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For a simple linear inhomogeneous ODE, it's easy to derive that the Green's function should satisfies $$L_xG(x) = \delta(x-x')\tag{1}$$ where $L_x$ is the differential operator. However, for the linear inhomogeneous wave equation $$\left(\frac{\partial^2}{\partial t^2}-c^2\frac{\partial^2}{\partial x_i^2}\right)u(x,t) = f(x,t)\tag{2}$$ where $c$ being the velocity of the signal, let's say it's a simple constant here.

I have a hard time showing why the Green's function for the wave equation Eq.(2) should satisfy $$ \left(\frac{\partial^2}{\partial t^2}-c^2\frac{\partial^2}{\partial x_i^2}\right)G(x,t;x',t') = \delta(x-x')\delta(t-t') \tag{3} $$ Here the Green function is the simplest free space Green function, namely it satisfies $G\to0$ as $|x-x'|\to\infty$, and it vanishes identically for all $t<t'$ (I think this is called the causal Green function with principle of causality).

All the references and materials that I can find simply state that the Green's function for the wave equation above should satisfy Eq.(3), but no one actually derive it.

So can anybody give me some hints?

Appreciate it!

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    $\begingroup$ What do your require of a Green's function $G$ for a (linear) operator $L$ other than that $L(G)=\delta$, where the $\delta$ is simultaneous in all the time-and-space-and-whatever variables? Can you clarify? $\endgroup$ – paul garrett Aug 29 '20 at 22:43
  • $\begingroup$ @paulgarrett The details are added, and yes, the $\delta$ is simultaneous in all the time-and-space. Please take a look. $\endgroup$ – TurbJet Aug 29 '20 at 23:53
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You have to use the idea of advanced and retarded Green's function.

Links: english, german

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