A description is given of a model developed for the prediction of the dissipation of energy in random waves breaking on a beach. The dissipation rate per breaking wave is estimated from that in a bore of corresponding height, while the probability of occurrence of breaking waves is estimated on the basis of a wave height distribution with an upper cut-off which in shallow water is determined mainly by the local depth. A comparison with measurements of wave height decay and set-up, on a plane beach and on a beach with a bar-trough profile, indicates that the model is capable of predicting qualitatively and quantitatively all the main features of the data.

A description is given of a model developed for the prediction of the dissipation of energy in random waves breaking on a beach. The dissipation rate per breaking wave is estimated from that in a bore of corresponding height, while the probability of occurrence of breaking waves is estimated on the basis of a wave height distribution with an upper cut-off which in shallow water is determined mainly by the local depth. A comparison with measurements of wave height decay and set-up, on a plane beach and on a beach with a bar-trough profile, indicates that the model is capable of predicting qualitatively and quantitatively all the main features of the data.

[1] In this paper the energy budget of wave group-induced subharmonic gravity waves in the nearshore region is examined on the basis of the energy equation for long waves in conjunction with analyses of a high-resolution laboratory data set of one-dimensional random wave propagation over a barred beach. The emphasis is on the growth of forced subharmonics and the deshoaling of the reflected free waves in the shoaling zone. The incident lower-frequency subharmonics are nearly fully reflected at the shoreline, but the higher-frequency components appear to be subject to a significant dissipation in a narrow inshore zone including the swash zone. The previously reported phase lag of the incident forced waves behind the short-wave groups is confirmed, and its key role in the transfer of energy between the grouped short waves and the shoaling bound waves is highlighted. The cross-shore variation of the local mean rate of this energy transfer is determined. Using this as a source function in the wave energy balance allows a very accurate prediction of the enhancement of the forced waves in the shoaling zone, where dissipation is insignificant. The phase lag appears to increase with increasing frequency, which is reflected in a frequency-dependent growth rate, varying very nearly from the free-wave variation $ h À1/4 (Green's law) for the lower frequencies to the shallow-water equilibrium limit for forced subharmonics $h À5/2 for the higher frequencies. This observed frequency dependence is tentatively generalized to a dependence on a normalized bed slope, controlling whether a so-called mild-slope regime or a steep-slope regime prevails, in which enhanced incident forced waves dominate over breakpointgenerated waves or vice versa.

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This paper deals with the following aspects of periodic water waves breaking on a plane slope breaking criterion, breaker type, phase difference across the surfzone, breaker height-to-depth ratio, run-up and set-up, and reflection. It is shown that these are approximately governed by a single similarity parameter only, embodying both the effects of slope angle and incident wave steepness. Various physical interpretations of this similarity parameter are given, while its role is discussed m general terms from the viewpoint of model prototype similarity.

[1] The growth rate, shoreline reflection, and dissipation of low-frequency waves are investigated using data obtained from physical experiments in the Delft University of Technology research flume and by parameter variation using the numerical model Delft3D-SurfBeat. The growth rate of the shoaling incoming long wave varies with depth with an exponent between 0.25 and 2.5. The exponent depends on a dimensionless normalized bed slope parameter b, which distinguishes between a mild-slope regime and a steep-slope regime. This dependency on b alone is valid if the forcing short waves are not in shallow water; that is, the forcing is off-resonant. The b parameter also controls the reflection coefficient at the shoreline because for small values of b, long waves are shown to break. In this mild-slope regime the dissipation due to breaking of the long waves in the vicinity of the shoreline is much higher than the dissipation due to bottom friction, confirming the findings of Thomson et al. (2006) and Henderson et al. (2006). The energy transfer from low frequencies to higher frequencies is partly due to triad interactions between low-and high-frequency waves but with decreasing depth is increasingly dominated by long-wave self-self interactions, which cause the long-wave front to steepen up and eventually break. The role of the breaking process in the near-shore evolution of the long waves is experimentally confirmed by observations of monochromatic free long waves propagating on a plane sloping beach, which shows strikingly similar characteristics, including the steepening and breaking.

[1] The cross-shore propagation of group-bound long waves is investigated. A detailed laboratory data set from Boers [1996] is analyzed using primarily the cross-correlation function for a sequence of closely spaced cross-shore locations, thus visualizing the propagation of the short-wave envelope and attendant low-frequency motion in detail. The results confirm the previously observed lag of the forced subharmonics behind the short-wave envelope that increases with decreasing water depth. The forced subharmonics are found to be released and reflected at the shoreline and to propagate in offshore direction as free waves. A theoretical, linear model for the forced wave evolution accurate to first order in the relative bottom slope is presented; it predicts a bottom-slope induced, spatially varying phase shift between the short-wave envelope and forced waves which is in good agreement with the observations. The phase shift has dynamical consequences since it allows energy transfer between the short-wave groups and the forced low-frequency response.

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