= Solution
Apply the <moving-interface conservation jump identity> first to mass, then to each momentum component. Write $j_i=\rho u_i$ and $\Pi_{ij}=\rho u_iu_j+p\delta_{ij}-\sigma_{ij}$, with all quantities understood piecewise on the two sides. Define the jumps of flux relative to the moving <shock wave> by
$$
Q=[\rho(\mathbf u-\mathbf v)\cdot\mathbf n],\qquad L_i=[\rho u_i(\mathbf u-\mathbf v)\cdot\mathbf n+p n_i-\sigma_{ij}n_j].
$$
The global distributional conservation equations are
$$
\partial_t\rho+\partial_ij_i=Q\delta_s,\qquad\partial_tj_i+\partial_j\Pi_{ij}=L_i\delta_s.
$$
Differentiate the first in time and subtract the divergence of the second. With $\rho'=\rho-\rho_0$ and the piecewise <Lighthill stress tensor> $T_{ij}=\rho u_iu_j+(p'-c_0^2\rho')\delta_{ij}-\sigma_{ij}$, this gives
$$
\boxed{(\partial_t^2-c_0^2\nabla^2)\rho'=\partial_i\partial_jT_{ij}+\partial_t(Q\delta_s)-\partial_i(L_i\delta_s).}
$$
Derivatives act on the complete distributions, including their moving support. The <acoustic quadrupole> term represents momentum-stress fluctuations throughout the volume. The time derivative of the surface mass-flux defect is an <acoustic monopole>, representing injection or removal of mass/volume. The divergence of the surface momentum-flux defect is an <acoustic dipole>, representing a force sheet. This is the <distributional acoustic analogy across a moving interface>.
For an actual freely propagating fluid <shock wave> with no singular mass or momentum supply, the <Rankine-Hugoniot conditions> give \b[$Q=0$ and $L_i=0$]. Such a shock does not acquire independent monopole and force-sheet sources merely because it is discontinuous. Its effects remain in the distributional derivatives of $T_{ij}$, including singular derivatives of its jump. Nonzero surface sources are appropriate for an interface with exchange/forcing or for a formulation that omits one side of the fluid.
\b[A shock does not, by itself, justify retaining the $m^4$ scaling.] That estimate required a low-<Mach number> stress $O(\rho_0U^2)$ varying on the slow time $\ell/U$. Fast shock motion, short time scales, or thermodynamic deviations can invalidate that estimate. If these same compact, slow-source assumptions remain valid for the integrated <Lighthill stress tensor>, its quadrupole estimate still follows, even distributionally. There is no universal replacement power deducible from the mere presence of a shock; nor should vanished physical flux defects be treated as additional independent radiation sources.
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