3 
The well-known wave pattern which accompanies a 
moving ship is complicated, as it consists of both diverging 
waves and transverse waves. We begin with the simpler 
wave formation which is obtained by drawing a long rod 
over the surface at a steady speed in a direction at right 
angles to the rod; we observe that the water surface behind 
the rod is undulating, with parallel ridges and hollows suc- 
ceeding each other regularly. The distance between conse- 
cutive ridges is called the wave-length; it is found that the 
waves have definite wave length and a definite height 
(a feet) above the mean water level for a given speed (v) 
of the rod. It can be proved that over the range where 
there are regular waves the mean energy of the wave motion 
is 4wa? foot-pounds per square foot of the surface, where 
w pounds is the weight of a cubic foot of water. 
What is the length of the train of regular waves behind 
the rod at any time? Its front is at the rod, and so moves 
forward with velocity v; its rear depends upon how and when 
the rod was started. Suppose the motion has been steady 
for a considerable time, so that the range of regular waves is 
large compared with the initial disturbances in getting up 
speed ; it can be shown that the rear of the train of waves 
moves forward at a certain speed (wv) less than v. This 
velocity of the rear is called the “group velocity ”; if we 
observe a group of waves advancing into still water we may 
notice the crests moving forward relatively to the group, 
so that the wave velocity is greater than the. group velocity. 
The result in the present connection is that the wave-train 
is increasing constantly in length at a certain speed (v-u); 
hence the energy in the wave motion is increasing at the 
rate }wa? (v-w) per foot-length of the rod. Energy must be 
supplied at this rate in order to maintain the constant speed 
v of the rod. If we write the rate of supply as Rv, then R 
is a force per foot of the rod and is called the wave-making 
resistance. We have then— 
R=} we? (v—-u)v . : } (1) 
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