linear effect of seakeeping which some of the theoretical methods are able to predict. So the methods whose validation is investigated in this article against the experimental results are mostly linear ones. We have to wait for the future work to confirm validity of the most sophisticated results of the nonlinear theories.

2. Linear Theory I

Correct modelling of the interaction of the steady disturbance on the water surface produced by the constant forward speed of a ship with the unsteady flow produced mainly by the ship-waves interaction is not straightforward in formulating a theoretical approach to predict the wave-induced motions and wave loads of the ship. The total flow produced by the ship in waves is obviously decomposed into the steady flow and the other flow naturally unsteady. It is appropriate to attempt the following form for the velocity potential Φ of the total flow.


The flow is described in the right-hand reference frame fixed to the ship moving at the constant speed U into the positive x direction; the x axis coincides with the ship’s center line and the x−y plane is the mean water surface; the z axis is taken vertically upward. The second term of (1) corresponds to the uniform flow relative to the ship, the third term represents the steady flow and the fourth term the unsteady flow component.

Wave elevation ζ is also a superposition of the steady component ηS and the unsteady component η.


Magnitude of the unsteady flow is determined by magnitude of the incident waves and the resulting ship motions. Magnitude of the steady flow is supposed to be dependent on ship geometry such as its slenderness. It means that rational basis of the linearization of will be independent of that of . When we discuss their interaction without introducing any assumption on the ship geometry (practical hull forms are never slender), the most reasonable choice of will be a full nonlinear solution. Yet we like to somehow avoid the full nonlinear solution and use physically correct but simpler solution; too complicated mathematical expression of might lead to unnecessary difficulty of any linear theory of the unsteady flow .

The choice of the steady flow model, if we have no mathematical basis, will inevitably be decided on physical argument or arbitrary; consequently various approaches will be possible to incorporate the effect of into . Their validity therefore must be carefully tested on ‘hydrodynamic’ experiment.

A popular way to account for the steady flow effect allowing the relatively easy formulation of the unsteady flow is the introduction of the low order solution of the steady flow. The simplest choice is to assume the steady disturbance is zero; the steady flow around the ship is assumed to be only the uniform relative flow U in deriving the free surface conditions.

We linearize the free surface conditions and the hull surface condition, with respect to the velocity potential and the corresponding wave elevation η. The wave-induced ship motions are assumed to be of the order Ο (η). will be a solution of the boundary value problem:





and the radiation condition.

(4) and (5) are the free surface conditions satisfied on the mean water surface z=0. (6) is the body boundary condition imposed on SB representing the ship hull surface immersed under z=0 at the ship’s mean position, a is the motion vector of a point r(x, y, z) on the hull surface; r(x, y, z) moves due to the wave-induced ship motions. V is the steady flow velocity around the ship given by


n is the unit vector normal to the boundary surface and directing outward from the fluid. The third term of the body boundary condition on SB is to correct the difference of the steady flow velocity on SB from that on the exact instantaneous hull surface. The fourth compensates the effect of the variation Δn of

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