Understanding and Prediction of Ship Motions 



Since (17a) depends on the value of the damping coefficient over the whole spec- 

 trum, one may expect that the added mass will be incorrectly predicted unless 

 b*i^(aj) is approximately correct over the entire frequency spectrum. The "Hi- 

 Fi" approximation espoused by Kotik and Mangulis has proper asymptotic limits, 

 and so this effect does not vitiate their calculations. However, an example to 

 the contrary may be found in slender body theory, where added mass and damp- 

 ing coefficient predictions both break down at high frequency. Equations (17a) 

 and (17b) cannot be used with predictions based on slender body theory. 



2. Throughout this survey, I assume that the amplitude of all disturbances 

 is bounded for all time, and this assumption is necessary for taking Fourier 

 transforms of (13')- If there is an instability such as static divergence (which 

 can occur in yaw with an inadequately controlled ship) or if there is negative 

 damping at any frequency (which can in principle occur at high speeds), then 

 these formulas are invalid for all modes unless the modes of difficulty are con- 

 strained to have zero amplitude. 



Exciting Forces 



One of the most difficult parts of using any equations of motion for analyti- 

 cal predictions of ship motions is calculating the forcing functions, that is, find- 

 ing the force and moment exerted by the incident waves on a restrained ship. 

 Quite often in the past, the practice has been advocated of using the pressure in 

 the undisturbed wave and integrating it over the actual surface of the ship. In 

 other words, it is assumed that the presence of the ship does not affect the pres- 

 sure in the water. This assumption, often referred to as the "Froude-Krylov" 

 assumption, is obviously not generally correct, although under certain circum- 

 stances it may not be grossly in error. Properly, one must formulate a bound- 

 ary value problem in which there are included both the incident waves and the 

 diffracted waves. The two systems of waves must be such that the total fluid 

 velocity on the ship surface satisfies the correct boundary condition there. 



In addition, there will be waves generated by the motions of the ship. From 

 a hydrodynamic point of view, this presents an easier problem than the incident- 

 diffracted wave problem, because the normal velocity component on the hull is a 

 fairly simple, known function, depending only on the shape of the hull and on the 

 six rigid body modes of motion. Therefore Haskind (1957) made a considerable 

 contribution to our problem when he showed that the forces due to incident waves 

 could be calculated from solutions of the forced oscillation problem. Specifi- 

 cally, he showed that if we can solve the hydrodynamic problems involved in the 

 oscillation of a ship in an otherwise calm sea, then we can also compute the 

 force and moment on a ship restrained in incident waves. 



Haskind proved his result only for the case of a ship at zero speed. His 

 solution is rederived in a paper by Newman (1962). Recently, Newman has 

 shown that an analogous result can be obtained for the case of a ship with for- 

 ward speed. However, there is a logical difficulty in such case, for it is not 

 certain that the diffraction-wave potential should satisfy the ordinary linear 

 free surface condition. This problem is discussed in the next chapter in con- 

 nection with thin ship theory. We shall limit our discussion here to the pub- 

 lished case of a ship at zero speed. 



43 



