The forces on a submerged spheroid moving in a circular path 300 
Expanding under the integral sign, we obtain the required expression in terms of 
spherical harmonics. Obtaining the resultant force involves multiplying the pressure 
by a surface harmonic of the first order and integrating over the sphere. Thus we 
only need the first-order term from (18), which is 
(=1)" 
27 
7 
iK,@ exp [— 2 qf] (sina cos f cosu+ sine sin f sinu—1 cos @) 
—7 
exp[—ik,hcosu]sinnudu. (19) 
In accordance with (16), this must be multiplied by 2 to get the correct operative 
value of the pressure. We insert these results in (17), multiply by sing sinf and 
integrate over the surface of the sphere. It is easily verified that this process gives 
the same expression (12) for the tangential resistance. 
5. For the resultant radial force outwards, we carry out the same process on the 
pressure derived from the first three terms of (6), noting that in accordance with 
(16), the first-order surface harmonic from the second and third terms of (6) must 
be multiplied by 3. We then multiply by sina cos/ and integrate over the surface 
of the sphere. The details of the calculation need not be given; after some reduction, 
we find for the resultant radial force outwards the expression 
3 a? K ens 
pari c(t — a + 47pa®Q? »y il Kang 
This expression does not lend itself readily to numerical computation. We notice, 
however, that the first term in (20) represents an effective mass 377pa3(1— 3a3/16f*), 
which is the first approximation for a sphere under a free surface, neglecting gravity. 
On the other hand, when the angular velocity is small the last term in (20) approxi- 
mates to t7pa®Q?/f3, since Ln?J,(Kh)J),(Kh) = txh. Thus for small velocity, the 
effective mass approximates to 27pa3(1+3a/16f%), as for a sphere under a rigid 
surface. 
It is of some interest to make calculations from (12), so as to obtain some idea of 
the nature and magnitude of the effect of curvature of the path upon the wave 
resistance. Curves showing the results are given in figure 1. The abscissae are values 
of hQ/ J (gf), so as to include rectilinear motion for comparison; the ordinates are 
values of R/M’g(a/f)?, where M' is the mass displaced by the sphere. Curve A is 
for steady rectilinear motion, that is, for the limiting case h/f—oo, and calculated 
from (13). Curve Bis for h = f. Even in this case the mean curve approximates to A, 
but it is of interest to note the hump and hollows due to wave interference when the 
sphere is making complete circles. For curve C we have taken h = 4f; it shows. how 
with increasing radius of the circular path these interference effects disappear and 
the wave resistance approximates quite closely to that for straight-line motion at 
the same linear speed. 
6. We consider now a prolate spheroid with its axis at a constant depth f below 
the surface, its centre C describing a horizontal circle of radius h with constant 
angular velocity Q, the axis of the spheroid remaining at right angles to the rotating 
radius through C. We use the same fixed axes as before, with cylindrical co-ordinates 
O(a, 0, z); and, when required, we use rotating axes C(x, y,z) with Cx along the axis 
of the spheroid in the direction of motion and Cz vertically upwards. 
nd, (Kh) J,(Kkh)dk. (20) 
557 
