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The solution exact solution of the Helmholtz equation for the scattering of waves by a sphere is relatively straightforward and has been known since the time of Lord Rayleigh.

The exact solution of the wave equation for scattering of waves by a sphere is much trickier and has been dealt with in two relatively recent papers - here and here.

I have been trying to determine whether there exists an exact solution for the wave equation in the two-dimensional case, i.e. for a disk/cylinder. It is not mentioned in the recent papers I listed above for the case of the sphere, and I haven't been able to find anything despite an extensive search of the literature.

Does anyone know if an exact solution of the wave equation for the scattering of waves by a disk/cylinder exists, or is this still an open problem?

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  • $\begingroup$ What do you mean by the "exact solution"? The papers you refer to are about numerical solutions. $\endgroup$ – Alexandre Eremenko Apr 4 '19 at 20:28
  • $\begingroup$ @AlexandreEremenko - Maybe 'explicit representation' is a better term than 'exact solution.' I used the term 'exact solution' as that is how these representations are described in the first paper I referred to. Equation (4.18) for $\phi(t)$ in this paper for instance is referred to as a component of the 'exact solution' $\phi(t)Y_n^m$ even though it contains summations and integrals. So I am wondering if there exist analogous explicit representations for the two dimensional case? $\endgroup$ – electroscience Apr 6 '19 at 9:37

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