The kinematic wave velocity C is the slope of the flow-concentration curve for 

 fixed x. This fact is known in the literature of flood waves as the Kleitz-Seddon law. 



20. Other fields of application of kinematic wave theory 



Other fields of application in which extensive use of kinematic wave theory 

 has proved possible are the theory of traffic flow on long crowded roads [23], where 

 one starts from the assumption (see Fig. 10) that at each point of the road the mean 

 velocity is a function of vehicle concentration (falling from a maximum velocity at 

 zero concentration to a zero velocity for a certain critical concentration k 3 — / for Jam); 

 and, again, the theory of sedimentation of small particles in a liquid, where also the 

 velocity falls as the concentration increases, this time as a consequence of Einstein's 

 law for the viscosity of suspensions. This theory was worked out in 1952 by Professor 

 G. J. Kynch [18], and it was only after Dr. Whitham and I had published our paper 

 on the general theory that we realized that Professor Kynch had anticipated many of 

 our results, at least as far as this application is concerned. 



I might remark also that at an early stage of our work we felt that the fact 

 that, in flood waves, the energy travels not at the gravity-wave velocity but at the slower 



100 



CONCENTRATION k 

 (VEHICLES PER MILE) 



(a) 



200 



f 



(b) 



Figure 70. (a) Mean velocity V and flow q plotted as a function of concentration k for rural roads 

 in U.S.A.: data due to B.D. Greenshields. Proc. Highw. Res. Bd. 14, 448 (Washington, 1935). 

 (b) Flow-concentration curve for flood waves, showing construction to determine speed of kine- 

 matic shock wave. 



speed of the kinematic waves has so much in common with the theory of group velocity 

 (in systems with what I have called frequency dispersion) that it might in some obscure 

 way be a special case of that theory. We could not sustain this view for long, because 

 no frequency analysis comes into the floodwave problem, but later we saw that the 

 two problems were differently related, both being in fact special cases of the general 

 kinematic wave theory. 



To see this for the case of wave propagation with frequency dispersion, one 

 has to apply a law of conservation of number of wave crests. This gives that the flow q 

 is the number of wave crests passing per unit time, which is the frequency (c/A), and 

 that k is the number per unit distance, namely 1/A, so that the frequency remains con- 

 stant along kinematic waves travelling at the speed 



dq d(c/\) 



dc 



dk d(l/X) 



= c 



dX 



(23) 



which is in fact the group velocity (2). Conservation of number of wave crests is not, 



33 



