Analytical continuation of two-dimensional wave fields

Wave fields obeying the 2D Helmholtz equation on branched surfaces (Sommerfeld surfaces) are studied. Such surfaces appear naturally as a result of applying the reflection method to diffraction problems with straight scatterers bearing ideal boundary conditions. This is for example the case for the classical canonical problems of diffraction by a half-line or a segment. In the present work, it is shown that such wave fields admit an analytical continuation into the domain of two complex coordinates. The branch sets of such continuation are given and studied in detail. For a generic scattering problem, it is shown that the set of all branches of the multi-valued analytical continuation of the field has a finite basis. Each basis function is expressed explicitly as a Green's integral along so-called double-eight contours. The finite basis property is important in the context of coordinate equations, introduced and utilised by the authors previously, as illustrated in this article for the particular case of diffraction by a segment

Diffraction of a plane wave by two ideal strips

Summary The problem of scattering of a plane wave by two strips lying in one plane and having ideal boundary conditions is studied. The following exact results are obtained. 1) The embedding formula is derived. This formula enables to express the far-field diagram, depending on two variables (the angle of incidence and the angle of scattering) as the combination of 4 functions depending on one variable. 2) The ordinary differential equation with respect to the spectral variable is derived for the components of the far-field diagram. 3) The evolution equations describing the dependance of the farfield diagram on the parameters of the problem (such as the coordinates of the edges of the scatterer) are derived. The results listed above are obtained by applying two independent approaches: the Wiener-Hopf functional equations approach and the diffraction (Schwarzschild's) series approach

A contribution to the mathematical theory of diffraction. Part II: Recovering the far-field asymptotics of the quarter-plane problem

We apply the stationary phase method developed in (Assier, Shanin \& Korolkov, QJMAM, 76(1), 2022) to the problem of wave diffraction by a quarter-plane. The wave field is written as a double Fourier transform of an unknown spectral function. We make use of the analytical continuation results of (Assier \& Shanin, QJMAM, 72(1), 2018) to uncover the singularity structure of this spectral function. This allows us to provide a closed-form far-field asymptotic expansion of the field by estimating the double Fourier integral near some special points of the spectral function. All the known results on the far-field asymptotics of the quarter-plane problem are recovered, and new mathematical expressions are derived for the secondary diffracted waves in the plane of the scatterer