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• W3098235920 abstract "Series expansions of unknown fields <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML overflow=scroll><mml:mi mathvariant=italic>Φ</mml:mi><mml:mo>=</mml:mo><mml:mo>∑</mml:mo><mml:mrow><mml:mrow><mml:msub><mml:mi>φ</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mi>Z</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:mrow></mml:mrow></mml:math> 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 Φ , leading to slowly convergent series, and rendering CMTs mild-slope approximations. In the present paper, the heuristic approach introduced in Athanassoulis & Belibassakis (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 Φ , valid for any smooth, non-planar boundaries and any smooth enough Φ . 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." @default.
• W3098235920 created "2020-11-23" @default.
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• W3098235920 date "2017-05-01" @default.
• W3098235920 modified "2023-10-09" @default.
• W3098235920 title "Exact semi-separation of variables in waveguides with non-planar boundaries" @default.
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• W3098235920 doi "https://doi.org/10.1098/rspa.2017.0017" @default.
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