Computing Hearing and Pitching Motions 505 
|y| $ B/2. Only a small error arises if in this second member of (24) 9 [(x-€),0,z] is writ- 
ten instead of 9 ,L(x—),¥; z]. Since it is well-known that circular waves are generated by a 
periodical singularity of which the dependence on the coordinate z is described by the fac- 
tor e”? (provided that the distance from the origin of the circular waves is not too small) it 
is suggested that the same dependence on the coordinate z holds for the second member of 
Eq. (24). For all singularities within this member have the same frequency and there are no 
singularities at x itself. It, therefore, may be permitted to further simplify the partial poten- 
tials in this second term, viz.: 
0, ((x- €).y,2] + e”* 9, [(x- €),0,0 (27) 
In this way the coordinates y and z are removed from the integrals. This term can then 
be described as a product of a function of x times the exponential function e””. This holds 
within the region of the ship body, i.e., for ordinates y which do not go far beyond B/2. 
Since the condition on the surface of the body is well satisfied by the first member of 
Eq. (24) the second member causes a disturbance of this boundary condition. To reduce the 
error an additional velocity U as function of x will be introduced. U may be explained as the 
velocity of an additional deformation of the water surface (depending on x and z for |y| $ B/2) 
so that a certain section of the ship body has the velocity (U, + U) relative to the water sur- 
face. This deformation is called “additional” because a deformation (depending on y and z) 
takes place already in the two-dimensional case. The additional deformation diminishes when 
the depth increases following the function e”* because the influence of singularities with 
circular frequency @ lying far outside of x will be eliminated. 
Therefore, the following formula instead of Eq. (22) will be chosen for the total poten- 
tial: 
wWx,y.2) = 4) if U4, (2) + UA, CE) 0, (x- 2.9.7] ac (28) 
n=0 L 
Two different distribution functions 4, , and A, . are introduced. The second indices 
of these functions denote the functions belonging to cases (a) and (c) treated in part I. 
Since the additional deformation U decreases when the depth increases, the distributions 
corresponding to the two-dimensional case (c) of the body “in longitudinal waves” will be 
chosen for this part since the motion of the water particles decreases with the same expo- 
nential function. This formula has been transformed in the same manner discussed previously 
into the identical formula: 
1 2 , +0 
O(x,y,2) = 5 a” {uc 454609 | On [(x- €),¥.2] dé 
n=0 
+00 +a 
On ((x-€),y.2] dé | U.ce4, (ES) 
- @ 
U(x) A(x) | 
-@ 
+ UE) A, CE) - Ux) ACD - Ux) A, CO] On (X-O).Y2 ae}. (29) 
