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

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.

Type
Papers
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