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On the Existence of Trapped Modes in Channels of Arbitrary Cross‐section
Author(s) -
Groves M. D.
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(199704)20:6<521::aid-mma868>3.0.co;2-2
Subject(s) - mathematics , eigenvalues and eigenvectors , boundary value problem , operator (biology) , mathematical analysis , domain (mathematical analysis) , dirichlet distribution , differential operator , section (typography) , boundary (topology) , cross section (physics) , channel (broadcasting) , neumann boundary condition , geometry , physics , quantum mechanics , telecommunications , biochemistry , chemistry , repressor , advertising , transcription factor , computer science , business , gene
Abstract This article establishes the existence of trapped‐mode solutions of a linearized water‐wave problem. The fluid occupies a symmetric horizontal channel that is uniform everywhere apart from a confined region which either contains a thin vertical plate spanning the depth of the channel or has indentations in the channel walls. A trapped mode corresponds to an eigenvalue of a non‐local Neumann–Dirichlet operator for an elliptic boundary‐value problem in the fluid domain. The existence of such an eigenvalue is established by generalizing previous results concerning spectral theory for differential operators to this non‐local operator. © 1997 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd.