in Whitham's treatment of flood waves [22]. I shall describe the further ramifications of 

 this result separately in the two cases. 



17. Conditions for the formation of tidal bores 



One of the problems of river waves which most need an improved mathematical 

 treatment, with the object of throwing light on what are the essential physical processes 

 involved, is that of the propagation of the influence of tidal motions in an estuary into 

 the upstream part of the river. Despite all the beautiful experimentation which has 

 been done with tidal models, the physical mechanisms governing what part of the tidal 

 energy passes upstream are still not fully clear. An interesting beginning, however, has 

 been made by Mr. M. R. Abbott [1], and I should like to describe his work. 



He uses a simplified method due to Dr. G. B. Whitham [32] for calculating 

 wave propagation with amplitude dispersion. In this method, one first calculates the 

 flow using the linearised equations for small disturbances. One then allows for ampli- 

 tude dispersion by taking the paths of individual waves as calculated, and modifying 

 each path by giving it at every point a wave velocity appropriate to the local depth 

 and flow speed in the presence of the wave. 



In this method, therefore, one must begin by considering the variation of wave 

 amplitude on linearised theory. This is a combination of attenuation due to friction 

 (equation (16)), and amplification due to reductions in the breadth and depth of the 

 river as the wave passes upstream. If only the attenuation were present the amplitude 

 would decay too rapidly for steepening to the extent of bore formation to occur. 

 Therefore the reductions in breadth and depth are essential. Their effect can be calcu- 

 lated only very roughly. If no energy is reflected back down the river as the channel 

 becomes narrower and shallower we have Green's law [12, 19] 



a 2 6i l/2 Ai l/4 



- = • (17) 



Oi &2 l/2 /l2 l/4 



This can give very substantial amplification. At the other extreme, if the narrowing 

 occurs suddenly, so that the maximum energy is reflected, then we have the much 

 smaller amplitude increase 



a-2 2 



- = " -= < 2 . (18) 



<2i bo \Zh 2 



1 + 



&i Vhi 



Intermediate values occur in intermediate conditions. The condition for Green's law 

 is that the length scale of the tidal wave is small compared with the length scale in 

 which the breadth or depth changes by a large fraction. Now, the basic time scale 

 ( period/ (2tt)) for tidal waves is 2 hours, so that for a typical wave in a river of 

 moderate depth the length scale would be around 20 miles. This is unlikely to be small 

 compared with the length scale for changes in breadth and depth. For greater rivers 

 both scales would be larger but the conclusion would be much the same. However, for 

 want of a better approximation, Abbott used Green's law of amplification combined 

 with the attenuation due to turbulent friction and deduced a condition for a bore to 

 form. This requires that the amplitude shall remain sufficiently high for sufficiently 

 long, assisted by narrowing of the river and in spite of turbulent energy loss — suffi- 

 ciently, I say, for neighboring waves to run into one another due to amplitude dis- 

 persion. The condition may be expressed in terms of the total variation of tide level 

 in the estuary, and for the Severn Abbott predicted bore formation if the tidal varia- 

 tion exceeds 39 feet. This is close to what is observed; the tidal variation is 41 feet at 

 the spring tides and 22 feet at the neap tides and bores appear only during a few days 

 before and after the highest tide. Again, Abbott's theory gives the position where the 



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