Journal of Fluid Mechanics


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


a1 Department of Mathematics, University of Strathclyde, 26 Richmond St, Glasgow G1 1XH, UK

a2 Division of Civil Engineering, University of Dundee, Dundee DD1 4HN, UK


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.

(Received July 24 2006)

(Revised July 20 2007)