Methods | Part IB, 2003

Consider the wave equation in a spherically symmetric coordinate system

2u(r,t)t2=c2Δu(r,t)\frac{\partial^{2} u(r, t)}{\partial t^{2}}=c^{2} \Delta u(r, t)

where Δu=1r2r2(ru)\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)=1r[f(r+ct)+g(rct)]u(r, t)=\frac{1}{r}[f(r+c t)+g(r-c t)]

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

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

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