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Homework Help: Wave equation

  1. May 2, 2010 #1

    fluidistic

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    1. The problem statement, all variables and given/known data
    Show that [tex]u(r,t)=\frac{f(r-vt)}{r}[/tex] is a solution to the tridimensional wave equation. Show that it corresponds to a spherical perturbation centered at the origin and going away from it with velocity v. Assume that f is twice differentiable.


    2. Relevant equations

    The wave equation: [tex]\frac{\partial ^2 u }{\partial t ^2}- c^2 \triangle u =0[/tex].

    3. The attempt at a solution
    I just used the wave equation and found out that [tex]\frac{\partial ^2 u}{\partial t^2} = v^2 u''[/tex].
    While [tex]\triangle u =\frac{1}{r} \left [ u''+ \frac{u}{r^2} - \frac{2u'}{r} \right ][/tex].
    So the wave equation is satisfied if [tex]u'' \left ( v^2-\frac{c^2}{r} \right ) + \frac{2 c^2 u'}{r^2} - \frac{u c^2}{r^3}=0[/tex].
    It's likely wrong so either I set up badly the problem, either I set it up OK but made some errors.
     
    Last edited: May 2, 2010
  2. jcsd
  3. May 2, 2010 #2

    vela

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    I think that's your problem. Where did you get this expression for the Laplacian?
     
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