256 Mr. J. McCowan on the 



Thus 



■-^{♦M-V^}-* • • • .« 



For the case of a channel of rectangular section, equations 

 (6) to (9) reduce to 



z = ¥ (x-tz l /k), (60 



*to = 3i/i, m 



* = F{^(3v^Hfe)*h .... (8') 



u = 2\/Jz-k (9') 



Equations (7) to (9) supply immediately the solution of two 

 very important general problems. Thus, if we are given the 

 form of the surface, say z = F(f), of a train of waves advancing 

 in a channel of given uniform cross section at any time, which 

 may conveniently be taken as zero, the form of the surface at 

 any subsequent time t will be given by (7) and (8), and the 

 velocity u by (9) . Again, if a procession of waves is propa- 

 gated along the channel by imposing a forced motion, say 

 z =f(t) , at some part of the channel, which may conveniently 

 be taken as the origin, then the form and motion generally of 

 the advancing waves will again be given by the equations (7) 

 to (9), (8) being modified to the form 



«=/*«-*/( **•*<*)-*) I. • • • • (io) 



*=/S'-f/(3 i/&-k)\, (10') 



if the section is rectangular. 



In both these solutions it will be seen that the constant k is 

 left undetermined ; by (9) we see that it will be determined 

 by the consideration of the normal velocity of flow in the 

 channel independent of the wave-motion. Thus, if when no 

 waves are being propagated along the channel, there is a 

 uniform flow in it in the direction of the axis of x, the velocity 

 being U, then, if we regard the wave-motion as such as not in 

 itself producing a permanent flow, we must have 



UT= Cudt 



Jo 



when the time T is taken sufficiently long. 



Thus, by (9), k will be determined by the condition 



or 



T= ^y~^^ ] --\/w)\ cu ~ m ' ■ • (n) 



