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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| Veröffentlicht in: | SIAM journal on applied mathematics Jg. 70; H. 7/8; S. 2308 - 2328 |
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Society for Industrial and Applied Mathematics
01.01.2010
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| ISSN: | 0036-1399, 1095-712X |
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| Abstract | 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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| AbstractList | 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. 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. [PUBLICATION ABSTRACT] |
| Author | MEYLAN, MICHAEL H. PETER, MALTE A. |
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| CitedBy_id | crossref_primary_10_1080_17455030_2022_2120220 crossref_primary_10_1007_s00773_011_0155_9 crossref_primary_10_1017_jfm_2018_207 crossref_primary_10_1017_S0022112010005653 crossref_primary_10_3846_mma_2020_10315 crossref_primary_10_1016_j_oceaneng_2024_117932 crossref_primary_10_1016_j_jfluidstructs_2017_10_002 crossref_primary_10_1016_j_oceaneng_2023_115368 crossref_primary_10_1016_j_apor_2014_02_001 crossref_primary_10_1007_s00033_016_0709_0 crossref_primary_10_1016_j_coastaleng_2022_104245 crossref_primary_10_3390_jmse12061004 crossref_primary_10_1007_s40997_023_00686_2 crossref_primary_10_1007_s00419_022_02342_8 crossref_primary_10_1007_s10665_011_9492_2 crossref_primary_10_1016_j_wavemoti_2024_103421 crossref_primary_10_1016_j_apor_2012_02_003 crossref_primary_10_1016_j_jfluidstructs_2016_11_012 crossref_primary_10_1016_j_oceaneng_2018_07_054 crossref_primary_10_1007_s00773_012_0191_0 |
| Cites_doi | 10.1137/0524084 10.1023/A:1004389503576 10.1016/j.engstruct.2003.10.001 10.1002/cpa.3160300303 10.1137/060665208 10.1017/S002211200900723X 10.1017/S0022112009007204 10.1007/BF00251419 10.1016/j.oceaneng.2004.02.003 10.1146/annurev.fl.27.010195.000555 10.1017/S0308210506000138 10.1146/annurev.fluid.35.101101.161153 10.1051/m2an:2007040 10.1007/BF00252896 10.1016/j.coldregions.2007.04.007 |
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| References | R5 R7 R8 R9 R10 Meylan M. H. (R19) 2003; 44 R20 Povzner A. Y. (R22) 1953; 32 R23 R11 R25 R24 R15 R26 R18 Friedman A. (R4) 1967; 17 R17 R1 |
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| SubjectTerms | Approximation Boundary conditions Buoyancy Computational fluid dynamics Eigenfunctions Eigenvalues Elastic plates Evolution Mathematical analysis Mathematical models Quadrants Singularities Studies Water waves Waves |
| Title | A GENERAL SPECTRAL APPROACH TO THE TIME-DOMAIN EVOLUTION OF LINEAR WATER WAVES IMPACTING ON A VERTICAL ELASTIC PLATE |
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