THE DAMPING OF 
both e and k, approximate to unity, further, it can be shown 
that the asymptotic value of the integral in (23) is 2/15 xg ae. 
Hence, as one would expect, the ratio (23) approaches unity for 
a sufficiently long narrow spheroid. Under the same conditions, 
k’ approaches unity and the asymptotic value of the integral 
in (24) is 2/105x ae; hence the ratio (24) approaches unity 
under these limiting conditions. 
For numerical computation we can obtain power series for 
the integrals by substituting the known expression for the square 
of a Bessel function and integrating term-by-term: thus we have 
7/2 
| Jp (Ky a e cos 6) sec? 6 dO 
Y 2 (— 1)" (m + 1)! (kp ae)?" +3 
mao (2m + 1) (2m + 3) (m!)2 (m + 3)! (25) 
7/2. 
| Jz) (ko ae cos A) sec? 6d A 
; 2 (— 1)” (m + 2) (Ko a e)2m+5 
= 2 
22,2 m + 3)(2m+5)m!(m +5)! ~ 5) 
The series can be computed readily for values of ky ae up to 
about 6. For higher values, the integrals were computed by 
direct quadrature, using intervals of 5 deg. throughout the range. 
Owing to the lack of suitable tables, the Bessel functions had 
to be evaluated separately in each case; however kp @ e was not 
taken larger than 10 as, with the degree of accuracy attempted, 
there was no appreciable difference then from the asymptotic 
value. 
6. We may extend the method to give approximate formulae 
for any long solid of revolution which is completely immersed. 
There is a well-known approximate solution for the transverse 
motion of a long solid of revolution in an infinite liquid, in which 
the flow is treated as two-dimensional; it consists of taking a 
distribution along the axis of two-dimensional dipoles of 
moment S/z per unit length per unit velocity, where S is the 
cross-sectional area at any point. 
We have seen in (11) that the transverse motion of a spheroid 
is given by a line distribution of three-dimensional dipoles along 
the axis from — ae to + ae, of moment per unit length per 
unit velocity 
1 
“4 
For a long spheroid, for which e is nearly unity, (27) is approxi- 
mately (1 + k) S/4 77; and to the same order we may take the 
distribution as extending over the whole of the axis. This 
suggests that for any elongated solid of revolution we might 
assume a distribution of three-dimensional dipoles along the 
axis of moment (1 + k) S/4 7 per unit length. Thus for heaving 
oscillations V cos o t of such a solid with its axis at depth f, 
2 
(+ ky) (@ 2 — 12) a (27) 
HEAVE AND PITCH 
we may apply (7) and (8). If 2/ is the length of the solid, and 
we take the origin at the centre of the axis, we have 
Ey = 
21 
aap ons +k? vena | (P2+.Q%)d8 (28) 
0 
vl 
with P+iQ= | Sietmorcovas oo ol 
—e 
Similarly, for pitching oscillations to the same approximation 
ED Te 
27 
2 paca igeGe | (P2 + Q2)d6 (30) 
0: 
l 
with P+iQ= | AS(permicnrah . 0 o Gil) 
e 
It may be noted that k and k’ are the virtual inertia coefficients 
for the solid as a whole; though, under the given condition 
they both approximate to unity. 
7. All the foregoing calculations are for a solid of revolution. 
With a view to removing this limitation, expressions were 
obtained for a general ellipsoidal form. 
For an ellipsoid with unequal axes, a > b > c, and with the 
a, b axes horizontal, the dipole distribution is in a horizontal 
plane and extends over the area enclosed by the elliptic focal 
conic. Application of (7) and (8) leads to expressions for the 
energy loss. 
If the larger transverse axis is vertical, c > 6, the distribution 
lies in a vertical plane, and within the elliptic focal conic; a 
simple modification of (7) and (8) gives the required results. 
It was decided eventually that it was not worth while carrying 
out computations; the expressions are of the same type as for 
a spheroid, though more complicated. It appeared that if the 
transverse axes b, c do not differ greatly, the main difference in 
the results as compared with a spheroid is a scale factor arising 
from the different values of the virtual inertia coefficients. 
References 
(1) WetnsLum, G., and Sr. DENIs, M.: ‘On the Motions of 
Ships at Sea,”’ S.N.A.M.E., 58, p. 84, 1950. 
(2) St. Denis, M.: ‘‘On Sustained Sea Speed,” S.N.A.M.E., 
59, p. 745, 1951. 
(3) Korvin-Kroukoysky, B. V., and Lewis, E. V.: “Ship 
Motions in Regular and Irregular Seas,’ Intl. Shipb. 
Progress, 2, p. 81, 1955. 
(4) Korvin-KRroukoysky, B. V.: “Investigation of Ship Motiang 
in Regular Waves,”’ S.N.A.M.E., 63, p. 386, 1955. 
(5) HaveLock, T. H.: “The Damping of the Heaving and 
Pitching Motion of a Ship,” Phil. Mag., 33, p. 660, 1942. 
614 
