Asymptotic solutions of the Helmholtz equation: generalised FriedlanderKeller ray expansions of fractional orderTools Tew, R.H. (2020) Asymptotic solutions of the Helmholtz equation: generalised FriedlanderKeller ray expansions of fractional order. European Journal of Applied Mathematics, 31 (1). ISSN 14694425 Full text not available from this repository.AbstractApplications of a WKBJtype `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 wideranging 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 strategicallychosen 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 leadingorder amplitude occurring as a preexponential factor. One particular choice of fractional power is considered in detail, and waves with specified radiallysymmetric or planar wavefronts are then analysed, along with a boundary value problem typifying twodimensional 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 smallscale perturbations to their underlying profile.
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