Theory of Short and Long Waves 23 



this restriction is abandoned, forms the subject of a classical research by 

 Stokes (1847). 



Stokes developed his theory by using Rayleigh's method by superposing 

 a wave disturbance upon a steady current. 



If in the case of infinite depth, neglecting small quantities of the order 

 A 3 /P, a stationary wave disturbance of the wave length I is superimposed on 

 a steady motion with the velocity c the velocity potential q> and the stream 

 function y will be: 



<P 



-x — Ae^s'mxx 

 c 



- = — z+Ae*- cos xx 

 c 



(11.15) 



The equation of the wave profile of the disturbed surface \p = is found 

 by successive approximations from z 



z = Ae* z cosxx = A (I + xz+lx 2 z 2 -{-...) cos xx 



= | xA 2 +A (1 + f x 2 A 2 ) cos xx + 1 xA 2 cos 2 xx + 1 x 2 A 3 cos 3kx + . . . . 



if 



A(l->x 2 A-) =a 



we obtain 



r\ = \ xa 2 + a cos xx + | *tf 2 cos 2x:x + 1 x 2 a 3 cos 3xx-\- ... (11.16) 



With increasing amplitude the wave profile difTers more and more from 

 the simple-harmonic type. The wave crests become narrower and steeper, 

 wave troughs wider and flatter. The wave profile as expressed in equation (11.16) 

 corresponds to the equation of a trochoid, in which the circumference of the 

 rolling circle is 2n\x = X and the length of the arm of the tracing point is a. 

 So far his profile corresponds to that of the Gerstner waves (see p. 26). 

 The ratio of 2a\l = r * B represents already a high value (see p. 48) and, 

 neglecting the terms of 3rd order in xa we omit in (11.16) a value less than 

 one thirtieth of the wave height. 



Burnside (1916) raised the question as to the convergence, both of the 

 series which form the coefficients of the successive cosines when the ap- 

 proximation is continued and of the resulting series of cosines. He even 

 doubted the possibility of waves of rigorously permanent type. Levi-Civita 

 (1925, p. 264; Geppert, 1929, p. 424) has proved the convergence and the 

 existence of waves of a permanent type in agreement with hydrodynamic 

 principles. The wave profile of a permanent wave as calculated by Levi-Civita 

 difTers very little from the profile computed by Stokes and Gerstner (trochoids), 

 and for small amplitudes there is practically no difference. With increasing 

 amplitude there is a gradual transition of the wave profile from the harmonic 

 type to the trochoidal form, and a further increase of the wave height will 

 change the profile again. If the trochoidal form were exact instead of being 



