[ 314 J 



XXXVIII. Note on the Theory of the Solitary Wave. 

 By Sir G. G. Stokes*. 



IN a paper on the Solitary Wave by Mr. J. McCowsn r 

 printed in the July number of the Philosophical Magazine, 

 for a copy of which I am indebted to the kindness of the author ? 

 he refers to a conclusion which I advanced in a paper written 

 long since, and reprinted full ten years ago, according to which 

 a solitary wave could not be propagated without change 

 of form. As I have known for the last ten years that this 

 conclusion was erroneous, and have published a paper in 

 which the motion of a uniformly propagated solitary wave 

 was considered, I am not concerned to defend it; but it may 

 be well to point out the true source of the error, respecting 

 which I cannot agree with Mr. McCowan, 



While the first volume of my Collected Papers was going 

 through the press, I was led to the conclusion (see p. 227) 

 that the highest possible waves of the oscillatory kind (the 

 motion being irrotational) presented a form in which the 

 crests came to wedges of 120°. On reflecting on the applica- 

 tion of this to very long waves propagated in water of which 

 the depth is small compared with the length of wave, I was 

 led to perceive that the conclusion above mentioned was 

 erroneous, and also that the source of the error was that it 

 was not sufficient, even though a solitary wave were very 

 long, to treat it as indefinitely long, and consequently to take 

 the horizontal velocity as the same from the surface to the 

 bottom. On speaking on the subject to Lord Rayleigh, he 

 referred me to the previous papers on the solitary wave by 

 M. Boussinesq and himself, with which I was not at the time 

 acquainted. The conclusion of a supplement to my paper on 

 oscillatory waves, which forms the last article in vol. i., shows 

 that I was then fully alive to the possibility of the propagation 

 of a solitary wave without change ; and in a short paper en- 

 titled " On the highest wave of uniform propagation (Pre- 

 liminary notice), " read before the Cambridge Philosophical 

 Society in 1883, and printed in the Proceedings (vol. iv. 

 p. 361), I have indicated a new method, depending on a 

 process of trial and error, for determining numerically the 

 circumstances of uniform propagation of waves, whether of 

 the oscillatory or solitary class, more especially in the extreme 

 case in which the crest comes to a wedge of 120°, so that the 

 wave is on the point of beginning to break. 



I cannot agree with Mr. Mc Cowan either that the form of 



expansion which I used is inadmissible, or that the form which 



he proposes at p. 58 to substitute, that of a series involving 



exponentials in which the coefficient of # in the index is 



* Communicated by the Author. 



