THE MOMENT ON A SUBMERGED SOLID OF 
REVOLUTION MOVING HORIZONTALLY 
By T. H. HAVELOCK (King’s College, Newcastle upon Tyne) 
[Received 20 February 1951] 
SUMMARY 
The moment, due to surface waves, on a submerged solid of revolution moving 
axially at constant depth below the surface of the water is examined in detail. 
1. A SUBMERGED solid of revolution moves axially with uniform velocity - 
and with its axis at a constant depth below the surface of the water. If the 
solid is such that the motion in an infinite liquid can be represented by a 
known source-sink distribution along the axis, the horizontal and vertical 
forces on the solid due to the wave motion can readily be obtained to the 
usual approximation; however, for the moment about a transverse hori- 
zontal axis it is necessary to obtain the velocity potential to a higher degree 
of approximation, a point which was noticed in an early paper on the 
circular cylinder (1) but which has sometimes been. overlooked. In the 
present note we consider a prolate spheroid, for which this extension can 
be carried out; the form of the additional term in the moment in 
this case suggests an approximation applicable to other elongated solids 
of revolution, such as a Rankine ovoid, generated by an axial source 
distribution. 
2. We suppose the spheroid to be held at rest in a uniform stream of 
velocity c in the negative direction of Oz, the axis being at a depth f below 
the free surface of the water. We take O at the centre of the spheroid, 
Oz along the axis, Oy transversely, and Oz vertically upwards. Using the 
known axial distribution for motion in an infinite liquid, the velocity 
potential is given by 
ae 
¢ = cx+Ac [ bale 
{y?--2?+ (x—k)*} 
(oo) 6 
ae sep | dé | KP ose) __ gait +iee de, (1) 
K—kKy Sec?0-+ iu sec 0 
0 
where 
AQ — 2e/(1—e?) —log{(1+e)/(t—e)}, @ = («—k)cos 0+ ysin 8, 
ko = g/c?, and the limit is taken as p> 0. 
{Quart. Journ. Mech. and Applied Math., Vol. V, Pt. 2 (1952)] 
5092.18 
575 
