Skip to main content

Research Repository

Advanced Search

Asymptotic solutions of the Helmholtz equation: Generalised Friedlander–Keller ray expansions of fractional order

Tew, Richard H.

Asymptotic solutions of the Helmholtz equation: Generalised Friedlander–Keller ray expansions of fractional order Thumbnail


Authors

RICHARD TEW richard.tew@nottingham.ac.uk
Associate Professor



Abstract

Applications of a WKBJ-type `ray ansatz' to obtain asymptotic solutions of the Helmholtz equation in the high{frequency limit are now standard, and underpin the construction of `geometrical optics' ray diagrams in many electromagnetic, acoustic and elastic reflection, transmission and other scattering problems. These applications were subsequently extended by Keller to include other types of rays - called `diffracted' rays - to provide an accessible and impressively accurate theory which is relevant in wide-ranging sets of circumstances. Friedlander and Keller then introduced a modified ray ansatz to extend yet further the scope of ray theory and its applicability to certain other classes of diffraction problems (tangential ray incidence upon an obstructing boundary, for instance), and did so by the inclusion of an extra term proportional to a power of the wavenumber within the exponent of the initial ansatz. Our purpose here is to generalise this further still by the inclusion of several such terms, ordered in a natural sequence in terms of strategically-chosen fractional powers of the large wavenumber, and to derive a systematic sequence of boundary value problems for the coefficient phase functions that arise within this generalised exponent, as well as one for the leading-order amplitude occurring as a pre-exponential factor. One particular choice of fractional power is considered in detail, and waves with specified radially-symmetric or planar wavefronts are then analysed, along with a boundary value problem typifying two-dimensional radiation whereby arbitrary phase and amplitude variations are specified on a prescribed boundary curve. This theory is then applied to the scattering of plane and cylindrical waves at curved boundaries with small-scale perturbations to their underlying profile.

Citation

Tew, R. H. (2020). Asymptotic solutions of the Helmholtz equation: Generalised Friedlander–Keller ray expansions of fractional order. European Journal of Applied Mathematics, 31(1), 1-25. https://doi.org/10.1017/s095679251800044x

Journal Article Type Article
Acceptance Date May 11, 2018
Online Publication Date Sep 18, 2018
Publication Date 2020-02
Deposit Date May 24, 2018
Publicly Available Date Mar 29, 2024
Journal European Journal of Applied Mathematics
Print ISSN 0956-7925
Electronic ISSN 1469-4425
Publisher Cambridge University Press
Peer Reviewed Peer Reviewed
Volume 31
Issue 1
Pages 1-25
DOI https://doi.org/10.1017/s095679251800044x
Public URL https://nottingham-repository.worktribe.com/output/932129
Publisher URL https://www.cambridge.org/core/journals/european-journal-of-applied-mathematics/article/asymptotic-solutions-of-the-helmholtz-equation-generalised-friedlanderkeller-ray-expansions-of-fractional-order/9E162EEF38941D0A01F69909CD01676E

Files





You might also like



Downloadable Citations