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Analytical solutions for tsunami runup on a plane beach: single waves, N-waves and transient waves

Published online by Cambridge University Press:  22 February 2010

PER A. MADSEN  and
HEMMING A. SCHÄFFER
PER A. MADSEN*
Affiliation:
Department of Mechanical Engineering, Technical University of Denmark, 2800 Kgs Lyngby, Denmark
HEMMING A. SCHÄFFER
Affiliation:
Department of Mechanical Engineering, Technical University of Denmark, 2800 Kgs Lyngby, Denmark
*
†Email address for correspondence: prm@mek.dtu.dk
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Abstract

In the literature it has so far been common practice to consider solitary waves and N-waves (composed of solitary waves) as the appropriate model of tsunamis approaching the shoreline. Unfortunately, this approach is based on a tie between the nonlinearity and the horizontal length scale (or duration) of the wave, which is not realistic for geophysical tsunamis. To resolve this problem, we first derive analytical solutions to the nonlinear shallow-water (NSW) equations for the runup/rundown of single waves, where the duration and the wave height can be specified separately. The formulation is then extended to cover leading depression N-waves composed of a superposition of positive and negative single waves. As a result the temporal variations of the runup elevation, the associated velocity and breaking criteria are specified in terms of polylogarithmic functions. Finally, we consider incoming transient wavetrains generated by monopole and dipole disturbances in the deep ocean. The evolution of these wavetrains, while travelling a considerable distance over a constant depth, is influenced by weak dispersion and is governed by the linear Korteweg–De Vries (KdV) equation. This process is described by a convolution integral involving the Airy function. The runup on the plane sloping beach is then determined by another convolution integral involving the incoming time series at the foot of the slope. A good agreement with numerical model results is demonstrated.

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Papers
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Journal of Fluid Mechanics , Volume 645 , 25 February 2010 , pp. 27 - 57
Copyright
Copyright © Cambridge University Press 2010

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Footnotes

‡

Present address: SchäfferWaves, Sortedam Dossering 59 D st., DK-2100 Copenhagen Ø, Denmark. Email address: Hemming@SchafferWaves.dk

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