The transient surge motion of a moored ship among random waves is studied with emphasizing the role of the time-memory effect and the initial condition, Since the wave damping of monochromatic waves with low frequency is extremely small, the contribution from the time-memory function to damping is vital to the motion response within the framework of potential theory. It is preferred to derive the time-memory function from the Fourier inversion of the wave damping for the sake of numerical accuracy. Formally the equation of surge motion represents a damped linear dynamic system and the response tends to a stable point attractor irrespective of the initial condition. But in the case of slow drift, there exists an inset due to the small amount of damping, which bifurcates two different attractors entirely depending on the initial condition of motion.
A moored ship in a sea is subjected to second-order wave forces as well as to linear oscillatory ones. The second-order force contains slowly-varying components, of which the characteristic frequency can be as low as the natural frequency of horizontal motions of the moored ship. As a consequence, the slowly-varying force can excite an unexpectedly large horizontal excursion of the ship, which may result in a serious damage on the mooring system. A lot of investigators have tackled with this problem under respective interests. To mention a few among them, Havelock considered second-order forces on a ship due to total reflection on one hand(1940) and relative linear motion on the other hand (1942). Maruo (1960) rigorously derived it in a general form using momentum theorem. In a similar fashion, Remery & Hermans (1972) developed a method in frequency domain. In these approaches, the low-frequency drift force has been approximated by the steady component at zero difference frequency.