THE FORCES ON A SUBMERGED BODY MOVING UNDER WAVES 
To determine 42 we have to expand ¢; in a suitable 
series of spheroidal harmonics so that condition (6) 
may be used. We shall take ¢, in the form 
dy ——— h CETKIHEKZ + ilex + ot + x) 
(7) 
where the real part is to be taken. 
We first expand exp (« z+ ix) on the surface of 
the spheroid in spherical harmonics. On ¢ = 0 = le, 
we have x = a p, z = b(1 — p22)? cos w, and we assume 
exp [« b (1 — p22)? cos w +ixap] 
=ZYDAPi(y)cossw . (8) 
By the usual process for determining the coefficients 
we have 
As 2n+1(a—s)! 
a De Grom 
27 1 
[eos a) desl et —wtcose +ixay Ps (u) dp . (9) 
) 1 
noting that A, is given by (9) with s=0, but with a 
factor 4 
Taking the integration in w first, we have 
27¢ 
[ew [x b(1 — p2)? cos w] coss wdw 
0 
=?) mI. b (1 — p2)] 
where I, is the Bessel function of that type. 
It can also be shown that 
1 
|e (4) I,[« b el = p22) jeixeu du 
=1 
= (2 m)# ()" Pi (Lo) Ina (« ae) (x ae) 
J denoting the ordinary Bessel function. 
Hence we have 
(10) 
(11) 
atl ! 
AS=(2 7) ()""5(2n + 1) “ = 
Pr (Co) Ina (k @e)/(k ae)? 
A,, being given by s = 0, with a factor 4 
Hence the required expansion is 
exp(kz + ik x)= x x AS PS (11) PS (C)/PS (fo) cos s w 
ee (n —s)! 
- (27) 3 30 Ont DT 
Tres (ka e) PS() Ps(2) cos sw 
with a factor 4 for the terms with s = 0. 
To satisfy (6) we take for ¢2 a similar series with the 
typical term 
— ASPs (1) Q; (2) [Ps (Lo)/PS (Co) Q;(Zo)] cos sw (14) 
Hence ¢; + ¢2 is given by the real part of 
—he ea Kd tiCot+a) 
woe OF(Lo) PS () — BS (£0) (2) 
2D An Pa (1) Be (Lo) QS (Lo) 
with the coefficients A given by (12). 
(13) 
cossw . (15) 
591 
3. Pressure Equation 
The variable part of the pressure, omitting the 
buoyancy term, is given by 
p=po¢g[dt—4Lp(qe +a, + 4G) 
with qz, qu, Go for component velocities in the three 
corresponding directions. On the spheroid, the normal 
velocity g, is zero. Also the tangential component q,, 
comes only from ¢; + ¢2 and is of the first order, and 
as usual in wave-theory we neglect g2. So we have only 
to consider the tangential component q,,, which is given 
on the spheroid, using (5), by 
qu = —[(l — piace (5 — ?)*] 0 Pf pw 
ee 
ae(G— p) 
fav f Q ey | 
n) 
SON yh +} 
The square of the term in V in (17) corresponds to the 
pressure on a spheroid in steady axial motion in an 
infinite liquid, and the resultant forces and moments 
due to this term are zero. The square of the second 
term in (17) is of the second order and is neglected, 
and we are left with only the product term. Hence we 
need only consider, on the spheroid, 
(16) 
(17) 
) 
P= py, (hi + $2) 
Vii+k) 1— 
Pp = to ae Git és) 
where we have simplified the form by using the fact 
that the axial virtual inertia coefficient ky, is given by 
= € Qi (£0)/Qi (£0) (19) 
Turning to (15) for the value of ¢; + ¢2 on the spheroid, 
we use the relation 
PQ) — PQs = (— 1)°*! (n+ 8) —3)! (G — 1) (20) 
Introducing the coefficients A from (12) we obtain, on 
the spheroid 
gi + $2 = 
with 
B= 3, y M2 Dae 1) Oe Ynan (a) 
Ps (12) cos s wl(& — 1) Q§ (60) 
with a factor 4 for the terms in s = 0. 
Thus the effective pressure for calculating the forces 
is given by 
27? 
p= (=) phce “4 
’ (se Ke) WC = ) O13) pep step 
[ioe - a2 (2 — 2) smal? 2) 
with B given by (21), and noting that eventually we 
take the real part of the expressions. 
(18) 
(2 afeaejthce 4Becit# 
(21) 
