A Fifth‐Order Stokes Theory for Steady Waves

Abstract

An alternative Stokes theory for steady waves in water of constant depth is presented where the expansion parameter is the wave steepness itself. The first step in application requires the solution of one nonlinear equation, rather than two or three simultaneously as has been previously necessary, In addition to the usually specified design parameters of wave height, period and ~ water depth, it is also necessary to specify the current or mass flux to apply any steady wave theory. The reason being that the waves almost always travel on some finite current and the apparent wave period is actually a Dopplershifted period. Most previous theories have ignored this, and their application has been indefinite, if not wrong, at first order. A numerical method for testing theoretical results is proposed, which shows that two existing theories are wrong at fifth order, while the present theory and that of Chappelear are correct. Comparisons with experiments and accurate numerical results show that the present theory is accurate for wavelengths shorter than ten times the water depth.

WAVECALC: an Excel-VBA spreadsheet to model the characteristics of fully developed waves and their influence on bottom sediments in different water depths

“…,"""1 .. 1>011",,, 10_"",,_ ", Furthermore, although the equations presented here are internally coherent and correspond well to existing theories (e.g. Airy, 1845;Stokes, 1847;1880;Boussinesq, 1871;Korteweg and De Vries, 1895;Dean, 1965;Cokelet, 1977;Miles, 1980;Sakkai and Battjes, 1980;Fenton, 1985;1988;Fenton and McKee, 1990) and published laboratory observations (e.g. Shore Protection Manual, 1984;Dean and Dalrymple, 1991), they have not yet been tested rigorously under field conditions, where complications may arise because of wave interference, different types of marine currents, wave reflection and refraction, as well as irregular bottom topography.…”

“…Nevertheless, although the same basic equations are used for all water depths, they appear to correspond well to most of the major theories (Airy, 1845;Boussinesq, 1871;Stokes, 1880;Cokelet, 1977;Dean, 1965;Fenton, 1985;1988;Fenton and McKee, 1990) within their applicable depth zones.…”

Higher harmonic wave loads on a vertical cylinder in finite water depth

“…For the direct numerical integration approach, we integrate F up to the instantaneous fifth order incident free surface using Stokes fifth order wave theory (see for instance the classical paper by Skjelbreia & Hendrickson (1960)), and add the F ψ from (2.3). A misprint in their paper pointed out, for instance, by Fenton (1985) should be noted: the sign ahead of the number 2592 in the expression of C 2 used in the fifth-order formula for the phase speed is wrong. Consistent with the derivation of the fifth-order Stokes wave theory, we use the following Taylor expansion to evaluate the quantities in (2.2) for 0 z |ζ|,…”

Composite modelling of subaerial landslide–tsunamis in different water body geometries and novel insight into slide and wave kinematics

“…Stokes solution (Fenton, 1985), a 2 nd order Cnoidal model (Wiegel, 1960), a higher order solitary wave theory (Munk, 1949) as well as stream function theory (Dean, 1965).…”