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The singularity expansion method and near-trapping of linear water waves

Published online by Cambridge University Press:  19 August 2014

Michael H. Meylan  and
Colm J. Fitzgerald
Michael H. Meylan*
Affiliation:
School of Mathematical and Physical Sciences, The University of Newcastle, Callaghan, NSW 2308, Australia
Colm J. Fitzgerald
Affiliation:
Department of Engineering Science, University of Oxford, Oxford OX1 3PJ, UK
*
†Email address for correspondence: mike.meylan@newcastle.edu.au
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Abstract

The problem of near-trapping of linear water waves in the time domain for rigid bodies or variations in bathymetry is considered. The singularity expansion method (SEM) is used to give an approximation of the solution as a projection onto a basis of modes. This requires a modification of the method so that the modes, which grow towards infinity, can be correctly normalized. A time-dependent solution, which allows for possible trapped modes, is introduced through the generalized eigenfunction method. The expression for the trapped mode and the expression for the near-trapped mode given by the SEM are shown to be closely connected. A numerical method that allows the SEM to be implemented is also presented. This method combines the boundary element method with an eigenfunction expansion, which allows the solution to be extended analytically to complex frequencies. The technique is illustrated by numerical simulations for geometries that support near-trapping.

JFM classification

Waves/Free-surface Flows: Surface gravity waves Waves/Free-surface Flows: Wave scattering
Type
Papers
Information
Journal of Fluid Mechanics , Volume 755 , 25 September 2014 , pp. 230 - 250
Copyright
© 2014 Cambridge University Press 

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Meylan and Fitzgerald supplementary movie

The exact (solid line) and SEM (dashed line) solution for the time shown with the variation in fluid floor plotted for illustration. Stills from the movie are shown in Figure 3.

Download Meylan and Fitzgerald supplementary movie(Video)
Video 6.4 MB

Meylan and Fitzgerald supplementary movie

The exact (solid line) and SEM (dashed line) solution for the time shown with the variation in fluid floor plotted for illustration. Stills from the movie are shown in Figure 4.

Download Meylan and Fitzgerald supplementary movie(Video)
Video 6.5 MB

Meylan and Fitzgerald supplementary movie

The exact (solid line) and SEM (dashed line) solution for the time shown with circular bodies plotted for illustration. Stills from the movie are shown in Figure 5.

Download Meylan and Fitzgerald supplementary movie(Video)
Video 10.1 MB

Meylan and Fitzgerald supplementary movie

The exact (solid line) and SEM (dashed line) solution for the time shown with the two docks plotted for illustration. Stills from the movie are shown in Figure 6.

Download Meylan and Fitzgerald supplementary movie(Video)
Video 9.6 MB

Meylan and Fitzgerald supplementary movie

The exact (solid line) and SEM (dashed line) solution for the time shown with the two docks plotted for illustration. Stills from the movie are shown in Figure 7.

Download Meylan and Fitzgerald supplementary movie(Video)
Video 9.8 MB

Meylan and Fitzgerald supplementary movie

The SEM (dashed line) solution for the time shown with the bodies and variation in fluid floor plotted for illustration. Stills from the movie are shown in Figure 11.

Download Meylan and Fitzgerald supplementary movie(Video)
Video 10 MB

Meylan and Fitzgerald supplementary movie

The SEM (dashed line) solution for the time shown with the bodies and variation in fluid floor plotted for illustration. Stills from the movie are shown in Figure 12.

Download Meylan and Fitzgerald supplementary movie(Video)
Video 9.8 MB