SHIP VIBRATIONS: THE VIRTUAL INERTIA OF A SPHEROID IN SHALLOW WATER 
Vertical Oscillations of a Floating Spheroid 
11. We suppose the spheroid to be floating with one- 
half immersed. As before, Ois at the centre of the 
spheroid, and Oz vertically downwards; the free surface 
of the water is the plane z = 0, while the bed is the plane 
z=flae. We have now to consider the condition at 
the free surface. For simple harmonic motion of 
frequency o we have the usual linearized condition 
¢+(g/o*)d¢/3z=0. Using this condition we should 
have to take into account the wave motion of the free 
surface, but that is beyond the scope of the present 
work. There are two limiting simplifications which may 
be made according to the conditions of the particular 
problem; we may take ¢ = 0, the free surface condition 
neglecting gravity, or we may take d ¢/d z = 0, the rigid 
surface condition. Taking into account the frequency 
of ship vibrations, the appropriate alternative seems to be 
¢ = 0; the measure of agreement between calculated and 
observed frequencies justifies this assumption as a working 
hypothesis. 
The conditions for the velocity potential are now 
(i) the given normal velocity over the submerged half of 
the spheroid, (ii) }¢/)z = 0 for z=fJae, (iii) 6=0 
for z= 0. This is the same as considering the complete 
spheroid in water contained between two parallel planes 
z= +/f/ae, with the normal velocity given over the 
whole surface. For a vertical vibration we begin, as 
before, with 
bo = Pr (4) Q; (4) sin w 
In order to satisfy conditions (ii) and (ili) we now have 
an infinite series of image systems alternatively positive 
and negative, associated with the points z= + 2s/f/ae. 
Hence we have 
(33) 
d= —2 & (— 1) 1 Pi) Q() sine, (34) 
Sa 
We have obtained in the previous sections, the value of 
¢, for any one of these image systems, and also its 
contribution to 6 T/T). Hence for the vertical vibra- 
tions we have 
Saye (One) (— 1)-"D,,/ 
n(n + 1) (G3 — 1) QL (Gy) Qh (Lo) 
with D,,, given by (19) with g = 2sf/ae. 
For instance, for n = 1, 5 T/Tp is given by (26) with 
(35) 
1D) = 2 (= Wea? 1D (36) 
1 
with D, given by (27) with q = 2s fJae. 
The limiting form of this result for a long spheroid is 
(2 b7/5 f?) Z(— 1)8-1 s-2, or a? 57/30 f?. 
For any given case, having computed and graphed the 
double integral involved as a function of the parameter gq, 
it is a simple matter to obtain from the graph the 
summation with respect to s. 
588 
Horizontal Vibrations of Floating Spheroid 
12. Deep water. If we retain the same condition at 
the free surface, the horizontal vibrations are a more 
difficult problem. With the given normal velocity on 
the immersed half of the spheroid, the conditions are 
now 
d PofdS = Pi (u)cosw; C=4; OK wer 
do =0; w=0 (37) 
This is equivalent to considering the complete spheroid 
in an infinite liquid with the conditions 
> dofd C= Pr(u) cos w; 0 < 
> dod 6 = —P, (H) cos w; — 
$9 =0; w=0 (38) 
To satisfy these conditions, we express the value of 
0 go/d ¢ in a suitable infinite series of Legendre functions, 
of the form 
XD AS, PS (u) sins w (39) 
Forming the series by the usual methods, it is seen 
that s must be even, and the coefficients are given by 
Mae (GARB AGE Psa so 
Tat Gone FOP wide (40) 
It follows that if m is even, the coefficients are only 
different from zero if m is odd; while if n is odd, m 
must be even. The velocity potential is then given by 
bo = DAN Qn (fo). PS, (4) Q5,(Q sins (41) 
This form of solution gives the assigned normal velocity 
on the spheroid for all points other than those for which w 
is actually zero, that is for points not actually on the water 
line. There is, in fact, a discontinuity in the normal 
velocity on crossing the water line; there will be a 
corresponding infinity in the tangential velocity at these 
points. However, as in similar problems involving 
what is effectively flow round a sharp edge, the usual 
surface integrals for the kinetic energy lead to a finite 
result. 
From (38) and (41) we may obtain the kinetic energy, 
and if we introduce the suitable factor according to the 
motion of the solid, the corresponding inertia coefficient 
can be calculated. 
13. Water of Finite Depth—tIf the water is of finite 
depth we should have, as in § 11, an infinite series of 
image systems in subsidiary spheroidal co-ordinates, each 
system being an infinite series of terms. Further, 
expanding any one term so as to obtain the image in the 
spheroid would involve infinite series. Finally, in con- 
trast to the conditions for vertical motion, from the 
form of the conditions all the terms in the series con- 
tribute to the kinetic energy. Thus, in general, the 
method becomes much too complicated. 
However, when we form the expression for the kinetic 
energy for cases with which we are dealing, it appears 
that the first few terms of the series account for much 
