  The extreme characteristics of long wave run-up are studied in this paper.
First we give a brief overview of the existing theory which is mainly based on
the hodograph transformation (Carrier & 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 resonant
regimes do exist for certain values of the frequency. In the setting 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 play a role in wave run-up, with steeper waves reaching higher run-up
values.
