Ogilvie 



free- surface condition and an ordinary kinematic body boundary con- 

 dition, as well as the outgoing- wave condition. This is a standard 

 problem which must generally be solved numerically with the aid of 

 a large computer; such programs exist. The force and moment were 

 expressed as integrals of added-mass-per-unit-length and damping- 

 per-unit-length, both of which could be found from the velocity 

 potential for the 2-D problem. Finally, the determination of the wave 

 excitation force and moment was carried out by application of the 

 Khaskind formula, which permits us to avoid the singular perturbation 

 problem involved in solving for the diffraction wave. 



3,4, Oscillatory Motion with Forward Speed 



The problem of predicting the hydrodynamic force on an 

 oscillating ship with forward speed is not fundamentally much differ- 

 ent from the same problem in the zero-speed case. It is considerably 

 more complex, to be sure, but no new assumptions are needed. 



The approach here is that of Ogilvie and Tuck [ 1969] . Alter- 

 native approaches have been devised by numerous other authors; 

 some of these were mentioned in the last section. The distinguishing 

 characteristics of the Ogilvie- Tuck approach are: 1) application of 

 the method of matched asymptotic expansions, and 2) assumption that 

 frequency is high in the asymptotic sense that co = 0(e"'^^), while 

 Froude number is 0(i), Also, the problem is broken down into a 

 series of linear problems by the use of a "motion- amplitude" param- 

 eter, 6, which is a measure of the amplitude of motion relative to 

 the size of ship beam and draft. 



The reference frame is assumed to move with the mean motion 

 of the center of gravity of the ship. Thus it appears that there is a 

 uniform stream at infinity, and we take this stream in the positive 

 X direction. The z axis points upward from an origin located in 

 the plane of the undisturbed free surface, and the y axis completes 

 the right-handed system, (Positive y is measured to starboard,) 



Let the velocity potential be written: 



<^(x,y,z,t) = Ux + Ux(x,y,z) + 4/(x,y,z,t), (3-35a) 



where U[ x +x(x,y,z)] is the solution of the steady-motion problem 

 discussed in Section 3,1. For the moment, we simply assume that 

 ilj(x,y,z,t) includes everything that must be added to the steady- 

 motion potential so that <|>(x,y,z,t) is the solution of the complete 

 problem. We shall also divide the free- surface deformation function 

 into two parts: 



C(x,y,t) = Ti(x,y) + e(x,y,t), (3-35b) 



758 



