Long Waves in Shalloiv Water. 109' 



showing that the drift will exist unless 



E 

 yu<+ A-7> =0. 



In the extreme case when «=1, the solution represents 

 waves of the solitary type. We shall then have either 

 X=l and /z = 0, or \= — 1 and fi=§. 



The solutions are therefore either 



o V 2(c a -^/3) A ' 



which represents Scott Russell's Solitary Wave, or 



/(ar + ty) = - g •— — (ar + ty) 



c V 2(c 2 -^A/3) A " ^ ; 



This last solution represents a comparatively slow-travelling 

 solitary wave of elevation, which is essentially accompanied 

 by a perceptible drift. As far as I know it has not been 

 observed, and the method of transmitting the wave by re- 

 flexion at the ends of a trough, employed in Scott Russell's 

 experiments, would not be likely to disclose it. 



The need of an accompanying drift for the progressive 

 transmission of these long waves, and the fact that long 

 waves can be observed which are not accounted for by this 

 method, seem to point to the probability that some more com- 

 plex motion of the fluid than a uniform drift may be necessary 

 for the exhibition of these phenomena, and to raise a doubt 

 whether the artifice of reducing the motion to one of steady 

 motion is attended by advantage in studying this species of 

 wave. 



I have ventured to describe these waves as of the type of 

 those discovered by Drs. Korteweg and de Vries because of 

 the similarity of the form of the free surface ; but the 

 relations (20), (21), and (22) do not lead to results identical 

 with those of the authors named. 



The equation which my results give, as a first approxima- 

 tion, for the free surface is 



*=£*\m 



(24) 



_<?-gh 



K^^i/^r-} 



(26)) 



