Solitary and periodic gravity—capillary waves of finite amplitude

Two-dimensional solitary and periodic waves in water of finite depth are considered. The waves propagate under the combined influence of gravity and surface tension. The flow, the surface profile and the phase velocity are functions of the amplitude of the wave and the parameters l = λ/H and τ = T/ρgH2. Here λ is the wavelength, H the depth, T the surface tension, ρ the density and g the acceleration due to gravity. For $\dot{\tau} >\frac{1}{3}$, large values of l and small values of the amplitude, the profile of the wave satisfies the Korteweg–de Vries equation approximately. However, for τ close to $\frac{1}{3}$ this equation becomes invalid. In the present paper a new equation valid for τ close to 1/3 is obtained. Moreover, a numerical scheme based on an integrodifferential-equation formulation is derived to solve the problem in the fully nonlinear case. Accurate solutions for periodic and solitary waves are presented. The numerical results show that the Korteweg–de Vries equation does not provide an accurate description of periodic gravity-capillary waves for $\tau < \frac{1}{3}$. In addition, it is shown that elevation solitary waves cannot be obtained as the continuous limit of periodic waves as the wavelength tends to infinity. Graphs of the results are included.