Shoaling of subharmonic gravity waves

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 similar toh(-1/4) (Green's law) for the lower frequencies to the shallow-water equilibrium limit for forced subharmonics similar toh(-5/2) 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 breakpoint-generated waves or vice versa.