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A fully dispersive weakly nonlinear water wave model is developed via a new approach named the multiterm-coupling technique, in which the velocity field is represented by a few vertical-dependence functions having different wave-numbers. This expression of velocity, which is approximately irrotational for variable depth, is used to satisfy the continuity and momentum equations. The Galerkin method is invoked to obtain a solvable set of coupled equations for the horizontal velocity components and shown to provide an optimum combination of the prescribed depth-dependence functions to represent a random wave-field with diversely varying wave-numbers. The new wave equations are valid for arbitrary ratios of depth to wavelength and therefore it is possible to recover all the well-known linear and weakly nonlinear wave models as special cases. Numerical simulations are carried out to demonstrate that a wide spectrum of waves, such as random deep water waves and solitary waves over constant depth as well as nonlinear random waves over variable depth, is well reproduced at affordable computational cost.

A different approach to the solution of the singular Rayleigh equation is presented in the context of the water wave growth problem as modelled by wind-induced shear instabilities. The approach is based on the analytical solution of a Bessel equation in the vicinity of the singular point, which is obtained from Rayleigh's equation with an arbitrary wind profile. Wave growth rates are computed using an integral expression derived from the dispersion relation of the air-sea interface. Computations of the present approach agree well with those of Conte & Miles (1959) for the special case of a logarithmic wind profile. Effects of the shape of the wind profile on the wave growth rate are investigated by using the 1/7-power law to represent the wind profile. Comparisons of the growth rates for the logarithmic wind profile and for the 1/7 profile reveal appreciable differences which must be investigated further, possibly using measured wind profiles within 10 m above the sea surface.