ABSTRACT
The evolutional characteristics of Stokes waves are shown in this study. They are illustrated with appropriate governing equations. In this consideration, the fifth order approximation was firstly obtained analytically. This was numerically studied and analyzed graphically. It was found to be nearest to the observed values when compared with previous empirical attempts. Using separately the third, fourth and fifth order approximations of wave profile functions respectively, expressions for wave - induced ocean current were derived. Thereby, it was suggested that wave - current speed has more impact on the surface of the ocean but, however, reduces exponentially downwards as the depth increases. The present analysis also suggests that the wave – current speed increases with increase in wave steepness and the higher the order, the higher the wave-current speed.
In the analysis of Stokes waves as a diffusion problem, the fifth order expansion was obtained using Korteweg - de Vries (KDV) equation with the diffusion term. This study suggests that the phase velocity grows with increase in wave steepness whilst the group velocity shows the opposite tendency. The peak potential energy lies between second and third order solutions while that of kinetic energy attains the peak at second order and then becomes fairly stable.
Further, a comparison of the profile of Stokes waves and Gerstner’s trochoidal waves expanded to fifth is made. Close study of the results shows that to third order, Stokes waves have similar characteristics to trochoidal waves in terms of wave profile, wave speed and wave – current speed. They differ significantly in fourth order. The essential difference is that Stokes waves are slightly higher at the crests. The phase speed of higher order Stokes waves increases as the order increases due to the effect of wave steepness.
