Electronic supplementary material (ESM) for the paper Exact semi-separation of variables in waveguides with nonplanar boundaries
2017-05-03T15:21:25Z (GMT) by
Series expansions of unknown fields <i>Φ=∑φ<sub>n</sub>Z<sub>n</sub></i> in elongated waveguides are commonly used in acoustics, optics, geophysics, water waves and other applications, in the context of coupled-mode theories (CMTs). The transverse functions <i>Z<sub>n</sub></i> are determined by solving local Sturm–Liouville problems (reference waveguides). In most cases, the boundary conditions assigned to <i>Z<sub>n</sub></i> cannot be compatible with the physical boundary conditions of <i>Φ</i>, leading to slowly convergent series, and rendering CMTs mild-slope approximations. In the present paper, the heuristic approach introduced in (Athanassoulis & Belibassakis 1999 <i>J. Fluid Mech</i>. <b>389</b>, 275–301) is generalized and justified. It is proved that an appropriately enhanced series expansion becomes an exact, rapidly convergent representation of the field <i>Φ</i>, valid for any smooth, nonplanar boundaries and any smooth enough <i>Φ</i>. This series expansion can be differentiated termwise everywhere in the domain, including the boundaries, implementing an exact semi-separation of variables for non-separable domains. The efficiency of the method is illustrated by solving a boundary value problem for the Laplace equation, and computing the corresponding Dirichlet-to-Neumann operator, involved in Hamiltonian equations for nonlinear water waves. The present method provides accurate results with only a few modes for quite general domains. Extensions to general waveguides are also discussed.