Polar Wave Equation

Hi!

I've been given the following problem to solve:

Consider the azimuthally symmetric wave equation:

[itex]\frac{∂²u}{∂t²}[/itex] = [itex]\frac{c²}{r}[/itex][itex]\frac{∂}{∂r}[/itex](r[itex]\frac{∂u}{∂r}[/itex]) where u(r,0)=f(r), u_t(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([itex]\frac{λ}{c}[/itex]r)+DY₀([itex]\frac{λ}{c}[/itex]r)
However, we know as r→0, the Y₀ function tends to -∞, thus
R=CJ₀([itex]\frac{λ}{c}[/itex]r)

So, u becomes:

u=Dsin(λt)J₀([itex]\frac{λ}{c}[/itex]r)+Ecos(λt)J₀([itex]\frac{λ}{c}[/itex]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([itex]\frac{λ}{c}[/itex]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 :smile: