SHIP VIBRATIONS: THE VIRTUAL INERTIA OF A SPHEROID IN SHALLOW WATER 
spheroid. Numerical computations have been made for 
a spheroid whose length 2. is just over 10 times the 
beam 2 5, and the results are shown in Fig. 1; the com- 
putations were troublesome, and a high degree of 
accuracy was not attempted. 
ie) 15 20 2:5 3:0 2:5 
Fic. 1.—RELATIVE INCREASE OF VIRTUAL INERTIA COEFFICIENT (OT/To) 
FOR RATIO OF DEPTH OF WATER TO DRAFT (f/b), VERTICAL VIBRATIONS 
(V), HORIZONTAL VIBRATIONS (#7). 
The ordinates are the relative increase in inertia 
coefficient, that is the ratio of the increase to the value 
in deep water; the abscissae are the values of //b, or the 
ratio of depth of water to the maximum draught. The 
curves in question are those marked V. Those marked 
OV and 1 V are for translation and rotation respec- 
tively; but we may regard the set of curves as repre- 
senting vertical vibrations specified by the number of 
nodes, 0, 1, 2, 3, respectively. From this point of view, 
it is of interest to note the varying influence of depth 
according to the type of vibration; it is clear, for 
instance, that using values derived from pure translation 
would give misleading results for 2- or 3-node vibrations. 
The curves V in Fig. 1 were obtained from the general 
results given in equation (35). These expressions have 
simple approximate forms when the spheroid is very 
long; the values are 0-658, 0-470, 0:439, 0-429 times 
bf? for n=1, 2, 3, 4 respectively. In the present 
case, for which the length-beam ratio is 10, the curves 
approximate fairly closely to these values for small depth 
of water. As regards actual measurements, there are 
no experimental results which are strictly comparable. 
Prohaska® has given a formula 2 Cy d?/f?, where Cg is 
the block coefficient and d is the mean draught. As the 
form indicates, this is based on two-dimensional theory, 
with the coefficient chosen to agree as well as possible with 
results from actual ship forms. The prolate spheroid 
40 
584 
is not a normal ship form, nevertheless it is of interest to 
note that this formula gives 0-466 b*/f?, which may 
be compared with the approximate values given above. 
4. The remaining sections of the work are devoted to 
the similar horizontal vibrations of the floating spheroid, 
dealing first with deep water. It is generally known that 
if inertia coefficients for horizontal motions are calcu- 
lated using the free surface condition, the values are 
much less than if the rigid surface condition had been 
used. Ifa circular cylinder, half immersed, is oscillating 
horizontally at right angles to its axis the inertia coeffi- 
cient is 4/7 compared with the usual value unity. Fora 
log of square cross-section, Wendel™ has calculated that 
the value for horizontal motion is about 0-337 of the value 
for vertical motion; for a general account of virtual 
inertia coefficients reference may be made here to a 
recent paper by Weinblum.© Calculations for three- 
dimensional motion do not seem to have been published, 
though no doubt the general nature of the results is 
known. We give in §12, general expressions for a 
prolate spheroid, half immersed, from which the inertia 
coefficients could be found for the various types of 
horizontal motion we have been considering; these 
include translation, rotation, and 2-node and 3-node 
vibrations. Approximate calculations have been made 
for the particular case of a length-beam ratio of 10, and 
these indicate that the values are of the order of 0-4 of 
the values for a deeply submerged spheroid. 
5. The last section deals with the same problem for 
water of finite depth. Here the mathematical difficulties 
are such as to preclude a general form of solution for 
the various types of vibration. However, taking the 
simplest type n = 1, an approximation is obtained in 
(53) for the relative increase in inertia coefficient due to 
the finite depth of water; it is considered that this 
approximation is sufficient to show the essential character 
of the effect of depth of water. Taking the same 
particular case of a length/beam ratio of 10, numerical 
computations have been made from this expression and 
the results are shown in the curve labelled OH in Fig. 1. 
The two curves to be compared are the curves OV and 
OH;; they are both for the same type of vibration, the 
former being vertical and the latter horizontal. The 
point of special interest is the remarkable difference 
between vertical and horizontal vibrations as regards 
the influence of shallow water. This difference is 
expressed simply if we take the approximate values for 
a long spheroid; in that case, it is easily shown that the 
expression (53) for horizontal motion is of the order of 
(b/f)*, while we have already seen that for vertical 
motion the approximation is of order (b/f)*. This may 
be confirmed by working out a simple two-dimensional 
case, a circular cylinder half-immersed. In this case the 
conditions of the problem may be satisfied to any 
required degree of accuracy in the ratio b/f; it may be 
sufficient to state the results here. If the motion is 
vertical, the inertia coefficient in deep water is unity; the 
relative increase in shallow water is given by 
0-8225(b/f)? + 0-3382(b/f)* + 0-1391(b/f)e +... 
