= Surface sources of a discontinuous Lighthill stress tensor
{title2=$\partial_i\partial_jT_{ij}$}
Write a piecewise smooth <Lighthill stress tensor> as $T_{ij}=T^-_{ij}+D_{ij}H(S)$, where $D_{ij}=T^+_{ij}-T^-_{ij}$ and the regular level surface $S=0$ is a <shock wave>. The <distributional derivative of the Heaviside step function> gives
$$
\partial_i\partial_jT_{ij}=H(S)\partial_i\partial_jT^+_{ij}+H(-S)\partial_i\partial_jT^-_{ij}+(D_{ij,i}S_j+D_{ij,j}S_i+D_{ij}S_{ij})\delta(S)+D_{ij}S_iS_j\delta'(S).
$$
Here repeated indices are summed, $S_i=\partial_iS$ and $S_{ij}=\partial_i\partial_jS$. Keep the coefficients as smooth extensions before multiplying distributions: prematurely replacing the coefficient of $\delta'(S)$ by its surface trace can lose a $\delta(S)$ contribution. The extra <surface delta distributions> and their derivatives are supported entirely on the discontinuity.
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