504 The Effect of Shallow Water on Wave Resistance. 
simple transverse waves is about 27/; this may be called the principal wave- 
making length of the disturbance, to use a term from the theory of ship 
resistance. Taking next the curve for p = 2, we can see indications of two 
maxima. The first occurs at about the point 0°97 on the velocity scale; it 
clearly corresponds to the deep water maximum, and cdmes lower down the 
scale, because waves of given length occur at a lower velocity as the depth 
diminishes. There is also a second maximum at a velocity of about 1:25 ; 
this is due to the other factor in the resistance, namely, the increased effect 
as the velocity approaches the velocity ,/(gh) of the so-called wave of 
translation, which in this case is at the point 1:41 on the velocity scale. 
From the next curves, p=143 and p=1, we see the increasing 
importance of the latter effect as the depth becomes less. For the curve 
p = 1-43, there is a maximum near the velocity 1:1, the corresponding value 
of (gh) on the scale being 1:2. There is no other actual maximum, but 
there is an enhanced resistance at about 0°92, followed by a flattening of the 
curve between that point and the point 1:05; we may take the increased 
effect at 0°92 to correspond to the deep water maximum in the lower 
curves. Similarly for the curve »=1, the corresponding values are: 
increased effect at about 0°81, diminished slope of curve between 0:82 
and 0:9, maximum at 0:97, velocity of wave of translation 1:0. The last 
curve, p = 0°75, shows that, as the depth becomes small, the second effect 
becomes the predominant feature; the excess resistance increases rapidly in 
magnitude, and occurs practically at the velocity ,/(gh). This effect is still 
more pronounced for p = 0°5, but the results are not shown in the figure. 
It is obvious that, as the ratio of 4/2 diminishes, the disturbance becomes 
more like that due to a line disturbance ; in simple calculations on the latter 
assumption, the resistance increases indefinitely at the velocity /(gh), and 
falls suddenly to zero above that velocity. It will be seen from the figure 
that in all cases the resistance falls after the velocity ,/ (gh), as, in fact, may 
be deduced directly from the expression (17). 
In a comparison between these results and the experimental curves of 
ship resistance described in § 1, it is advisable to consider in each case the 
difference between the resistance in water of a given depth and that in deep 
water; in this sense the results agree in character. Thus the first effect of 
finite depth may be regarded as due to the lowering of the chief wave- 
making velocity; it is only when the depth of water becomes of the same 
order as the beam of the ship that the critical velocity is practically that of 
the wave of translation. 
In describing the experimental curves, it was stated that the excess 
resistance has a maximum value at a certain critical velocity. But there is 
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