MOMENTUM 

 .08- 



ENERGY 



Figure 9. Energy-momentum diagram for steady flows, including stationary-wave motions. 



broken-line barrier. In these, obviously, much energy must be lost in spray and turbu- 

 lence. Brooke Benjamin has recently shown that many transitions to wavy flow regimes 

 resulting from changes in depth or breadth of the stream can also be understood by use 

 of this diagram. 



16. Attenuation of long waves by turbulent friction. 



I must pass now to flows which extend over much greater lengths of river than 

 those we have considered so far — over lengths so great, in fact, that wave attenuation 

 by turbulent friction becomes important. If one begins by writing down the equation 

 for long waves of small amplitude on a stream of subcritical velocity, V, resisted by a 

 frictional force / • pV 2 per unit area, one finds that for either direction of propagation 

 they are damped approximately like 



g-fVt/h ( 16 ) 



so that for / zz 0.005, say, their amplitude is reduced by e~ x in the time taken for the 

 stream to travel 200 times its depth. This result appears for upstream propagation in 

 Abbott's treatment of the formation of tidal bores [1] and for downstream propagation 



29 



