THE FORCES ON A SUBMERGED BODY MOVING UNDER WAVES 
We also replace J3,. in (41) in terms of Jj). and Isp 
and after some reduction we obtain, AGS: putting 
o=«(V +c), the form 
M=(2 7)? ¢pab?he—"4(« ae) + 
{[1+ (4) 7 jhe | “14 (G4) & 
—3(1+kh)0+ K)| Js. + 3 (1 + ka) 
(1 — 2 ko) Jin} co (ot + a) (44) 
where the argument of the J functions is «ae, and 
B = length-beam ratio = a/b. 
5. Summary of Notation and Results 
We may express these results more conveniently in the 
following notation, in which we also define suitable force 
and moment coefficients. 
L = Length of spheroid = 2 a. 
e = Eccentricity = (1 — 6?/a’)}. 
8 = Length-beam ratio = a/b. 
D = Displacement = $ 7 g pa b2. 
ky, k2, k’ = Axial, transverse, rotational virtual inertia 
coefficients, as defined and evaluated, for 
instance, in Ref. 3. 
V = Speed of advance (positive against the waves). 
f = Froude number = V/(g L)?. 
d = Depth of axis below the surface. 
h = Amplitude of waves = half wave height. 
A = Wavelength = 2 a/k. 
c = Wave velocity = (g/x)?. 
2 m/o = Period of encounter. 
o=«(V +c). 
6 = 2 7 (h[A) e774 = Maximum effective wave 
slope at depth of axis. 
X, Z = Resultant forces in directions O x, O z. 
M = Resultant moment about O y. 
C, = Surging force coefficient = X (max)/D @. 
C, = Heaving force coefficient = Z (max)/D 0. 
C,, = Pitching moment coefficient 
= M (max)/D L @. 
In this notation, the results obtained are 
X= —D20C,cos(ot+«);Z= — DOC,sin (ct + «); 
M=DL90C,,cos(ot + «) (45) 
3/2 A322 eL 
C=7,en tht) In (>) - 49) 
By? 2aL 
C= or [pte +0(A —) ‘(—4)] 
Oe yp Te a). (47) 
593 
4,5 -aeen(i) {1+ (4) # | In (A) 
A ) {hi + (GS = N Kk’ 
—2(+h)0+ K)| Isp (S— 
meL 
H+ e)C = 2h) (“)h} G8) 
6. General Discussion 
The phase relations between the waves and the forces 
can be seen from a comparison of (4) with (45). It is 
of interest that these relations are unaltered by the 
improved theory: there is, for instance, a difference of 
phase of 90 deg. between the heaving force and the 
pitching moment. It is also confirmed that the period 
of the forces and moment is the period of encounter 
with the waves. 
From (46) we have the unexpected result that the 
surging force coefficient is independent of the speed of 
advance. The coefficient is small and oscillating in 
value for small values of A/L; for a long spheroid, the 
highest zero position is at about A/L = 0-7. The graph 
of C,, except for a scale-factor, is the same as the curve 
labelled f= 0 in Fig. 1. For large values of A/L, the 
surging force X approximates to 
—(1 +k) D @cos(ct + a) 
assuming the wave slope to be kept constant. 
Similarly from (47), the heaving force Z approximates 
to —(1 + k2) D @sin (ot + «) for large values of A/L. 
In general C, varies with the speed of advance due to 
the difference between k; and ky. We take for illustra- 
tion the case of a long spheroid with e approximately 
unity and k,, ky approximately 0, 1 respectively. Fig. 1 
shows C, on a base of A/L for zero speed and for 
f=0-5 and — 0-5; we note that f positive is for motion 
against the waves and f negative for motion with the 
waves. 
2:5 
2:0 
1:8 
LO tea yr 2Om2:2a4 
Fic. 1.—HEAVING FORCE COEFFICIENT FOR VARYING A/L; f POSITIVE 
AGAINST WAVES, NEGATIVE WITH WAVES 
There are several points of interest in the pitching 
moment coefficient (48). In passing, it may be remarked 
