= € mD + e -mD_ 9 ( € mD_ e -«»D) . , ( 43 ) # 



Waves in Flowing Water. 527 



wave-motion of arbitrary magnitude, and arbitrarily chosen 

 position for one of the zeros, with wave-length such that the 

 velocity of wave-propagation is U, and the direction of motion 

 such as to cause the progression of the wave to be up-stream. 

 The wave-motion thus instituted constitutes a set of free 

 stationary waves, and the superposition of this upon the case 

 of motion represented by our symmetrical solution consti- 

 tutes the general solution of the problem of single-ridge 

 steady motion. To find the arbitrary addition which we must 

 thus make to our symmetrical solution to find the general 

 solution, put (13) into the following form : 



2H 



& ~~~ "^ mU 2 



This shows that if H = 0, .£) may have any value (that is to 

 say, we may have stationary waves of any magnitude over 

 a plane bottom) if 



e mT) + 6 -m-D ^j 2 ( € ^-€- mD )=0 . . (44). 



This is in fact the well-known equation to find the velocity 

 U relatively to the water, of periodic waves of wave-length 

 2ir/m in a canal of depth D. For us at present equation (44) 

 is to be looked upon as a transcendental equation for deter- 

 mining the wave-length corresponding to U a given velocity 

 of progress ; and it has, as we have seen, only one real root 

 when U < VgT) ; but no real root when U > \/gT). Putting 

 now in (43) ~U 2 =gb, and comparing with (32), we see that 

 mD = (T 1 ; and going back to equation (2) above we see that 



_ cr^Cx — a) ,,_. 



£cos lv p ...... (45); 



where <£) and a are arbitrary constants, is the addition which 

 we must make to (39) to give the general solution for the case 

 b < D. Putting together this and (39) and (40), we accord- 

 ingly have for the general solution of the single-ridge steady- 

 motion problem, for the case of U < ^D, 



&=Ocos < ^ + (C'+ 1 M^. <ri N 1 ) S in^+ 1 ^.|ftN ie 4 I ' 



when x is positive, and 

 u /-. °"l# ,ru iA/D XT x . cr Y x , JA/D » Ji* } (46); 



9=0 cos jy +(C- f-L^ . a.NOsm ^ + o_^ . ^ t e j> 



when x is negative 

 where C and C denote arbitrary constants. 



