A GENERAL SPECTRAL APPROACH TO THE TIME-DOMAIN EVOLUTION OF LINEAR WATER WAVES IMPACTING ON A VERTICAL ELASTIC PLATE

We present a solution in the time-domain to the two-dimensional linear water-wave problem, in which a semi-infinite fluid region is bounded on one side by a vertical elastic plate. The problem is solved using a generalized eigenfunction expansion from the solutions for single frequencies, and we beg...

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Bibliographic Details
Published in:SIAM journal on applied mathematics Vol. 70; no. 7/8; pp. 2308 - 2328
Main Authors: PETER, MALTE A., MEYLAN, MICHAEL H.
Format: Journal Article
Language:English
Published: Philadelphia Society for Industrial and Applied Mathematics 01.01.2010
Subjects:
Approximation
Boundary conditions
Buoyancy
Computational fluid dynamics
Eigenfunctions
Eigenvalues
Elastic plates
Evolution
Mathematical analysis
Mathematical models
Quadrants
Singularities
Studies
Water waves
Waves
ISSN:0036-1399, 1095-712X
Online Access:Get full text
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Summary:We present a solution in the time-domain to the two-dimensional linear water-wave problem, in which a semi-infinite fluid region is bounded on one side by a vertical elastic plate. The problem is solved using a generalized eigenfunction expansion from the solutions for single frequencies, and we begin with a novel solution of the single-frequency problem. By formulating the problem using the acceleration potential, we find an inner-product space, in which the evolution operator with continuous spectrum is self-adjoint. This inner-product space is required for the generalized eigenfunction solution, which allows us to prescribe arbitrary initial water-surface and plate displacements and velocities. Furthermore, using the generalized eigenfunction expansion, the solution is approximated by deforming the contour of integration and using the contributions from the singularities of the analytic continuation. Numerical experiments show that the long-time behavior in certain situations can be closely approximated by this method.
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ISSN:0036-1399
1095-712X
DOI:10.1137/090756557