Diffraction by a Right-Angled No-Contrast Penetrable Wedge: Analytical Continuation of Spectral Functions

We study the problem of diffraction by a right-angled no-contrast penetrable wedge by means of a two-complex-variable Wiener-Hopf approach. Specifically, the analyticity properties of the unknown (spectral) functions of the two-complex-variable Wiener-Hopf equation are studied. We show that these spectral functions can be analytically continued onto a two-complex dimensional manifold, and unveil their singularities in $\mathbb{C}^2$. To do so, integral representation formulae for the spectral functions are given and thoroughly used. It is shown that the novel concept of additive crossing holds for the penetrable wedge diffraction problem and that we can reformulate the physical diffraction problem as a functional problem using this concept

Diffraction by a right-angled no-contrast penetrable wedge: recovery of far-field asymptotics

We provide a description of the far-field encountered in the diffraction problem resulting from the interaction of a monochromatic plane-wave and a right-angled no-contrast penetrable wedge. To achieve this, we employ a two-complex-variable framework and use the analytical continuation formulae derived in (Kunz $\&$ Assier, QJMAM, 76(2), 2023) to recover the wave-field's geometrical optics components, as well as the cylindrical and lateral diffracted waves. We prove that the corresponding cylindrical and lateral diffraction coefficients can be expressed in terms of certain two-complex-variable spectral functions, evaluated at some given points

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 by a Right-Angled No-Contrast Penetrable Wedge Revisited: A Double Wiener-Hopf Approach

In this paper, we revisit Radlow's innovative approach to diffraction by a penetra ble wedge by means of a double Wiener-Hopf technique. We provide a constructive way of obtaining his ansatz and give yet another reason for why his ansatz cannot be the true solution to the diffraction problem at hand. The two-complex-variable Wiener-Hopf equation is reduced to a system of two equations, one of which contains Radlow's ansatz plus some correction term consisting of an explicitly known integral operator applied to a yet unknown function, whereas the other equation, the compatibility equation, governs the behaviour of this unknown function

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

Diffraction by a right-angled no-contrast penetrable wedge: recovery of far-field asymptotics

We provide a description of the far-field encountered in the diffraction problem resulting from the interaction of a monochromatic plane-wave and a right-angled no-contrast penetrable wedge. To achieve this, we employ a two-complex-variable framework and use the analytical continuation formulae derived in (Kunz & Assier, QJMAM, 76(2), 2023) to recover the wavefield’s geometrical optics components, as well as the cylindrical and lateral diffracted waves. We prove that the corresponding cylindrical and lateral diffraction coefficients can be expressed in terms of certain two-complex-variable spectral functions, evaluated at some given points

SPECTRAL STUDY OF THE LAPLACE-BELTRAMI OPERATOR ARISING IN THE PROBLEM OF ACOUSTIC WAVE SCATTERING BY A QUARTER-PLANE

This is the author accepted manuscript. It is currently under an indefinite embargo pending publication by Oxford University Press.The Laplace-Beltrami operator on a sphere with a cut arises when considering the problem of wave scattering by a quarter-plane. Recent methods developed for sound-soft (Dirichlet) and sound-hard (Neumann) quarter-planes rely on an a priori knowledge of the spectrum of the Laplace-Beltrami operator. In this paper we consider this spectral problem for more general boundary conditions, including Dirichlet, Neumann, real and complex impedance, where the value of the impedance varies like $\textit{α/=r, r}$ being the distance from the vertex of the quarter-plane and α being constant, and any combination of these. We analyse the corresponding eigenvalues of the Laplace-Beltrami operator, both theoretically and numerically. We show in particular that when the operator stops being self-adjoint, its eigenvalues are complex and are contained within a sector of the complex plane, for which we provide analytical bounds. Moreover, for impedance of small enough modulus |α|, the complex eigenvalues approach the real eigenvalues of the Neumann case.R.C. Assier would like to acknowledge the support by UK EPSRC (EP/N013719/1)

DIFFRACTION OF ACOUSTIC WAVES BY A WEDGE OF POINTSCATTERERS

This article considers the problem of diffraction by a wedge consisting of two semi-infinite periodic arrays of point scatterers. The solution is obtained in terms of two coupled systems, each of which is solved using the discrete Wiener--Hopf technique. An effective and accurate iterative numerical procedure is developed to solve the diffraction problem, which allows us to compute the interaction of thousands of scatterers forming the wedge. A crucial aspect of this numerical procedure is the effective truncation of slowly convergent single and double infinite series, which requires careful asymptotic analysis. A convergence criteria is formulated and shown to be satisfied for a large class of physically interesting cases. A comparison to direct numerical simulations is made, highlighting the accuracy of the method.Comment: 24 pages, 11 figures (12 image files