THEORY OF SHIP WAVES AND WAVE RESISTANCE. 15 
curves with experimental results to see whether absolute values are 
reasonably of the right order of magnitude; we cannot expect more 
when we remember the simplified form of the model and the other 
limitations of the theory. 
The three dotted curves in Fig. 8 are experimental curves of 
residuary resistance, the number marking each curve being the ratio 
of draught to length. The curves 0°0475 and 0:0385 have been drawn, 
on the scales used in Fig. 8, from those given by R. E. Froude for 
ships of 400 feet length of 5,390 and 4,090 tons displacement respec- 
tively ; while the curve 0'083 has been deduced from some results given 
by J. L. Kent. We notice at once how much more prominent the 
interference effects are on the theoretical curves; this is probably due 
chiefly to the neglect of fluid friction, whose indirect effect may be 
equivalent to an altered distribution of velocity in the present calcula- 
tions. The effect of differences of form, other than that expressed by 
the ratio of draught to length, is also obvious from the dotted curves. 
When we remember that the calculated curve, say that marked 0°05, 
is for a simple form not specially fitted to any actual model, the 
general agreement of order of magnitude over a considerable range of 
velocity is sufficient at least to justify the fundamental assumptions of 
the theory. 
It is perhaps needless to add that we are very far indeed from 
being able to predict or to calculate in advance the wave resistance of 
an actual ship. Nevertheless our chief aim will have been achieved 
if we have gained more insight into the nature of the problem ; for 
in this respect at least, the pursuit of theoretical investigations, even 
if apparently remote from practical requirements, is essential to a 
complete and scientific solution of the various problems of ship motion. 
NOTES AND REFERENCES. 
1.—The effect of a travelling surface pressure can be obtained by regarding it as a 
succession of applied impulses and by integrating suitably the expressions for the effect 
of a single impulse. Take axes Ox and Oy in the undisturbed water surface and Oz 
vertically upwards; let ¢ be the surface elevation and let O move with uniform velocity 
c in the direction Ox. If the pressure distribution is symmetrical round O and is 
given by 
p=F (r), 7? = x2 + y?, 5 E 0 0 5 5 (ili) 
the surface elevation can be obtained in the form 
gps = -| e 7 By [100 Jo [ «v{e + cu) + 7] sin (kVu)k?dx, . > @) 
0 “0 
where /4 is to be made zero after the integrals have been evaluated ; further V?=g/x, and 
F(k) = [re Jo(xa)ada, . 5 = 5 en) 
J being the Bessel Function of zero order. 
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