2 GWVTIII'.K, On the Propagation of a Solitary Wave. 



conditions arc satisfied at a suitably chosen series of 

 points alon^^ the surface is hardly practicable, we are 

 obliged to base a solution on the fulfilment of the surface 

 conditions, with close approximation in some parts of the 

 wave, and examine the degree of approximation obtained 

 over the rest of the wave. Since the conditions appear 

 from experiment to be more readily satisfied over the 

 outskirts of the wave than in the neighbourhood of the 

 crest, it is natural to base the approximate solution upon 

 the enforcement of the surface-condition upon the outskirts, 

 and to examine the results in the neighbourhood of the 

 crest. It must, however, be admitted that this condition 

 is not very well fitted for comparison with results of 

 experiment performed in a tank. 



For this purpose it is necessary to take the form of 

 the approximate solution such as will represent some- 

 thing, roughly at any rate, of the form of the wave in 

 question. 



Consider the motion, by the usual artifice, reduced to 



the state of steady motion, and write, with the usual 



notation, 



X + iy=/{,i> + i\P) 



= ^ ^ +Atanh?/i --- ^ . . . (i) 



c c 



Let j3 denote the depth of the fluid, and ^ = t-)3 the free 



surface, the maximum elevation h will be found when 



xp = cf5, (p = o ; thus /i = X tan 7;/j3. 



Hence 



ip , • \ 2in4> I , , 2?fi(l> iv , 



X = i + h smh — ^- /(cosh 1 + cos 2inp) Uin wp, 



c c j c 



_y = /(3 -I A(l + COS2W/))/ (cosh" ^ + COS2W/j) , 



at the free surface. 



These equations indicate a form of free surface having 



the general character of that of the solilar)- wave, and, in 



