= Solution
Put $\rho'=\rho-\rho_0$, and take the <retarded acoustic Green function> for $\partial_t^2-c_0^2\nabla^2$:
$$
G_R(x-y,t-\tau)=\frac{\delta(t-\tau-|x-y|/c_0)}{4\pi c_0^2|x-y|}.
$$
This selects the causal, outgoing <acoustic density perturbation>, with no additional incoming homogeneous <wave equation> solution. Convolving the <acoustic dipole> forcing with this <Green function> and moving its spatial <derivative> outside the integral gives
$$
\rho'=-\partial_{x_i}\int d\tau\int d^3y\,
\frac{F_i(y,\tau)\delta(f(y,\tau))|\nabla_y f(y,\tau)|}{4\pi c_0^2|x-y|}
\delta\left(t-\tau-\frac{|x-y|}{c_0}\right).
$$
The <surface delta distribution> converts the spatial integral to the moving surface. At fixed surface labels $(p,q)$, set
$$
r(\tau)=|x-y(p,q,\tau)|,\quad
\widehat r=\frac{x-y}{|x-y|},\quad v=\partial_\tau y(p,q,\tau),\quad
M_r=\frac{\widehat r\cdot v}{c_0}.
$$
The <radial Mach number> enters the <moving-surface retarded Jacobian>, because $dr/d\tau=-\widehat r\cdot v$ and hence
$$
\frac d{d\tau}\left(t-\tau-\frac{r(\tau)}{c_0}\right)=-(1-M_r).
$$
Let $\tau^*$ be a root of the <retarded time> equation
$$
\boxed{t=\tau^*+\frac{|x-y(p,q,\tau^*)|}{c_0}.}
$$
The delta change-of-variable rule now gives
$$
\boxed{\rho'(x,t)=-\partial_{x_i}\iint
\left[\frac{F_i(y,\tau)h_ph_q}{4\pi c_0^2|x-y|\,|1-M_r|}\right]_{\tau=\tau^*}dp\,dq.}
$$
Here $h_p h_q$ is the surface-area factor from the orthogonal surface coordinates. For a subsonic surface, $|v|<c_0$, the retarded equation is monotone in $\tau$ and has one root when the motion is defined for the required past times. For more general motion, sum the displayed contribution over all simple retarded roots. A root with $1-M_r=0$ requires a separate limiting treatment; the simple-root formula does not apply there.
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