Exact semi-separation of variables in waveguides with nonplanar boundaries

TL;DR

This paper introduces an exact, rapidly-convergent series expansion method for solving waveguide problems with nonplanar boundaries, enabling precise semi-separation of variables in complex geometries.

Contribution

It generalizes and justifies a heuristic series expansion approach, providing an exact semi-separation of variables applicable to any smooth, nonplanar boundary, improving convergence and accuracy.

Findings

Series expansion converges rapidly for nonplanar boundaries.

Method accurately computes Dirichlet-to-Neumann operators.

Few modes suffice for precise solutions in complex domains.

Abstract

Series expansions of unknown fields $\Phi=\sum\varphi_n Z_n$ 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 $Z_n$ are determined by solving local Sturm-Liouville problems (reference waveguides). In most cases, the boundary conditions assigned to $Z_n$ cannot be compatible with the physical boundary conditions of $\Phi$, leading to slowly convergent series, and rendering CMTs mild-slope approximations. In the present paper, the heuristic approach introduced in (Athanassoulis & Belibassakis 1999, J. Fluid Mech. 389, 275-301) is generalized and justified. It is proved that an appropriately enhanced series expansion becomes an exact, rapidly-convergent representation of the field $\Phi$, valid for any smooth, nonplanar boundaries and any smooth enough $\Phi$. 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.