of course, accurately true; but, once sufficient dispersion has ocurred to render small 

 the frequency change in a single wavelength, then the appearance or disappearance of 

 crests through the occurence of horizontal points of inflexion must become very rare. 



21. Kinematic shock waves 



Now kinematic waves themselves do not suffer frequency dispersion, but they 

 do suffer amplitude dispersion, and it is the consequences of this fact that Dr. Whitham 

 and I have felt were inadequately covered in the literature, making a new treatment 

 necessary. Kinematic waves are. as we have seen, totally different from long gravity 

 waves, and are important for problems with time scales over which long gravity waves 

 would be completely damped out, but they have this property of amplitude dispersion 

 in common. Thus, the wave velocity C is a function of the flow q carried by the wave, 

 as well as of the position x. Since the mean velocity V increases with concentration, 

 the graph of q = kV against k is concave upwards, and C =r dq/dk increases with 

 k, and also with q. Hence waves carrying high values of the flow travel faster than 

 waves carrying low values, and it is possible for the former to overtake the latter, 

 giving discontinuities in the flow q. We suggested the name "kinematic shock waves" 

 for these discontinuities, since their process of formation is exactly like that of shock 

 waves in a gas. It is similar also to the process of bore formation in long gravity waves 

 which we discussed earlier. 



Now, if a discontinuous wave moves downstream with velocity W, it is easily 

 seen that the flow across the moving wave is q — Wk. and it follows by continuity that 

 the law of motion of such a kinematic shock wave requires that this expression be 

 continuous across it, giving 



52 - qi 



W = . (24) 



rC2 — Kl 



This speed W of the kinematic shock wave is the slope of the chord (Fig. 10) joining 

 the points {k x , q x ) and (k 2 , q. 2 ) on the flow-concentration curve representing the 

 upstream and downstream conditions. In the limit of a weak shock wave it coincides 

 with the slope of the tangent, that is, dq/dk, the velocity of continuous waves. 



Now one must not suppose from the use of the word "discontinuity" that these 

 kinematic shock waves are very concentrated things, similar to bores. Actually, owing 

 to the time lag needed for adjustments of flow and concentration to catch up with one 

 another, such a wave, though discontinous on the simple theory, may extend over many 

 miles. It is in fact the so-called "monoclinal flood wave" which in the special case of 

 uniform flow upstream and downstream has been computed for special river models 

 by Thomas [31] and Dressier [7] as well as by Whitham [22]. Our suggestion for treating 

 more general cases is that a first approximation to the flow be obtained by kinematic 

 wave theory, and that the results be then improved by replacing such discontinuities as 

 arise by calculated profiles of this kind, choosing those appropriate to the computed 

 strengths of the kinematic shock waves (see lower diagram in Fig. 11). The justifica- 

 tion for this is that it is only in the places where the flood-wave profile has been con- 

 siderably steepened by amplitude dispersion that the time lags for adjustments of flow 

 and concentration to catch up with one another will be important. 



22. A new method for calculating flows with kinematic shock waves 



It is unnecessary to give here any details of special calculations involving kine- 

 matic waves, but I will describe a new method of determining the position of the kine- 

 matic shock wave as a function of time, which is simpler than the methods given in 

 our original paper. 



To fix the ideas, suppose that the flow q is measured as a function of time at a 

 particular station x=x 1 in a river, and that we want to predict what the flow variation 



34 



