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Generalized Eigenfunctions for Half‐Plane Diffraction: Rawlins' Problem with Real Wave Numbers
Author(s) -
Rottbrand K.
Publication year - 1998
Publication title -
zamm ‐ journal of applied mathematics and mechanics / zeitschrift für angewandte mathematik und mechanik
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.449
H-Index - 51
eISSN - 1521-4001
pISSN - 0044-2267
DOI - 10.1002/zamm.19980781595
Subject(s) - mathematics , factorization , eigenfunction , mathematical analysis , orthonormal basis , pure mathematics , mathematical physics , eigenvalues and eigenvectors , physics , quantum mechanics , algorithm
This paper deals with a famous diffraction problem for Σ : × < 0, y = 0 as an obstacle for some time‐harmonic plane incident wave field. RAWLINS [5] was the first to solve the corresponding mixed (D/N) b. v. p. to the scalar Helmholtzian in 1975. And he was the first to solve the equivalent pair of coupled Wiener‐Hopf equations explicitly by factoring [6] their discontinuous 2 × 2 Fourier matrix symbol in 1980. Although for real wave numbers k the factorization procedure fails it will serve as the basis: Following the lines given by ALI MEHMETI [1] in his habilitation thesis (1994) for the (D/D) b. v. p. we combine the idea of integral path deforming onto the branch cuts of ✓ζ 2 ‐k 2 in MEISTER's book [3] (1983) with the modern Wiener‐Hopf method solution derived by SPECK [9] (1989) in a H 1+ ϵ, ϵ 0 Sobolev space setting. The symmetry of the intermediate spaces H 3 , H ‐5 , |s| < 1/2, which is due to generalized factorization plays a key role in order to get Laplace integral representations of the generalized eigenfunctions of the problem which allow real wave numbers k. As a remarkable fact 0 < ϵ < 1/4 must hold.