Generalization of Rayleigh's high-frequency theory for the 2D Helmholtz equation in a half-space subject to a radiation condition at infinity and a Dirichlet condition on a 1D periodically-uneven boundary
Abstract
The 2D Helmholtz equation, radiation condition and Dirichlet boundary condition, are the translation, in mathematical terms, of (at least) three physical 2D problems for the prediction of the total scalar wavefield on one side of an impenetrable 1D periodically uneven boundary when: a) a plane TE electromagnetic wave propagating in the vacuum strikes the boundary the other side of which is occupied by a perfectly-conducting medium, b) a plane SH elastic elastic wave strikes a rigid boundary, c) a plane acoustic wave strikes a pressure-release boundary. The first attempt to solve such problems in a non-heuristic manner was made by Lord Rayleigh (in his book, 'The Theory of Sound' which appeared in 1896). My task will be to revisit Rayleigh's theory of diffraction by a sinusoidal-shaped, impenetrable boundary, and more specifically, his perturbation method for obtaining a mathematically-explicit solution to the diffraction problem in the high-frequency regime. In so doing, I shall correct and generalize Rayleigh's method to obtain solutions for arbitrary angles of incidence as well as for 1D periodic impenetrable boundaries of quite-general shape.
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BibTeX
@misc{wirgin2026generalization,
title = {{Generalization of Rayleigh's high-frequency theory for the 2D Helmholtz equation in a half-space subject to a radiation condition at infinity and a Dirichlet condition on a 1D periodically-uneven boundary}},
author = {Armand Wirgin},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.14135},
eprint = {2607.14135},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.14135},
note = {CC0},
}
RIS
TY - UNPB
TI - Generalization of Rayleigh's high-frequency theory for the 2D Helmholtz equation in a half-space subject to a radiation condition at infinity and a Dirichlet condition on a 1D periodically-uneven boundary
AU - Armand Wirgin
PY - 2026
DA - 2026/10/30
PB - arXiv
DO - 10.48550/arxiv.2607.14135
UR - https://arxiv.org/abs/2607.14135
AB - The 2D Helmholtz equation, radiation condition and Dirichlet boundary condition, are the translation, in mathematical terms, of (at least) three physical 2D problems for the prediction of the total scalar wavefield on one side of an impenetrable 1D periodically uneven boundary when: a) a plane TE electromagnetic wave propagating in the vacuum strikes the boundary the other side of which is occupied by a perfectly-conducting medium, b) a plane SH elastic elastic wave strikes a rigid boundary, c) a plane acoustic wave strikes a pressure-release boundary. The first attempt to solve such problems in a non-heuristic manner was made by Lord Rayleigh (in his book, 'The Theory of Sound' which appeared in 1896). My task will be to revisit Rayleigh's theory of diffraction by a sinusoidal-shaped, impenetrable boundary, and more specifically, his perturbation method for obtaining a mathematically-explicit solution to the diffraction problem in the high-frequency regime. In so doing, I shall correct and generalize Rayleigh's method to obtain solutions for arbitrary angles of incidence as well as for 1D periodic impenetrable boundaries of quite-general shape.
ER -