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Water waves with moving boundaries

Published online by Cambridge University Press:  26 October 2017

Athanasios S. Fokas  and
Konstantinos Kalimeris Open the ORCID record for Konstantinos Kalimeris [Opens in a new window]
Athanasios S. Fokas*
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 0WA, UK Viterbi School of Engineering, University of Southern California, Los Angeles, CA 90089-2560, USA
Konstantinos Kalimeris
Affiliation:
Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Altenberger Straße 69, 4040 Linz, Austria
*
†Email address for correspondence: [email protected]
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Abstract

The unified transform, also known as the Fokas method, provides a powerful methodology for studying boundary value problems. Employing this methodology, we analyse inviscid, irrotational, two-dimensional water waves in a bounded domain, and in particular we study the generation of waves by a moving piecewise horizontal bottom, as it occurs in tsunamis. We show that this problem is characterised by two equations which involve only first-order derivatives. It is argued that under the assumptions of ‘small amplitude waves’ but not of ‘long waves’, the above two equations can be treated numerically via a recently introduced numerical technique for elliptic partial differential equations in a polygonal domain. In the particular case that the moving bottom is horizontal and under the assumption of ‘small amplitude waves’, but not of ‘long waves’, these equations yield a non-local generalisation of the Boussinesq system. Furthermore, under the additional assumption of ‘long waves’ the above system yields a Boussinesq-type system, which however includes the effect of the moving boundary.

JFM classification

Mathematical Foundations: General fluid mechanics Geophysical and Geological Flows: Ocean processes Waves/Free-surface Flows: Surface gravity waves
Type
Papers
Information
Journal of Fluid Mechanics , Volume 832 , 10 December 2017 , pp. 641 - 665
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Copyright
© 2017 Cambridge University Press 

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