Figure 6. Integral curves of equation (9), showing spiral singularity and transition between curves 



by means of a hydraulic jump. 



The two curves (Fig. 4) for the different kinds of breaking meet at a point 

 representing the solitary wave of greatest height computed by McCowan [24]. The 

 linearised theory leading to sinusoidal waveforms is valid only if the amplitude is small 

 compared with that giving breaking. Conversely, it is only if the amplitude is large 

 compared with that of the solitary wave of given \/h that the so-called shallow-water 

 theory which neglects frequency dispersion and predicts steepening and the formation 

 of a hydraulic jump or bore is valid. 



12. Occurrence of hydraulic jumps in steady flow 



We must now discuss these bores in more detail. They occur also, of course, 

 as stationary jumps in steady streams wherever they decelerate from supercritical to 

 subcritical velocity. Then the equations of steady flow 



dv I dh 



vh = Q , v — = gia j - 



dx \ dx/ h 



fpi' 2 

 h 



(8) 



where Q is the volume flow per unit breadth, a is the slope of the bottom and / a 

 coefficient of friction, can be written 



dh ah 3 - fh c s 



dx 



h z - h c 3 



(9) 



where h c is the critical depth 3 -\/ Q-/g for which vz y (gh), and the flow is sub- 

 critical when h > h c (which for constant h means « < /, a gradual slope) and super- 

 critical when h < h c (which for constant h means a > /, a steeper slope giving the 

 so-called "torrent flow"). At a point where a decreases through the critical value /, 

 the integral curves of (9) have a spiral singularity (Fig. 6) and no continuous solu- 

 tion exists.* The curves specified by the upstream and downstream conditions (the 

 latter being the so-called backwater curve determined by the water-level in the down- 



* On the other hand, at a point where a increases through the critical value /, the 

 integral curves have a saddle-point singularity, and a solution representing continuous acceler- 

 ation through the critical speed exists. 



25 



