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    Temam, Roger and Petcu, Madalina 2013. An interface problem: The two-layer shallow water equations. Discrete and Continuous Dynamical Systems, Vol. 33, Issue. 11/12, p. 5327.

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    Didenkulova, Ira and Pelinovsky, Efim 2011. Nonlinear wave evolution and runup in an inclined channel of a parabolic cross-section. Physics of Fluids, Vol. 23, Issue. 8, p. 086602.

    PEDERSEN, GEIR K. 2011. Oblique runup of non-breaking solitary waves on an inclined plane. Journal of Fluid Mechanics, Vol. 668, p. 582.

    Petcu, Madalina and Temam, Roger 2011. The one-dimensional shallow water equations with transparent boundary conditions. Mathematical Methods in the Applied Sciences, p. n/a.

    ANTUONO, M. and BROCCHINI, M. 2010. Solving the nonlinear shallow-water equations in physical space. Journal of Fluid Mechanics, Vol. 643, p. 207.

    Matteo, Antuono and Brocchini, Maurizio 2008. Maximum run-up, breaking conditions and dynamical forces in the swash zone: a boundary value approach. Coastal Engineering, Vol. 55, Issue. 9, p. 732.

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  • Journal of Fluid Mechanics, Volume 591
  • November 2007, pp. 413-436

The near-shore behaviour of shallow-water waves with localized initial conditions

  • DOI:
  • Published online: 25 November 2007

We consider the behaviour of solutions to the nonlinear shallow-water equations which describe wave runup on a plane beach, concentrating on the behaviour at and just behind the moving shoreline. We develop regular series expansions for the hydrodynamic variables behind the shoreline, which are valid for any smooth initial condition for the waveform. We then develop asymptotic descriptions of the shoreline motion under localized initial conditions, in particular a localized Gaussian waveform: we obtain estimates for the maximum runup and drawdown of the wave, for its maximum velocities and the forces it is able to exert on objects in its path, and for the conditions under which such a wave breaks down. We show how these results may be extended to include initial velocity conditions and initial waveforms which may be approximated as the sum of several Gaussians. Finally, we relate these results tentatively to the observed behaviour of a tsunami.

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M. Antuono & M. Brocchini 2007 The boundary value problem for the nonlinear shallow water equations. Stud. Appl. Maths 119, 7393.

A. G. Dawson & S. Shi 2000 Tsunami deposits. Pure Appl. Geophys. 157, 875897.

I. S. Gradshteyn & I. M. Ryzhik 2000 Table of Integrals, Series and Products, 6th edn.Academic.

S. T. Grilli , I. A. Svendsen & R. Subramanya 1997 Breaking criterion and characteristics for solitary waves on slopes. J. Waterway Port Coastal Ocean Engng 123 (3), 102112.

P. A. Guard , T. Baldock & P. Nielsen 2005 General solutions for the initial runup of a breaking tsunami front. In Intl Symp. on Disaster Relief on Coasts. Monash University, Australia.

V. Heller , J. Unger & W. Hager 2005 Tsunami runup – a hydraulic perspective. J. Hydraul. Engng 131 (9), 743747.

F. Hoefel & S. Elgar 2003 Wave-induced sediment transport and sandbar migration. Science 299, 18851887.

U. Kânoğlu & C. Synolakis 2006 Initial value problem solution of nonlinear shallow water-wave equations. Phys. Rev. Lett. 97, 148501.

F. Lavigne , C. Gomez , M. Giffo , P. Wassmer , C. Hoebreck , D. Mardiatno , J. Prioyono & R. Paris 2007 Field observations of the 17 July 2006 tsunami in Java. Nat. Haz. Earth Sys. Sci. 7, 177183.

P. A. Luccio , S. I. Voropayev , H. J. S. Fernando , D. L. Boyer & W. N. Houston 1998 The motion of cobbles in the swash zone on an impermeable slope. Coastal Engng 33, 4160.

R. E. Meyer 1986 aOn the shore singularity of water waves. I. The local model. Phys. Fluids 29, 31523163.

R. E. Meyer 1986 bRegularity for a singular conservation law. Adv. Appl. Math. 7, 465501.

J. Nott 2003 Waves, coastal boulder deposits and the importance of the pre-transport setting. Earth Planet. Sci. Lett. 210, 269276.

D. H. Peregrine 1972 Equations for water waves and the approximations behind them. In Waves on Beaches and Resulting Sediment Transport (ed. R. E. Meyer ), chap. 3, pp. 95121. Academic.

J. Polet & H. Kanamori 2000 Shallow subduction zone earthquakes and their tsunamigenic potential. Geophys. J. Intl 142, 684702.

S. Tadepalli & C. E. Synolakis 1994 The runup of N-waves on sloping beaches. Proc. R. Soc. Lond. A 445, 99112.

V. V. Titov & C. E. Synolakis 1998 Numerical modelling of tidal wave runup. J. Waterway Port Coastal Ocean Engng 124 (4), 157171.

H. Yeh 2006 Maximum fluid forces in the tsunami runup zone. J. Waterway Port Coastal Ocean Engng 132 (6), 496500.

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Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
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