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Time-dependent linear water-wave scattering in two dimensions by a generalized eigenfunction expansion

Published online by Cambridge University Press:  27 July 2009

MICHAEL H. MEYLAN
MICHAEL H. MEYLAN*
Affiliation:
Department of Mathematics, The University of Auckland, New Zealand
*
†Email address for correspondence: meylan@math.auckland.ac.nz
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Abstract

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We consider the solution in the time domain of the two-dimensional water-wave scattering by fixed bodies, which may or may not intersect with the free surface. We show how the problem with arbitrary initial conditions can be found from the single-frequency solutions using a generalized eigenfunction expansion, required because the operator has a continuous spectrum. From this expansion we derive simple formulas for the evolution in time of the initial surface conditions, and we present some examples of numerical calculations.

JFM classification

Waves/Free-surface Flows: Wave-structure interactionsWaves/Free-surface Flows: Wave scattering

Information

Type
Papers
Information
Journal of Fluid Mechanics , Volume 632 , 10 August 2009, pp. 447 - 455
Copyright
Copyright © Cambridge University Press 2009

References

REFERENCES

Friedman, A. & Shinbrot, M. 1967 The initial value problem for the linearized equations of water waves. J. Math. Mech. 17 (2), 107–180.Google Scholar
Hazard, C. & Lenoir, M. 2002 Surface water waves. In Scattering (ed. R. P. P. Sabatier), pp. 618–636. Academic.CrossRefGoogle Scholar
Hazard, C. & Loret, F. 2007 Generalized eigenfunction expansions for scattering problems with an application to water waves. Proc. R. Soc. Edinburgh 137A, 995–1035.CrossRefGoogle Scholar
Hazard, C. & Meylan, M. H. 2007 Spectral theory for a two-dimensional elastic thin plate floating on water of finite depth. SIAM J. Appl. Math. 68 (3), 629–647.Google Scholar
Ikebe, T. 1960 Eigenfunction expansions associated with the Schroedinger operators and their applications to scattering theory. Arch. Ration. Mech. Anal. 5, 1–34.Google Scholar
Linton, C. M. & Evans, D. V. 1991 Trapped modes above a submerged horizontal plate. Q. J. Mech. Appl. Math. 44 (3), 487–506.CrossRefGoogle Scholar
Linton, C. M. & McIver, P. 2001 Handbook of Mathematical Techniques for Wave/Structure Interactions. Chapman & Hall/CRC.Google Scholar
McIver, M. 1996 An example of non-uniqueness in the two-dimensional water wave problem. J. Fluid Mech. 315, 257–266.CrossRefGoogle Scholar
Mei, C. C. 1989 The Applied Dynamics of Ocean Surface Waves. World Scientific.Google Scholar
Meylan, M. H. 2002 Spectral solution of time dependent shallow water hydroelasticity. J. Fluid Mech. 454, 387–402.CrossRefGoogle Scholar
Meylan, M. H. 2008 a Time-dependent solution for linear water waves by expansion in the single-frequency solutions. In Proceedings of the ASME 27th Intl Conf. on Offshore Mechanics and Arctic Engineering, OMAE 2008.Google Scholar
Meylan, M. H. 2008 b Wikiwaves. http://www.wikiwaves.org.Google Scholar
Meylan, M. H. & EatockTaylor, R. Taylor, R. 2009 Time-dependent water-wave scattering by arrays of cylinders and the approximation of near trapping. J. Fluid Mech. 631, 103–125.Google Scholar
Meylan, M. H. & Sturova, I. V. 2009 Time-dependent motion of a two-dimensional floating elastic plate. J. Fluids Struct. 25, 445–460.CrossRefGoogle Scholar
Newman, J. N. 1977 Marine Hydrodynamics. MIT Press.CrossRefGoogle Scholar
Ogilvie, T. F. 1964 Recent progress towards the understanding and prediction of ship motions. In Proceedings of the Fifth Symposium Naval Hydrodynamics, pp. 3–97. Office of Naval Research.Google Scholar
Povzner, A. Y. 1953 On the expansions of arbitrary functions in terms of the eigenfunctions of the operator −Δu + cu (in Russian). Mat. Sbornik 32 (74), 109–156.Google Scholar
Stoker, J. J. 1957 Water Waves: The Mathematical Theory with Applications. Interscience.Google Scholar
Wilcox, C. H. 1975 Scattering Theory for the d'Alembert Equation in Exterior Domains. Springer-Verlag.CrossRefGoogle Scholar

Meylan supplementary movie

Movie 1. The evolution of an initial displacement given by equation (4.1) for two docks on the fluid surface -1.75 < x &lt-1 and 1 < x < 1.75. This is also shown in in figure 2. Note that the docks are shown in black as a schematic.

Download Meylan supplementary movie(Video)
Video 2 MB

Meylan supplementary movie

Movie 2. The evolution of an initial displacement given by equation (4.1) for two docks on the fluid surface -1.5 < x < -1 and 1 < x < 1.5. Note that the docks are shown in black as a schematic.

Download Meylan supplementary movie(Video)
Video 2.3 MB

Meylan supplementary movie

Movie 3. The evolution of an initial displacement given by equation (4.1) for two docks on the fluid surface -1.25< x < -1 and 1< x < 1.25. Note that the docks are shown in black as a schematic.

Download Meylan supplementary movie(Video)
Video 1.7 MB

Meylan supplementary movie

Movie 4. The evolution of an initial displacement given by equation (4.1) for two submerged docks occuping z=-0.1 and -1.75< x < -1 and 1< x < 1.75. This is also shown in in figure 3. Note that the docks are shown in black as a schematic.

Download Meylan supplementary movie(Video)
Video 2.4 MB

Meylan supplementary movie

Movie 5. The evolution of an initial displacement given by equation (4.1) for two submerged docks occuping z=-0.2 and -1.5< x < -1 and 1< x < 1.5. This is also shown in in figure 4. Note that the docks are shown in black as a schematic.

Download Meylan supplementary movie(Video)
Video 1.7 MB

Meylan supplementary movie

Movie 6. The evolution of an initial displacement given by equation (4.1) for two submerged docks occuping z=-0.3 and -1.75< x < -1 and 1< x < 1.75. Note that the docks are shown in black as a schematic.

Download Meylan supplementary movie(Video)
Video 1.2 MB