forward velocity is different from that of wave propagation. Then the ship 

 undertakes oscillations of six degrees of freedom around its mean position, 

 among which surge, heave, and pitch are longitudinal oscillations and 

 sway, yaw, and roll are lateral oscillations. Since the viscosity effect 

 in the longitudinal oscillations is very little, the potential flow 

 theory seems to offer a fairly accurate prediction, while in the lateral 

 oscillations, effect of viscosity plays an important role, so that we 

 cannot expect any reliable results by theories without taking account of 

 the viscosity. Therefore, we will confine our discussions in the longi- 

 tudinal oscillations hereafter. The general formulation for the os- 

 cillation with six degrees of freedom is described in Appendix A. 



On developing the perturbation analysis, we take the ratio 6 of wave 

 amplitude to wavelength and the ratio £ of beam-to-length of the ship as 

 basic parameters. Since they are mutually independent, we can expand the 

 velocity potential with respect to 6 first. The first term is independent 

 of 6 and represents the fluid motion when the ship moves with uniform 

 velocity in still water. The linearized theory takes terms up to the first 

 order with respect to 6. If we are concerned with the ship motion in 

 regular waves, the term which is linear to 6 is a simple harmonic and 

 represents the oscillatory part of the velocity potential. The next stage 

 is the expansion of the above portions of the velocity potential, which 

 have been linearized already by 6, by the slenderness ratio £. There is a 

 term which is independent of £ in the oscillatory potential. It represents 

 the incident waves which may be assumed as simple harmonic too. The other 

 part represents the disturbance by the ship. Consider a relative motion 

 with respect to the coordinates moving with the average forward velocity U, 

 and take the axis of x in the direction opposite to the forward velocity of 

 the ship and the axis of z vertically upwards. Then the velocity potential 

 can be written in the form Ux + <J>, and (j) satisfies the Laplace equation 



V 2 cf> = (89) 



in the space occupied by the fluid. The boundary conditions satisfied by 

 the velocity potential are those on the hull surface and on the free 



38 



