Third order approximation to capillary gravity short crested waves with uniform currents

A third-order analytical solution for the capillary gravity short crested waves with uniform current (the main current direction is along the vertical wall) in front of a vertical wall is derived through a perturbation expanding technique. The validity and advantage of the new solution were proved by comparing the results of wave profiles and wave pressures with those of Huang and Jia [H. Huang, F. Jia, The patterns of surface capillary gravity short-crested waves with uniform current fields in coastal waters, Acta Mech. Sinica 22 (2006) 433–441] and Hsu [J.R.C. Hsu, Y. Tsuchiya, R. Silvester, Third-order approximation to short-crested waves, J. Fluid Mech 90 (1979) 179–196]. The important influences of currents on the wave profiles, wave frequency, the ratio of maximum crest height to the total wave height, and wave pressure are investigated for both small-scale (for example, the incident wave wavelength is 9.35 cm) and larger-scale (for example, the incident wave wavelength is 5 m) short crested wave. By numerical computation, we find wave frequency of short crested wave system is greatly affected by incident wave amplitude, incident angle, water depth, surface tension coefficient and the strength of the currents for small-scale incident wave. Furthermore, for the larger-scale short crested wave system, the higher-order solution with uniform current is particularly important for the prediction of wave profile and wave pressure for different water depth and incident angle. Computational results show that with the increase of the current speed, the crests of wave profile and wave pressure become more and more steep. In some cases, the crest value of wave pressure with strong current would be larger about six times than that of no current. Therefore, ocean engineers should consider the short crested wave-current load on marine constructs carefully.