Shallow-water waves

Abstract

The physical problem is to calculate the profile and velocity components of a wave, subject to the conditions that the effects of viscosity and surface tension can be neglected. The flow is required to be steady, irrotational, and two dimensional, and the depth of the water is assumed to be small and constant. The shallow‐water approximation, first proposed by Friedrichs, is employed, and two additional orders of approximation, beyond the three calculated by Keller, are obtained. The formulas describing the solitary wave are obtained as a special case, and the velocity of the solitary wave so obtained is found to agree almost exactly with the Boussinesq‐Rayleigh formula c=gd(1+H/d) , where c is the wave velocity, d is the depth, and H is the wave height. This velocity differs by no more than 2 per cent from the experimental values of Stephan and Dailey.

Strongly Nonlinear Waves

“…Higher-order cnoidal wave theories have been obtained by Laitone (1960) and Chappelear (1962) to second and third order respectively. A systematic method using power series expansions in terms of shallowness, (Dj'A.…”

Sediment entrainment under fully developed waves as a function of water depth, boundary layer thickness, bottom slope and roughness

“…The characteristics of such shoaling, non-linear waves have been modeled by many authors (e.g. Bretschneider, 1960;Latoine, 1960;Skjelbreia and Hendrickson, 1961;Chappelear, 1962;Latoine, 1962Latoine, , 1965Ippen, 1966;Peregrine, 1972;Cokelet, 1977;Phillips, 1977;Fenton, 1985;Chakrabarti, 1987;Fenton, 1988;Dean and Dalrymple, 1991;Mei, 1991). However, the simplified but fully integrated equations of Le Roux (2007aRoux ( ,b, 2008a are used here, because they eliminate the discrepancies produced when using different wave theories for different water depths.…”

Evolution of basic equations for nearshore wave field

“…7) Laitone (1960) 8) derived a second-order solution, and a third-order solution was given by Chappelear (1962). 9) …”