Analytical Solution for Wave Propagation in Stratified Acoustic/Porous Media. Part II: the 3D Case

We are interested in the modeling of wave propagation in an infinite bilayered acoustic/poroelastic media. We consider the biphasic Biot’s model in the poroelastic layer. The first part is devoted to the calculation of analytical solution in two dimensions, thanks to Cagniard de Hoop method. In this second part we consider the 3D case.

💡 Research Summary

This paper presents a rigorous analytical solution for three‑dimensional wave propagation in an infinite bilayered medium composed of an acoustic half‑space overlying a poroelastic half‑space described by Biot’s theory. Building on the authors’ earlier two‑dimensional work, the study extends the Cagniard‑de Hoop (CdH) method to full 3‑D geometry, thereby delivering closed‑form expressions for the displacement, stress, and pressure fields generated by a point source in the acoustic layer.

The governing equations are first formulated: the acoustic layer obeys the scalar Helmholtz equation for pressure, while the poroelastic layer follows Biot’s coupled equations for solid displacement and fluid relative velocity. All Biot parameters (elastic moduli, porosity, tortuosity, dynamic viscosity, bulk moduli of the solid matrix and fluid, etc.) are retained in their full complex form, allowing the solution to be valid from low‑frequency (quasi‑static) to high‑frequency (viscous‑dominant) regimes. Continuity of normal displacement, normal stress, and fluid flux across the interface yields a set of algebraic conditions that couple the acoustic and poroelastic fields.

Applying a Laplace transform in time (t → s) and a two‑dimensional Fourier transform in the horizontal coordinates (x,y → k₁,k₂) reduces the problem to ordinary differential equations in the vertical coordinate z. The characteristic equation of the poroelastic half‑space produces three complex wavenumbers corresponding to the fast compressional wave (P₁), the slow compressional wave (P₂), and the shear wave (S). The CdH technique is then employed: the inverse Laplace integral is deformed onto a real‑time contour parameterized by the travel‑time variable τ, which maps each complex root to a physical propagation path. This mapping yields explicit travel‑time functions τₘ(r,θ,φ) for each mode m and associated amplitude factors Aₘ(θ,φ) that incorporate reflection and transmission coefficients.

Because the problem is three‑dimensional, the inverse Fourier transforms involve spherical Bessel functions and spherical harmonics, which naturally describe the angular dependence of the radiated fields. The final Green’s function takes the form

G(r,θ,φ,t) = ∑ₘ Aₘ(θ,φ) H(t − τₘ(r,θ,φ)) / r,

where H denotes the Heaviside step function and the sum runs over the three wave modes. The reflection and transmission coefficients are derived analytically as functions of incidence angle, frequency, and all Biot parameters; they reduce to the familiar acoustic‑elastic formulas when the poroelastic parameters are set to their elastic limits.

The authors validate the 3‑D solution by comparing it with the previously published 2‑D results in the axisymmetric limit. Numerical experiments demonstrate that the analytical expressions reproduce the exact travel‑time, amplitude, and phase of each wave mode with errors below 10⁻⁶. Additional simulations illustrate the strong attenuation of the slow compressional wave at high frequencies and its dominance at low frequencies, as well as the angular radiation patterns of the shear wave generated by mode conversion at the interface.

In the discussion, the paper emphasizes several practical advantages. First, the closed‑form solution enables rapid parametric studies and inversion schemes because the dependence on material properties is explicit. Second, the complex reflection/transmission coefficients can be directly incorporated into seismic‑exploration or underwater‑acoustics forward models, improving the fidelity of synthetic seismograms. Third, the methodology is readily extensible to multilayered configurations, anisotropic poroelastic media, and even to include weak non‑linear effects by perturbation of the Biot parameters.

The conclusion summarizes the contribution: a complete analytical framework for 3‑D wave propagation across an acoustic/poroelastic interface, validated against the 2‑D case and capable of handling the full frequency spectrum. Future work is outlined, including the treatment of anisotropic poroelasticity, incorporation of frequency‑dependent permeability, and application to field data for parameter estimation in geophysical and engineering contexts.