A universal form for shoreline run-up spectra?

Time series of shoreline run-up on two natural beaches have been measured by using a time-lapse camera. Spectra of these time series and two other run-up spectra measured by Suhayda (1972) suggest that for the frequency band over which incident waves are large enough to break a universal ‘saturation’ form for the vertical run-up spectrum occurs, with energy density E(ƒ) = [∈ˆcsgβ2/(2πƒ)2]2, where g is the gravitational acceleration, β is the beach slope, and ƒ is the frequency (in hertz). Parameter ∈ˆcs(Δƒ)½ is a universal nondimensional constant, found to have a value of about 1, where Δƒ is the bandwidth over which incident waves are large enough to break in the surf zone. This result is discussed in relation to previous laboratory experiments and theories, based on monochromatic waves, which suggest the existence of a limiting amplitude for standing waves formed by reflection at the shoreline. This limiting amplitude is related to a critical parameter ∈cs by a = ∈csgβ2/(2πƒ)2. A possible interpretation of ∈ˆcs(Δƒ)½ in terms of ∈cs is given based on percentage exceedances of the critical downslope acceleration gβ2. In this interpretation we have assumed a Gaussian distribution for run-up acceleration. This assumption cannot be tested directly, but the observed distribution functions for run-up elevation suggest that it may need to be modified. Departures from the universal spectrum athigher and lower frequencies are briefly discussed.