Computing Hearing and Pitching Motions 487 
condition will be satisfied by this velocity together with Eq. (8)? This potential, for which 
Eq. (2) will be applied putting U = 1, can be defined as a transverse deformation of the 
longitudinal wave. At the boundary of the contour the following condition will be used: 
dy. =. \e" 7 dy. (10) 
The boundary conditions (3), (6), and (10) cannot be satisfied exactly (except for w = 0 
or @ = 0), It has been proved that the following procedure is well converging. 
Into the boundary conditions (3), (6), or (10), Eqs. (2), (5), or (2), respectively, will be 
introduced. Within these equations the coordinates of the boundary of the contour can be 
replaced by Eq. (1). Then only one coordinate appears, viz.,9. The equations are written 
as follows: 
[oo] 
B. (F+ 6) + ) B, sin (2n 8) = 0, for -7 <6'S0. (1) 
n=1 
Of course, the coefficients A are linearly included in the coefficients B. 
The boundary condition is satisfied if all coefficients B vanish in Eq. (11). In Eqs. 
(2) or (5) the series are cut off after N terms so that N unknown coefficients A are included. 
Then Eg. (11) is also cut off after N terms and N linearized equations are: obtained: 
B = 1) B. =80: (12) 
By Eq. (12) 2N equations are represented since the coefficients A or B, respectively, 
are complex numbers. Solving these equations the potential is found. The boundary condi- 
tion is nearly satisfied. An error remains which changes sign on the contour several times 
and this more frequently the larger the N that is chosen. 
Having computed the unknown coefficients A the problem is now to determine the hydro- 
dynamic force. To obtain the force in a vertical direction the following integration around 
the contour is required: 
f@® dy. (13) 
In case (a) this integration yields only the hydrodynamic force. It is possible to add 
both the hydrostatic and the inertial force of the body and then to deal with the total force 
which is responsible for the heaving motion. 
In case (b) only a hydrodynamic force exists. Therefore the integration yields immedi- 
ately the force in a vertical direction which is caused from the wave on the restrained body. 
In case (c) the force which follows from the pressure in the undisturbed wave is added 
to the force from (13). This is necessary since (13) yields only the force which arises from 
the deformation of the wave. 
