Abstract
Summary 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 depending on two complex variables. The aim of the present work is to construct an analytical continuation of these functions onto a well-described Riemann manifold and to study their behaviour and singularities on this manifold. In order to do so, integral formulae for analytical continuation of the spectral functions are derived and used. It is shown that the Wiener–Hopf problem can be reformulated using the concept of additive crossing of branch lines introduced in the article. Both the integral formulae and the additive crossing reformulation are novel and represent the main results of this work.
Highlights
Introduction and literature reviewTowards a generalisation of Sommerfeld’s solution
An important step of the usual 1D Wiener–Hopf method is to draw conclusions about the analyticity properties of unknown functions originally defined by half-range Fourier transforms
We drew some conclusions about the domain of analyticity of unknown functions originally defined by 1/4 and 3/4 range Fourier transforms
Summary
The integrand contains a Green’s function for the LaplaceBeltrami operator on a sphere with a cut, which can be obtained, generally, by solving an integral equation Within this approach, the concept of the oasis domain has been introduced, i.e. of the domain in the 3D space where there are no scattered waves except the spherical wave diffracted by the tip of the cone. A formal description of where his proof went wrong (the solution does not satisfy the boundary condition) was given in (52), and, more recently, other technical reasons have been given in (53), showing along the way that in the far-field and in the case of Dirichlet conditions, Radlow’s solution led to surprisingly accurate results To conclude this introduction, we can say that getting a closed-form solution of the quarter-plane diffraction problem is still a challenging theoretical and practical task. We call this behaviour additive crossing and this allows us to rewrite the spectral formulation in a very different way to that of Section 2, which we will use in further work in order to derive specific results
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