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Help with a proof (2-D wave propagation stuff)

  1. Aug 14, 2007 #1
    I'm reading through an older paper on acoustics with one of those "it can be readily seen that" assertions. Of course, to save my life I can't verify the author's assertion. Here goes: starting with the Helmholtz equation [tex]\nabla^2\psi+k^2\psi=0[/tex], decompose [tex]\psi[/tex] with the 2-D cylindrical basis functions [tex]\psi_n = H^{(1)}_n(kr)\cos n\theta[/tex]. Then with the surface integral definitions

    \hat{Q}_{mn} = \frac{1}{4}\int n\cdot \mbox{Re}(\psi_m)\nabla\psi_ndS, \qquad
    Q_{mn} = \frac{1}{4}\int n\cdot \nabla(\mbox{Re}(\psi_m))\psi_ndS

    he asserts that by applying the divergence theorem to the difference [tex]\hat{Q}-Q[/tex] we end up with [tex]i[/tex] times the identity matrix. Pretty quickly I reach the expression


    without using any of the properties of [tex]\psi[/tex], where the point [tex]r=0[/tex] is not included in the volume of integration. But in the next step, where I substitute [tex]k^2\psi_n[/tex] for [tex]\nabla^2\psi_n[/tex], all I end up doing is 'proving' that the whole expression is equal to zero, which makes no physical sense. Can anyone see what I'm missing here?
  2. jcsd
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