 Solution

Substitution gives the displayed relation with $c_P^2=(\lambda +2\mu )/\rho$ and $c_S^2=\mu /\rho$. Resolving $A$ parallel and perpendicular to $k$ gives a longitudinal P mode of speed $c_P$ and two transverse polarizations of speed $c_S$, conventionally SV in and SH normal to the propagation plane.