Computing Hearing and Pitching Motions 503 
Il. THREE-DIMENSIONAL PROBLEMS 
Heaving and Pitching Motion in Smooth Water 
The motion of the body is determined by the velocity U,(x) in a vertical direction. This 
velocity is constant and independent of x relative to the heaving motion. Relative to the 
pitching motion this velocity grows linearly when x increases. The time factor e!?is 
omitted. 
In Ref. 11 the formula 
®(x,y,z) = 3 ae i UCLA Ge) 0. (x- €).y.z] dé (22) 
is chosen for the flow potential. The functions 9,[(~—€),y,2] within the integrals are de- 
noted partial potentials and are expressed as follows: 
0, [(x- Jz 
co (oa) 
Vm? +K* z 
= A) cos |[m(x - €)| in | SUMO EL ACOs) ak for 0 
0 0 fm? + K%* - vy + ip 
(23) 
Q, [(x- &.y.2] 
2(n-1 
ape | : } 322 ~ toyz 
ea [(x- €)? + y2 + z2| ae [(x- €)? + y* + z2| dis 
This formula satisfies both the condition of continuity and the condition on the free 
water surface outside the ship body. When applied to the limit case of an infinitely long 
two-dimensional body this formula is identical with Eq. (2). The transition to the limit is 
performed as follows: 
1. The distribution functions A,(x) become constants A, which are placed before the 
integrals. 
2. These coefficients A, are chosen such that they are identical with those computed 
in part I. 
3. The integrations are carried out from —~ to +. 
The formula furnishes a nearly exact solution in this limit case. Relative to a ship of 
finite length the distribution functions A (x) are chosen identical at each section x with 
those distributions 4 (x) which hold for the corresponding motion of the two-dimensional 
body. In Eq. (22) there are, therefore, no unknowns. It may also be expected that the 
