Non-separable solutions of Helmholtz' equation revisited

Even if the idea is already very old (proposed first by Moseley in 1965), it offers analytical solutions of Helmholtz equations for curved waveguides. One of the key elements in their successful use is the computation of recursion formulas to find a numerical solution to the problem of analysis of generally shaped waveguides. They will be derived in this paper. Even if the original theory includes a generalisation to 3 dimensional cavities we will limit ourselves to 2 dimensional waveguide cases.