The Wave-making Resistance of Ships. 277 
a be the amplitude of the waves, and w the weight of unit volume of water ; 
then the mean energy of the wave motion per unit area of the water surface 
is }wa’. Imagine a fixed vertical plane in the rear of the disturbance ; the 
space in front of this plane is gaining energy at the rate 4wa?v per unit time. 
But on account of the fluid motion, energy is supplied through the imaginary 
fixed plane to the space in front, and it can be shown that the rate of supply 
is $wa*u, where w is the group-velocity corresponding to the wave-velocity ». 
The nett rate of gain of energy is dwa?(v—u), and this represents the part of 
the power of the ship which is needed, at uniform velocity, to feed the 
procession of regular waves in its rear. An equivalent method of stating 
this argument is to regard the whole procession of regular waves from the 
beginning of the motion as a simple group; then the rear moves forward 
with velocity ~ while the head advances with velocity v, and the whole 
procession lengthens at the rate v—w. If we write Rv for the rate at which 
energy must be supplied by the ship, we call R the wave-making resistance, 
and we have 
R = fwa? (v—u)/v. (2) 
We notice that R is the wave-making resistance in uniform motion; it is 
only different from zero because w differs from v, that is, because the velocity 
of propagation depends upon the wave-length, 
In deep water, wv is $v, so that R is }wa?. In the application of this to 
a ship at sea, it is assumed that the transverse waves have a certain average 
uniform breadth and height, and, further, that the diverging waves may be 
considered separately and as having crests of uniform height inclined at 
a certain angle to the line of motion; if the amplitude is taken to vary as 
the square of the velocity, it follows that R varies as v4. Several formule 
of the type R= Av‘, or R = Avt+Bv®, have been proposed ; although these 
may be of use practically by embodying the results of sets of experiments, 
they are not successful from a theoretical point of view. Recently many 
such cases have been analysed graphically by Prof. Hovgaard ;* the general 
result is that a fair agreement may be made for lower. velocities with an 
average experimental curve neglecting the humps and hollows due to the 
interference of bow and stern wave systems, but at higher velocities the 
experimental curve falls away very considerably from the empirical curve. 
The method used here consists in considering the ship, in regard to its 
wave-making properties, as equivalent to a transverse linear pressure 
distribution travelling uniformly over the surface of the water. Taking 
a simple form of diffused pressure system and making some necessary 
* W. Hovgaard, ‘Inst. Nav. Arch. Trans.,’ vol. 50, p. 205, 1908. 
35 
