The Elastic Wave Equation
Seismic wave propagation in a homogeneous, isotropic elastic medium is described by
\[\nabla^2 \mathbf{u} = \frac{1}{v^2}\frac{\partial^2 \mathbf{u}}{\partial t^2}\]where $\mathbf{u}$ is the displacement field and $v$ is the wave speed. Two independent wave types emerge from the Helmholtz decomposition of $\mathbf{u}$ into scalar and vector potentials:
| Wave type | Motion | Velocity |
|---|---|---|
| P (compressional) | Parallel to propagation | $v_P = \sqrt{(\lambda + 2\mu)/\rho}$ |
| S (shear) | Perpendicular to propagation | $v_S = \sqrt{\mu/\rho}$ |
Here $\lambda$ and $\mu$ are the Lamé parameters and $\rho$ is density. Since $\mu > 0$ always, $v_P > v_S$.
Snell’s Law and Critical Angles
At an interface between two media, the angle of incidence $\theta_1$ and transmission $\theta_2$ satisfy
\[\frac{\sin\theta_1}{v_1} = \frac{\sin\theta_2}{v_2} = p\]where $p$ is the ray parameter, conserved along any ray path. Total internal reflection occurs when $\theta_2 = 90°$, giving the critical angle $\theta_c = \sin^{-1}(v_1/v_2)$. Head waves (refractions) travel along the interface at $v_2$ and are the basis of seismic refraction surveys.
Surface Waves
Surface waves are confined near a free surface and decay exponentially with depth. Love waves (SH particle motion, dispersive) and Rayleigh waves (elliptical retrograde motion) travel at speeds below $v_S$. Their dispersion relation links phase velocity $c(\omega)$ to depth-averaged shear modulus, making them ideal for near-surface $v_S$ profiling.