Solution

Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 78 / 4 / i / Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 2 / 40C / a / Solution

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
$$
c_P=\sqrt{\frac{\lambda+2\mu}{\rho}},
\qquad
c_S=\sqrt{\frac{\mu}{\rho}}.
$$

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 Incoming links: Elastic wave in an isotropic solid