THE FORCES ON A SUBMERGED BODY MOVING UNDER WAVES 
giving a surface elevation 
n =hsin[«(xcosy +ysiny)+ot+a] . 
provided o=x(Vcosy+c); c2=g/k 
To obtain ¢2 we have now to expand 
exp«[z +i(x cos y + ysin y)] 
in a series of spheroidal harmonics. It is found that 
instead of I(10) we require the two integrals 
27 
[er — p2)§(cosw + isin y sina) cosswdw 
= 27 C, 1, [kb (1 — p?)* cos y] (4) 
and the same integral with sin s w for cos s w, equal to 
(2) 
(3) 
S,I, [« 6 (1 — p2)! cos y] (5) 
in which 
C,=4[( + sin y)s + (1 —sin y)*]/cos y 4 
Sp 3 [@ =f sin’y)> —@— sin'y)s]/coss yy 6) 
Proceeding as in I Section 2, the only alteration required 
is to replace kae by kaecosy and Ajcossw by 
As (C, cos s w +78, sins w). 
Thus, as in I Section 3, we have for the effective part 
of the pressure 
2 7 
$ 
Peal Cac caky) peers 
) Gl se Ken) WL fe) OV) cee 
jics (Ti) = ei(ot+a) (7) 
with 
Un t4 (K Ze COS y) 
B= DICnL howe 4 
(% — 1) Qi (4) 
PS (u) (C, cos s w + 7S, sin s w) (8) 
with a factor 4 for the terms with s = 0. 
The forces X, Z and the pitching moment M are 
given by I(25); and, owing to the presence of terms in 
sins w, we have a swaying force Y and a yawing 
moment M’ given by 
QT 1 
Y= — #2 o(— [sino [ pPi dude 
0 —1 
27 1 
M = —18e(2 — D> | sin w | pP\Wdude 
0 —1 
ai aes (9) 
The evaluation of the forces and moments follows as 
in I Section 4, noting that we must introduce the 
appropriate values of the coefficients given in (6) and 
also that o =x (Vcos y + c). 
2. We add to the notation specified in I Section 5 
C, = swaying force coefficient = Y (max)/D 6 
C,, = yawing moment coefficient = M’ (max)/D L @ 
The components are given by 
X= —D2@C,cos(ct + a); 
Y = —D0@C,cos (ct + «); 
Z= —D0C,sin(ct +a); 
M=DL@GC,,cos (ot + a); 
M’ = DL@C,,sin(ot + «) 
These relations give the connection between the phases 
of the components and the phase of the surface waves; 
if in varying the angle of attack y from 0 deg. to 180 deg. 
any coefficient C passes through zero and changes sign, 
the corresponding force or moment changes in phase 
by 180 deg. 
The force and moment coefficients are given by 
Cs Tee iS -) @ (Cae te) ap Ge gos ) 
C, = Csi 
B/D p Nh OP 
yee G cos =) 
E AL fy te ce = “feos (ke —k)| Jo (= 1098 y) 
C= G, sinvy, 
3/2 A 24] 
2D i To ) {fi 1 e = i) «| 
San(“o e087) + 2 
Peale 
Lal) Lea = 24a) ia(“-057) || 
(11) 
3. We note that the surging force is independent of 
the speed of advance except, of course, for the alteration 
in the period. Except for the surging force, all the 
components can be derived from the expressions in 
I Section 5 by replacing the wavelength A by A/cos y 
and the speed f by fcos y. Further, the coefficients 
C, and C,, differ from C, and C,, respectively only by 
a factor sin y. Putting 180 deg. — y for y in the expres- 
sions (60), we see that at zero speed C, and C, are 
symmetrical about the middle position y = 90 deg., 
while C,, C,, and C,, are anti-symmetrical. Taking 
account of the speed of advance removes this element 
of symmetry, for the effect is different according as the 
waves are from ahead or from astern. 
When y = 90 deg. the solid is moving parallel to the 
wave crests and this case is of some interest. The results 
can be obtained by taking the limiting values of (11) 
as y is made equal to 90 deg., or can be worked out 
independently. 
We have for y = 90 deg. 
C,=0; G=G=lt+hk 
PY feos 7 
eL 
1+k)04 k:)| Isp (0087) 
GQ a=C= (=) nee 28). 
In this position the forces are independent of the speed 
of advance. As might be expected, the moments are 
zero at zero speed; but it is specially interesting that 
pitching and yawing moments are developed when the 
solid is advancing parallel to the wave crests. No doubt 
