Mr. J. M c Cowan on the Solitary Wave. 57 



We have seen that to a first approximation the volume of 

 the wave per unit breadth of channel is 2ah } and to the same 

 order we easily obtain also 



tf=2a7i=-\i7 =4 W*73 > • • • (53) 



the approximations being fair for low waves. 



Now as the wave rolls in its volume remains constant, and 

 therefore its height increases and length diminishes as the 

 depth diminishes, or exactly 



% =reF ; (54) 



so that the elevation varies inversely and the length directly 

 as the cube of the depth. As the wave becomes higher it will 

 be necessary to take the more exact formulas instead of (53) . 

 By (52) when the wave is on the point of breaking mh = l 

 and ma =2/3, and so using the more exact formula (31) for 

 the volume we get roughly 



v = l'5h 2 or h = '8 \fv, 



which gives the depth in which the wave will break. 



Thus the big waves will break first, and the depth in which 

 they break will vary as the square root of their volume. 



12. The Views of Sir George Stokes — Conclusion. 



Having thus examined in some detail the approximation 

 to the solitary wave which is obtained by taking the first term 

 only of (2), and having seen that even this first approximation 

 satisfies to a high degree of accuracy the condition for the 

 propagation of the wave without change, we are naturally 

 led to examine the argument given by Stokes, in his paper 

 " On the Theory of Oscillatory Waves," already cited, against 

 the possibility of the propagation without change of form or 

 velocity of any other form of wave than the infinite train of 

 " oscillatory " waves which he there discusses — to a degree of 

 approximation not quite so close as that with which we have 

 been occupied in the foregoing sections. 



Having found (§4) for the velocity of propagation U of 

 any wave form in liquid of depth h, 



JJ 2 =gm~ l tanh mh, 



and having previouslv (§ 2) put aside " imaginary " values 

 of the m as inadmissible, he infers that, since this will give 

 only one value ( + ) of m for a given value of U, there can 



