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SHIP VIBRATIONS: THE VIRTUAL INERTIA OF A SPHEROID IN SHALLOW WATER 
By PRorEssor Sirk THOMAS H. HAVELOCK, M.A., D.Sc., F.R.S. (Honorary Member and Associate Member of Council) 
Summary 
It is known that certain motions of the surface of a spheroid expressed by spheroidal 
harmonics are similar to flexural 2- and 3-node vibrations, and can be used to obtain 
virtual inertia coefficients for motion in an infinite liquid. These calculations are now 
extended so as to include the effect of a plane boundary, and are given in a general form 
which includes translation and rotation as well as the flexural vibrations. 
Consideration is given, in particular, to the vertical and horizontal vibrations of a float- 
ing spheroid, half-immersed, in water of given depth. Graphs are obtained for the variation 
of the relative increase of inertia coefficient with the depth of water. 
These show how 
the variation depends upon the type of vibration, and a result of special interest is the 
striking difference between horizontal and vertical vibrations; the relative increase is less for 
the horizontal vibrations, and decreases much more rapidly with increasing depth of water. 
PART I 
1. In this part we give a general account of the work, 
leaving details of the analysis to Part II. 
In calculating the frequencies of the natural flexural 
vibrations of a ship, allowance has to be made for the 
added inertia due to the surrounding water. This is 
usually carried out by a two-dimensional strip method 
which consists in obtaining a suitable expression for an 
elementary transverse section and integrating longitudi- 
nally; an empirical factor is then added to allow for the 
fact that the motion of the water is three-dimensional. 
The only direct three-dimensional calculations which 
have been made are for a prolate spheroid deeply im- 
mersed, or in an infinite liquid. It was shown by 
Lewis,” and about the same time independently by 
Lockwood Taylor,™ that certain motions of the surface 
of the spheroid expressible by spheroidal harmonics are 
approximately the same as for the 2-node and 3-node 
flexural vibrations, and so can be used to give an esti- 
mate of the increase of kinetic energy due to the 
surrounding water. 
Recently the influence of depth of water upon the 
added inertia has become of interest. Here, again, the cal- 
culations have been made by the two-dimensional method 
extended to allow for finite depth of water; reference 
may be made, in particular, to work by Prohaska.© 
In the present paper no attempt is made to examine 
afresh the general theory of the natural vibrations of a 
solid which is partially, or wholly, immersed in water, 
although a more complete theory is much to be desired; 
nor is any attempt made to deal explicitly with solids of 
ship form. Although the analysis may have wider 
applications, the main object of the paper is to carry 
out three-dimensional calculations for a prolate spheroid 
so as to include the effect of finite depth of water, and, 
in particular, to examine the vertical and horizontal 
vibtations of a spheroid floating in water of finite depth. 
583 
2. After a brief summary of the analysis for a spheroid 
in an infinite liquid (§ 6), we proceed to the case of 
finite depth of water. We consider a prolate spheroid, 
major axis 2 a and transverse axis 2 b, wholly immersed 
in water with its axis horizontal and at a height f above 
the bed; in the first place we suppose the water deep 
enough so that we can ignore any effect of the upper 
free surface. The surface of the spheroid is given a 
prescribed motion and we calculate the kinetic energy of 
the resulting fluid motion. Naturally, an exact solution 
is not obtained, and the degree of approximation may 
be indicated by reference to known simple cases. If a 
circular cylinder is moved transversely to its length, 
either parallel to the boundary or at right angles to it, 
the approximate relative increase in the virtual inertia 
coefficient is b?/2 f?. For a three-dimensional case, the 
only known result appears to be the similar approxima- 
tion given by Stokes for a sphere; if the motion is 
parallel to the boundary the relative increase is 3 57/16 f3, 
while for motion at right angles to the boundary it is 
3 b3/8 f3. We obtain the corresponding approximation 
for a prolate spheroid. The analysis is given in general 
form for motion of the spheroid specified by a harmonic 
of order n, for motion both parallel to the boundary 
and at right angles to it; particular cases of the solution 
include translation and rotation of the spheroid and also 
2- and 3-node vibrations. 
3. We turn next to the more interesting problem of a 
floating spheroid, which we suppose to be half immersed 
in the water. For a complete theory we should include 
the surface waves produced by the vibrations, but we 
neglect these meantime; having in view application to 
ship vibrations we adopt what seems to be the appro- 
priate simplification, the so-called free surface condition 
neglecting gravity. A modification of the previous 
section gives expressions for the relative increase in 
inertia coefficient for the various types of motion and in 
§ 11 we consider the vertical vibrations of the. floating 
