width gravity and friction are balanced, leading (say, for the Chezy resistance law) to a 

 relation 



v = K yjhu 



(28) 



where a, the slope of the free surface, is constant across the river and can be determined 

 by integration from the given total flow rate q. This approach has a solid basis for 

 channels of uniform section. 



However, in rivers one often observes a quite different kind of flow, which is 

 actually reversed in the regions of shallow water (Fig. 12). This is connected with the 

 general tendency of flow to separate, not only from solid obstacles, but also from local 

 areas of high resistance, for example a row of trees used as a wind break. 



The explanation is that if separation did not occur there would still be only a 

 slow flow in the area of high resistance. On the other hand, the velocity just outside 

 it (on the other side of the river) needs to be greater than its values upstream and 

 downstream if the total flow rate q is to be passed. As this high velocity slows down 

 (farther downstream) a rise in water level must occur, by Bernoulli's equation, and this 

 can bring to rest the slower fluid which is in the area of high resistance. Hence separa- 

 tion and reversed flow set in. 



It would be possible to treat these flows by equations of the usual boundary- 



figure 12. Separated flow resulting from presence of region (shown dotted) of shallow water. 



layer type, because the condition of constant water elevation across the river (corre- 

 sponding to the condition of constant pressure across a boundary layer) is very nearly 

 fulfilled. The only difference from the usual boundary-layer equations would be the 

 occurrence of a turbulent friction term. The condition for separation should emerge 

 from such a treatment, but to my knowledge this has not so far been carried out. 



Similar reversed flows occur in estuaries, where at high tide the presence of a 

 submerged sandbank is often revealed by a line of foam which has collected along the 

 locus where the forward and reversed flows meet. 



25. Improvements of kinematic wave theory 



At this stage it can be reasonably objected that I am advocating the use of a 

 kinematic-wave approach to the prediction of flood movement, based on an experi- 

 mental flow-concentration relationship, but am giving no suggestions on what should 

 be done if changes are taking place somewhat too rapidly for the time lag between 

 adjustments of flow and concentration to be accurately negligible. This serious criticism 

 I must now try to meet. 



The situation is similar to one that has arisen in the aerodynamics of rarefied 

 gases, when one tries to apply the theory of sound waves of finite amplitude and shock 

 waves to problems with characteristic length scales of only a few mean free paths, so 

 that such shock waves as occur are really thick. Important advances in this subject 

 were made recently by Prof. Eberhard Hopf [13] and Dr. lulian Cole [5], and I gave a 

 survey of it as my contribution to the G. I. Taylor anniversary volume [21]. There the 

 effect of lag in establishing equilibrium can in many cases be treated by taking a second 

 approximation to the flow equations in the form of the Burgers equation, which has a 



37 



