= Solution
Let $D_{ij}=T^+_{ij}-T^-_{ij}$, so the <Lighthill stress tensor> is $T^-_{ij}+D_{ij}H(S)$. Write $S_i=\partial_iS$ and $S_{ij}=\partial_i\partial_jS$. The <distributional derivative of the Heaviside step function> gives $\partial_iH(S)=S_i\delta(S)$. Applying the <product rule> a second time yields the <surface sources of a discontinuous Lighthill stress tensor>:
$$
\begin{aligned}
\partial_i\partial_jT_{ij}={}&H(S)\partial_i\partial_jT^+_{ij}+H(-S)\partial_i\partial_jT^-_{ij}\\
&+\bigl(D_{ij,i}S_j+D_{ij,j}S_i+D_{ij}S_{ij}\bigr)\delta(S)
+D_{ij}S_iS_j\delta'(S).
\end{aligned}
$$
Repeated indices are summed. The first line gives the two bulk <acoustic quadrupole> distributions. The second line consists of additional <surface delta distributions> and their derivatives, all supported on the <shock wave> $S=0$. For a symmetric <Lighthill stress tensor>, the first two coefficients of $\delta(S)$ combine to $2D_{ij,i}S_j$ after relabelling indices.
\b[A discontinuity therefore adds both a surface-delta source and a surface-delta-derivative source.] The level function need not be a signed distance: retaining its derivatives makes the expression invariant under a smooth reparametrization of the same surface. If the two bulk fields are only continuously differentiable, their second derivatives in the first line are read weakly; no classical second derivative is being assumed. The coefficients multiplying $\delta'(S)$ are kept as extensions, not replaced prematurely by traces.
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