Classes of Progressive Long Waves. 215 



- When /" and f vanish together, the constant in (8) is null, 

 and the solution is completed by 



/= a tanh mx, 



(& ft J i 



where ma= — , and 



c 



iW = 



Ht 



a* 

 A 2 





and c s must be slightly greater than g/i, whereas in the har- 

 monic periodic wave previously considered it was necessarily 

 slightly less. 



This case represents a low solitary wave of Scott Russell, and 

 the expression for (/> + iyfr, now obtained from the equations, 

 is that assumed by Mr. McCowan * in his investigation of 

 the properties of that wave. 



A point of interest is that the relation c 2 =gh, found by 

 the received method as giving the velocity of a long wave, 

 corresponds to a simple case of the cnoidal wave (8). 



If we retain the term in f 3 in (6) we obtain the most 

 general case of the elliptic function waves. In the case of the 

 solitary wave, 



2(c 2 -t//0 



/ 7 = == J .... (9) 



c + s/' gh cosh 2mx ' 



where, as before, 



)n z /r— - — ?— 



2 



(*-£)" 



From this the form of the free surface is readily found 



>y(2). t 



The form of/ is also easily found to be 



yv. ] t / 2 _ gh i g ^c^Wgh + Vc- s/ gh tanh mx < 

 V ^ y/c+ s/gh — v'c — s/gh tanh. mx 



The highest wave of this type can be determined by the 

 method iSir George Stokes has taught us. We must deter- 

 mine the relation between the parameters c and /* in order 



* Phil. Mag., July 189 J. 

 Q2 



