On wave set-up in shoaling water with a rough sea bed

In very gradually shoaling coastal water the energy of incident waves appears to be absorbed not by breaking at the upper surface, but predominantly by turbulent dissipation near the rippled sea bed. The question has been asked whether there can be then any wave set-up, that is any increase in the mean water level, at, or close to, the shoreline. To answer this question the equations of wave energy and momentum in water of slowly varying depth are generalized so as to include the presence of a dissipative boundary layer at the bottom. It is then shown that the resulting equation for the mean surface slope can be integrated exactly, to give the mean surface depression (the ‘set-down’) in terms of the local wave amplitude and water depth, outside the surf zone. In the special case of a uniform beach slope $s$, a closed expression is obtained for the wave amplitude in terms of the local depth, under two different sets of conditions: (i) when the thickness of the boundary layer at the bottom is assumed to be constant, and (ii) for waves over a rippled bed, when the boundary-layer thickness corresponds to the measured dissipation of energy in oscillatory waves over steep sand ripples. In both cases it is found that there exists a maximum bottom slope $s$ below which the wave amplitude must diminish monotonically towards the shoreline. This maximum value of $s$ is of order $10^{-3}$. The waves can indeed penetrate close to the shoreline without breaking, and the corresponding wave set-up is negligible. An example of where suitable conditions exist is on the continental shelf off North Carolina.