The Wiener–Hopf–Hilbert method for diffraction problems

Abstract

A powerful method of solving diffraction problems is described. In it, a system of Wiener–Hopf equations is transformed into a system of Hilbert problems. The latter apparently can be solved fora wider range of problems than the former. As an example, the diffraction problem of a plane wave incident on a half-plane with different face impedances is considered. This problem is not solvable by ordinary Wiener–Hopf techniques, but yields to the new method. Possible applications to some unsolved problems are mentioned.

On the Solution of Wiener--Hopf Problems Involving Noncommutative Matrix Kernel Decompositions

“…which, when µ = 1, is the problem studied previously by various authors, including Rawlins [31], Hurd [21], and Daniele [14]. This kernel has a commutative factorization, which is perhaps best written in the Khrapkov form [26],…”

“…Khrapkov [26,27], in articles concerned with the stresses in elastostatic wedges with notches, was the first author to express the commutative factorization in a form which indicates the subalgebra associated with this class of kernels. Many other authors have also examined the commutative cases, and, in particular, reference should be made to the ingenious approach to the simplest nontrivial commutative example by Rawlins [31], the direct scheme of Daniele [14], and the Hilbert problem technique of Hurd [21]. The latter approach, which transforms the factorization problem to a pair of uncoupled Hilbert problems defined on a semi-infinite branch cut, is generalizable to an interesting range of cases [32], one example of which is explored in [1].…”

A brief historical perspective of the Wiener–Hopf technique

“…Matrix Wiener-Hopf kernels are fundamentally distinct from their scalar counterparts in that there is no algorithmic approach to determining the factorization (4) of the transformed kernel [16]. Exact factorization can be achieved for matrices with certain special features: those that are upper (or lower) triangular in form; those that are of Khrapkov-Daniele, i.e., commutative, form (see [17][18][19][20]); those whose elements comprise meromorphic functions [21,22]; kernels with special singularity structure that allows the Wiener-Hopf equation to be recast into uncoupled Riemann-Hilbert problems [23][24][25]; and N × N matrices with special algebraic or group structure [26][27][28]. For more details on exact matrix kernel factorization the interested reader is referred to the last mentioned article and to references cited in [29].…”

Diffraction of an obliquely incident plane wave by a two‐face impedance half plane: Wiener‐Hopf approach

“…In the case of different face impedances the Wiener-Hopf formulation for perpendicular incidence leads to a system of two coupled fi•nctional equations that cannot be decoupled trivially. The factorization of the 2 x 2 matrix filnction was first achieved by Hurd [1976] Extensions of some of these factorization methods from 2 x 2 matrices to matrices of higher order are in principle possible, but the explicit procedures are cumbersome and impractical in many cases. Some particular systems of 4 x 4 order that can be decoupied into two 2 x 2 systems have been considered by Daniele [1984aDaniele [ , 1984b [Senior, 1959[Senior, , 1975a[Senior, , 1975b[Senior, , 1978[Senior, , 1986[Senior, , 1989[Senior, , 1991 It will be shown that consideration of the range of the mapping from the tangential to the normal field components and vice versa is equivalent to this conventional physically based remsoning of the removal of unwanted pole contributions.…”