3 Sir Thomas Havelock 
3. We suppose a sphere, half-immersed in water, to be given small periodic 
oscillations, the velocity of the centre being cos at. We take the boundary condition 
on the sphere to be satisfied at the mean position; thus, for all ¢, 
_ 0g 
Or me 
We shall assume that the velocity potential can be expressed in terms of a series of 
functions (8) together with a suitable periodic source at the origin. Hence we take 
1(u)cosot (r=a; 0<0<4n). (9) 
2k le K sin KZ + Ky COS KZ 
0 K2+ Ke 
K (ko) ar 
1 
= @?\—= —71Ky Yo(kyw) e 07 
? r oXo(ko ) T , 
x (C cos ot + Dsin ot) + 71k) a*Jo(Ky@) e “0 (C sin ot — D cos ot) 
+ Sam (Ro Pane =H!) , Pan(te = (A, cosot+B, sinct). (10) 
on yan y2ntl 
This expression satisfies the boundary condition (1) and also reduces to outward 
circular waves as woo. After some reduction, we obtain (9) in the form 
L(C cosat+ Dsin ot) + M(C sin ot — D cos ct) 
+ > {Pf Pon_1(4) + (2n + 1) Py, (4)} (A, cos ot + B, sin ot) = P()cosot, (11) 
1 
for all ¢ and for 0<6<4n. In (11) we have put £ = kya = o7a/g, and 
L = 1—7 {cos 0Y,(f sin @) + sin 0Y,( sin A)} e-4 © ? 
2h [? 2uwsin (fu cos 6) + (1 —u?) cos (fu cos 8) 
: J 0 (i+w? 
M = 7f?{cos OJ) f sin @) +sin 6S,( f sin 0)} eF 089. (13) 
K,(fusiné)du, (12) 
7 
The coefficients C, D, A,,, B,, are to be determined from (11). The functions defined 
by (8) are not orthogonal, but it turns out to be convenient to follow the usual 
procedure with (11) to give an infinite set of equations for the coefficients. Thus we 
multiply both sides of (11) by Py, 4(“) + (2m + 1) P,,,() and integrate with respect 
to « from 0 to 1; we take P,(u) for the case m = 0. 
We use the notation, with L given by (12), 
1 
Ly =| Ldp, B= {B Pom (#4) + (2m + 1) Py, (u)} Ly, (14) 
0 
with a similar notation for MW, derived from (13). Taking the terms in cos ot and 
sin ot separately, we obtain in this way a set of equations of which the first eight are 
I)C-M,D+4£A,—-1$A,+7%fAst+-.. = 4, 
LyC-MD+ +BY A+HPA,—sehAgt. = 84H 
LC — M,D+i8fA, + (22 +2304+4 6?) A.+ 3445+... =—-—de 
I,C —M, De PAy + posh 40 as + sceP + ih) 43h-- = 18: 
(Similar equations in D, —C, By, By, ... = 0.) (16) 
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