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Polar Wave Equation

  1. Apr 14, 2013 #1
    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), 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([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:
     
  2. jcsd
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