convenient exact solution for any initial conditions. It may be that a development along 

 these lines, involving a suitable approximation to the non-linear terms in the equation 

 of continuity, and the incorporation of a second-derivative term to represent the effect 

 of lag, will prove practicable. 



Alternatively, there is the solution given in our original paper, based on making 

 a direct observation of how the flow varies with the rate of increase of concentration 

 at a point for given concentration. We showed that if the ratio of the change of con- 

 centration to the change of time-derivative of concentration required to produce a given 

 flow change is taken constant (say as an average of the observed values), then such an 

 approach would necessarily give the kinematic shock waves their correct position and 

 velocity. 



As a final alternative, it may be that one who wants real accuracy rather than 

 just a bird's-eye view of the picture may always need to go, as at present, to models, 

 with the detailed characteristics of each stretch of the river reproduced in detail in the 

 model. 



26. Roll waves 



As my last topic, I come to what Dr. Whitham found [22] when he applied the 

 theory of the attenuation of gravity waves by turbulent friction to torrents. If 

 F = J 7, y gh is the Froude number of the stream, then on the model described 

 earlier the attenuation factor 



e - f vt/h (29) 



has to be replaced by 



r^-x 



F) 



(30) 



so that when F reaches 2 attenuation ceases. This is due to the fact that on the par- 



3 

 ticular model used the kinematic wave velocity C — — V with which the main part of 



the flow is carried becomes equal to the gravity-wave velocity V + \/(gh), so that 

 gravity waves thereafter carry the flow without attenuation. It seems that for any 

 system this will happen when C reaches V + yj ' (gh) . 



What is not so clear at a first glance is what happens when the kinematic-wave 

 velocity actually exceeds the gravity-wave velocity. However, Whitham's analysis corn- 



Figure 13. Roll waves (after Dressier [73 ). 



38 



