E. Meister

A contribution to the quarter-plane problem in diffraction theory

Abstract The three-dimensional analog of Sommerfelds famous half-plane problem for a screen Σ1: x1 > 0, x2 > 0, x3 = 0 is treated by an operator theoretic approach. This paper shows that the Dirichlet problem is well-posed for any complex wave-number k ∉ R, if we look for a solution u ∈ H1 (R3) of the Helmholtz equation (Δ + k2) u = 0 in e 7mid; ∑ 1 =g ∈ H 1 2 (∑ 1 ) with u ¦ Σ 1 = g ϵ H 1 2 (σ 1 ) . An explicit representation formula for the solution is established by use of general Wiener-Hopf operator theory, Sobolev space properties, and Hilbert space methods. The corresponding Neumann problem can be solved by analogy.

Diffraction problems with impedance conditions

The two- or three-dimensional electromagnetic diffraction problem for a half-plane impedance or reactance sheet belongs to a class of elliptic transmission problems of mixed type. Sobolev spaces of order 1 and ±1/2 are naturally involved according to the energy norm and the trace theorem, respectively. This operator theoretic approach presents the equivalence to systems of Wiener-Hopf equations and their solution in the sense of a well-posed problem with respect to the spaces under consideration. Slightly different impedance numbers for the two banks of the screen lead to a perturbation problem. All results yield direct a priori estimates for the solutions.

Sommerfeld diffraction problems with third kind boundary conditions

This paper continues earlier work on diffraction problems with first and second kind boundary conditions. New operator theoretic difficulties appear for third kind conditions corresponding to different behavior of the Fourier symbol matrix of the boundary operators at infinity. Compatibility conditions force another function space setting in order to obtain closed operators. Then well-posed problems can be solved by explicit Wiener–Hopf factorization using Khrapkov’s method, which also yields the asymptotics near the origin.

Double-knife EDGE problems with elastic/plastic type screens

A problem of diffraction of a wave by a pair of semi-infinite screens is considered. The screens are lined with two different wave bearing materials that can support surface waves. This type of problem arises in the propagation and, scattering of acoustic and electromagnetic waves by surface wave guides. To be specific, we shall couch our problem in terms of acoustics. These diffraction problems for two parallel wave bearing screens lead to boundary value problems which are governed by the Helmholtz equation, and some specific third kind boundary conditions. Such problems are shown to be well-posed for finite energy space solutions. Their representation is given by means of the canonical factorization of a non-rational matrix function.

Some Solved and Unsolved Canonical Transmission Problems of Diffraction Theory

The study of the refraction and scattering of waves by penetrable objects has been of large interest to physicists and mathematicians since about one century. Many problems coming from microwave techniques, non-destructive testing theory, geophysics, and other fields of engnineering raised new questions particularly with respect to the asymptotic behavior of scalar and rectorial wave-fields near geometrical singularities, like edges and vertices, on one side, and of the far-field patterns, on the other side. Most recently, the small- and long-time behavior of periodic time-dependent scattered fields is in the center of mathematical interest. In this talk an overview of the state of the art will be given and a number of unsolved problems listed.

Elastodynamic Scattering and Matrix Factorisation

The scattering of a time-harmonic elastic wave by a half-plane-shaped crack ∑ is modelled by a Dirichlet boundary value problem. This is shown to be equivalent to a system of Wiener-Hopf equations. The reduced symbol matrix function admits a right canonical factorisation that yields the correctness and explicit solution of the problem. Our factoring technique is based on the spirit of paired singular operators and decomposing commutative but non-rational matrix function algebras.

Integral Equations arising from Instationary Flows around Thin Wings

Within the scope of a three-dimensional linear theory we analyse a mixed screen boundary value problem for the Helmholtz equation (Δ + k2)Φ=0 where Φ is a perturbation velocity potential, generated by the presence of an oscillating wing in a basic flow.

Multiple-Part Wiener-Hopf Operators with Some Applications in Mathematical Physics

The diffraction of electro-magnetic waves by a finite number of semi-infinite scatterers leads to multiple part Wiener-Hopf equations with N elliptic operators acting on a Banach space. In the case of two complementary bounded projectors the problem of inversion is equivalent to that for a Toeplitz operator acting on the range of one of the projectors. Sufficient conditions for the invertibility of N-part WHOs are derived and an algorithm for the reduction of N-part operators to (N-1)-part operators on subspaces is displayed. In the special case of three-part operators they may be reduced also to a system of reciprocal WHOs like for the slit problem in diffraction theory.