• Corpus ID: 215828051

Diffraction by a quarter-plane. Links between the functional equation, additive crossing and Lam\'e functions

  title={Diffraction by a quarter-plane. Links between the functional equation, additive crossing and Lam\'e functions},
  author={Raphael Assier and Andrey V. Shanin},
  journal={arXiv: Analysis of PDEs},
In our previous work (Assier & Shanin, QJMAM, 2019), we gave a new spectral formulation in two complex variables associated with the problem of diffraction by a quarter-plane. In particular, we showed that the unknown spectral function satisfies a condition of additive crossing about its branch set. In this paper, we study a very similar class of spectral problem, and show how the additive crossing can be exploited in order to express its solution in terms of Lame functions. The solution… 

Analytical continuation of two-dimensional wave fields

Wave fields obeying the two-dimensional Helmholtz equation on branched surfaces (Sommerfeld surfaces) are studied and it is shown that such wave fields admit an analytical continuation into the domain of two complex coordinates.

Diffraction by a quarter–plane. Analytical continuation of spectral functions

  • R. AssierA. Shanin
  • Mathematics
    The Quarterly Journal of Mechanics and Applied Mathematics
  • 2018
The problem of diffraction by a Dirichlet quarter-plane (a flat cone) in a 3D space is studied. The Wiener–Hopf equation for this case is derived and involves two unknown (spectral) functions

A surprising observation on the quarter-plane problem

In this paper, we revisit Radlow's highly original attempt at a double Wiener--Hopf solution to the problem of wave diffraction by a quarter-plane. Using a constructive approach, we reduce the

Spectral study of the Laplace–Beltrami operator arising in the problem of acoustic wave scattering by a quarter-plane

© 2016 Published by Oxford University Press 2016. The Laplace-Beltrami operator (LBO) on a sphere with a cut arises when considering the problem of wave scattering by a quarter-plane. Recent methods

Diffraction by a quarter plane, the exact solution, and some numerical results

The exact solution for diffraction by a perfectly conducting quarter plane has been previously determined. This solution is general and needs some simplification before it can become a useful tool

Higher Transcendental Functions [Volumes I-III]

The work of which this book is the first volume might be described as an up-to-date version of Part II. The Transcendental Functions of Whittaker and Watson's celebrated "Modern Analysis". Bateman

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A general formalism is described whereby some regular singular points are effectively removed and substantial simplifications ensue for a class of Fuchsian ordinary differential equations, and