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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

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 interactions Waves/Free-surface Flows: Wave scattering
Type
Papers
Information
Journal of Fluid Mechanics , Volume 632 , 10 August 2009 , pp. 447 - 455
Copyright
Copyright © Cambridge University Press 2009

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References

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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