S. Beji et K. Nadaoka, A TIME-DEPENDENT NONLINEAR MILD-SLOPE EQUATION FOR WATER-WAVES, Proceedings - Royal Society. Mathematical, physical and engineering sciences, 453(1957), 1997, pp. 319-332
Citations number
19
Journal title
Proceedings - Royal Society. Mathematical, physical and engineering sciences
A weakly nonlinear and dispersive water wave equation, which in linear
ized form yields a new version of the time-dependent mild-slope equati
on of Smith & Sprinks (1975), is derived. The applicable spectral widt
h of the new wave equation for random waves is found to be more satisf
actory than that of Smith and Sprinks (1975). For very shallow depths
the equation reduces to the combined form of Airy's nonlinear non-disp
ersive wave equations; if the lowest-order dispersion is retained it p
roduces the combined form of Boussinesq's equations. In the deep-water
limit the equation admits the second-order Stokes waves as analytical
solutions. Furthermore, by introducing a right-moving coordinate tran
sformation, the equation is recast into a unidirectional form, renderi
ng the KdV equation in one limit while reproducing the second-order St
okes waves in the other.