Dispersion of surface waves

Remarks

Example: a layer of thickness $h$, density $\rho_1$ and S-speed $\beta_1$ over a half-space with $\rho_2$ and $\beta_2 > \beta_1$ carries Love waves with $\beta_1 < c < \beta_2$. With shear moduli $\mu_j = \rho_j \beta_j^2$, the dispersion relation is $$\tan\Big(kh\sqrt{c^2/\beta_1^2 - 1}\Big) = \frac{\mu_2 \sqrt{1 - c^2/\beta_2^2}}{\mu_1 \sqrt{c^2/\beta_1^2 - 1}}.$$ As $kh \to \infty$ the fundamental branch has $c \to \beta_1$, and as $kh \to 0$ it has $c \to \beta_2$, so the waves are dispersive. Non-example: a homogeneous half-space carries a Rayleigh wave whose speed is a fixed number below $\beta$, about $0.92\beta$ when $\lambda = \mu$. The medium has no length scale, so $c$ cannot depend on $k$, and the Rayleigh wave is not dispersive.