Second-order low-frequency wave forces on a vertical circular cylinder

Using the multiple-scale perturbation method, the diffraction of a nonlinear nearly periodic wavetrain by a vertical circular cylinder is investigated. The envelope of the incident wavetrain is assumed to modulate slowly in the direction of wave propagation. The relationship between the envelopes of incident and scattered waves is derived. It is shown that second-order scattered set-down waves propagate only at the long-wave velocity (gh)^½. The formula for low-frequency wave forces acting on the cylinder is presented. The low-frequency wave forces, which are second-order quantities, are caused by set-down waves beneath the wavetrain and the results of the self-interactions of the leading-order first harmonic wave components. Numerical solutions are presented for the case where the wave envelope varies sinusoidally.