Lamb waves

Definition

Let a homogeneous isotropic linear elastic plate occupy $\set{x \in \R^3 : \abs{x_3} < h}$, with P-wave speed $c_P$ and S-wave speed $c_S$, and with both faces $x_3 = \pm h$ free of traction. A Lamb wave is a nonzero elastodynamic solution without body force of the form $$u(x, t) = \operatorname{Re} \left[ \left( U_1(x_3), \, 0, \, U_3(x_3) \right) e^{i (k x_1 - \omega t)} \right],$$ with wave number $k$ and angular frequency $\omega$. Write $p^2 = \omega^2 / c_P^2 - k^2$ and $q^2 = \omega^2 / c_S^2 - k^2$. Such solutions exist exactly when $(k, \omega)$ satisfies one of the Rayleigh–Lamb equations:

1. symmetric modes, with $U_1$ even and $U_3$ odd in $x_3$: $$\frac{\tan(q h)}{\tan(p h)} = -\frac{4 k^2 p q}{(q^2 - k^2)^2};$$
2. antisymmetric modes, with $U_1$ odd and $U_3$ even in $x_3$: $$\frac{\tan(q h)}{\tan(p h)} = -\frac{(q^2 - k^2)^2}{4 k^2 p q}.$$