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3D wave equation - spherically symmetric transformations

  1. Sep 18, 2009 #1

    Applied Partial Differential Equations (Richard Heberman) 4ed.

    Consider the three dimensional wave equation

    [tex]\partial^{2}u/\partial t^2 = c^2\nabla^2 u[/tex]

    Assume the solution is spherically symetric, so that

    [tex]\nabla^2 u = (1/\rho^2)(\partial/\partial\rho)(\rho^2\partial u/\partial\rho) [/tex]

    (a) Make the transformation [tex]u = (1/\rho)w(\rho,t)[/tex] and verify that

    [tex] \partial^2w/\partial t^2 = c^2(\partial^2w/\partial \rho^2)[/tex]

    (b) Show that the most general sphereically symmetric solution of the wave equation consists of the sum of two sphereically symmetric waves, one moving outward at speed c and the other inward at speed c. Note the decay of the amplitude.

    I really have no idea how to do this. Any and all help (hopefully oriented to the level of someone not all that comfortable with PDEs) would be greatly appreciated.
  2. jcsd
  3. Sep 19, 2009 #2


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    is this a question for a class? I'm assuming it is not as it is posted in this forum, but just in case I will be stingy with my hints until I hear otherwise.

    for part a, all you have to know how to do is differentiate a product. If you are uncomfortable with this then you need to spend serious time reviewing calculus.

    for part b, you should have seen almost all of this when you studied the 1-D wave equation. Review that material and you should find what you need.

    good luck

  4. Sep 20, 2009 #3
    Its a suggested problem for our class; a learning exercise, not for marks.

    I'm pretty sure I've got part a, but b is still eluding me. For the record, we were never taught the wave equation before; it was part of a prerequisite course, but was cut out, and this course wasn't altered to reflect it.
  5. Sep 20, 2009 #4


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