= Solution
Apply the wave ansatz to the oscillatory <displacement> about a static body-force balance, or equivalently to the zero-body-force homogeneous equation. The parallel and perpendicular polarizations have speeds
$$
\boxed{c_p=\sqrt{\frac{\lambda+2\mu}{\rho}},\qquad
c_s=\sqrt{\frac\mu\rho}.}
$$
The first is longitudinal and the second transverse. A wave family is dispersive when its phase speed $\omega/|\mathbf k|$ varies with <wavenumber>. In each of these homogeneous bulk families, $\omega=c|\mathbf k|$, so \b[both are nondispersive] and group and phase speeds coincide.
The positivity assumptions imply $\lambda>-2\mu/3$, hence $\lambda+\mu>\mu/3>0$. Therefore $c_p^2-c_s^2=(\lambda+\mu)/\rho>0$: \b[<longitudinal waves> travel faster].
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