Suppose we have the 3D wave equation which, in terms of Green's functions, can be written as

$$ \left( \frac{1}{c^2}\frac{\partial^2}{\partial t^2} - \nabla \right)G(\bar{x},t) = \delta(\bar{x})\delta(t) $$

which yields the solution

$$G(x, t) = \frac{c}{4\pi x}\delta(x-ct) = \frac{c}{2\pi}\delta(x^2 - c^2t^2)$$

Now suppose $f(\bar{x},t)$ describes a pulse of duration $T$ from the origin. We have

$$f(\bar{x},t) = \delta(\bar{x}) , \, \, \, 0 < t < T $$

and we can therefore write the solution of the wave equation as

$$u(x,t) = \int^{cT}_{-cT} d^3x' \int^T_0 G(\bar{x}:\bar{x}';t:t')f(\bar{x}',t')dt'$$ $$ = \int^{cT}_{-cT} \delta(\bar{x}')d^3x' \int^T_0 \frac{c}{2\pi}\delta(x^2 - c^2t^2)dt'$$

Can anyone tell me if this is correct? If so, can I solve this integral or is this the simplest form it can be in?



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