Past exam of the mathematics course of the University of Cambridge / 2010 / ii / Paper 2 / 38A (Waves)

Taking the divergence and curl of the elastic equation givesSubstitution of the plane wave gives ${\omega }^{2}A=c_P^{2}k(k\cdot A)-c_S^{2}k\times (k\times A)$. Resolve $A$ parallel and perpendicular to $k$. A P wave has longitudinal polarization and ${\omega }^2=c_P^2|k{|}^2$; an S wave has transverse polarization and ${\omega }^2=c_S^2|k{|}^2$, with two transverse polarizations.

For the Rayleigh wave, take displacement potentials in the $x,z$ plane,where $\alpha =\sqrt{1-c^2/c_P^2}$, $\beta =\sqrt{1-c^2/c_S^2}$. For $0<c<c_S<c_P$, both waves decay into $z<0$. These are an evanescent longitudinal wave and an evanescent vertically polarized shear wave. At $z=0$, the Cauchy stress tensor ${\sigma }_{ij}=\lambda (\nabla \cdot u){\delta }_{ij}+\mu (u_{i,j}+u_{j,i})$ givesThe third traction component is already zero. The two remaining stress-free conditions have a nonzero solution $(A,B)$ exactly when their determinant vanishes:Substituting $\alpha ,\beta$ gives the required Rayleigh wave equation. The nonzero subsonic root supplies a self-sustained surface wave; no existence proof for that root is required. The formal zero-speed root is degenerate and does not describe a propagating wave.