Manchester Memoirs, Vol. xliv, (19CX)), No. JK 



IX. On the Conditions for the Propagation ot a 

 Solitary Wave. 



By R. F. GWYTIIER, M.A. 



Received and read February 6th, igoo. 



The characteristics of the solitary wave as experi- 

 mentally obtained by Scott Russell, and of which a full 

 mathematical investigation is still wanting, are: — it consists 

 of a single elevation, which is not necessarily small in 

 comparison with the undisturbed depth of the water, and 

 it has not been found possible to form a corresponding 

 wave of depression ; there is a limit to the extent of the 

 elevation in water of given depth, but the precise relation 

 between the greatest elevation and the depth has not been 

 determined : and, when the limit at which the conditions 

 requisite for the propagation of the wave are succeeded, 

 the wave breaks near the crest. 



The form of the wave has been examined by Boussinesq 

 and Lord Rayleigh, and Mr. McCowan has investigated 

 the conditions of the wave-propagation in a manner more 

 allied to that which I here propose, and the expression 

 found by him for the velocity of propagation is identical 

 with that which I obtain. The references to these investi- 

 gations are given in Art. 234 of Lamb's Hydrodynamics. 



The difficulty of satisfying the surface-conditions over 

 the wide extent of the solitary wave makes the complete 

 solution of the question difficult. As it is improbable that 

 any expression can be obtained from which these con- 

 ditions can be completely satisfied everywhere, and the 

 plan of seeking an approximate expression for which the 



May 4th, I goo. 



