with time will be at some other station x = x 2 far downstream. Then each value q of 

 the flow at x x will remain constant along a wave which will take a time 

 f* a dx f x 2 dk d f* s 



T{q) = I = / —dx = — k(q,x)dx (25) 



J*i C(q,x) >! dq dqj\ 



to reach x 2 . This T(q) can be calculated in various ways, but one way is to notice that 

 T(q) is the derivative with respect to q of the total volume of water between x x and x 2 

 when the river is flowing steadily at the rate q. 



The deformation of the curve is now easily sketched (Fig. 11). Each point 

 (q,t) on the flow-time curve at *, becomes (q,t + T(q)) at x 2 . We then have to 

 answer the question: what happens if waves carrying high values of q overtake those 

 carrying low values, leading to a curve of the form illustrated? The simple answer is 



KINEMATIC 

 SHOCK WAVE 



(a) 



IN.B. TIME SCALE OF ORDER A DAY) 



(b) 



Figure 7 7. (a) Construction to determine position of kinematic shock wave. Lobes shown must be 

 of equal area so that total area under (q, f) curve remains unaltered. 



(b) Replacements of kinematic shock wave by calculated "monoclinal flood wave" profile. 



that the kinematic shock wave (or discontinuous jump in q) must be fitted in at that 

 value of t for which the total area under the (q,t) curve, that is the total flow across 

 jc=* 2 , remains unaltered by the substitution. 



To see this, one need observe only that the (q,t) curve has been deduced from 

 a theory based on the equation of continuity. Therefore the area under it, or total flow 

 across x = x. 2 in a certain time interval must equal the total flow across xzzx x in the 

 same time interval, plus the volume of water between the stations at the beginning, 

 minus the volume of water left at the end.* However, in the real motion the total flow 



Mathematically this says only that 



ff/dk dq\ 



(qdt - kdx) = I / 1 dxdt 



JJs\dt dx) 



where C is a curve in the (x.t) plane and S its interior. 



35 



(26) 



