617 T. H. Havelock. 
plane. Subtracting (13) from (12), we find that the rate at which energy is 
being propagated less the rate of work reduces to the simple expression 
ua 
npe* | {f (0)}? cos? 6 d0. (14) 
0 
It may be noted that if we take mean values of (12) and (13) we have as the 
mean rate of flow of energy 
2 cos® 8 
amod |” (f(OP (15) 
and as the mean rate of work 
p08 (i (f (®)} 5 a = = 46. (16) 
The connection indicated in (15) and (16) is a generalization of the well-known 
result for simple plane waves that the mean rate of work is half the mean 
rate of flow of energy. 
3. Consider now the forced wave pattern produced by a body moving through 
the liquid, or by a localized pressure disturbance. The complete surface 
elevation may be separated into a local disturbance and a wave pattern. Ina 
frictionless liquid a possible solution is one in which the wave pattern extends 
to an infinite distance in advance of the body as well as in the rear. The 
determinate practical solution is that for which the wave pattern vanishes at 
a great distance in advance, and we may suppose this obtained by superposing 
over the whole surface a suitable free wave pattern. In that case, considering 
the flow of energy and rate of work across two fixed vertical planes, one far in 
advance and the other far in the rear, we see that (14) is equal to Re, where R 
is the wave resistance. Hence we have 
apes ie {f (8)}? cos? 6 d8, (17) 
when the wave pattern at a great distance to the rear approximates to the form 
(4). 
For example, the forced wave pattern produced by a submerged sphere, or 
more precisely by a horizontal doublet of moment M at depth f, approximates 
at a great distance behind the disturbance, to the free wave pattern 
42M 3 
C= : | zs sect § en "of 3°" 8 gin {ko (a! cos § + y sin 6) sec? 6} dé. (18) 
7 
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