across station 2 must also equal the total flow across station 1 plus this difference in 

 the volume between the stations at the beginning and end of the time interval. There- 

 fore the true discontinuous flow must have the same area under its (q,t) curve as the 

 calculated continuous flow which is replaces. 



By this rule the development of the flood as it progresses downstream is easily 

 calculated, and the kinematic shock wave is found first to grow in strength, and later 

 on to decay (as the total length of the wave increases). 



23. Mechanisms governing the flow-concentration relation 



I should like to discuss now the problem of what the flow-concentration curve 

 should be taken to be in the kinematic-wave theory of flood movements; this question 

 is the same as asking: what is the steady regime in the river for different uniform values 

 of the flow? Here the point I want to make is one which was already shown clearly 

 by James Seddon in his article [29] of 1900, namely, that this question can only be 

 answered observationally. The reasons need not be discussed at length here as the 

 principal arguments are clearly set out in Seddon's very readable article as well as 

 summarised in the paper by Dr. Whitham and myself. 



The main point is that the equations of hydraulics [11, 27], which were developed 

 largely for flow in man-made conduits of regular cross-section, are misleading for natural 

 rivers. For a conduit one could certainly apply kinematic wave theory, using for example 

 the Chezy law of frictional resistance for relating velocity to mean depth, which gives 

 for a rectangular section 



dq q 



q oc fc3/ 2j C = — = %- = %V (27) 



dk k 



5 

 Or one could use the Manning frictional law, which gives C/V — — or with 



either law and a different shape of cross-section one can get different values of C/V. 

 But the resulting theory could not be applied to the irregular bed shape of any real 

 river. As a matter of fact, if one knew precisely the dynamics of the flow there would 

 be little point in using a kinematic-wave theory, except as a rough guide, because one 

 could put the complete equations of motion on to an electronic computer. 



However, the variations in bed shape along the river contribute many mecha- 

 nisms controlling the flow rate which are ignored by the hydraulic equations; for 

 example, local flows similar to that through a vertical slit, or over a submerged weir. 

 To illustrate the point further I shall give details of just one of the many difficult 

 problems arising in steady river hydrodynamics, that of the occurrence of reversed-flow 

 regions. My object in discussing it is mainly to emphasize the complication of the 

 subject and to suggest that at present a dynamic theory is too difficult, but that a theory 

 like kinematic wave theory, based on one assumed relation which is especially suscepti- 

 ble to direct observational determination, is the best point from which to start in the 

 problem of flood waves. 



24. Reversed-flow regions in steady river flow 



Now, the standard works on hydraulics often consider flows in which the depth 

 h varies considerably across the width of the river, but they do not usually discuss clearly 

 what is the associated distribution of the flow velocity v across the width. Sometimes 

 it is tacitly assumed that v is approximately uniform, but for small ratios of depth to 

 width the bottom effectively prevents the existence of eddies which are anything like big 

 enough to diffuse a nearly uniform velocity over the width of river (in the way that 

 happens, say, in a pipe). An alternative hypothesis is that at each point across the 



36 



