## Abstract

A two-dimensional boundary-value problem for a porous half-space with an open boundary, described by the widely recognized Biot's equations of poroelasticity, is considered. Using complex analysis techniques, a general solution is represented as a superposition of contributions from the four different types of motion corresponding to P1, P2, S and Rayleigh waves. Far-field asymptotic solutions for the bulk modes, as well as near-field numerical results, are investigated. Most notably, this analysis reveals the following: (i) a line traction generates three wave trains corresponding to the bulk modes, so that P1, P2 and S modes emerge from corresponding wave trains at a certain distance from the source, (ii) bulk modes propagating along the plane boundary are subjected to geometric attenuation, which is found quantitatively to be *x*^{−3/2}, similar to the classical results in perfect elasticity theory, (iii) the Rayleigh wave is found to be predominant at the surface in both the near (due to the negation of the P1 and S wave trains) and the far field (due to geometric attenuation of the bulk modes), and (iv) the recovery of the transition to the classical perfect elasticity asymptotic results validates the asymptotics established herein.

## 1. Introduction

Studies of acoustic wave processes in porous media are motivated by a number of applications, for example in the field of seismic prospecting in petrophysics, the area of non-destructive testing of concrete and other porous construction materials, sound-absorbing material acoustics, the testing of surface coating by nanomaterials and medicine. Depending on the major practical application involved, frequency bands may vary greatly. For example, while low-frequency seismic prospecting focuses on frequencies of approximately 50 Hz, medical applications allow for frequencies up to approximately 3 MHz whereas testing of nanomaterials requires frequencies of approximately 100 MHz.

Mathematical approaches to acoustic wave propagation in porous media include the classical Biot's model (Biot 1956*a*,*b*), often reformulated for the particular areas of interest, for example for sound-absorbing materials (Allard 1993), thermodynamics-based theory with a balance equation for porosity (Wilmanski 1996) and a linearized version of the theory of porous media equations (de Boer 2005).

Boundary-value problems for a poroelastic half-space in the framework of Biot's theory have been studied extensively. In particular, the poroelastic Lamb's problem, the counterpart of the Lamb's problem in perfect elasticity (Lamb 1904), was considered in the works of Dey & De (1984), Philippacoupoulos (1988), Seimov *et al*. (1990) and Valiappan *et al*. (1995). The solutions for the case of axial symmetry were discussed in the works of Halpern & Christiano (1986), Seimov *et al*. (1990) and Molotkov (2002*a*,*b*). A closed form Cagniard solution (Aki & Richards 2002) was derived for the transient response problems in the high-frequency limit in the work of Paul (1976) and, for the case of a fluid–porous solid interface, by Feng & Johnson (1983).

Surface or poroelastic Rayleigh waves were studied extensively in a series of works by Deresiewicz, e.g. Deresiewicz & Rice (1962); results for the fluid–porous solid interface can be found in the works of Feng & Johnson (1983; high-frequency limit) and Gubaidullin *et al*. (2005; general case). Purely numerical results can be found, for example, in the works of Mesgouez *et al*. (2005; finite-element formulation) and Schanz & Struckmeier (2005; boundary-element formulation). Asymptotic results for the contact stresses can be found in the work of Gomilko *et al*. (2002).

It is important to distinguish between viscous attenuation due to the viscous interphase interaction and geometric attenuation along a plane boundary. Physically, waves propagating along the plane boundary of the porous half-space consist of superimposed bulk modes (P1, P2 and S waves) and a surface mode, or a poroelastic Rayleigh wave (Deresiewicz & Rice 1962; Seimov *et al*. 1990). While in the presence of dissipation all the waves in porous media exhibit viscous attenuation, it is logical to assume that, similarly to the perfectly elastic case (Lamb 1904), porous bulk modes will also exhibit geometric attenuation.

Unlike the previous studies, the primary aim of the present work is to carry out an in-depth analysis of the comparative contribution of each of the four wave types to general response. Consequently, analysis of the resulting expressions will allow the characterization of the geometric attenuation of the bulk modes in a quantitative manner.

In the present work, a two-dimensional boundary-value problem for a porous half-space, described by the widely recognized Biot's equations of poroelasticity, is considered. In this poroelastic version of Lamb's problem, the surface of a porous half-space is subjected to a prescribed line traction. A formal analytical solution of the problem in the Fourier–Laplace space is obtained by the application of the standard Helmholtz potential decomposition, which reduces the problem to a system of wave-type equations for three unknown potentials. These potentials correspond to two dilatational waves (of the first kind or P1 wave and Biot's slow wave of the second kind or P2 wave, which has no counterpart in elastic wave theory) and one shear wave, or S wave (Biot 1956*a*; Bourbie *et al*. 1987; Seimov *et al*. 1990).

Analysis of the formal solution based on branch cut integration in the complex slowness plane allows the representation of the response at the surface as a superposition of three wave trains, each containing P1, P2, S and Rayleigh waves. While the Rayleigh wave contribution is found in the closed form of a residue, the solution for the three bulk wave trains is given in the form of well-behaved integrals for which far-field asymptotic results are obtained. The properties of the bulk and surface modes propagating along the surface are subsequently discussed in detail.

The remainder of the present article is organized as follows: §2 contains a detailed introduction of Biot's formalism, gives necessary preliminaries and establishes the notation; §3 provides the derivation of the formal solution of the initial-value Lamb's problem, often referred to as Green's function; however, it is only used further to obtain the solution for a simpler harmonic line traction problem; §4 suggests the inversion of the obtained formal solution into physical space based on branch cut integration, which leads to the desired decomposition of the signal into contributions related to different wave types; subsequently, §5 provides asymptotic results for each wave train. Finally, the results are summarized in §6.

## 2. Model description

### (a) Governing equations

Consider Biot's set of equations for a full frequency range (Biot 1956*b*),(2.1)where ** u** and

**represent unknown solid and fluid displacement fields, respectively;**

*U**ρ*

_{ij}denote reference phase densities, which can be expressed in terms of the solid matrix density

*ρ*

_{s}, the saturating fluid density

*ρ*

_{f}, the tortuosity parameter

*a*and the porosity

*ϕ*as follows:

*ρ*

_{11}=(1−

*ϕ*)

*ρ*

_{s}+

*ϕρ*

_{f}(

*a*−1),

*ρ*

_{12}=

*ϕρ*

_{f}(1−

*a*),

*ρ*

_{22}=

*aϕρ*

_{f}; the damping factor

*b*is given by

*b*=

*ϕ*

^{2}

*η*/

*k*, where

*k*,

*η*denote the permeability of the medium and fluid viscosity, respectively; and

*λ*,

*Q*and

*R*are generalized poroelastic parameters which can be related to the porosity

*ϕ*, the bulk modulus of the solid

*K*

_{s}, the bulk modulus of the fluid

*K*

_{f}, the bulk modulus of the porous drained matrix

*K*

_{b}and the shear modulus

*μ*of both the drained matrix and the composite (Biot & Willis 1957):

*λ*=

*K*

_{b}−2

*μ*/3+

*K*

_{f}(1−

*ϕ*−

*K*

_{b}/

*K*

_{s})

^{2}/

*ϕ*

_{eff},

*Q*=

*ϕK*

_{f}(1−

*ϕ*−

*K*

_{b}/

*K*

_{s})/

*ϕ*

_{eff},

*R*=

*ϕ*

^{2}

*K*

_{f}/

*ϕ*

_{eff},

*ϕ*

_{eff}=

*ϕ*+

*K*

_{f}(1−

*ϕ*−

*K*

_{b}/

*K*

_{s})/

*K*

_{s}.

An expression for the stress tensor *σ*_{ij} and the pore pressure *p*_{f} can be written down in terms of the components of a Cauchy strain tensor and ∇.** u**, ∇.

**,(2.2)**

*U*(2.3)

The characteristic (or roll-over) frequency *f*_{c} is defined as (Biot 1956*a*)(2.4)The normalization frequency used in the present work is defined similarly as(2.5)

The frequency correction factor (see Biot (1956*b*) and Johnson *et al*. (1987) for discussion) is neglected in the following for simplicity. For the particular results of the present work, the adjustments due to the introduction of the correction factor were found not to be significant.

### (b) Helmholtz potential decomposition

Expansion of the displacement field into irrotational and solenoidal parts yields(2.6)and results in the following scalar and vector set of equations in Laplace space (Seimov *et al*. 1990):(2.7)

The first two equations in (2.7) may be rewritten in the matrix form(2.8)where is the inverse of the rigidity matrixand the components of mass matrix are given by(2.9)

#### (i) Dilatational waves (P waves)

It can be shown that with a similarity transformation of the matrix the equation (2.8) decouples into two wave equations in an eigenvector reference system(2.10)where satisfy the following quadratic equation (Biot 1956*a*):(2.11)with non-dimensional parameters defined as(2.12)

The above equations describe P1- and P2-wave behaviour, respectively, with phase velocities *β*_{1,2} given by(2.13)The P1 wave corresponds to the case when solid and liquid displacements are in phase, while the P2 wave describes out-of-phase motion (Biot 1956*a*; Bourbie *et al*. 1987). Moreover, waves of the first kind propagate faster and attenuate much slower than the wave of the second kind. The connection between the reference systems is given by the eigenvector matrix (2.14)where the components *M*_{1,2} can be found straightforwardly as(2.15)so that finally(2.16)

#### (ii) Shear waves (S waves)

The last two equations in (2.7) can be rewritten using Biot's non-dimensional parameters as(2.17)(2.18)where

So that, finally, we arrive at the wave equation(2.19)which defines the shear wave phase velocity *β*_{3} in the following way:(2.20)

## 3. Green's functions

### (a) General Laplace–Fourier solution

Consider a poroelastic half-space with an open boundary occupying the region *z*>0. At time *t*=0 the porous half-space is subjected to an impulsive external line traction −*Pδ*(*x*)*δ*(*t*) at the surface (instantaneous compression). The boundary conditions for the governing equations (2.1) and stress–strain relation can be represented in the following way (*z*=0):(3.1)

In the case of a two-dimensional problem, introduction of the three scalar potentials *Φ*_{1}, *Φ*_{2} and *Ψ*_{s} (*Ψ*_{1}=(0, *Ψ*_{s}, 0), *Ψ*_{2}=−*M*_{3}*Ψ*_{1}) is sufficient, so that(3.2)and thus the two-dimensional problem reduces to the solution of the wave equations in Laplace space(3.3)Equations (3.3), written down in Fourier space, become (here and henceforth, transformed solutions will be indicated by the arguments)(3.4)so that, taking into account far-field conditions, solutions of the above wave equations can be expressed in the form(3.5)where (*i*=1, 2, 3), with Re *ξ*_{i}>0 to satisfy radiation conditions; and *A*_{1,2}(*k*, *s*) and *B*(*k*, *s*) are unknown coefficients to be determined from the boundary conditions (3.1).

In the Laplace–Fourier space, the expressions for the stress tensor and the pressure in terms of the potentials (3.3) can be written down as(3.6)

Application of the boundary conditions (3.1) to the expressions (3.6) gives a linear algebraic system to determine the three unknown coefficients *A*_{1,2}(*k*, *s*) and *B*(*k*, *s*) as(3.7)whereand *F*(*k*, *s*) is the dispersion relation of the surface Rayleigh waves (Deresiewicz & Rice 1962; Seimov *et al*. 1990)(3.8)

Consider, as an example, the vertical component of solid displacement, *u*_{z}. As follows from (3.2):(3.9)so that using (3.5) and (3.7) the above expression becomes(3.10)and finally one can get the solution in the Laplace space(3.11)

(3.12)

(3.13)

Similarly, it is possible to obtain the exact analytical solutions for the stress tensor components, as well as the horizontal components of the displacement fields.

Solution of Lamb's problem for the perfectly elastic medium can be used as a benchmark solution, in the sense that one can show that the limiting case of the solution (3.11) recovers the analogous perfectly elastic case (Lamb 1904; Seimov *et al*. (1990); Graf 1991). Detailed discussion of the limiting case will be followed in §5*b*.

### (b) Harmonic line traction. Formal solution

Despite the different form of the time dependence, the solution (3.11) can be used to derive the solution for a harmonic line source. In this particular case, we assume harmonic time dependence for the displacements as well as for the components of the total stress tensor and pore pressure. Thus, the first equation in (3.1) reads: and one gets, for example, the following expression for the normal solid-phase displacement at the surface (*z*=0) in the physical domain:(3.14)where (*i*=1, 2, 3) and *F*(*k*, *ω*) is obtained from the Rayleigh wave secular equation *F*(*k*, *s*) (3.8) by the substitution *s*=i*ω*.

The change of variable *k*=*ωp*/*β*_{S} is made in (3.14), where *p* is the non-dimensional slowness parameter (Lamb 1904; Aki & Richards 2002) and(3.15)represents the high-frequency S-wave phase speed limit. The introduction of the following non-dimensional quantities:(3.16)leads to expressions for the non-dimensional vertical displacements, evaluated at the surface, and pore pressure, in the interior of the half-space. Henceforth tildes are omitted, and quantities discussed are assumed to be non-dimensional unless otherwise indicated, as well as the factor e^{it}, so that, without loss of generality, time-independent integrals are considered(3.17)(3.18)(3.19)where (*i*=1, 2, 3), *F*(*p*, *ω*) is the non-dimensionalized Rayleigh wave equation (3.8)(3.20)

The above multivalued integrals represent the formal solution of the problem, thus evaluation must be carried out taking into account far-field conditions , which define the values of the integral unambiguously.

### (c) Numerical results

Numerical results for the vertical solid and fluid displacements according to (3.17)–(3.19) are presented in figures 1–3 for the case of water-saturated Berea sandstone (table 1). The following dimensional frequencies are used in calculations: *ω*=*ω*_{c}, 10*ω*_{c}, 100*ω*_{c}. Further decrease or increase of the source frequency gives results similar to figures 1 and 3, respectively.

While straightforward numerical evaluation of the integral (3.19) does not encounter any computational difficulty as the integrand decays exponentially for large values of *p*, evaluation of the expressions (3.17) and (3.18) requires special care for sufficiently large values of *x*, as the results for the displacements at the surface are represented in the form of slowly decaying and highly oscillating, though convergent, integrals when *x*≠0. At the point *x*=0, where the traction is applied, an integrable singularity is present, which disappears, for example, in the case of the uniformly distributed stripe load. Indeed, one can show that the asymptotics of the integrands in (3.17), (3.18) takes the form cos(*px*)/*p* as *p*→∞, so that the convergence of the above integrals follows from the convergence of the integral cosine function *Ci*(*x*) (e.g. Abramovich & Stegun 1974).

Alternatively, the numerical evaluation of the above integrals can be conducted using branch cut integration (see §4*a*), in which case the resulting integrals along the hyperbolic branch cuts pose no computational complications. Results of the straightforward numerical integration are obtained for the moderate values of *x*. Despite the above-mentioned disadvantages of this approach, including the error introduced while we bound an improper integral, both approaches have been confirmed to yield matching results for the *x* values employed herein.

Numerical results shown in figures 1–3 illustrate the influence of the source frequency on the character of the spatial oscillations in both displacements and pore pressure. For relatively low frequencies, displacements are observed to be almost in phase, with approximately the same amplitudes (figure 1), while an increase of the source frequency leads to a weakening of the viscous coupling effect and, as a consequence, solid and fluid displacements can be of different amplitude and phase (figures 2 and 3) or, in fact, nearly out of phase in the high-frequency range for certain materials. Indeed, as follows from the governing equations, in the case when the characteristic frequency lies near unity, the inertia and viscous terms are approximately of the same order (Biot 1956*b*), so that an increase of the source frequency makes inertial terms dominant over viscous terms.

While these numerical results are only the first step and serve to illustrate the response at different frequencies, a more intriguing task is to decompose the obtained general solution into components related to the four different wave types. This can be achieved by means of contour integration in the upper complex slowness half-plane, so that the numerical results (for moderate values of *x*) presented in figures 1–3 will serve as a benchmark solution.

## 4. Poroelastic acoustic wave trains

### (a) Branch cut integration

The integrands in (3.17)–(3.19) contain six branch points located at *p*=±*β*_{1}, ±*β*_{2}, ±*β*_{3}, such that Im(*β*_{i})≠0, when *ω*≠0, and two poles at *p*=±*p*_{R}, Re(*p*_{R})>0, satisfying *F*(*p*_{R}, *ω*)=0, which correspond to the Rayleigh wave contribution. The necessary hyperbolic branch cuts (figure 4) in the complex *p*-plane (*p*=*ζ*+i*η*) can be selected according to(4.1)where *α*_{i}=Re(*β*_{i}), *λ*_{i}=−Im(*β*_{i})>0 (some discussion of hyperbolic branch cuts can be found in Graf (1991)).

Consider in detail, for example, the integral expression for the solid-phase vertical displacement component *u*_{z} (3.17)(4.2)

Conducting contour integration according to the scheme shown in figure 4, we note that for *x*>0 the closure is in the upper half-plane, while for *x*<0 one should consider the lower half-plane. Because the contribution of the integral along the semicircle of infinite radius vanishes according to Jordan's lemma, the above integral can be represented in the following way:(4.3)where the Rayleigh pole, *p*=−*p*_{R}, always has a positive imaginary part when *ω*≠0 and the values of the radicals along , and are prescribed uniquely according to the far-field conditions as follows: along and −*ξ*_{i}(*p*) along (*i*=1, 2, 3).

Thus, introducing the parametrization(4.4)one arrives at (4.5)where the residue and three contour integrals can be found as(4.6)(4.7)(4.8)(4.9)where

### (b) Wave trains

Now the solution for *u*_{z} (3.17) is represented in the form of a sum of contributions of three wave trains (4.7)–(4.9) and, obviously, the contribution of the Rayleigh wave (4.6), where the location of the pole (*p*_{R}) should be first found numerically to evaluate an expression for the residue. Summarizing the results obtained with the branch cut integration, one can write the following decomposition for the displacement field:(4.10)where , and are the expressions of the type (4.7)–(4.9), taken with opposite sign, and *R* denotes the contribution of the Rayleigh wave (4.6).

Similarly, representations in the form (4.10) can be derived for the rest of the components (though these cumbersome expressions are not provided explicitly, the results for the fluid phase *U*_{z} will be used further for numerical evaluation). Integrals of the type (4.7)–(4.9) pose no computational difficulties for numerical evaluation as their integrands do not contain any singularities; the integration path is limited, moreover the integrands take zero values at the endpoints.

Numerical results illustrating the above introduced decomposition are presented in figure 5 for dimensional frequency *ω*=100*ω*_{c}. Each wave train (4.7)–(4.9) represents the waves propagating with the phase slownesses and attenuations in the range [−*α*_{i},0], [*λ*_{i},∞), respectively, so that each wave train contains P1, P2 and S waves, and the waves in the wave trains propagate faster than the corresponding bulk mode. Moreover, these waves are also viscously attenuated at a faster rate, so that at a certain distance from the source one can expect solely P1, P2 and S modes along with the predominant Rayleigh wave.

While detailed discussion of the above results will follow in §6, we will next pursue the asymptotics of integrals of the type (4.7)–(4.9). The evolution of the wave trains in the far field and the emergence of the P1, P2 and S waves will be subsequently discussed in §5, where asymptotic results for each wave train are sought.

## 5. Asymptotic solutions

### (a) Oscillatory Laplace-type integrals

Integrals of the type (4.7)–(4.9)(5.1)with an appropriate change of variable can be approximated by the following model integral for sufficiently large *x*:(5.2)where the oscillatory term in (5.2) is approximated by the first two terms in the expansion of the expression around .

The integral in the expression (5.2) is an oscillating Laplace-type integral of the form (A 1), discussed in the appendix A (equation (A 3)), and satisfies the conditions of Watson's lemma reformulated for this particular type of integral (see appendix A). An expression for the leading-order term follows from the general asymptotic expansion (A 3):(5.3)so that, in order to determine the unknown terms *γ* and *a*_{0}, it is necessary to investigate the asymptotic behaviour of the function as or, in other words, the behaviour of the integrands (4.7)–(4.9) as .

It is easy to see that simple manipulations with the integrands allow the factoring out of the terms *ξ*_{i}(*p*_{i}) which completely determine the asymptotic behaviour as *τ*→−*α*_{i}. For example, in (4.7), the expression in the square brackets equals(5.4)The factor *ξ*_{i}(*p*_{i}) in some vicinity of *τ*=−*α*_{i} can be approximated by(5.5)or in terms of as(5.6)

Taking into account (5.2) and (5.3), the general asymptotic results for all three integrals can be summarized in the form(5.7)

It is convenient to introduce an upper index to denote the phase, while the lower index corresponds to the wave type. Thus and will denote the value of coefficient _{i} for the fluid and solid phases, respectively. These coefficients are provided for the solid-phase displacements in appendix B.

Figure 6 represents an example of the exact and asymptotic solutions for the P1 and S wave trains, and , respectively (vertical solid-phase displacement is considered). Asymptotic solution appears to be accurate and in phase with the numerically calculated exact solution. Moreover, comparison of these illustrates the process of emerging of the wave from the corresponding wave train.

### (b) Limiting case. Benchmark solution

It should be mentioned that the far-field solution of the form (4.10), (5.7) in the appropriate limiting case, exactly recovers classical asymptotic results, first derived by Lamb (1904), for the analogous elasticity problem.

Indeed, the transition to perfect elasticity (see, for example Bourbie *et al*. 1987) follows from the vanishing of the poroelastic parameters: *Q*→0; *R*→0; and *ϕ*→0, so that *ρ*→*ρ*_{s}, *ρ*_{12}→0, *ρ*_{22}→0, *n*_{12}→0, *m*_{12}→1/2, *β*_{1}=*β*_{2}, *β*_{3}=1 and *λ*_{i}→0.

The solution for the far-field bulk modes for the vertical component of the solid-phase displacement, *u*_{z}, in dimensional form in accordance with (4.10), (5.7) is given by (tildes are reintroduced to indicate non-dimensional quantities)(5.8)Using and the values of the coefficients provided in appendix B one getswhere , and *c*_{p}, *c*_{s}, are longitudinal and transverse phase velocities, respectively. Introducing *h*=*ω*/*c*_{p} and *k*=*ω*/*c*_{s}, so that and , one arrives at the following expression:(5.9)The values of the coefficients in the limiting case can be found to be(5.10)so that, finally, using Lamb's (1904) original notation(5.11)

## 6. Summary and final remarks

The present article is dedicated to the detailed investigation of the wave processes at the boundary of a porous half-space subjected to a harmonic line traction in the framework of Biot's theory. Formal analytical and closed form far-field asymptotic solutions have been established herein, and these allow analysis of the response at different source frequencies, as well as the investigation of the basic properties of each wave type travelling along the plane boundary.

Examples of numerical results have been shown (figures 1–3) and these illustrate the influence of the source frequency on the character of the spatial oscillations in both displacements and pore pressure. These may be summarized as showing a weakening of the viscous coupling between phases as frequency is increased, with higher frequency regimes leading to oscillations that are of different amplitude and phase in the fluid and solid. As has been pointed out by Biot (1956*b*), this follows from the governing equations. Indeed, in the case when the characteristic frequency lies near unity, the inertia and viscous terms are approximately of the same order, so that a further increase of the source frequency makes inertial terms dominant over viscous terms.

An oscillating line source generates three bulk wave trains, containing P1, P2 and S waves, respectively, and a surface poroelastic Rayleigh wave. Decomposition of the formal general solution into contributions of the four wave types reveals the following: bulk modes propagating along the surface exhibit *x*^{−3/2} attenuation in addition to attenuation due to viscous interphase interactions (terms of the form in expression (5.7)). Results for the spatial attenuation are found to be similar to those known from the classical elastic wave theory (Lamb 1904). Moreover, the classical elastic wave theory asymptotic results can be exactly recovered in the appropriate limiting case from the asymptotic solution obtained herein for the poroelastic waves.

Analysis of the formal solution shows that the waves in the bulk wave trains consist of waves that propagate faster and also exhibit greater viscous attenuation than the corresponding bulk modes. Thus, at a certain distance from the source one can observe solely P1, P2 and S modes or, in other words, the emergence of these modes from the corresponding wave trains (figure 6).

It is known that the P2 effect is difficult to measure at low frequencies (Nagy 1999). Indeed, owing to the rapid viscous attenuation of the slow P2 wave, high-frequency oscillations are necessary to capture this effect. In the present work, relatively high dimensional frequency *ω*=100*ω*_{c} was used to emphasize the P2-wave contribution. In this particular case, the P2 effect is observed only in the vicinity of the source and is found to be more pronounced in the fluid phase (figure 5).

Poroelastic Rayleigh waves are found to be predominant at the surface in both the near field, due to the negation of the P1 and S wave trains, and the far field, due to the geometric attenuation of the bulk modes. Thus, the frequency-dependent character of spatial oscillations and relative vertical fluid–solid motion, or flux, at the surface will mostly be determined by the properties of the poroelastic Rayleigh waves in both the solid and the fluid phases. Taking advantage of the closed-form solution for the Rayleigh waves, some of their basic properties have been investigated. In particular, it is found that, unlike the wave in the solid phase, the amplitude–frequency variation of the Rayleigh wave in the fluid phase is more rapid and exhibits a distinct minimum (figure 6).

Finally, solutions established herein can be used further in the analysis of the influence of the frequency correction factor (Biot 1956*b*; Johnson *et al*. 1987), which is left beyond the scope of the present work. Moreover, these results provide the necessary foundation to explore the questions related to the energy characteristics of different types of poroelastic acoustic waves.

## Footnotes

- Received June 25, 2007.
- Accepted November 7, 2007.

- © 2007 The Royal Society