An integral equation technique is employed to obtain eigenvalues and eigenmodes of the homogeneous Holmholtz equation for a two‐ or three‐dimensional closed region of arbitrary shape with arbitrary first‐order homogeneous boundary conditions. The method is described for general (i.e., nonseparable) geometries, with a discussion of the simplification introduced by having a separable geometry given in an Appendix. A numerical example is given for a nonseparable geometry, i.e., a two‐dimensional right triangle of arbitrary enclosed angle with Neumann boundary conditions. Results for the special case of an isosceles right triangle agree very well with a known analytical solution.

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