= P-wavefront discontinuity transport
{c}
{title2=$\nabla\cdot[\rho\alpha(a^{[k]})^2\mathbf n]=0$}
Let $t=T(\mathbf x)$ be a smooth <wavefront> and suppose the first jump in time <derivatives> of <displacement field> is $[\partial_t^k\mathbf u]=a^{[k]}\mathbf n$, with $k\geq2$. Substitution of $\mathbf u=a^{[k]}\mathbf n(t-T)_+^k/k!+\cdots$ into the elastic equation first gives the <eikonal equation> $|\nabla T|=1/\alpha$ and longitudinal polarization $\mathbf n=\alpha\nabla T$. Projecting the next-order compatibility condition onto $\mathbf n$ gives $2\rho\alpha\mathbf n\cdot\nabla a^{[k]}+a^{[k]}\nabla\cdot(\rho\alpha\mathbf n)=0$, hence the displayed conservation law. Coefficients and <wavefront> must be sufficiently differentiable for this classical transport equation; its local form fails at a <caustic>.
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