Rayleigh wave

Definition

Let a homogeneous isotropic linear elastic solid fill the half-space $\set{x \in \R^3 : x_3 < 0}$, with density $\rho$, Lamé parameters $\lambda, \mu$, and P-wave and S-wave speeds $c_P > c_S > 0$. A Rayleigh wave is a solution of the elastodynamic equations without body force of the form $$u(x, t) = \operatorname{Re} \left[ \left( U_1(x_3), \, 0, \, U_3(x_3) \right) e^{i k (x_1 - c t)} \right], \qquad k > 0,$$ such that:

1. $U_1(x_3) \to 0$ and $U_3(x_3) \to 0$ as $x_3 \to -\infty$,
2. the surface $x_3 = 0$ is free of traction, $\sigma_{13} = \sigma_{33} = 0$ there, and
3. $(U_1, U_3)$ is not identically zero.

Such solutions exist exactly when $c = c_R$, the unique root in $(0, c_S)$ of the Rayleigh equation $$\left( 2 - \frac{c^2}{c_S^2} \right)^2 = 4 \sqrt{1 - \frac{c^2}{c_P^2}} \sqrt{1 - \frac{c^2}{c_S^2}}.$$