4.I.2D

Consider the wave equation in a spherically symmetric coordinate system

$\frac{\partial^{2} u(r, t)}{\partial t^{2}}=c^{2} \Delta u(r, t)$

where $\Delta u=\frac{1}{r} \frac{\partial^{2}}{\partial r^{2}}(r u)$ is the spherically symmetric Laplacian operator.

(a) Show that the general solution to the equation above is

$u(r, t)=\frac{1}{r}[f(r+c t)+g(r-c t)]$

where $f(x), g(x)$ are arbitrary functions.

(b) Using separation of variables, determine the wave field $u(r, t)$ in response to a pulsating source at the origin $u(0, t)=A \sin \omega t$.

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