Wave Equation Spherical Perturbations

[fluidistic]

Homework Statement

Show that $$u(r,t)=\frac{f(r-vt)}{r}$$ is a solution to the tridimensional wave equation. Show that it corresponds to a spherical perturbation centered at the origin and going away from it with velocity v. Assume that f is twice differentiable.

Homework Equations

The wave equation: $$\frac{\partial ^2 u }{\partial t ^2}- c^2 \triangle u =0$$.

The Attempt at a Solution

I just used the wave equation and found out that $$\frac{\partial ^2 u}{\partial t^2} = v^2 u''$$.
While $$\triangle u =\frac{1}{r} \left [ u''+ \frac{u}{r^2} - \frac{2u'}{r} \right ]$$.
So the wave equation is satisfied if $$u'' \left ( v^2-\frac{c^2}{r} \right ) + \frac{2 c^2 u'}{r^2} - \frac{u c^2}{r^3}=0$$.
It's likely wrong so either I set up badly the problem, either I set it up OK but made some errors.

 

[vela]

While $$\triangle u =\frac{1}{r} \left [ u''+ \frac{u}{r^2} - \frac{2u'}{r} \right ]$$.

I think that's your problem. Where did you get this expression for the Laplacian?