241 Prof. T. H. Havelock. 
The work is arranged in the following order: a general expression for the 
wave resistance as a function of the time, an exact solution for a certain 
waveless system, a comparison of this solution and the group approximation, 
and an approximate solution for certain systems which leave regular waves 
in their rear. 
2. Consider, first, the effect of a single impulse applied to the surface of 
deep water, with no initial displacement of the surface. Take the axis of y 
vertically upwards, the axis of x horizontal, and the origin in the undisturbed 
surface. If the impulse is given by F(z), and if the Fourier method is 
applicable, the elevation at any time ¢ is given by 
=aipn) = [ev sin (Vt) dic | “F@) exe) dg, (1) 
where V = (g/«)3, and the real part of the integral is to be taken. The 
effect of a pressure system, whether stationary or moving, can be obtained by 
integrating (1) suitably with respect to the time. For the pressure system 
may be considered as a succession of impulses; to each impulse there 
corresponds a fluid motion with definite velocity potential, and the velocity 
potential of the fluid motion at any instant is the sum of the velocity 
potentials due to all the previous impulses. Similarly, the corresponding 
surface elevations are simply superposed, and we obtain the required solution 
by an integration. 
For a pressure system moving with uniform velocity c, we have to 
substitute a-+ct for « in (1) and then integrate with respect to ¢ between 
the limits 0 and ¢. But the solution so obtained is indeterminate to a 
certain extent, for we can superpose on it any infinite train of waves of 
wave velocity c. The so-called practical solution is found by choosing the 
amplitude of this train so as to annul the main regular waves in front of 
the travelling system. The integrals are, in fact, indeterminate, and are 
evaluated by taking their principal value, in Cauchy’s sense of the term. 
Another way of avoiding this difficulty is to introduce small frictional terms 
proportional to the velocity. The integrals are then determinate, though 
more complicated in form ; however, the final results, after the analysis is 
completed, can be simplified by taking the frictional coefficient as small as 
we please. We shall use this method, and it is sufficient for our purpose to 
write, instead of (1), 
—o 
we oy f ® erbet eV sin (« Vt) dic I F (a) eda, (2) 
(0) 
where, ultimately, is to be considered small.* 
* Compare Lamb, ‘Hydrodynamics,’ 1932 edn. p. 348. 
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