Wave Equation In Spherical Coordinates

**mathematicalbagpiper** · May 4th 2011, 12:11 PM

Let $u$ be the solution of the Initial Value Problem:

$\Delta u-u_{tt}=0$
$u(x,0)=\phi (r)$
$u_t(x,0)=\psi (r)$

Where $\phi$ and $\psi$ are continuous. Show that the solution can be represented in the region ${r>0, t>0}$ as:

$u(r,t)=\frac{1}{2r}[(r+t)\phi(r+t)+(r-t)\psi(r-t)+\int_{r-t}^{r+t}\tau\psi(\tau)d\tau]$ .

I honestly have no idea where to start. :-(

**TheEmptySet** · May 4th 2011, 12:37 PM

Originally Posted by mathematicalbagpiper

Let $u$ be the solution of the Initial Value Problem:

$\Delta u-u_{tt}=0$
$u(x,0)=\phi (r)$
$u_t(x,0)=\psi (r)$

Where $\phi$ and $\psi$ are continuous. Show that the solution can be represented in the region ${r>0, t>0}$ as:

$u(r,t)=\frac{1}{2r}[(r+t)\phi(r+t)+(r-t)\psi(r-t)+\int_{r-t}^{r+t}\tau\psi(\tau)d\tau]$ .

I honestly have no idea where to start. :-(

Hint:

Use the change of coordinates

$\alpha=r-t \quad \beta=r+t$

and rewrite the PDE using

$u(\alpha,\beta)$

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