THE FORCES ON A SUBMERGED BODY MOVING UNDER WAVES 
that it reduces to zero for a sphere, as should be the 
case. The variation with the length-beam ratio is shown 
by the quantity in square brackets in (48); for instance, 
taking the corresponding values of the inertia coefficients 
this quantity reduces to 
1-823 Js + (2 7 LIA)* f (0-485 Jey — 0-279 Jy); B = 5 
1-883 Jsj. + (27 LJA)} f (0-549 Igy. — 0-313 Jyjo); B=10 
2 Isp + (2 7 LIA)* f O- 666 Js. — 0:333 Jy); B > © 
We choose the last case for numerical calculation; that 
is, we consider a long solid of revolution for which 
ky, k2, k’ are approximately 0, 1, 1 respectively, and 
we obtain C,, from (48) with these values. It is inter- 
esting that there are values of A/L for which the pitching 
moment is independent of the speed of advance; for 
this case, these are the roots of the equation 
2 Ss (7 LIA) — Sip (@LPA)=0 . . (49) 
The two highest roots are approximately A/L = 0-53 
and \/L = 1-51. 
This point is brought out in Fig. 2, which shows 
Cy on a base A/L for zero speed and for f=0:5, 
= — 0-5; the curves show how the effect of speed of 
“45 
-40 
“35 
+30 
N pees Se 
“4 “6 8 QO) 12 1:4 1-6 
Fic. 2.—PITCHING MOMENT COEFFICIENT FOR VARYING A/L; 
f POSITIVE AGAINST WAVES, NEGATIVE WITH WAVES 
fe) pL, L 
1:83992;O2-252:492:6 
advance on the pitching moment, with or against the 
waves, differs according to the value of A/L. The same 
result is also shown in Fig. 3, which gives C,, on a 
base f for given wavelengths. As this is a linear relation, 
the graphs are straight lines; those for A/L = 0-53, 1-15 
are parallel to the base line. 
OLE ee minal 
-O-1 ese | 3 
-O°5 -O-25 e) 025 O:5 
Fic. 3.—PITCHING MOMENT COEFFICIENT FOR GIVEN A/L, VARYING 
SPEED OF ADVANCE 
The results which have been given in this paper were 
obtained by direct calculation for a spheroid of any 
ratio of length to beam; nevertheless, in the form in 
which they are given in Section 5, they are probably a 
good approximation for any fairly long solid of revolu- 
tion. The conclusions are not directly applicable to 
surface ships; however, they may possibly indicate the 
nature of the difference to be expected when speed of 
advance and other factors are taken into account. 
The present analysis can be extended to include 
motion of the submerged solid obliquely to a train of 
waves, and it is hoped to examine the various forces 
and moments in subsequent work. 
References 
(1) HaveLock, T. H.: Quart. Journ. Mech. App. Math.,, 
V, 1952, p. 129. 
(2) WicLey, W.C.S. Trans. I.N.A., 1953, p. 268. 
(3) Lams, H.: Hydrodynamics, pp. 141, iss [6th edi- 
tion (1932)]. 
Part H. MOTION OBLIQUE TO THE WAVE CRESTS 
Summary 
Expressions are obtained for the forces and moments on a submerged spheroid moving 
under waves at any angle of attack. Graphs are given to show how these quantities depend 
upon the wavelength and upon the speed and direction of advance of the body. 
It appears 
that, when account is taken of the speed, pitching and yawing moments are developed when 
the body is moving parallel to the wave crests. 
1. The results given in Part I (which we shall refer 
to as I) can be extended to cover motion in any direc- 
tion. As the analysis is carried out by the same 
steps as in I, we need only indicate the necessary 
modifications. The solid moves in the direction O.x, 
and the wave train moves at an angle y to Ox; with 
y between 0 deg. and 90 deg. the motion is against the 
waves, while if it is between 90 deg. and 180 deg. the 
motion is with following waves. 
We have equation I(5) as before; and it is readily 
verified that we have now 
di = —hee-*4* cos [x (x cos y + ysin y) + ot + a] 
(1) 
