418 SURFACE WAVES. [CHAP. IX 



or $ = C&amp;lt;P ^ (xiv), 



where k = &amp;lt;r 2 /&amp;lt;7 = 2?r/X . . . . . (xv). 



Hence, finally, 



(xvi), 



where 



-*-(& 



The addition of a constant to b merely changes the position of the origin and 

 the value of C \ we may therefore suppose that 6 = for the limiting cycloidal 

 form of the path. This makes C=k~ l . 



If the time be reckoned from some instant other than that of passage 

 through a lowest point, we must in the above formula} write a + ct for ct, where 

 a is arbitrary. If we further impress on the whole mass a velocity c in the 

 direction of ^-negative, we obtain the formulae (1) of Art. 232. 



234. Scott Russell, in his interesting experimental investiga 

 tions*, was led to pay great attention to a particular type which 

 he calls the solitary wave. This is a wave consisting of a single 

 elevation, of height not necessarily small compared with the depth 

 of the fluid, which, if properly started, may travel for a consider 

 able distance along a uniform canal, with little or no change of 

 type. Waves of depression, of similar relative amplitude, were 

 found not to possess the same character of permanence, but to 

 break up into series of shorter waves. 



The solitary type may be regarded as an extreme case of 

 Stokes oscillatory waves of permanent type, the wave-length 

 being great compared with the depth of the canal, so that the 

 widely separated elevations are practically independent of one 

 another. The methods of approximation employed by Stokes 

 become, however, unsuitable when the wave-length much exceeds 

 the depth ; and the only successful investigations of the solitary 

 wave which have yet been given proceed on different lines. 



The first of these was given independently by Boussinesqf and Lord 

 RayleighJ. The latter writer, treating the problem as one of steady motion, 

 starts virtually from the formula 



where F(x) is real. This is especially appropriate to cases, such as the 



* &quot;Keport on Waves,&quot; Brit. Ass. Rep., 1844. 



t Comptes Eendus, June 19, 1871. 



t &quot;On Waves,&quot; Phil. Mag., April, 1876. 



