SHIP VIBRATIONS: THE VIRTUAL INERTIA OF A SPHEROID IN SHALLOW WATER 
If the motion is horizontal the inertia coefficient in deep 
water is 4/7”, and the relative increase in shallow water 
is given by 0-6314(b/f)* — 0-2190(0/f)§ +... If we 
graph these two expressions we obtain curves of the 
same character as the curves OV and OH in Fig. 1. 
As regards observed results for actual ship vibrations, 
it has been stated that there is no measurable change of 
frequency of horizontal vibrations in shallow water, in 
striking contrast to the observations on vertical vibra- 
tions. If that should prove to be the case, it would 
confirm the assumptions underlying the present analysis; 
however, it would be of value to have a direct examina- 
tion of the problem under conditions which would allow 
both of theoretical calculation and of precise experi- 
mental determination. 
PART I 
Infinite Liquid 
6. Consider a prolate spheroid, of semi-axes a,b and 
eccentricity e, in an infinite liquid. We take axes with O 
at the centre, Ox along the axis of the spheroid, Oy 
transversely, and Oz vertically downwards. We shall 
use non-dimensional space-co-ordinates, giving the ratio 
of any distance to the length ae. We have then 
spheroidal co-ordinates (u, ¢, w) with 
x= pl; y= (1 — #)(0 — 1) cosa; 
z= (1 — )”? — 1)fsinw (1) 
The spheroid is given in these co-ordinates by ¢ = 
= l/e. Consider the fluid motion given by the velocity 
potential 
$ = CP(u)Q)(C) sin w cos ot (2) 
This motion would be produced by a distribution of 
normal velocity over the surface of the spheroid given by 
— offv = — (C/ae)[(1 — e%)f(1 — eu2)]* 
P!(wJQU(L) sinw cos ot . (3) 
where the dot denotes differentiation, a notation we 
shall use throughout. We make the usual approxima- 
tion for vibrations of the spheroid of small amplitude, 
assuming this to be equivalent to a distribution of normal 
velocity given over the spheroid in its mean position. 
It is well known that, with suitable values of the 
constant C, for n = 1 or n = 2 the fluid motion given 
by (2) can be produced by motion of the spheroid as a 
rigid body; if n = 1, this motion is translation parallel 
to Oz, while if n= 2 it is rotation round Oy (eg. 
Lamb, Hydrodynamics, p. 141). For higher orders of 
harmonics, deformation of the spheroid is necessary 
The present application is, chiefly, to the transverse 
flexural vibrations of a spheroid of large ratio of length 
to beam. We may then regard the deformation as a 
simple shear of transverse sections of the spheroid. It 
can be shown that the normal velocity (3) is produced 
by such a transverse motion with the velocity distribution 
along the axis proportional to P,(x/a). For instance, 
985 
with n= 3 the nodes of the vibration are given by 
x/a = + 4/5, while for n=4 we have a 3-node 
vibration with nodes at x/a = 0, + »/3/7. It is 
possible to improve this approximation to the natural 
vibrations, as pointed out by Lewis” and by Lockwood 
Taylor,” by taking combinations of spheroidal harmonics 
or by other refinements. But the additional complica- 
tion is not worth while for the present purpose; we are 
concerned not so much with the absolute value of the 
inertia coefficient as with its relative increase in shallow 
water. 
From (2) and (3) we obtain the kinetic energy of the 
fluid by integrating over the surface of the spheroid; and 
we have 
T = —4p|b¢)ov) dS (4) 
= —7pa(l — ee. [n(n + DI2n +1] 
C? Qi (Lo) Qi(So) cos?a t . (5) 
The kinetic energy of the spheroid can be obtained from 
the corresponding velocity distribution in the solid, and 
hence the virtual inertia coefficient; but these results are 
already known. 
Semi-Infinite Liquid, with Rigid Boundary 
7. Let the axis of the spheroid be parallel to a plane 
rigid boundary given by z=f/ae. If To is the kinetic 
energy of the fluid for a given type of motion when the 
spheroid is in an infinite liquid, and 6 T is the increase 
in kinetic energy due to the boundary, we are concerned 
with the ratio 6 T/T, which is, of course, the relative 
increase in the corresponding virtual inertia coefficient. 
If we imagine this quantity expressed in powers of the 
tatio b/f, the approximation at which we aim is the 
leading term in such an expression. This can be 
obtained in the following way. Let 49 give the motion 
in an infinite liquid with the given normal velocity over 
the spheroid. Let ¢, be the image of this in the 
boundary, giving zero normal velocity over the boundary; 
and let ¢, be the image of ¢; in the spheroid, so that the 
normal velocity over the spheroid is unaltered. Then, 
using ¢o + ¢, + ¢, in the usual surface integral for the 
kinetic energy, we obtain this approximation. 
It is convenient to give here some formulae which are 
used throughout the analysis. 
We require the expansion of the inverse distance 
between two points whose spheroidal co-ordinates are 
(uw) and (1, €, w,); this is (Hobson) 
pla z@ n + 1) P, (14) Q, (61) P, (4) PD) 
2 n n— s)!]? 
+22@n49 3 (0s 5)| 
PS (444) Q5 (C1) PS (4) Ps (6) cos s(w, — w) (6) 
an expansion which is valid for ¢, > ¢. 
We also need the relation 
POGVO-POw® 
=(— I)! [@+5Y¥a —s) YC — 1) 
(7) 
