2 results
Analytical solutions for tsunami runup on a plane beach: single waves, N-waves and transient waves
- PER A. MADSEN, HEMMING A. SCHÄFFER
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- Journal:
- Journal of Fluid Mechanics / Volume 645 / 25 February 2010
- Published online by Cambridge University Press:
- 22 February 2010, pp. 27-57
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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.
Infragravity waves induced by short-wave groups
- Hemming A. Schäffer
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- Journal:
- Journal of Fluid Mechanics / Volume 247 / February 1993
- Published online by Cambridge University Press:
- 26 April 2006, pp. 551-588
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A theoretical model for infragravity waves generated by incident short-wave groups is developed. Both normal and oblique short-wave incidence is considered. The depth-integrated conservation equations for mass and momentum averaged over a short-wave period are equivalent to the nonlinear shallow-water equations with a forcing term. In linearized form these equations combine to a second-order long-wave equation including forcing, and this is the equation we solve. The forcing term is expressed in terms of the short-wave radiation stress, and the modelling of these short waves in regard to their breaking and dynamic surf zone behaviour is essential. The model takes into account the time-varying position of the initial break point as well as a (partial) transmission of grouping into the surf zone. The former produces a dynamic set-up, while the latter is equivalent to the short-wave forcing that takes place outside the surf zone. These two effects have a mutual dependence which is modelled by a parameter K, and their relative strength is estimated. Before the waves break, the standard assumption of energy conservation leads to a variation of the radiation stress, which causes a bound, long wave, and the shoaling bottom results in a modification of the solution known for constant depth. The respective effects of this incident bound, long wave and of oscillations of the break-point position are shown to be of the same order of magnitude, and they oppose each other to some extent. The transfer of energy from the short waves to waves at infragravity frequencies is analysed using the depth-integrated conservation equation of energy. For the case of normally incident groups a semi-analytical steady-state solution for the infragravity wave motion is given for a plane beach and small primary-wave modulations. Examples of the resulting surface elevation as well as the corresponding particle velocity and mean infragravity-wave energy flux are presented. Also the sensitivity to the variation of input parameters is analysed. The model results are compared with laboratory experiments from the literature. The qualitative agreement is good, but quantitatively the model overestimates the infragravity wave activity. This can, in part, be attributed to the neglect of frictional effects.