Wave Equation In Spherical Coordinates

**mathematicalbagpiper** · May 4th 2011, 01:11 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, 01:37 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

$u(\alpha,\beta)$

Math Help - Wave Equation In Spherical Coordinates

LinkBack

Thread Tools

Display

Wave Equation In Spherical Coordinates

Similar Math Help Forum Discussions

Please help in transforming rectangle coordinates to spherical coordinates.

wave equation in polar coordinates

Spherical Coordinates

Identify the surface whose equation is given in spherical coordinates

Spherical coordinates!

Search Tags