In a constant depth, suppose wave motion is interacting with a steady unidirectional current in the x-direction. The velocity potential may be expressed as the sum of the wave motion and the current, that is
n which Φ is the velocity potential for wave-current interaction, Φ* the velocity potential for wave only, and � the constant current velocity. (a) Find the governing equations and the boundary conditions in terms of Φ, �, and the free surface elevation �. (b) In case the current is much stronger than the wave-induced water particle motions, what are the corresponding linearised governing equations and the boundary conditions? (c) If � = �(�), is the flow field irrotational? Please explain.