Slow-drift motion of a two-dimensional block in beam seas

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Abstract
Because of the inherent nonlinearity in the boundary conditions on the free surface, water waves with frequencies from neighbouring parts of the sea spectrum interact and force low-frequency oscillations at the second order. Since the physical phenomenon involves vastly different timescales, the perturbation method of multiple scales is applied here to a rectangular cylinder in beam seas. It will be shown that at the second order there are two kinds of long waves; one is locked to the envelopes of the incident, reflected and transmitted short waves, while the other propagates away from the body at the long-wave velocity (gh)½. The latter contributes to the damping of slow-drift oscillations of the body. Analytical results for the displacement amplitudes of the slow sway and for the radiated long waves are derived. The transient evolution due to incident envelopes of finite and semi-infinite duration is also predicted.

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