Modelling criteria for long water waves

Abstract

Model equations which describe the evolution of long-wave initial data in water of uniform depth are tested to determine explicit criteria for their applicability. We consider linear and nonlinear, dispersive and non-dispersive equations. Separate criteria emerge for the leading wave and trailing oscillations of the evolving wave train. The evolution of the leading wave depends on two parameters: the volume (non-dimensional) of the initial data and an Ursell number based on the amplitude and length of the initial data. The magnitudes of these two parameters determine the appropriate model equation and its time of validity. For the trailing oscillatory waves, a local Ursell number based on the amplitude of the initial data and the local wavelength determines the appropriate model equation. Finally, these modelling criteria are applied to the problem of tsunami propagation. Asymptotic (t → ∞) linear dispersive theory does not appear to be applicable for describing the leading wave of tsunamis. If the length of the initial wave is approximately 100 miles, the leading wave is described by a linear non-dispersive model from the source region until shoaling occurs near the coastline. For smaller lengths (∼ 40 miles) a linear dispersive (but not asymptotic) model is applicable. The longer-period oscillatory waves following the leading wave, which can induce harbour resonance, apparently require a nonlinear dispersive model.