THE COUPLING OF HEAVE AND PITCH DUE TO SPEED OF ADVANCE 
tp U2? —$p U2 (1 +)? 0 — pyfe( — pw?) - (15) 
The integrals can be evaluated, and the result for this vertical 
force can be reduced to the simple form 
$apabU(l +k)? a(a + 2b)/(a + 6) —1] . 
This result was given in an equivalent form in a previous 
paper, which dealt with the sinkage of a general ellipsoidal 
form at low speeds. To estimate the magnitude of this 
effect we may equate (15) to mgpabs, and call s the 
equivalent sinkage. If, for instance, a/b = 10, we find 
s = 0-0157 U2/g = 0-314 f? b, with f as the Froude number. 
The effect in the present problem means simply an alteration 
of the origin O; but, as it is small except for high speeds, 
we Shall neglect it in what follows. 
9. Turning now to the term pd ¢/d1¢ in the pressure, it 
is easily seen from the various expressions which have been 
given, that the term from F, gives no contribution to the 
vertical force; and we have for this part of the vertical force 
(16) 
1 
ae? Cy (Q — D'phi| sino de | P! (w) Fy (4, Co, w) d 
0 =i 
(17) 
Substituting for F, from (3) and carrying out the integrations 
we can express this vertical force upwards as —k,M h, 
where M = a p.ab* = mass of spheroid and 
2n+ 1 (n = s)! Q@ (Zo) (Cs)? 
@—IP@F)(@—Na®eG) " 
ky 55753 
Wem ® 
(18) 
This expression is essentially positive, and k, is the virtual 
inertia coefficient for broadside motion of the submerged 
spheroid under the assumed condition of a rigid water 
surface. 
Similarly, putting the pressure term p 0 g/d t in (14), we 
find that F, gives no contribution to the moment, and we 
have for this part of the moment 
ud 1 
tpae(Q— 1) al sino de | P} (ju) Fo (tu, Co, w) dp 
0 = 
(19) 
The moment of inertia of the spheroid about Oy is 
I=47pab*(a +6’). We find that (19) can be ex- 
pressed as — k’ I ps with 
; 10 a? e3 2n+1 (n—s)! 
‘ 32a? + BS) UU (2 Zeal) (rates) 
sun (Co) (60) (Ds)? (20) 
(43 — 1) Q; (Co) 
k’ is the virtual inertia coefficient for rotation about Oy 
under the assumed surface condition. 
10. Lastly, from (8) and (12), the remaining terms in the 
pressure are 
1 — p? 
az e2 (C2 — 
(— ADF pw + PdFp/d yw) 
(21) 
With this in (13) and (14), it is seen that the only contribu- 
tion: to Z comes from the term in F, and the only con- 
tribution to M from the term in F,. Putting in the 
pail 
Lk) U 
ED 2) 
600 
expressions for F, and F, and carrying out the integrations, 
the results can be expressed as an upward vertical force 
pMU ¢ with 
2 20s) S 2 2p) C= 
me nate=ol(s2 cae 1)2 (n +s)! 
ade CDG Ds Es (22) 
(ZG — 1) Q, (Lo) 
and a moment — gq M UA, with 
220 sala) SS Zac | Go 
77 @ n=2 s=0 (s? oe 1)2 (n ar s)! 
aC AC UE oH (08) 
(Z — 1) Q5 (Go) 
where we have written 
1 
=| 4 =e > PH (uw) Ps (uw) ds 
n B= > 
1 
Re ae 1 . 
EF, = |e$ a 5 P§ (1) Ps(u) dw. (24) 
Summing up these results we get the equations of motion 
of the spheroid, with m as the metacentric height, 
1 +k)Mh—pMUp+amgpabh ei. an 
tk) 1p +qMUh+Mgmyp=0 
where k, k’, p, q are positive coefficients given by (18), (20), 
(22), and (23). It should be noted again that in these 
expressions s is even, the terms in s = 0 having a factor 4; 
further, in expressions involving the coefficients C and F, 
n is even, in those with D and E, z is odd. 
11. In the coupling terms in (25) the coefficients p and q 
are, in general, of unequal value numerically, but in the 
corresponding terms in Haskind’s equations they are equal. 
Haskind denotes these terms by — cUd and +cUh, 
with c defined by a double surface integral. On examination 
it appears that this expression for c does not involve the 
wave motion, but involves only the velocity potential due to 
the oscillations determined as if the free surface condition 
were ¢ =0. Further, it is based on replacing the solid by 
a source distribution over the surface of density o where 
470 equals the normal surface velocity, and this is then 
contracted to a distribution over the vertical plane section; 
this is a simplification which is appropriate when the form 
approximates to a thin disc. 
Turning to equation (2) the functions F, and F, were 
determined to satisfy the surface condition 0 ¢/dz=0. 
Suppose, for a moment, that we determine F, and F, from 
the surface condition ¢ = 0; then we should have 
¢=Ux—aeUP, (HQ ()/Q) (&o) 
eal ieICos a 
( — 1)! QI (G) 
a ers : 
so ESP TMV OL] 
1G= ea) = pan. 
make this velocity potential consistent and satisfying, 
P! (2) Q! (Z) sin w 
(26) 
