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Time-dependent water-wave scattering by arrays of cylinders and the approximation of near trapping

Published online by Cambridge University Press:  17 July 2009

MICHAEL H. MEYLAN  and
RODNEY EATOCK TAYLOR
MICHAEL H. MEYLAN*
Affiliation:
Department of Mathematics, The University of Auckland, New Zealand
RODNEY EATOCK TAYLOR
Affiliation:
Department of Engineering Science, Oxford University, UK
*
†Email address for correspondence: [email protected]
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Abstract

We consider the solution in the time domain of water-wave scattering by arrays of bottom-mounted cylinders. It has already been shown that near trapping occurs for certain arrangements of cylinders and we are especially focused on this phenomenon. We begin with the well-known single-frequency solution to the problem of a group of cylinders, and the extension of this solution to complex frequencies. It has been shown that singularities (scattering frequencies or resonances) occur for certain values of the complex frequency and these singularities are associated with the near-trapped mode. We show that it is possible to approximate the solution near these singularities, and produce a modal shape which is associated with the near-trapped mode. We then consider the time-dependent problem, beginning with the well-known incident plane wave packet solution. We also show how the problem of an arbitrary initial displacement can be found using the single-frequency solutions. This latter result relies on a special inner product which gives a generalized eigenfunction expansion (because the operator has a continuous spectrum). We then consider the approximation of the time-dependent motion using special mode shapes associated with the scattering frequencies. This approximation relies on the scattering frequencies lying close to the real axis. We present numerical results which show that this approximation is accurate for sufficiently large time.

JFM classification

Waves/Free-surface Flows: Wave-structure interactions Waves/Free-surface Flows: Wave scattering
Type
Papers
Information
Journal of Fluid Mechanics , Volume 631 , 25 July 2009 , pp. 103 - 125
Copyright
Copyright © Cambridge University Press 2009

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References

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Meylan and Taylor supplementary movie

Movie 1. The near trapped mode for four cylinders associated with the scattering frequency at 2.7641 - 0.0122i.

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Video 490 KB

Meylan and Taylor supplementary movie

Movie 2. The near trapped mode for nine cylinders associated with the scattering frequency at 2.7114 - 0.0041i

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Video 818.7 KB

Meylan and Taylor supplementary movie

Movie 3. One of the near trapped modes for nine cylinders associated with the scattering frequency at 2.7635 - 0.0086i

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Video 779.8 KB

Meylan and Taylor supplementary movie

Movie 4. One of the near trapped modes for nine cylinders associated with the scattering frequency at 2.7635 - 0.0086i

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Video 721.9 KB

Meylan and Taylor supplementary movie

Movie 5. The near trapped mode for nine cylinders associated with the scattering frequency at 2.8284 - 0.0102i

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Video 840.2 KB

Meylan and Taylor supplementary movie

Movie 6. The true (a) and approximate (b) solution for four cylinders for a plane incident wave given by equation (4.7).

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Video 2.3 MB

Meylan and Taylor supplementary movie

Movie 7. The true (a) and approximate (b) solution for nine cylinders for a plane incident wave given by equation (4.7).

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Video 3.8 MB

Meylan and Taylor supplementary movie

Movie 8. The true (a) and approximate (b) solution for four cylinders for a plane incident wave given by equation (4.8). We do not start the approximate solution until t=0.

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Video 4.1 MB

Meylan and Taylor supplementary movie

Movie 9. The true (a) and approximate solution (b) for an incident displacement given by equation (4.10) for four cylinders.

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Video 2 MB

Meylan and Taylor supplementary movie

Movie 10. The true (a) and approximate solution (b) for an incident displacement given by equation (4.10) for nine cylinders.

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Video 2.1 MB