I have a linear acoustic wave propagation originated from a monopole source, written as

\begin{align} \mathcal{L}p(\mathbf{x},t) = S_m(\mathbf{x},t), \quad \mbox{in } \Omega \end{align}

where the wave operator $\mathcal{L} \equiv \left(\frac{1}{c_0^2}\frac{\partial^2}{\partial t^2} - \nabla^2 \right)$. I want to solve it using hard-wall Green's function, $G(\mathbf{x},\mathbf{y},t-\tau)$, to take the complex boundary condition into account (the solution of which I got from numerical Green's function evaluation). The complementary problem can be written as \begin{align} \mathcal{L}G(\mathbf{x},\mathbf{y},t-\tau) = \delta(\mathbf{x} - \mathbf{y})\delta(t-\tau), \quad \mbox{in } \Omega \end{align} Now, what is the condition that guarantees that the Green's function is differentiable in time after the initial impulse $t = \tau$, if any? My physical intuition tells me that is probably true, as the wave is travel on some characteristic line that guarantees the smoothness, but mathematically how do I show that? Thanks.