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Generalized herglotz domains
Author(s) -
Colton D.,
Wimp J.,
Hsiao G. C.
Publication year - 1986
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.1670080129
Subject(s) - mathematics , mathematical analysis , eigenvalues and eigenvectors , scattering , dirichlet boundary condition , boundary value problem , domain (mathematical analysis) , plane (geometry) , boundary (topology) , function (biology) , fourier transform , field (mathematics) , cylinder , pure mathematics , geometry , physics , optics , quantum mechanics , evolutionary biology , biology
Let F (θ k, α) be the far field pattern arising from the scattering of a time harmonic plane acoustic wave of wave number k and direction a by a sound‐soft cylinder of cross section D . Suppose F has the Fourier expansionwhere a n = a n ( k , . Then if ϰ 2 is a Dirichlet eigenvalue for D , sufficient conditions are given on D for the existence of a nontrivial sequence | b n | where the b n are independent of such that for all directionsDomains for which this is true are called generalized Herglotz domains. The conditions for a domain to be a generalized Herglotz domain are given either in terms of the Schwarz function for the analytic boundary ϱ D or in terms of the Rayleigh hypothesis in acoustic scattering theory and examples are given showing the applicability of these conditions.