10 WAVE PATTERNS AND WAVE RESISTANCE. 
all this work we are assuming the liquid to be frictionless; or, rather, we suppose that 
frictional resistance and the effects of viscosity have been treated separately and so eliminated 
from the wave problem in order to make it more amenable to calculation. It is true that 
the most direct idea of wave resistance is to regard it as what it is in fact, namely, the 
combined backward resultant of the fluid pressures taken over the hull of the ship; but this 
is by no means the simplest method for purposes of calculation. 
On the other hand, by a direct application of the method of energy and work, we 
shall see that we only need to know the wave pattern at a great distance in the rear 
of the ship. 
Denote by S the position of the ship at any instant, by A and B two infinite vertical 
planes in given fixed positions at right angles to the direction of motion of the ship, the 
plane A being in advance of the ship and the plane B to the rear. 
Consider the rate of increase of the energy of the fluid in the region between the 
surface of the ship and these two planes, and consider also the forces operating at the 
boundaries of this portion of fluid. The fluid possesses kinetic energy due to its motion 
and potential energy arising from alterations in the surface elevation. Calculate the rate 
at which total energy, kinetic and potential, is flowing into the region in question across the 
plane B and call this E(B). A similar calculation would give E(A) for the rate at which 
total energy is flowing out of this region across the plane A. At any point of the plane B 
let p be the fluid pressure and w the component fluid velocity inwards at right angles to 
this plane. The fluid to the left of B is doing work on the fluid to the right at a rate pw 
per unit area at each point of the plane; summing up for the whole plane, we call W (B) 
the rate at which work is being done on the fluid in question across the plane B. Similarly, 
— W (A), calculated in the same way for the plane A, is the rate of work across that plane 
upon the fluid between the two planes. Finally, if R is the resultant resistance to the 
motion of the ship and c¢ its velocity, the ship is doing work on the fluid at a rate Re. 
Hence, equating the total rate of work upon this portion of fluid to the rate of increase 
of its total energy, we deduce a general expression for R, 
Re = E(B) — W(B) — {E(A)— W(A)}. . . . (17) 
This holds for ary two fixed planes, one in advance of the ship and the other to the 
rear. If we take plane A further and further in advance, the quantities E(A) and W (A) 
approximate to zero, since the ship is advancing into still water. And if we take B further 
and further to the rear, the disturbance approximates to a free wave pattern such as we 
have considered in the previous sections and we can calculate the quantities E and W for 
any plane of that free wave pattern. Thus we have finally 
Cie We ce ene Pe ere eee OS) 
where E and W are calculated from the free wave pattern to which the disturbance approxi- 
mates at a great distance in the rear of the ship. 
11. This method is familiar in its application to plane waves with straight parallel 
crests. It is probable that the first calculations of wave resistance were those made in this 
way for plane waves, the argument being usually expressed in terms of group velocity. 
For simple harmonic waves of height h the average total energy is }gph® per unit area of 
surface; thus the quantity E of (18) is }gph?c per unit length parallel to the crests. 
The quantity W is exactly one-half of this amount; or, as it is usually expressed, the group 
velocity is one-half the wave velocity. Hence from (18) we have R = }gph?, where R is 
the wave resistance per unit length of the cylindrical body to whose motion the waves 
are due. 
It is rather curious that this method has not been used for obtaining the wave resistance 
from the wave pattern produced by ordinary ship forms. The formule in use at present 
have been developed by other methods. In some cases they have been found from the 
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