WAVE RESISTANCE THEORY AND APPLICATION 15 
The limiting form as — © gives the steady state 
which is ultimately established for uniform mo- 
tion in a straight line; namely, 
AIC (gl 2) le oe reese 
Fale =e if e—ko(S—2) se? sin (kosec 0)x 
xX cos (koy sin 6 sec? 0) sec? 0 dé — “er 
7 © p—k(fi=2) 
Pf, /1 Bes eao f e cos (kx cos Deen 0) dk 
0 k — Ro sec? 0 
where ky = g/c?, and the origin O is now a moving 
origin vertically over the source. 
Calculating the surface elevation from this ex- 
pression, it is found that the wave pattern at a 
great distance to the rear approximates to the 
form 
Akom fe 
& = eé 
( —1/2 
cos [ky sec? 6 (x cos 6 + y sin @)] sec? 6 da 
—kof sec29 
From these results for a single source we can 
derive expressions for other singularities, or for 
any distribution of sources and sinks. Knowing 
the wave pattern at a great distance to the rear, 
we can, from energy considerations, write down 
the corresponding wave resistance of the solid 
body which is represented by the given distribu- 
tion. It may be remarked that the forces and 
moments on the submerged body can be calcu- 
lated as the resultant of the fluid pressures on its 
surface, but in that case the approximation must 
be carried to the next stage, that is, to the stage 
oo + di + de in the notation used here; this is 
necessary in order to satisfy the condition at the 
surface of the solid to the required degree of ap- 
proximation and it is a point which has sometimes 
been overlooked. 
We leave this brief description of fundamental 
theory with the remark that nearly all the work 
on such problems has been limited to uniform mo- 
tion in a straight line. More recently, Sretensky 
has given some formulae for accelerated motion; 
and Brard has examined the motion of a source in 
a straight line, the strength of the source being 
subject to periodic variation, with a view to ap- 
plying the results to the interesting problem of 
the pitching of a ship under way. 
Using the integration method outlined in the 
foregoing, I have worked out the case of a sphere 
moving with uniform velocity in a circular path 
at constant depth below the surface. If a is the 
radius of the sphere, / the radius of the circular 
path, f the depth of the center of the sphere, and 
c the linear velocity in the path, the wave resistance 
is given by 
Ar? 4n*pasc* 
gh phy In? (He) e— (antes /ah?) 
R= 
J, denoting the Bessel function. 
If we make / tend to infinity, keeping c con- 
stant, this reduces to the known result for a sphere’ 
in linear motion with uniform velocity c, namely 
Ro = 4rpa®ko'c? bia e—2kof see? B sec’ B dB 
These expressions can be evaluated numeri- 
cally, and Fig. 2 shows some results. 
R 
M'g(a/f)® 
Curve A is the resistance-velocity curve for 
linear motion with constant velocity. Curve B is 
for circular motion with h = f; we notice here 
the humps and hollows due to the motion of the 
sphere in the waves produced by previous revolu- 
tions in the path. Forcurve C, h = 4f and we see 
how quickly these effects diminish with increasing 
radius of the path and the resistance approxi- 
mates quite closely to that for linear motion at 
the same speed. A more interesting case is that 
of a prolate spheroid whose center is describing 
a circular path. The motion of the spheroid in- 
volves both translation in the direction of the axis 
at each instant and rotation about a vertical 
axis; the analysis is rather complicated but ex- 
pressions were obtained for the wave resistance, 
the radial force outwards and the couple on the 
spheroid. 
Fig. 3 shows calculations of the wave resistance 
for a spheroid whose length is 24 times the maxi- 
mum breadth, the radius of the path being equal 
to the length of the spheroid. Curve A is for 
linear motion and eurve B for motion in this cir- 
cular path; even in this extreme case the wave re- 
sistance is not much affected by the curvature of 
the path. These problems are, no doubt, chiefly 
of academic interest in themselves; but the de- 
velopment of such work may have a bearing on 
questions of great practical interest in the theory 
of steering, stability and so forth. 
565 
