484. O. Grim 
In the method proposed by the author the advantages of the two methods shall be com- 
bined and their disadvantages shall, if possible, be avoided. A strip method will be applied 
for the distribution of the singularities; i.e., for each section of the ship the singularity will 
first be chosen to satisfy the condition on the surface of a two-dimensional body of the sec- 
tion in question. The distribution of singularities so obtained is not yet accurate, however, 
even if an accurate representation of the two-dimensional cases is taken as a basis, because 
the representation of the three-dimensional ship body requires a somewhat different distribu- 
tion of singularities. An improved distribution will be obtained with the help of an integral 
equation. 
The method will be described here for the speed V = 0 only. For V £0 it is necessary 
to complete the formulae for the flow potential and, in addition, to allow for the speed in the 
boundary condition. These extensions have been worked out; since, however, numerical re- 
sults are not yet available, this extension of the method will not be discussed here. 
The following problems will be treated: the heaving motion, the pitching motion, and 
the forces generated in a vertical direction by waves. An ideal fluid free from friction will 
be assumed and, further, the problem will be linearized. It should be mentioned that the 
boundary condition at the body cannot be satisfied exactly. These approximations as well 
as additional simplications will be discussed in the course of the paper. 
I. TWO-DIMENSIONAL PROBLEMS 
Introduction 
Since the treatment of the complete three-dimensional problem requires solutions of two- 
dimensional problems to be known, it is necessary to briefly discuss the latter. The method 
used is the same, in principle, as deduced by the author in 1953 [10] on the basis of which 
later published results were computed [12]. As now an electronic computor is available it 
was possible to carry out the computations both more precisely and for a sufficient large 
number of transverse section contours. 
Computations have been made for three problems, viz., 
(a) for the periodical heaving motion of the body in smooth water, 
(b) for the vertical force generated on the restrained body by a transverse surface wave, 
and 
(c) for the hypothetical case of a two-dimensional body in a “longitudinal wave” which 
cannot be realized physically. For this case which is important relative to the 
method described under II results have been given by Abels [9]. 
Description of the Method 
The system of coordinates is fixed in space. Its origin lies in the plane of the waterline 
when at rest and in the midst of the contour, the y-axis being horizontal, the z-axis being 
vertical. The computations have been carried out for transverse profiles which can be rep- 
resented by the transformation formula of Lewis. On the basis of this formula the coordinates 
of the profile can be expressed by means of the parameters a and b and of the coordinate 0 
running from 0 to —7. 
