Waves due to a floating sphere making heaving oscillations 6 
On the sphere we have 
¢@ = aL'(C cosot + Dsinot) + mfaJ( sin 0) e4°? (C sin ot — D cos ot) 
ar x a © Py ait) + P,,(/) (A, cos ot+ B,, sin ot), (29) 
where, after using (21), (23) and (25), 
L’ = 1—4nfe-#es6 {H,( B sin 0) + Yo(fsin 6)} — BA e Fos’, (30) 
We obtain Z in the form 
Z = 2npatko sin ot — 37pa2ho cos ct, (31) 
with Pp = PO aA CYP) AAA esa lg= cons (32) 
2h = 11 D+nBM,C+ (6 +4) B,-— ag Bo t+ 73533 —--- (33) 
In (32) and (33) we have put 
1 1 
T= [UR (ude, Mi; =| (sin dyereo? Rly) dy. (34) 
0 0 
The velocity of the sphere being cos ot, the first term in (31) represents an addition 
to the effective mass, the virtual inertia coefficient being & as given by (32). The 
0-8 
k 
0-6 
Oe 2h 
0-2 
0 0-4 08 12 1-6 2-0 
oa/g 
Figure 1. Variation of virtual inertia coefficient k and damping 
parameter 2h with frequency 
second term in (30) being proportional to the velocity, the quantity / as given by 
(33) may be called a damping parameter; it gives some estimate of the damping 
factor if the motion were unforced damped periodic motion. We may obtain an 
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