stream reservoir [27]) must be joined by a hydraulic jump at which h changes discon- 

 tinuous!}' but the momentum flow 



Qt 



S = v *h + %gh* = h %gh* ( 10) 



h 



is continuous. Such a jump is possible because 5 has a minimum value for h =z h c . 



13. Mechanics of energy loss in bores 



Thus, classical bore theory (due to Rayleigh [26]) assumes conservation of 

 volume flow Q across the bore, and conservation of momentum flow S (more pre- 

 cisely, the decrease in momentum flow v 2 h is balanced by the difference in horizontal 

 pressure force V2gh- upstream and downstream). Energy, however, is lost at the bore; 

 the total head Viv 2 + gh decreases by 



(h 2 - /ii) 3 



jig , (11) 



hih-2 



and Rayleigh suggested this was due to friction. 



Now it is certainly true that vigorous turbulent dissipation and churning-up of 

 the flow is the main mechanism of energy loss in strong bores, but weaker bores have 

 a different structure. They carry a train of waves behind them, whose position is 

 stationary relative to the bore, and Favre [10] found that these waves exhibit no break- 

 ing, and that the flow appears perfectly smooth (Fig. 7) for depth ratios h 2 /h 1 < 1.28. 



to 



Figure 7. Week bore with train of waves behind. 



As an example, the tidal bore formed on the British river Severn often carries such a 

 train of waves along behind it. It is clear that the energy dissipation then takes place 

 principally by radiation through this stationary wave train at a speed equal to the 

 difference between the wave velocity and the group velocity — exactly as in the waves 

 generated by a two-dimensional obstacle in a sub-critical stream. A calculation of 

 amplitude by Lemoine [20], assuming sinusoidal waves, was based on the amount of 

 energy lost per unit time (which is pQ times (11) per unit span), and gave 



a /h 2 - h x \ X (h 2 - hA-i'* 



-=1.15- -), - = 3.0— - (12) 



h 2 \ h, ) h 2 \ hi I 



The reason why fairly long waves are produced is that their velocity must equal the 

 flow speed behind the bore, which is only just subcritical. As a result, a substantial 

 amplitude is needed to carry away a fairly small amount of energy, simply because 

 the difference between group velocity and wave velocity is small for such long waves. 

 In support of this theory of Lemoine there is the observation that the waves form one 

 by one behind the bore after it is first created — as the energy transmitted by the group 

 of waves travels backwards. 



26 



