Heaving and Pitching Motion of a Ship. 671 
2 mf 2 \* 
as muh ( —) eof cos(pt+7 or). 5 - 6 (17) 
g TK o” 
The rate of transmission of energy outwards is obtained from the rate 
of work of the fluid pressure over the surface of a vertical cylinder of 
radius 7, that is, from 
DY Go 
<f ge tem de, S Aehiyed ln Space ea LS) 
Using (16), we obtain for E, the mean rate of transmission of energy 
outwards, 
Deo Pore joe Aw, 5 5 6 8 oo (IG) 
This result may be generalized to cover any given distribution over a 
surface S in the liquid. Let m cos pt be the source strength per unit area 
at a point (2’, y’, 2’) on this surface ; we have to substitute for 7 in (15) 
the quantity 
(7? 2rax’ cos @—2ry’ sin 0+ x’2+-y’2)?, ae <eean((20) 
and then integrate over the distribution. 
It is readily seen that we only need the approximation for r large, 
namely 
8 + 
pm ( ms) te sin (+4 —«¢"] -+Q cos (+7 ~«a)} > (i) 
where P+1Q= {J Oe, Of, NE Oe CT OG 5 no (BD) 
From these expressions, we obtain for the mean rate of outflow of 
energy 
270 
H—2npeop |, (ESOP, 65 0 4 4 5 CR 
7. Consider a circular cylinder, of radius 6, immersed with its axis 
vertical to a depth f and making forced vertical oscillations given by 
asin pt. As in the two-dimensional problem, if 27g/p? is larger than the 
diameter of the cylinder, we may assume the wave motion to be due to 
an alternating source m cos pt at a depth f, with 47m=b2pa. Hence, 
from (19), we have 
Bay nt gfe) oe ke eo a eS) 
Assuming that this may be used to evaluate N for the natural damped 
vibrations when the cylinder is floating freely, we obtain 
2, 
Nay Avi. fee Phe Pees HAIN) 
From the usual hydrostatic theory, o*=g/f. Without attempting to 
evaluate the effective mass in this case, we write M’=(1+-.)M=(1-+y)zpb?f. 
Hence, with these values, the logarithmic decrement is 
Nl Gre 
~ oM’ (1+ pf?” 
For instance, with f=4b and neglecting x. we should have 6=0-04. 
(26) 
497 
