THEORY OF SHIP WAVES AND WAVE RESISTANCE. 2 
is simply that the latter problem can be solved mathematically in 
certain cases. 
It is clear that the wave resistance is the resultant of the surface 
pressures when resolved in a direction opposite to that of the motion. 
These calculations have been carried out (Note 1), but we shall only 
consider here the graphical form of the results 
Fig. 1 shows the variation of wave resistance R with the velocity. 
The pressure system is of a certain localized type, symmetrical round 
a centre which moves over the surface with constant velocity c; the 
quantity f is a length which may be called the effective radius of the 
applied pressure system. There are various points of interest in this 
curve, but I shall only mention one or two which have their analogues 
in ship resistance. Notice that the wave resistance is very small at 
low speeds. Then it begins to increase rapidly and reaches a maximum 
when the speed c is about equal to / (gf); this means that the wave 
resistance is a maximum when the length of the transverse waves 
produced is of the same order as the length of the pressure system. 
After this stage the resistance deereases gradually to zero. A little 
consideration will show that this last result might have been anticipated ; 
it may be described as a sort of planing or smoothing action of the 
pressure system when the velocity becomes very large. 
Shallow Water.—Before we leave this elementary pressure system 
we may use it in another interesting problem. We have assumed 
so far that the water is very deep, but we can examine the effect of 
shallow water by adding the condition that at the bottom of the water 
ithe vertical velocity must vanish. The work becomes more difficult 
but formal solutions can be obtained and calculations made from them 
(Wote 2). We know that on water of depth h the speed of transverse 
waves cannot exceed the value ,/(gh), which is the speed of the so-called 
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