Highest Waves in Water, 431 



The work of Prof. Stokes fills the gap between these and 

 the infinitesimal waves. 



Waves in Deep Water. 



Suppose at first the water so deep that it may be considered 

 infinitely so. Make the motion steady by moving the water 

 forward bodily with the backward velocity Y of the wave. 

 Take rectangular axes, x horizontal, y vertically downwards, 

 in a plane of motion, with the origin at a wave-summit. Let 

 <j>, t/r be the velocity and stream functions, the surface of the 

 water being yjr = and the bottom tJt = go . 



Call the velocity q, the inclination of the wave-line to the 

 horizontal at <£, 0, the curvature at the same point K. 



At the summit the curvature and the velocity are zero, and 

 the ratio of the two is finite and not far from constant along 

 the whole wave-line, as appears from the investigation fol- 

 lowing. 



Accordingly we assume, taking <p = 0, $=tt as consecutive 

 summits, 



K = g(a + aicos2<£ + a 2 cos4<£ + . . .), 



where a v a 2 . . . are small compared with a . 

 Since K = ^-T7? 



— - = a + a 1 cos 2<£> + a 2 cos 4$ + . . . . 

 acp 



In order to find the connexion between (f> + i-^r = w and 

 x + iy=z throughout the liquid, we proceed in the manner of 

 Riemann and Schwarz, that is, we find by means of the 

 assumed surface-condition a function of z which is real over 

 the surface and possesses only simple poles in the liquid. 

 This function can then be extended continuously in its range, 

 throughout the plane w, its value at (<£, — i|r) being the con- 

 jugate of that at ($, ^). We then have a function throughout 

 the plane w whose singularities are confined to simple poles, 

 and whose form can be written down according to the 

 principles of Cauchy. 



Put , dz TT 



l0 ^= u - 



Along the surface, or ^ = 0, 



U = log e i9 /q = — log q + W, 

 and therefore, along the surface, 



dJJ _dU_ dlogq .dO t 



dw d<f> dcf> d<{> ' 



