To investigate the effects of the low-frequency oscillations on the nonlinear wave loads, the interaction between the slow surge oscillation of a uniform circular cylinder and the ambient diffraction wave field is considered. The frequency of the slow oscillation is assumed to be much smaller than the incident wave frequency. A frame moving with the low-frequency oscillation is adopted to describe the problem. Perturbation analysis is based on two time scales and two small parameters, i.e. the incident wave slope and the ratio of two frequencies. The formulated boundary value problems are solved by means of the Green's theory and semi-analytical solutions are presented for a uniform circular cylinder with a draft equal to the water depth. The radiation condition is proposed for each order of potentials to ensure the existence of the unique solution. The restriction on the validation of the solutions is discussed. The so-called wave-drift damping and added mass are filled out from the nonlinear wave loads, which are calculated by the integration of the hydrodynamic pressure over the instantaneous wetted body surface, depending on whether the phase to be in accordance with the velocity or the acceleration of the slow oscillation.
As well known, triggered by the slowly varying nonlinear wave loads, moored ocean vessels may oscillate at a low frequency in the horizontal plane, i.e. in surge, sway and yaw. These low-frequency oscillations will in turn affect the wave loads acting on the body. As a first step to estimate this effect, the nonlinear wave-drift force is separated into two parts: one proportional to the velocity of the lowfrequency oscillations while the other one is proportional to the acceleration. It is now commonly accepted that the wave-drift damping plays a key role in determining the amplitude of low-frequency drift oscillations at resonance.