# Polar Wave Equation

1. Apr 14, 2013

### DesZoras

Hi!

I've been given the following problem to solve:

Consider the azimuthally symmetric wave equation:

$\frac{∂2u}{∂t2}$ = $\frac{c2}{r}$$\frac{∂}{∂r}$(r$\frac{∂u}{∂r}$) where u(r,0)=f(r), ut(r,0)=g(r), u(0,t)=1 and u(L,0)=0.

Use the separation of variables method to find the solution to this PDE.

Using a substitution of u(r,t)=T(t)R(r), I obtained the following answers:

T=Asin(λt)+Bcos(λt)
R=CJ0($\frac{λ}{c}$r)+DY0($\frac{λ}{c}$r)
However, we know as r→0, the Y0 function tends to -∞, thus
R=CJ0($\frac{λ}{c}$r)

So, u becomes:

u=Dsin(λt)J0($\frac{λ}{c}$r)+Ecos(λt)J0($\frac{λ}{c}$r)

My problem is I'm unsure how to use my boundary conditions to solve for what λ should be. Substituting the boundary conditions in, I get:

J0($\frac{λ}{c}$L)=0
and
Dsin(λt)+Ecos(λt)=1

Does anyone have any suggestions/hints on how to solve for λ here? I don't necessarily need a full answer, even just a hint will do, and then I should be able to figure the rest out myself