[ 45 ] 



VII. On the Solitary Wave. By J. M c Cowan, M.A., B.Sc, 



Assistant Lecturer on Natural Philosophy, University 

 College, Dundee*. 



IN his Report on Waves to the British Association in 1844, 

 Scott Russell gave an account of experiments he had 

 made on the propagation, along the surface of still water in 

 a straight channel with rectangular cross-section, of a wave 

 consisting of a single elevation or depression, and which he 

 called a w T ave of translation or solitary wave to distinguish it 

 from waves forming part of a train. From these experiments 

 he concluded that the solitary wave was unique, having 

 characteristics entirely it's own : — it had a definite form de- 

 pending only on the depth and the volume of the water 

 composing it, and this form, in the case of a wave of elevation, 

 appeared to be propagated with constant velocity and without 

 any change except such degradation in height as might 

 reasonably be attributed to frictional and other disturbances. 



In 1845 Earnshawj sought to give a theory of these waves, 

 but it was unsatisfactory as involving a discontinuity in the 

 pressure within the liquid. 



In his Report on Recent Researches on Hydrodynamics to 

 the British Association in 1846, Stokes, commenting on 

 Russell's experiments and Earnshaw's theory, concludes 

 that the observed degradation of the wave is not to be attri- 

 buted wholly, nor even chiefly, to friction, but is an essential 

 characteristic of the motion ; and, again, in 1847, in a paper 

 " On the Theory of Oscillatory Waves "J, he reiterates this 

 opinion and offers a proof involving, however, an oversight 

 which I shall be able to point out. 



It has been thought by some that the solitary wave is in- 

 cluded in the general theory of long waves, but this is 

 certainly only so to a very rough approximation, for its 

 velocity does not agree closely with that of the long wave, 

 nor does it gradually increase in steepness in front as the long- 

 wave does, the change which does take place in it being 

 simply a diminution in height and consequent increase in 

 length such as might be caused by a dissipation of its energy 

 by friction, &c. 



The first sound approximate theory of the wave was given 

 by Boussinesq in 1871 §, who obtained an equation for its 



* Communicated by the Author, having been read before the Edinburgh 

 Mathematical Society, May 8, 1891. 

 t Trans. Canib. Phil. Soc. vol. viii. 

 X Trans. Camb. Phil. Soc. vol. viii. 

 § Comptes Rendas, torn, lxxii. 



