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

David Pritchard, Laura Dickinson

Research output: Contribution to journalArticlepeer-review

16 Citations (Scopus)


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.
Original languageEnglish
Pages (from-to)413-436
JournalJournal of Fluid Mechanics
Publication statusPublished - 2007


  • nonlinear shallow-water equations
  • fluid mechanics
  • statistics

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