Third-Order Analytical Solutions for Bichromatic Waves on Linearly Varying Currents

Abstract

A third-order analytical solution for bichromatic waves on currents with constant vorticity is derived by using perturbation method Unlike the derivation of monochromatic waves moving-frame method cannot be used in the case of bichromatic waves because there are multiple waves with different celerities and the flow can in no way be steady-state Also for shear currents which is rotational flow velocity potential cannot be defined However with the consideration of the wave part of the fluid motion remaining irrotational (Tsao 1959 and Kishida and Sobey 1988) some of the terms in the expanded boundary conditions can be ignored thus the derivations can be further processed As a result the third-order explicit expressions of the stream function the velocity potential and the surface elevation are obtained The nonlinear dispersion relation is also derived to account for the interacting wave components with different frequencies and amplitudes The obtained solutions including the nonlinear dispersion relation are verified by reducing to those of previous results in the case of monochromatic waves and uniform currents of Chen and Juang (1990) Madsen and Fuhrman (2006) and Kishida and Sobey (1988) The comparisons between the solutions are shown to be in good agreements The influence of current velocity and vorticity on the wave characteristics such as wavelength and maximum particle velocity is illustrated Comparisons between different wave and current conditions are also made Finally the influence of shear currents on the intensities of bound long wave components induced by the nonlinear wave-wave interaction of bichromatic waves are also discussed

Date of Award	2020
Original language	English
Supervisor	Shih-Chun Hsiao (Supervisor)