Run-up amplification of transient long waves

The extreme characteristics of the run-up of transient long waves are studied in this paper. First, we give a brief overview of the existing theory which is mainly based on the hodograph transformation (Carrier and Greenspan (1958)). Then, using numerical simulations, we build on the work of Stefanakis et al. (2011) for an infinite sloping beach and we find that resonant run-up amplification of monochromatic waves is robust to spectral perturbations of the incoming wave and that resonant regimes do exist for certain values of the frequency. In the canonical problem of a finite beach attached to a constant depth region, resonance can only be observed when the incoming wavelength is larger than the distance from the undisturbed shoreline to the seaward boundary. Wavefront steepness is also found to affect wave run-up, with steeper waves reaching higher run-up values.