Change of Form of Long Waves. 435 



which the first is due to a true change of form of the wave- 

 surface, whilst the second may be attributed to a small 

 advancing motion of the wave, which is leftafter the addition 

 of the uniform motion with velocity Qo=\/gl' To this effect 

 we have still at our disposal the quantity a, whose close con- 

 nexion with the uniform motion, which we have added in 

 order to make the wave nearly stationary, has been indicated 

 above. 



One of the best ways to obtain the desired separation is 

 certainly to make stationary the highest point of the wave, 

 and this is effected by fulfilling the condition 



2(« + 2<7p 2 ) = 3(4<7p 2 -A), 

 or 



ct = 4c<rp 2 — f/i ; 



for in that case equation (32) is simplified to 



V _ qg)_ ^ 4(7 ^2_^ sech 2 ^' . tanh^ju? ; 



dt I 



and then, for # = 0, 



(33) 



b. 



~dv -v dv 



d# "dt . , ,, .,, ~dv 



- = - is zero together witn ^— . 



In discussing this equation (33), we see at once that a 

 solitary wave (31) is stationary when h—^crp 2 ; and this is in 

 accordance with the equation (17) of the stationary solitary 

 wave which we have obtained above. When h > ^ap 2 , the 

 change of form of the wave, calculated from (33), is shown 

 by the dotted line in fig. 2. 



Ffc. 2. 





Here the wave becomes steeper in front *, whilst for 

 h < 4crp 2 the figure would show the opposite change of form, 

 when, contrary to the opinion expressed by Airy and others, 

 the wave becomes less steep in front and steeper behind. 



* The left side of the figure is the front side of the wave, because the 

 wave has been made stationary by the application of a positive velocity 

 (t. e. from left to right) to the fluid. 



