. The traction-free surface evolves as a solution of the two-dimensional wave equation (the membrane equation) (2.13) with wave speed *c*_{R}, while *at each instant* the displacements carried by it (Achenbach 1998) are readily obtained by spatial differentiation of a single harmonic function *Φ*(*x*,*y*,*Z*;*t*) whose normal derivative at *Z*=0 simply equals at that very instant *t*. It is readily checked that, for plane-strain motions with no displacement along O*y*, this solution reduces to a superposition of two non-distorting solutions described by Chadwick (1976) propagating in the positive and negative directions along O*x*, respectively. Moreover, the usual pair of harmonic conjugate functions within that representation are just *Φ*_{x} and *Φ*_{Z}, two derivatives of a single harmonic function. In general, solutions of equation (2.13) do not propagate without change of form—this property applies only to uni-directional waves.

## 4. Omni-directional Stoneley waves

The steps leading, in §2, to equations (2.13) and (2.14) apply equally to surface waves and to interfacial waves guided by a plane *z*=0 between two isotropic homogeneous elastic media (Stoneley waves; Stoneley 1924). Consequently, the steps in §3 may also be suitably modified to seek both travelling waves having speed and, more generally, waves with displacements satisfying the modification of equation (3.3)
with normal displacement at the interface also satisfying the membrane equation (2.13). Indeed, using *ρ*_{±}, *λ*_{±} and *μ*_{±} to denote the densities and Lamé coefficients in *z*>0 and *z*<0, respectively, leads straightforwardly to the modification of equations (3.6) for components *u*^{±}, *v*^{±} and *w*^{±} of **u**=**u**^{±} in *z*>0 and *z*<0, with parameters
4.1
respectively. The boundary conditions (3.5) are replaced by continuity of both displacement and surface traction at *z*=0, namely
4.2
The decay conditions are as and as .

As for Rayleigh waves, solutions independent of *y* (travelling waves) must have , which possesses no time-harmonic solutions satisfying also the traction continuity condition −*μ*^{+}*β*^{+}_{1}=*μ*^{−}*β*^{−}_{1}. Thus, Stoneley waves travelling parallel to O*x* must be sagittally polarized (i.e. *v*≡0). Seeking solutions
to equation (3.6) uncouples the equations governing and from those governing and and reduces each to the Cauchy–Riemann pairs
4.3
so allowing the representations
4.4
with *Z*=*β*_{p}*z*,*p*=1,2, and where the relevant domain is *Z*>0 for and *Z*<0 for . The displacement continuity conditions (at *Z*=0) become
while, since , the conditions of traction continuity (at *Z*=0) may be written as
and

Compatibility of the interface conditions may be shown either by first considering time-harmonic solutions of equation (4.4) with and or, more generally, by taking
4.5
in terms of a single arbitrary harmonic function *Φ*(*θ*,*Z*) defined in *Z*≥0 and decaying as . Then, the interface conditions become
4.6

First eliminating *b*_{−} between each of the pairs (4.6)_{1}, (4.6)_{2} and (4.6)_{3}, (4.6)_{4} and writing *κ*≡2(*μ*_{+}−*μ*_{−}) gives
and
from which *a*_{−} may be eliminated to give the ratio *a*_{+}:*b*_{+} from
4.7
Similarly, eliminating *b*_{+} and then *a*_{+} gives
4.8
These agree with the expressions for *n* and *n*′ given by Chadwick (1976), who writes the ratios of parameters as *a*_{+}:*b*_{+}:*a*_{−}:*b*_{−}=−*n*:1:*mn*′:*m*. Also, equations (4.6)_{1} and (4.6)_{3} give two separate expressions for *m* as
as in eqn (3.8) of Chadwick (1976). Consistency between these two gives the secular equation determining the wave speed *c*_{St} in the form due to Stoneley (1924)
4.9
Another form, obtained directly from equations (4.6), is
4.10
and shows how the limit *μ*_{−}=0 recovers the secular equation for Rayleigh waves in *z*>0.

The range of values of *μ*_{+}/*μ*_{−}, *ρ*_{+}/*ρ*_{−}, *μ*_{+}/(*λ*_{+}+2*μ*_{+}) and *μ*_{−}/(*λ*_{−}+2*μ*_{−}) for which equation (4.9) or (4.10) has real roots *c*_{St} (the Stoneley wave speed) is restricted, but was investigated by Schölte (1947). Within this range, in the expressions for the displacements
4.11
the ratio of parameters is
while *Φ*(*θ*,*Z*) is *any* harmonic function defined in *Z*≥0, satisfying *Φ*_{θθ}+*Φ*_{ZZ}=0 and decaying as .

Then, just as surface waves travelling in all directions are described in terms of a single solution to the membrane equation through equations (3.11)–(3.13) (which reduce to equation (2.11) for *Φ*=*Φ*(*x*−*c*_{R}*t*,*Z*) with *ϕ*=−(1−*β*_{1}^{2})*Φ*_{x}, *ψ*=(1−*β*_{1}^{2})*Φ*_{Z}, so that *ϕ*_{x}=*ψ*_{Z} and *ϕ*_{Z}+*ψ*_{x}=0), there is a generalization of equation (4.11) in terms of a single harmonic function *Φ*(*x*,*y*,*Z*;*t*) and a solution to the membrane equation.

Let *Φ*=*Φ*(*x*,*y*,*Z*;*t*) satisfy Laplace’s equation (3.8) in *Z*>0 for each *t*, with as for all *x*,*y*,*t*. Then, it is readily confirmed that the displacement field
4.12
satisfies all of the field equations (3.1) (with ± corresponding to *z*>0 and *z*<0, respectively), the continuity equations (4.2) for displacements and tractions at *z*=0 and the decay as , provided that at *each* value of *z*, the potential *Φ* also satisfies the membrane equation with speed *c*_{St}. Here, **∇**_{2}≡**e**_{1}∂_{x}+**e**_{2}∂_{y}. In particular, the normal displacement at the surface *z*=0,
satisfies the membrane equation in the form
4.13

Thus, just as solutions of the membrane equation (1.2) with speed *c*_{R} describe Rayleigh waves propagating in all directions, solutions to equation (4.13), which is merely the membrane equation with Stoneley speed *c*_{St} determined from equation (4.10), describe omni-directional Stoneley waves. The complete representation (4.12) is further simplified by the choice of normalization of such that
Then, *Φ*(*x*,*y*,*Z*;*t*) is the solution to Laplace’s equation ∇^{2}*Φ*=0 in *Z*>0, which decays as and which has normal derivative at each instant *t* given by (i.e. equations (3.12) and (3.13)). It may be observed that additionally *Φ* satisfies .

## 5. Schölte–Gogoladze waves

Schölte (1947) and Gogoladze (1948) independently described time-harmonic waves propagating without attenuation in one direction along the interface between an inviscid fluid and an isotropic elastic solid. In the solid, taken to occupy *z*<0, equations (3.1) and (3.4) are unchanged. In the region *z*>0, linearizing a fluid motion about the rest state having density *ρ*_{f} gives
where is the density perturbation, **v** is the fluid velocity and *p* is the pressure, which is related to by . Then, since , the pressure *p* satisfies the wave equation
5.1
where is the acoustic speed. Associated (small) displacements **u** satisfy **u**_{t}=**v**, so that the traction in the fluid at *z*=0 is . Thus, the governing system in *z*>0 is equation (5.1) with as and the interface conditions reduce (as observed by Craster 1996) to the system (4.2) with *μ*_{+}≡0 and *λ*_{+}≡*ρ*_{f}*c*_{f}^{2}.

In describing uni-directional waves by
(i.e. so that *ϕ*^{(1)}≡0, *ψ*^{(1)}≡0), notation has been simplified by replacing *μ*_{−},*λ*_{−},*ρ*_{−},*β*^{−}_{1} and *β*^{−}_{2} by *μ*,*λ*,*ρ*,*β*_{1} and *β*_{2}, while taking . Then, in the interface conditions, there is no equivalent to equation (4.6)_{1}, since tangential velocity components *u*^{+}_{t} and *v*^{+}_{t} are not constrained. Moreover, as , the parameter *β*^{+}_{1} is neither real nor finite, but, since , the interface conditions are replaced by
5.2
Solving these gives the secular equation for Schölte waves
5.3
It is known that many combinations of fluids and solids do not permit solutions of equation (5.3) giving real speeds *c*=*c*_{Sch}, but for combinations that do permit non-attenuated Schölte waves, the amplitude parameters are in the ratios *b*_{+}:*a*_{−}:*b*_{−}=*β*_{2}(1−*β*_{1}^{2}):−2*β*_{f}*β*_{2}:*β*_{f}(1+*β*_{1}^{2}).

Then, just as equation (4.11) may be generalized to give omni-directional Stoneley waves in the form (4.12), the uni-directional non-evanescent Schölte waves are readily generalized to give omni-directional waves in the form
5.4
which, following the *choice* *b*_{+}=*β*_{f}^{−1}, involves just a single solution *Φ*(*x*,*y*,*Z*;*t*) of equation (3.12), chosen at each instant to have normal derivative *Φ*_{Z}(*x*,*y*,0;*t*) equal to the normal displacement . Moreover, evolves as a solution to the membrane equation with speed *c*=*c*_{Sch} given as a real solution of equation (5.3).

## 6. Conclusion

Rayleigh, Stoneley and Schölte–Gogoladze waves are non-dispersive since each is a solution to a linear boundary-value problem that contains no natural scale of length or time. Chadwick (1976) showed how the first two problems allow waves in which the normal displacement at the free surface or interface is an arbitrary non-distorting waveform, and that these may be described in terms of a single function harmonic in a half-plane. Those waves travel in a single direction across the surface, or interface. In this paper, we show that for all three problems, whenever the material behaviour is invariant under rotations about the surface normal and allows uni-directional surface or interface waves to propagate without attenuation, this representation may be generalized to describe surface and interface waves travelling simultaneously in all directions. All components of material displacement may be written in terms of a single function *Φ*(*x*,*y*,*Z*;*t*), harmonic in a half-space and with normal derivative being equal to the normal displacement at the surface. Moreover, that surface displacement evolves as a solution to the membrane equation, with wave speed being equal to that of the appropriate uni-directional case (surface, or interface). In fact, *each* Cartesian displacement component at *each* depth *z* evolves according to the same membrane equation, while the algebraic coefficients and the scalings of *Z* within the representation of displacements are simply those found in standard analysis of time-harmonic, uni-directional waves. Although details are given only for isotropic materials, extension of a representation appearing in Parker & Kiselev (2009) yields similar results for transversely isotropic materials.

The unifying role of the membrane equation is emphasized. It is seen that, just as in two dimensions, surface- and interface-guided waves simultaneously satisfy the one-dimensional wave equation and possess a representation in terms of a single harmonic function decaying within a half-plane, then, in three dimensions, they may be expressed in terms of a single solution to the membrane equation. At each instant, the displacements at all depths are connected through the decaying solution of Laplace’s equation in a half-space, with boundary data being equal to the solution of the membrane equation at that instant. This highlights the dual hyperbolic/elliptic character of surface-guided waves as noted, for example, by Kaplunov *et al.* (2006). Although all the underlying pdes are hyperbolic, the boundary conditions cause the displacement field at each instant to be governed by an elliptic system, yet the time evolution of disturbances is governed by a hyperbolic equation (the membrane equation). In the uni-directional case, the membrane equation reduces to the one-dimensional wave equation, for which the general solution is a superposition of two non-distorting profiles, as in the D’Alembert representation. The Chadwick (1976) treatment concentrates on the right-propagating travelling wave of arbitrary form, demonstrating that it propagates without change of profile and has an associated displacement field that may be expressed in terms of a single harmonic function.

The fact that the membrane equation (2.13) possesses exact beam-like and particle-like solutions (e.g. Kiselev & Perel 2000, 2002; Kiselev 2007) allows construction of Rayleigh, Stoneley and Schölte–Gogoladze waves exhibiting similar properties regarding the lateral variables. This will be demonstrated elsewhere, along with the construction of uni-directional, non-stationary waves with polynomial dependence upon *x* and *y* (as yet published only for linear dependence; Kiselev 2004).

## Acknowledgements

D.F.P. is grateful for the support of a Leverhulme Emeritus Fellowship while undertaking this research. A.P.K. acknowledges the support of the Engineering and Physical Sciences Research Council under grant EP/G000972. The authors are indebted to Maria Perel for a helpful discussion and to anonymous referees for their observations and suggestions.

## Footnotes

- Received November 10, 2009.
- Accepted January 29, 2010.

- © 2010 The Royal Society