The Stroh formalism is widely used in the study of surface waves on anisotropic elastic half-spaces, to analyse existence and for calculating the resultant wave speed. Normally, the formalism treats complex exponential solutions. However, since waves are non-dispersive, a generalization to waves having general waveform exists and is here found by various techniques. Fourier superposition yields a description in which displacements are expressed in terms of three copies of a single pair of conjugate harmonic functions. An equivalent representation involving just one analytic function also is deduced. Both these show that at the traction-free boundary just one component (typically the normal component) of displacement may be specified arbitrarily, the others then being specific combinations of it and its Hilbert transform. The algebra is closely related to that used for complex exponential waves, although the surface impedance matrix is replaced by a transfer matrix, which better embodies the scale invariance properties of the waves. Using the scale-invariance property of the boundary-value problem, a further derivation is presented in terms of real quantities.
For analysis of the structure and existence of guided waves at the surface of an elastic half-space or at the interface between two such uniform half-spaces of general anisotropy the formalism due to Stroh  has proved useful, as evidenced by the extensive treatments by Barnett & Lothe [2,3], Chadwick & Smith  and Ting  and more recently through extensions to other geometry by Fu et al. . Since much of this work concerns non-dispersive guided wave solutions (i.e. with wave speed independent of wavelength), a generalization is readily available. Indeed, any instance of non-dispersive waves should immediately suggest that the governing equations define no natural length scale.
The usual Stroh formalism treats only sinusoidal (i.e. complex exponential) solutions to the governing partial differential equations and boundary conditions. However, since the work of Friedlander  and Chadwick , it has long been known that a representation of guided surface and interface waves of general form is available in terms of a single analytical function of a complex variable bounded in an abstract half space (or, equivalently, in terms of a pair of harmonic conjugate functions). This account generalizes the Stroh formalism to waveforms of arbitrary profile and shows how features usually associated with the real and imaginary parts of the complex exponential wave have close parallels with those of the real and imaginary parts of a single analytic function or, equivalently, of a harmonic function and its harmonic conjugate. It also helps in clarifying how much freedom there exists in assigning the profile of an arbitrary surface wave—a property that applies in other physical situations without a length scale, such as piezoelectric surfaces, elastic interfaces and a tangential discontinuity in magneto-hydrodynamics.
In §2, the standard Stroh treatment  of elastic surface waves is reformulated in terms of a displacement gradient vector and the traction vector, since together they satisfy two coupled first-order partial differential equations. This emphasizes the scale invariance [10,11] of the governing boundary-value problem, the fact of which causes sinusoidal surface waves to be non-dispersive, with deformation distribution scaling inversely with the wave number k. (It may be noted that Stroh  expressed all displacements and three stress functions as linear combinations of just three analytic functions, but in treating surface waves he restricted attention to complex exponential functions). The account is largely parallel to standard treatments, but leads to replacement of the surface-impedance matrix by a transfer matrix, which is simply related to it. In §3, Fourier superposition of wavelengths shows how general disturbances travelling uniformly at specified speed v may be written in terms of the real and imaginary parts of three analytic functions which decay away from the boundary of a half space. Moreover, imposing the boundary condition of zero traction shows that the three analytic functions are identical apart from scaling by the components of a (complex-valued) eigenvector and produces algebra simply adapted from that as in  for complex-exponential waveforms. Indeed, the speed v of surface waves must make a transfer matrix singular, a condition which has almost identical form to the equivalent condition in  for the surface-impedance matrix. Also, a representation for displacements within the travelling wave is obtained, expressing them as a linear combination involving three differently mapped copies of a single pair of conjugate harmonic functions. In §4, the representation is reinterpreted to emphasize that these waves may have one component of displacement at the boundary (usually the normal component) arbitrarily specified. Choosing this as the boundary data for the harmonic function which is the real part of the required analytic function then completely determines that analytic function, since it must decay away from the boundary of a half-plane. Since the boundary values of the two conjugate harmonic functions are related as Hilbert transforms, the new representation shows that, in general, at most one component of displacement at the boundary may be chosen as a localized function (i.e. having compact support). This illustrates the, now familiar, observation that surface-guided waves have both elliptic character and hyperbolic character.
Section 5 concludes with some comments on how this representation may be generalized to piezo-electric waves and to interface waves, including evanescent cases.
2. A generalized Stroh formalism for elastic surface waves
Using a standard notation for linear elasticity in which ui (i=1,2,3) are the components of the displacement vector u, while cijkl are the elastic coefficients and x1,x2 and x3 are the cartesian coordinates, the governing field equations and boundary conditions for disturbances in the homogeneous half space x2>0 are 2.1where ρ is the density, ti (i=1,2,3) are the components of traction on any surface of constant x2, where commas denote partial derivatives and superposed dots denote derivatives with respect to time t. Here, the elastic coefficients satisfy the symmetry conditions and the ellipticity condition cijklξijξkl>0 for all non-zero, real, symmetric tensors ξ. Surface-guided waves travelling parallel to Ox1 at speed v additionally have displacement components ui(x1−vt,x2) depending only on x1−vt and x2 and decaying as . Within these waves, , so that the governing system (2.1) becomes 2.2together with 2.3Here, following usual practice  in using the Stroh formalism, the coefficients Tik≡ci2k2, Rik≡ci1k2 and Qik≡ci1k1 have been introduced. Then, when again following normal practice by treating the traction components ti as primary variables, equation (2.2)1 becomes 2.4Normally, the system is analysed to derive an impedance relation between ti and ui but, observing that equation (2.1)1 is homogeneous of degree 2 in derivatives, while the boundary condition (2.1)2 is homogeneous of degree 1, we use as supplementary variable the displacement gradient vector p≡u,1, rather than u (noting that the surface wave problem is scale-invariant, a feature very significant in nonlinear theory (Hunter  and Parker and Hunter )). Then, using matrix notation in which T has elements Tij, etc., the definition (2.2)2 of the traction vector t on each plane x2=const. 2.5may be rearranged as a formula for the remaining deformation-gradient components as 2.6This may be substituted into (2.4) in its matrix form t,2+[Ru,2+(Q−ρv2I)p],1=0 to give 2.7Since equation (2.5) may be differentiated to give t,1=Tp,2+RTp,1, equation (2.7) should be accompanied by 2.8to form a first-order system of equations for the two vectors p and t.
Systems (2.7) and (2.8) may be written in a form similar to that used by Fu et al.  as 2.9where, apart from the contribution ρv2I within N3, the 3×3 submatrices N1≡−T−1RT, N2≡T−1 and N3≡RT−1RT−Q+ρv2I are formed entirely from material constants. Equation (2.9) possesses special sinusoidal wave solutions 2.10with k real, where c is an eigenvector and p the corresponding eigenvalue of the 6×6 real matrix N arising in (2.9) and defined by 2.11For specified v, the possible values for p are either real or occur as complex conjugate pairs, while the solutions (2.10) decay as only if Im[p]>0 for k>0 and Im[p]<0 for k<0. Any real values for p correspond to disturbances (2.10) which are plane waves inclined to the Ox1 axis and so do not decay away from x2=0. They cannot contribute to a surface-guided wave, but do not occur for [1,2,9], where is the limiting velocity defined by in which vb is a speed of a bulk mode propagating in the direction of the unit vector within the Ox1x2 plane. Then, those for which Im[p]>0 may be labelled as 2.12with α(q) and β(q) their real and imaginary parts and with β(q)>0 (so that the remaining eigenvalues and eigenvectors are p(q)*=α(q)−iβ(q)≡p(q+3) and c(q)*=c(q+3), say). This ensures that, for each q=1,2,3 and for each positive wave number k, the solution to equation (2.9) 2.13is sinusoidal in x1 and decays as . Moreover, the superposition 2.14is a travelling sinusoidal solution, for all complex constants dq (q=1,2,3). This may be split, as in (2.10), into displacement gradient vectors and traction vectors as Thus, for all choices of the vector d≡(d1,d2,d3)T and all choices of speed , the displacement gradient vector and traction at the surface are related through 2.15where the matrices P and C are formed from the upper and lower halves and of c(q)(q=1,2,3) as 2.16Then, using the transfer matrix defined as L≡CP−1 shows how the surface traction t(x1,0,t) may, for any d, be directly related to the surface displacement gradient vector p(x1,0,t) given by (2.15) through . This transfer matrix L depends upon v. If, for some value v, it is singular, the traction will vanish when p(x1,0,t) is given by (2.15), where Pd is chosen as the null vector of L (so that d is the null vector of C). The condition selecting v is analogous to the condition that the surface-impedance matrix is singular  and, moreover [2,12] is known to define a unique speed v of Rayleigh waves for arbitrary orientations of wave propagation in arbitrary anisotropic materials.
A characterization of L is available, without the need to identify the eigenvalues p(q) and the corresponding eigenvectors of N. It is akin to that obtained by Fu & Mielke . For each p(q), equation (2.11) gives the pair of equations 2.17Since and for each q=1,2,3, where e1,e2 and e3 are the standard unit vectors and since P−1 exists, these give so that 2.18After allowing for differences of notation and upon writing M=iL, this becomes the algebraic Riccati equation (1.2) of Fu & Mielke  defining the surface-impedance matrix M of . This is hardly surprising, since equation (2.10) shows that the surface traction corresponds to the surface displacement , when .
In §3, the Stroh formalism for (complex) exponential waves is extended to surface waves with general surface elevation profile travelling at the Rayleigh wave speed v. In this extension, the transfer matrix is preferred to the surface-impedance matrix, since it provides a scale-invariant relationship between the surface displacement gradient vector p(x1,0,t) and the surface traction (i.e. independent of k).
Before describing this treatment for waves with general waveform, a further analogy with  is noted. Firstly, equations (2.17) are rewritten as Then, the eigenvalue problem (2.11) defining p=p(q) and may be written as 2.19which gives a sixth degree polynomial in p having real coefficients for each v2. When all six roots p occur as complex conjugate pairs, selecting those with and so defining D=diag(p(1)p(2)p(3)), equation (2.14) may be written so that 2.20Note that, in (2.18), the factor (say) has the property (cf. equations (3.8) and (3.10) of ). Moreover, has eigenvectors with eigenvalues p(q), so that , while, by substituting from (2.18) for Q−ρv2I, equation (2.19) gives which is equivalent to (3.11) of .
3. General waveforms and the Stroh formalism
For each q=1,2,3 and each complex-valued function F(q)(k) defined on , two real conjugate harmonic functions ϕ(q)(X,Y) and ψ(q)(X,Y) may be defined through so that they satisfy the Cauchy–Riemann equations 3.1Using these, a general Fourier superposition of each travelling sinusoidal solution Y(q) of (2.13) may be written as which may be put into the form 3.2in terms of the vectors a(q) and b(q) which are the real and imaginary parts of c(q), and moreover Z(q) satisfies (2.9). Thus, Z(q) is written in terms of a pair of harmonic conjugate functions ϕ(q) and ψ(q) in which the arguments are related to x1−vt and x2 through the real and imaginary parts of the complex eigenvalue p(q), with multiplicative factors defined by the corresponding eigenvector.
An alternative confirmation of this representation is to use the eigenvalue property (2.11) in the form Na(q)=α(q)a(q)−β(q)b(q) and Nb(q)=β(q)a(q)+α(q)b(q) after differentiating (3.2) and so to show that, for Y=Z(q), while, since Na(q)=α(q)a(q)−β(q)b(q) and Nb(q)=β(q)a(q)+α(q)b(q), then with all functions evaluated at (X,Y)=(x1−vt+α(q)x2,β(q)x2).
Each of the three solutions Z(q) of (3.2) involves a pair of conjugate harmonic functions ϕ(q)(X,Y) and ψ(q)(X,Y) which may be written in terms of a single harmonic function Φ(q)(X,Y) as and . Then, the general disturbance travelling parallel to Ox1 at any subsonic speed v is a superposition of these and so has a representation in terms of three separate functions Φ(q)(x1−vt+α(q)x2,β(q)x2) each harmonic in Y ≡β(q)x2>0 and decaying as . It then has the form: 3.3and so is a superposition of vector multiples of the three pairs of harmonic functions of appropriate variables x1−vt+α(q)x2 and β(q)x2.
An alternative representation uses three complex-valued functions G(q)(X+iY)=Φ(q)(X,Y)+iΨ(q)(X,Y) which are analytic in Y >0, which decay as and in which . Then, equation (3.3) may be rewritten as 3.4It should be noted that Stroh  implied an equivalent relation for the three displacements and for three stress functions in the case of general subsonic, travelling disturbances, but in treating surface waves restricted attention to complex exponential functions. Observe that, associated with the three analytic functions G(q)(X+iY), the material displacements u(x1,x2,t) which satisfy u,1=p take the form: 3.5Then, if v is chosen so as to make C singular with d a null vector and if the analytic functions G(q)(X+iY) are chosen to be proportional to each other, so that on the real axis they are G(q)(X)=dqG(X) for arbitrary G(X), the surface displacement is given by 3.6while the surface traction vanishes, since Cd=0 in 3.7The corresponding displacement gradient vector at all x2≥0 is and so, on the surface x2=0, may be written as 3.8where , with real part and imaginary part , is the null vector of L=CP−1 and where Φ(X,Y) is the real and Ψ(X,Y) is the imaginary part of the single analytic function G(X+iY). Moreover, G(X+iY) is defined throughout Y >0 by the boundary values of its real part, the harmonic function Φ(X,Y), while is its Hilbert transform .
Correspondingly, the displacement at the boundary is 3.9When the null vector d of C is scaled so that the second component of the null vector of L=CP−1 has the real value +1, the surface elevation has the simple representation 3.10while the other components of displacement are 3.11This shows that all three displacement components at x2=0 are linear combinations of and its Hilbert transform, the linear combinations being determined by the real and imaginary parts of the null vector . Of course, it is also possible to describe u2(x1,0,t) and u3(x1,0,t) in terms of by the substitution . For example, this choice gives 3.12since .
4. Representation of the general travelling wave in terms of real functions
Corresponding to an arbitrary bounded function on there exists a unique bounded, harmonic function Φ(X,Y) in y≥0 having boundary values and which decays as , together with a bounded conjugate harmonic function Ψ(X,Y) satisfying ∂Ψ/∂Y =∂Φ/∂X and ∂Ψ/∂X=−∂Φ/∂Y . The boundary values of Φ and Ψ are related as Hilbert transforms through where denotes the principal-value integral. These are just the identities provided by the Cauchy integral formulae relating the values on the real axis of the real and imaginary parts of an analytic function bounded in Y ≥0 which decays so that as .
For arbitrary , it is possible, without using Fourier superposition, to represent the displacements u(x1,x2,t) within an elastic surface wave travelling at speed v and having normal displacement at the boundary in terms of Φ(X,Y) and Ψ(X,Y). The algebra required in constructing the representation may simply be related to that required within the construction of the standard sinusoidal (complex exponential) solutions , which corresponds to when U2(0)=+1.
By seeking solutions in which the 6-vector Y=(p,t)T of §2 depends only upon x1−vt and x2, it follows that ∂Y/∂x2=N∂Y/∂x1 (i.e. (2.9)). Solutions may then be constructed as a superposition of three special solutions in each of which the two functions ϕ(q) and ψ(q) decay so that , as . Then by requiring that 4.1(which is the eigenvalue problem Nc(q)=p(q)c(q) (2.11) where c(q)=a(q)+ib(q) with p(q)=α(q)+iβ(q) and ) it is found that, for each q=1,2,3, ϕ(q) and ψ(q) may be written in terms of conjugate harmonic functions , of X=x1−vt+α(q)x2 and Y =β(q)x2. Furthermore, it is possible to write , in terms of the conjugate harmonic functions Φ(q)(X,Y) and Ψ(q)(X,Y). By choosing these to be expressed in terms of a single pair of conjugate harmonic functions (Φ(X,Y),Ψ(X,Y)) through 4.2with d+q and the real and imaginary parts of dq defined by the conditions LPd=0 and with , the representation (3.3) for p and t is equivalent to (3.5) in the form 4.3Moreover, this gives the boundary values (equivalent to (3.9)) 4.4and is consistent with , since (Pd)2=+1 and where . It also recovers the relations (3.11).
Although derivation of (4.3) and (4.4) requires some algebra involving complex eigenvalues and eigenvectors, the condition det L=0 which defines the propagation speed v the matrix L may be found directly from (2.18). The complex vectors forming the columns of P and the eigenvalues p(q)≡α(q)+iβ(q) may be found from (2.19) and have β(q)>0 for each q=1,2,3. The dq are just the components of the complex vector d expressing the null vector of L in terms of the .
It may be noted from (4.4) that, in general, the three components of displacement at the boundary are inter-related. The longitudinal and transverse components are combinations of the normal component and its Hilbert transform, with multipliers determined as identically those which arise for the corresponding components within a sinusoidal travelling wave. Indeed, that standard wave solution has , so that the general surface elevation profile and its Hilbert transform are the generalizations in (4.4) of the real and imaginary parts of . Thus, the real vectors and in (4.4) are most readily identified from the standard complex exponential solution.
5. Concluding comments
The Stroh formulation has been extended from the standard treatment of complex exponential elastic waves on uniform anisotropic half-spaces to surface waves in which the elevation profile is arbitrary. Much of the standard treatment readily transfers to this general case, since surface waves on uniform media are non-dispersive (even if wave speed may depend strongly on propagation direction). Using the (complex-valued) transfer matrix L rather than the (related) surface impedance matrix, identifying the speed for which it is singular then constructing its null vector and finding three vectors and values pq from (2.19) allows the general displacement field to be written as (4.3). In this, it is seen that real and imaginary parts usually associated with displacements having the respective factors and are merely the factors required to multiply an arbitrary harmonic function Φ and its harmonic conjugate Ψ, both decaying with x2.
In fact, a more concise derivation follows from the ansatz in terms of a single analytic function G≡Φ+iΨ (i.e. (3.4) with G(q)=dqG) which, when inserted into (2.9), leads to equation (2.11) for each c(q)=c, p=p(q). The traction-free boundary condition t(x1,0,t)=0 then gives Cd=LPd=0 directly. It is clear from this derivation that the fact that all derivatives in (2.9) are first derivatives (i.e. scale invariance) is crucial.
Generalizations of the current treatment to piezo-electric surface waves, Stoneley waves and to Schölte waves are possible. All involve linear partial differential equations and scale-invariant boundary-value problems in one half-space or two adjacent half-spaces. The eigenvalue problems determining the wave speed and deformation field of piezo-electric surface waves were computed in [13,14] in the context of nonlinear evolution effects. The cases both of an electrically earthed boundary and of free space adjoining the material were studied. For Stoneley waves (at the interface between two elastic solids) and Schölte waves (at a solid/fluid interface), the fact that general waves in many directions simultaneously have a simple representation in terms of harmonic functions when materials are isotropic , indicates that, in anisotropic cases, waves propagating in one direction with arbitrary waveform may be treated by the generalized Stroh formalism. The possibility of treating also evanescent cases is suggested by Parker .
The author thanks a referee for the observation that in 1962 Stroh used a representation which could readily have led to these results.
- Received May 9, 2013.
- Accepted August 27, 2013.
- © 2013 The Author(s) Published by the Royal Society. All rights reserved.