L08 THE MECHANICS OF THE EARTH'S ATMOSPHERE. 



Since C 2 must be positive it follows that 



0.66537 <cos 2 £<0.68615; 



so that values of cos 2 s that are smaller than 0.643 are thereby excluded. 

 But when we consider that for values of C that are larger than 1, the 

 above-given series for the coordinates of the boundary surface are no 

 louger convergent, there results a lower limit that is still higher than 

 the preceding, which corresponds to the value 



cos 2 s> 0.67264= -h + v 7 ^- 



For this value the altitude of the wave will still be finite, namely : 



H= £- x 2.5112= A 0.39967. 



But the fact that the value of the coordinates can no longer be de- 

 veloped in converging series, according to the powers of cos (ad) and 

 sine (ad), shows that a discontinuity or an ambiguity of the coordinates 

 must have come into existence. In fact the equations (la) also show 

 that for small values of h 



h sin 6 



tang (ny) = 2 



te v a ' cos 0— cos e 



e 2nx_ a 2( CQS #_C0S f) 2 . 



From the first of these it follows that wherever tan (n y) has a finite 

 value then cos must be nearly equal to cos £, and only at the points 

 where tan (n y) is very small and passes through zero can 6 increase 

 and rapidly pass through the interval to the next point, where cos 6 

 approaches again the value, cos €. 



Now for such values of h the diminution of the terms in the series 

 expressing the value of the pressure will not be rapid enough, in order 

 to express the value of the function sufficiently well by using only the 

 first three terms of the series, and the true form of the wave curve for 

 such values of h can only be obtained by further approximations. 

 However, these relations show that waves which rise too high lose the 

 continuity of their surfaces. But sharp ridges can not occur on the 

 surfaces of the waves except when they are at rest relatively to the 

 medium into which they protrude. For when the medium flows around 

 the edge there would occur infinite velocity and infinite pressure at 

 the place in question, which must violently draw up the other liquid, 

 as in fact is occasionally observed in high and foaming waves. 



Iu the case of waves that advance with the same velocity as the 

 wind the summits can in fact have a ridge of 120° before they break 

 into foam. 



