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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: t.fokas@damtp.cam.ac.uk
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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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References

Ablowitz, M. J., Fokas, A. S. & Musslimani, Z. H. 2006 On a new non-local formulation of water waves. J. Fluid Mech. 562, 313–343.CrossRefGoogle Scholar
Ablowitz, M. J. & Haut, T. S. 2008 Spectral formulation of the two fluid euler equations with a free interface and long wave reductions. Anal. Applics. 6 (04), 323–348.Google Scholar
Ablowitz, M. J. & Haut, T. S. 2009 Asymptotic expansions for solitary gravity-capillary waves in two and three dimensions. Proc. R. Soc. Lond. A, rspa20090112.Google Scholar
Ambrose, D. M. & Nicholls, D. P. 2014 Fokas integral equations for three dimensional layered-media scattering. J. Comput. Phys. 276, 1–25.Google Scholar
Ashton, A. C. L. & Fokas, A. S. 2011 A non-local formulation of rotational water waves. J. Fluid Mech. 689, 129–148.Google Scholar
Burini, D., De Lillo, S. & Skouteris, D. 2015 On a coupled system of shallow water equations admitting travelling wave solutions. Math. Prob. Engng 2015, 197978.Google Scholar
Davis, C. R. & Fornberg, B. 2014 A spectrally accurate numerical implementation of the Fokas transform method for Helmholtz-type PDES. Complex Var. Elliptic Equ. 59 (4), 564–577.Google Scholar
Deconinck, B. & Oliveras, K. 2011 The instability of periodic surface gravity waves. J. Fluid Mech. 675, 141–167.CrossRefGoogle Scholar
Deconinck, B. & Trichtchenko, O. 2014 Stability of periodic gravity waves in the presence of surface tension. Eur. J. Mech. (B/Fluids) 46, 97–108.CrossRefGoogle Scholar
Fokas, A. S. 1997 A unified transform method for solving linear and certain nonlinear PDEs. Proc. R. Soc. Lond. A 453, 1411–1443.CrossRefGoogle Scholar
Fokas, A. S. 2000 On the integrability of linear and nonlinear partial differential equations. J. Math. Phys. 41 (6), 4188–4237.Google Scholar
Fokas, A. S., Flyer, N., Smitheman, S. A. & Spence, E. A. 2009 A semi-analytical numerical method for solving evolution and elliptic partial differential equations. J.  Comput. Appl. Maths 227 (1), 59–74.CrossRefGoogle Scholar
Fokas, A. S. & Nachbin, A. 2012 Water waves over a variable bottom: a non-local formulation and conformal mappings. J. Fluid Mech. 695, 288–309.Google Scholar
Fornberg, B. & Flyer, N. 2011 A numerical implementation of Fokas boundary integral approach: Laplace’s equation on a polygonal domain. Proc. R. Soc. Lond. A, rspa20110032.Google Scholar
Fulton, S. R., Fokas, A. S. & Xenophontos, C. A. 2004 An analytical method for linear elliptic PDEs and its numerical implementation. J. Comput. Appl. Maths 167 (2), 465–483.Google Scholar
Hashemzadeh, P., Fokas, A. S. & Smitheman, S. A. 2015 A numerical technique for linear elliptic partial differential equations in polygonal domains. Proc. Math. Phys. Engng Sci. 471 (2175), 20140747.Google Scholar
Kânoğlu, U. & Synolakis, C. E. 2006 Initial value problem solution of nonlinear shallow water-wave equations. Phys. Rev. Lett. 97 (14), 148501.Google Scholar
Nicholls, D. 2016 A high–order perturbation of surfaces (hops) approach to Fokas integral equations: three-dimensional layered-media scattering. Q. Appl. Maths 74 (1), 61–87.Google Scholar
Oliveras, K.2009 Stability of periodic surface gravity water waves. PhD thesis, University of Washington.Google Scholar
Saridakis, Y. G., Sifalakis, A. G. & Papadopoulou, E. P. 2012 Efficient numerical solution of the generalized Dirichlet–Neumann map for linear elliptic PDEs in regular polygon domains. J. Comput. Appl. Maths 236 (9), 2515–2528.CrossRefGoogle Scholar
Synolakis, C. E. & Okal, E. A. 2005 1992–2002: perspective on a decade of post-tsunami surveys. In Tsunamis, pp. 1–29. Springer.Google Scholar
Tadepalli, S. & Synolakis, C. E. 1994 The run-up of N-waves on sloping beaches. Proc. R. Soc. Lond. A 445, 99–112.Google Scholar
Vasan, V. & Deconinck, B. 2013 The inverse water wave problem of bathymetry detection. J. Fluid Mech. 714, 562–590.Google Scholar