A function to determine wavelength from deep into shallow water based on the length of the cnoidal wave at breaking

Abstract

Various methods exist to determine the wavelength of shoaling waves. However, many are applicable to only part of the depth range, while most require tedious calculations or graphical resolutions. It is shown here that the wavelength immediately seaward of the breaker, L b, is equal to two-thirds of the deepwater wavelength L o for fully developed waves breaking over a nearly horizontal bottom. It is also given by T w√[ g(0.5 H b + d b)] for any H o/ L o ratio or bottom slope, where T w is the wave period, g the acceleration due to gravity, H b the breaker height, d b the breaker depth, and H o the deepwater wave height. This corresponds to within 4.5% with the wavelength at this depth according to cnoidal theory, for which simplified equations are presented. These relationships yield a continuous function of the wavelength L w at any depth d, viz. L w = {[ L b T w][ g(0.5 H b + d)] 0.5} 0.5, where d has a maximum value of L o / 2.965. At this depth, L w coincides with the deepwater wavelength given by the standard Airy equation gT w 2 / 2π for any wave period. The wave celerity C b just seaward of the breaker, given by L b / T w, is also equal to the surf bore velocity U s = √[ g(0.5 H b + d b)] immediately after wave breaking, thus maintaining continuity into the surf zone.