THE FORCES ON A SUBMERGED BODY MOVING UNDER WAVES 
4. Resultant Forces 
If p is the fluid pressure at any point of the spheroid; 
(Z, m, n) the direction-cosines of the normal; (X, Z) the 
resultant forces in the directions Ox, Oz; M the 
resultant moment round O y, we have 
xe —|] pias; Vb, == —|] pnds; 
M=—|| pz—nxds (23) 
taken over the surface of the spheroid. In the spheroidal 
co-ordinates (2) we have 
T= (GS DG ae 
n= Co(1 — p?)? cos w/(f5 — y?)3; 
dS=@ e(@ —1)(G —wdpuda 
Hence we have 
(24) 
27 1 
X= —@e(G —1) [oranda 
0-1 
or 1 
Z= — ae (Co (GQ — 1)? cos w | p Pl (W) dude; 
0 =i 
Qn 1 
M = 343 63 (C5 — D! [cos w | pPLQ ded 
0 =i 
(25) 
For the horizontal force X, taking account of the 
integration in the angular co-ordinate w, we see that 
we only need the terms in B independent of w; and for 
the general term from the second part of (22) we require 
the value of the integral 
1 
[ta — p?)[(G — p)] Pi (uw) P, (uw) due 
=i 
It is easily seen that (26) is zero if n is odd; and if n is 
even, and equal to 2 m, it can be shown that the value 
of (26) is 
(26) 
= 2 (@ — 1) Op, (Co) (27) 
Hence from (21) and (25) we have 
X = (2 7)3? a? e? (CG — 1) phc(kae)te“4 
[o Jsol(S — 1) Qi (So) — Cl + x) (V/a e?) 
DH Gm + Tonle? . C8) 
m=1 
the argument of the Bessel functions being « a e. 
From the properties of these functions, the sum of 
the series in (28) is simply — (« ae) J3j2 (k ae). 
Using properties of the spheroidal harmonics such 
as (20) it can be deduced from (19) that 
1 + ky = —ef(@ — 1) Qi (4) (29) 
Thus the quantity in square brackets in (28) reduces to 
e'[—co(Q+h)+«V(l + ky)| Jap (k ae) (30) 
Noting that o =«(V +c), this simplifies further; and 
we obtain, taking the real part, 
X = —(2 7)? gp b2he—4 (1 + ky) (ka e3)-? 
Iso (« @e) COs (a t + a) (31) 
592 
From the expression for the vertical force in (25), we 
have to evaluate for the general term the integral 
1 
| [G = 2G — -2)] PEW) PL) du. G2) 
This integral is zero if nm is odd. If n is even and equal 
to 2 m, it can be shown to have the value 
(2/ £0) (G Ta 13? Qhn (Zo) 
From (21) and (25) we obtain 
Z=(2 7)? ab(kae)* phce*4 
[27 o Ispof(S — 1) Qi + iV Ge)" (1 + kx) (G — 1)? 
3 (HS WF Ea + ) yall ee? (34) 
the argument of the J functions being «ae, and that 
of Q being Co. 
The series in (33) sums as in the previous case; further 
if kz is the virtual inertia coefficient for transverse 
motion of the spheroid, we have 
(33) 
ko = — fo Qi (Lo) — 1) Qi (Lo) (35) 
from which we can deduce 
(Z —1)? G +k) =2/(G =) QI Gs 6s) 
Thus the quantity in square brackets in (34) has J). as 
a factor, and another factor is 
dd +k) WV +c) —C +k) V 
Collecting these results we obtain finally 
Z=(2 7)? gp b?he-*4 (x ae) + 
[1 + ke + (ka — kx) V/c] Ign (x ae) sin (ot + a) . (37) 
Similarly, for the moment M from (21) and (25) we 
have to evaluate the integral 
1 
[to PLB de . G8) 
=i 
In this case, (38) is zero when n is even. When n is 
odd and equal to 2m + | it has the value 
6 (G — 19? Qam+1 (So) (39) 
for all values of m except m = 0; when m = 0, (38) has 
the special value 
6 ( — 1) QI (Lo) — 8 
With these Values in (21) and (25) we have 
M = (2 7)3? a 3 (k ae) + (G —1)thce*4 
{2 o Ispl(& — 1) Q + (1 + ka) (Via e2) 
[= 4 Japl(@ — 1) Qi + (G — DF 
¥ (— 1)" (4m + 3) Tom gap} ett? (41) 
0 
The series of Bessel functions has the sum k ae J, (k ae). 
Also we substitute from (36) for Qj in terms of ky. We 
may also introduce the virtual inertia coefficient for 
rotation; k’ is defined as the relative increase in moment 
of inertia of the spheroid rotating like a solid of density p. 
It can be shown that 
ki = —QY(S)/6o(G — D2 — 1) Q (4) 42) 
Using (20) and the expressions for P} and P; we deduce 
2G —1)Q) = &(B—-MH1+AB—-wR] 43) 
(40) 
