198] 



HIGHEST WAVES IN WATER. 



533 



are imaginary. This limiting value of B is that which makes the 

 highest position of the straight line, for which sin kx = 1, tangent 

 to the exponential curve. We then have 



j -y V -y -| 



jy = kX for the curve, equal to y r = ^- for the line, 

 from which 



The upper and lower branches of the curve 151) then come together, 

 and the wave -profile has an angle. Waves cannot be higher than 

 this without breaking. By differentiation of 151) we find for the 



summit, -^- = + 1, so that the angle between the two sides of the 



wave is a right angle (Fig. 170). As a matter of fact, before 



the waves are as 



high as this, the \ 



equation 141) is \ / 



no longer satisfied \ / 



with sufficient \ / 



approximation 

 for the waves to 

 have the form in 

 question. By an 

 elaborate system 

 of approximation, 



Michell 1 ) has 



shown that the highest waves have a height .142 A, while the equa- 

 tion 151) gives .2031. It was shown by Stokes 2 ) that at the crest 

 the angle was not 90, but 120, as follows. 



In the stationary wave, in order to have an edge, the velocities u 

 and v for a particle at the surface must both vanish together, for 

 if v alone vanishes, there will be a horizontal tangent. Consequently, 

 if we place the origin at the crest, equation 139) becomes 



Fig. 170. 



gy 



o. 



But if we represent the surface by a development of the form of 

 equation 128), on account of symmetry there will be only sine terms, 

 and if in the neighborhood of the origin we retain only the most 

 important term, we may put 



1) Michell, The highest Waves in Water. Phil. Mag. 36, p. 430, 1893. 



2) Stokes, On the Theory of Oscillatory Waves. Trans. Cambridge Philo- 

 sophical Society, Vol. VIII, p. 441, 1847. Math, and Phys. Papers, Vol. I, p. 227. 



