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The Impedance Boundary Value Problem for the Helmholtz Equation in a Half‐Plane
Author(s) -
ChandlerWilde S. N.
Publication year - 1997
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/(sici)1099-1476(19970710)20:10<813::aid-mma883>3.0.co;2-r
Subject(s) - helmholtz equation , mathematics , mathematical analysis , fredholm integral equation , boundary value problem , integral equation , uniqueness , plane (geometry) , diffraction , scattering , geometry , physics , optics
We prove unique existence of solution for the impedance (or third) boundary value problem for the Helmholtz equation in a half‐plane with arbitrary L ∞ boundary data. This problem is of interest as a model of outdoor sound propagation over inhomogeneous flat terrain and as a model of rough surface scattering. To formulate the problem and prove uniqueness of solution we introduce a novel radiation condition, a generalization of that used in plane wave scattering by one‐dimensional diffraction gratings. To prove existence of solution and a limiting absorption principle we first reformulate the problem as an equivalent second kind boundary integral equation to which we apply a form of Fredholm alternative, utilizing recent results on the solvability of integral equations on the real line in [5]. © 1997 B. G. Teubner Stuttgart–John Wiley & Sons Ltd.