* 
16 WAVE RESISTANCE THEORY AND APPLICATION 
1.0 
hO/V(gF) 
Fic. 3 
Another matter of interest is the question of 
accelerated motion and effective mass or effective 
moment of inertia in such cases. There has been 
some discussion, for instance, about the suitable 
condition to take at the free surface of the water 
for an approximate estimate of effective inertia 
in ship problems; we may, on the one hand, 
neglect the wave motion completely and take the 
surface to be rigid, or, on the other hand, we may 
neglect gravity and treat it as a free surface. 
Very little work has been done in this field and it 
seemed worth while to attempt a more detailed 
examination of some simple case which could be 
carried far enough to allow of numerical calcula- 
tion. I have worked out the problem to a certain 
stage for a circular cylinder moving with con- 
stant acceleration and starting from rest; some 
of the results are shown in Fig. 4. 
BERS Raa 
AAS OCG aS 
EB SSeS 72bNa 
ND 
AS SEAS 
i | 
The abscissae give the velocity acquired from 
rest with the given acceleration, and the total 
resistance at each speed has been divided into 
two parts. For the first case, the acceleration is 
g/7 and the two parts of the resistance are shown 
by the curves A, and A». From the way in which 
the two parts emerge from the calculations, it is 
convenient to call A, the wave resistance, and for 
comparison the curve Ry shows the wave resist- 
ance for uniform motion at each speed. At uni- 
form speed the ordinates of A, would, of course, 
be zero; in accelerated motion it is appropriate 
to call A, the inertia part of the resistance, al- 
though, as the curve shows, it depends upon the 
velocity as well as upon the acceleration. If we 
had used the approximations of treating the sur- 
face of the water (1) as rigid, or (2) as free but 
neglecting gravity, the part A, would be zero and 
A, a straight line of constant ordinate; for (1) the 
ordinate would be 0.15.0n the diagram, and for (2) 
it would be 0.135. It is interesting to note how, 
in fact, A. varies between these extremes as the 
velocity increases. The curve B, is the wave re- 
sistance part for a greater acceleration; namely, 
1.276g. The corresponding curve Bz is similar in 
character to A, but is not shown on the diagram 
as its ordinates would be nine times those of Ag. 
These results are obviously not of much value 
for direct application; but they may serve to show 
the need for further work in a region which has 
been somewhat neglected, in which there are 
problems which could be studied both theoreti- 
cally and experimentally with a view to practical 
applications. 
Floating Bodies of Ship Form. If the solid is 
only partially immersed in the water we have a 
much more difficult problem, even when we as- 
sume the water to be frictionless. In the usual 
theory of wave motion we neglect the square of 
the fluid velocity. Further, except in special 
circumstances, the first two or three terms of an 
approximation similar to that for a submerged 
solid may be inadequate. 
Then there are also complications arising from 
the intersection of the solid and the water, with 
the different conditions over the two surfaces; 
and, in general, any solution which has been ob- 
tained involves a mathematical infinity in the 
vertical component of fluid velocity at the bow 
and stern. Meantime most of the work on ship 
forms has been limited to cases of small ratio of 
beam to length where these difficulties may be 
neglected in the first place, and further approxima- 
tions made later to improve the theory. How- 
ever, a more direct approach is much to be de- 
sired, so as to give an adequate theory of wave re- 
sistance for a floating solid. In particular, a de- 
tailed study of simple forms would be valuable. 
for instance, a vertical circular cylinder, or a 
sphere or spheroid half immersed in water. One 
may apply to such problems a remark made by 
Kelvin in regard to the motion of a wholly sub- 
merged circular cylinder, which was solved some 
years later by Lamb; after suggesting the prob- 
lem he left it with the remark, “‘it is a mathemati- 
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