7 Waves due to a floating sphere making heaving oscillations 
alternative expression for h from energy considerations. The motion as 7-00 is 
given by 
3 
b>71k,a* (= =| eo? {C'sin (ot —Kyo +47) —Dcos(ot—Kyu+}m)} (35) 
0 
The average rate of flow of energy outwards is 7?poa3(C?+ D?); equating this to 
2 3 
#npoa*h we have h = 3nB(C2 + D?). (36) 
For numerical evaluation, the L’ and M’ quantities were computed by methods 
similar to those used for the L and M quantities in §5. Asan example, from the values 
given above for # = 0-4, we find & = 0-656; from (33) we obtain h = 0-174, while 
(36) gives h = 0-177. It will be appreciated that the values for the B coefficients are 
more liable to error than for the A coefficients; however, the two values for h were 
in fair agreement. Although the numerical computations for k and h were only made 
approximately the results were sufficiently consistent to be represented by smooth 
curves; these are shown in figure 1. The virtual inertia coefficient k begins from a 
limiting value of 0-828, rises to a maximum of about 0-88, falls to a minimum of 0-38 
and it then, presumably, rises slowly to the limiting value of 0-5. In order to use the 
same ordinate scale, the damping parameter 2/ is shown in figure 1; this rises to 
a maximum of about 0:35, the largest values of the damping parameter occurring 
in the frequency range in which the virtual inertia coefficient varies most rapidly. 
REFERENCES 
Grim, O. 1953 Jahrb. Schiff. Ges. 47, 277. 
John, F. 1950 Comm. Pure Appl. Math. 3, 45. 
Ursell, F. 1949 Quart. J. Mech. Appl. Mech. 2, 218. 
Ursell, F. 1953 Proc. Roy. Soc. A, 220, 90. 
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