Ship Waves. 469 
carried out on an actual model, would no doubt involve other changes which 
would have to be considered in a theory capable of taking exact account of 
actual dimensions ; but meantime we may isolate the effect of this particular 
change. 
For convenience we consider separately the effect on the local disturbance 
and on the wave motion to the rear. Taking the former, we see from (6) 
and (10) that the difference amounts to replacing + Qy (koq,) by 
a {Q1 (Kod2) — Q, (Ko%1)}- (13) 
This can be shown in a form applicable to various velocities and to various 
ranges of 7, — x, by graphing the quantity 
Zz {Q (p + ) — Q (p)} (14) 
on a base p, for several values of k. These curves are shown in fig. 2. 
Fia. 2.—Curves of {Q (p + k) — Q, (p)}/4k for different values of k. 
In applying these curves to actual distances along the ship model, we note 
that p= Kyv = ga/u?, where wu is the velocity; and similarly k = gd/u’, 
where d is the range over which the original sudden change in slope has been 
distributed. Thus the relative importance of the effects depends upon the 
ratio gd/u?, or upon the ratio of d to A, the wave-length of straight water waves 
for velocity uw. In the diagram, k = 0 denotes the curve for the sharp corner ; 
the bow of the model is to the right of the diagram and the stern to the left. 
Apart from the general smoothing effect, the chief point to notice in these 
curves is the raising of the profile forward of the point in question and a lower- 
ing to the rear of it. This is due to taking the range d entirely to the rear of 
the original sharp corner. If, on the other hand, the corner is taken at the 
middle of the range d in each case, by a suitable relative displacement of the 
curves, it is easily seen that the smoothing of the corner does not make any 
appreciable difference to the local disturbance except within the range d 
itself. 
364 
