Thread: Wave Equation In Spherical Coordinates

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

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

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

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

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

Originally Posted by mathematicalbagpiper

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

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

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

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

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

Hint:

Use the change of coordinates

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

and rewrite the PDE using

$\displaystyle u(\alpha,\beta)$