bore is formed correct])' as 12 miles upstream of the estuary; for comparison, one 

 obtains only 4 miles in a theory neglecting friction. The agreement on both points is 

 really far better than it should be in view of the fact that the tidal length scale is too 

 great for Green's law to be applicable. 



However, there is another observed fact which may explain this mystery. 

 The theory indicates that where the bore forms the local depth 77 has a vertical tangent 

 but has already been rising for some time. What is observed is that the bore when it 

 arrives is the initial increase of depth from the low-water mark. This indicates that 

 perhaps the wave as a whole has not obeyed Green's law, but that only the steepened 

 part, where the effective length scale has been greatly reduced by amplitude dispersion, 

 has been capable of extracting the full Green's-law amplification out of the reductions 

 in breadth and depth. The rest of the wave has been left behind, its energy being 

 reflected downstream (as well as dissipated by turbulence), and only the concentrated 

 part of the disturbance has reached the upper river. This hypothesis would explain 

 why the use of Green's law for this concentrated part gives a good result for the 

 criterion, and predicted position, for bore formation. However, it is clear that a more 

 exact analysis of this problem is greatly needed. 



18. Effect of friction on the downstream propagation of flood waves 



I come now to the effect of the turbulent frictional dissipation of long gravity 

 waves on the downstream propagation of flood waves; with this problem of how the 

 effect of variations over a period of days in the runoff into the river is felt far down- 

 stream we reach perhaps the extreme long-wave end of the spectrum of river waves. 



Now, it is clear that any local increase in flow will cause a signal to be propa- 

 gated downstream at the velocity -\f (gh) of long gravity waves. However, we have 

 seen that this signal is damped by a factor of e- 1 in the time that it takes the river to 

 travel a distance of 200 times the depth, say in 10 minutes or so. Here there will in 

 general be no compensating amplification, by reductions in breadth or depth, and 

 therefore in about an hour the signal moving downstream at the speed of gravity waves 

 will be completely damped out. The question then arises: what has happened to the 

 additional fluid put in at the beginning? It must be found somewhere in the river! 

 (Note in passing that such an appeal to continuity could not be made in the upstream 

 propagation problem, where a rise in the estuary water-level can perfectly well be 

 accommodated through a steady flow in the river, the so-called "backwater curve." 

 No steady flow, however, can pass down the river a flow quantity whose value at one 

 upstream point is varying) . 



The answer to the question can be obtained in any particular case by getting 

 a complete solution of the equations for small disturbances. This has been done for 

 a channel of uniform breadth and slope and a velocity-squared resistance law by Whit- 

 ham, who included it as section 3 of our joint paper on kinematic waves [22]. This 



, . 3 

 proves that the maximum of the disturbance travels downstream at the velocity — V 



(where V is the stream velocity) ; this of course is in general far less than the velocity of 



gravity waves V + \/{gh). Such a theory shows that, although the characteristics of 



the partial differential equations represent propagation at the speeds V + \/(g/z) and 



V — \/(gh), nevertheless signals propagated along those characteristics are so damped 



that for changes extending over really long periods they are unimportant, and only the 



3 

 propagation speed — V, with which the bulk of the energy travels, is vital to the 



problem. 



This propagation speed is that of the Kleitz-Seddon flood waves, which can 

 however be treated separately under far less restrictive conditions both as to the 

 character of the river and the size of disturbance which is treated, and which therefore, 



31 



