Elastic wave in an isotropic solid
= Elastic wave in an isotropic solid
For density $\rho$ and <Lamé parameters> $\lambda,\mu$, the displacement satisfies
$$
\rho\mathbf u_{tt}
=(\lambda+\mu)\nabla(\nabla\cdot\mathbf u)+\mu\nabla^2\mathbf u.
$$
Its longitudinal and transverse wave speeds are
$$
c_P=\sqrt{\frac{\lambda+2\mu}{\rho}},
\qquad
c_S=\sqrt{\frac{\mu}{\rho}}.
$$