522 O. Grim 
motion with a large frequency, the forces exerted by the ship upon the water are in the same 
phase along the whole length of the ship. At each section of the ship an oscillatory force 
acts upon the water. The force at each section excites an elementary wave. If the ship 
makes a heaving motion with a large frequency, the resulting wave is small due to the phase 
relation between the elementary waves. However, for the ship in head seas the sum of all 
the elementary waves, which gives the deformation of the original head wave, is largely due 
to the phase relation between the elementary waves, which is different from the phase rela- 
tion for heaving motion. This is the reason why the deformation is large for the ship in 
head seas and why the deformation of the surface of the water in the longitudinal direction 
is small for heaving motion with large frequencies. 
J. B. Keller (New York University) 
My comment is based on the following consideration. Dr. Grim’s work begins with the 
solution of a certain two-dimensional problem and then he shows how to utilize a solution 
of the two-dimensional problem to solve the three-dimensional problem. I would like to 
point out another method of using two-dimensional problems to solve a three-dimensional 
problem. The method I have in mind is especially useful for short waves, i.e., waves that 
are short compared to the dimensions of a ship. The method is that of geometrical optics. 
I would like to suppose that Fig. D3 is the waterline of a ship which we are looking at from 
the top and I would like to consider the forces exerted on this ship by a wave coming in at 
some oblique direction. Then, according to the principals of geometrical optics, the wave 
can be thought to travel along rays and each ray will hit the waterline and be reflected ac- 
cordin g to the law of reflection and give rise to a reflected ray. The calculation of the re- 
flection coefficient, that is the amplitude and phase of the reflected wave, is a local affair 
and depends only upon the geometry of the ship in the neighborhood of the point of reflec- 
tion. Furthermore, since waves of short wavelength penetrate a very short distance into the 
water, it is only the geometry of the ship very near the surface at the point of reflection that 
determines the reflection coefficient. Since the geometry of a ship at a single point near 
the waterline can be described in terms of the radius of curvature of the vertical section and 
the slope of the ship at the waterline, and also, of course, the curvature of the waterline it- 
self, those three quantities — the two radii of curvature and the slope — will determine com- 
pletely the reflection properties for this particular ray. The radius of curvature of the water- 
line will be taken into account because neighboring 
rays get reflected in slightly different directions. 
The lateral divergence of the reflected rays will ac- 
count for the curvature of the waterline. In order to 
compute the reflection coefficient for a given ray, it 
suffices to solve a two-dimensional problem, namely, 
reflection from a circular cylinder with the same 
curvature as the ship has in a vertical section at the 
waterline. The circular cylinder should cut the water 
with the same slope the ship has at that place. That 
is a two-dimensional problem that hasn’t been solved, 
but never mind, [ just promised to tell how to relate 
a two-dimensional problem to a three-dimensional 
WATERLINE one. If we could solve that two-dimensional problem 
for each position along the ship, then by such a geo- 
eee metrical construction we could calculate the re- 
flected wave in the three-dimensional problem. The 
Fig. D3. Geometric optics analogy result for the two-dimensional problem might also be 
REFLECTED 
RAYS 
