SHORELINE PROFILE OF THE STOKES-MODE EDGE WAVES by

Based on the assumptions of inviscid and incompreoible fluid, irrotational flow, and 01 infinitesimal wave amplitude, Stokes (1846) found a solution to the water-wave problem with a uniformly sloping impermeable boundary (beach). The solution, often termed the Stokes-mode edge wave, can be written in terms of velocity potential, , as To-MN € (x,y,zt) =Ag sin J0 e- ky cms 0 + kz si P sin (kx- wt), (1) I where A is the amplitude of wave runup distance along the beach surface, co is the wave angular frequency, k is the wave number in the longshore direction, P is the beach slope from the horizontal, and the coordinates (x, y, z) point to the alongshore, offshore, and vertically upward directions, respectively, as shown in Fig. 1. Equation (1) indicates that edge waves propagate parallel to the shoreline with their crests pointing offshore. The maximum wave amplitude occurs at the shoreline, y = 0, and decays exponentially offshore with an e-folding distance of (k cos The corresponding dispersion relation is given by co2 = gk sin 3. (2) This dispersion relation indicates that edge waves are dispersive, i.e. the group velocity, Daok, is a function of k. Rotating the coordinates from the horizontal about the longshore axis by 03, (1) and (2) can be expressed by A(x,Y,t) = e- kY sin (kx-cot, (3