corresponds of course to waves (cnoidal waves, in fact). In the extreme case of curve 

 A the amplitude becomes zero and we have a steady subcritical stream. In the other 

 extreme case of curve C we have two solutions, either a steady supercritical stream or an 

 oscillation taking an infinite time — this corresponds to the solitary wave. For positions 

 of the curve beyond A or C there is no solution; at least, the depth would become zero 

 in a finite distance. 



Now in the bore problem we start upstream with a uniform supercritical flow, 

 so we see that if R and S remain unchanged the only possible downstream flow is the 

 solitary wave. To this extent Lemoine is proved wrong. However, you observe that 

 it is necessary to reduce the total head R by viscous dissipation by only the slightest 

 amount to raise the curve to one of type B. A reduction by the full value given by 

 Rayleigh would of course raise it to one of type A, representing a uniform subcritical 

 flow; no greater loss of head is possible without simultaneous loss of momentum. But 

 if energy intermediate between zero and the classical Rayleigh value is dissipated, then 

 the curve becomes one of type B and cnoidal waves will be present behind the bore. We 

 find that these will have the amplitude calculated by Lemoine if 20% of the classical 

 energy dissipation is accomplished by friction, but that amplitudes from zero to 1.3 

 times Lemoine's value are possible for other values of this percentage. These considera- 

 tions give a reason for the wide scatter of the experimental results. 



75. Energy-momentum diagram for steady flows 



Next, following up these general ideas, Brooke Benjamin and I and Mr. S. C. 

 De [2, 6] constructed a diagram (Fig. 9) giving the values of R and S for different 

 kinds of steady flow with given volume flow Q. The abscissa and ordinate on this dia- 

 gram are R/R c and S/S c , where R c and S c are the total head and momentum flow for 

 a critical stream of volume flow O; actually, both R and S have minimum values for 

 such a uniform stream with v zz y (gh). On this diagram the cusped curve repre- 

 sents uniform streams with the Froude numbers v/yj (gh) as marked, subcritical on 

 the left, supercritical on the right. The broken-line barrier represents "waves of greatest 

 height." Waves are possible in the thin region between this and the other two barriers. 

 Weak, sinusoidal waves occur only near the left-hand barrier, cnoidal waves occur 

 in the region below Z, Stokes waves in the region above Z. The solitary wave lies on 

 the right hand branch of the cusped curve below Z, since this can appear out of a 

 uniform supercritical stream without any loss of momentum or energy. The point Z 

 represents the solitary wave of greatest height. For every point in the diagram the 

 amplitude and wavelength have been computed (those for cnoidal waves by Benjamin 

 and myself and those for Stokes waves by De). Hence the diagram can be used to 

 find the characteristics of the waves generated by adding or subtracting any given 

 amounts of energy or momentum from a given uniform flow. 



In the problem of the weak bore, for example, one starts from a point below 

 Z in Fig. 9, representing uniform supercritical flow, and moves to the left in proportion 

 to the energy removed. If some momentum is lost also by friction at the bottom one 

 moves also slightly down. Again, in the problem of wave resistance to a two-dimen- 

 sional obstacle, one can use the diagram to obtain a theory for large-amplitude waves 

 by starting at a point on the left-hand curve representing uniform subcritical flow and 

 moving down (if the obstacle is streamlined so that only momentum is lost), or down 

 and to the left (if dissipation occurs so that energy also is lost). In any problem, if 

 one lands beyond the broken-line barrier the waves have exceeded their greatest pos- 

 sible height and must lose energy by breaking, which carries the point away to the left. 

 Strong bores represent a transition from the isolated solutions above Z, representing 

 uniform supercritical streams, to the range of physically possible flows to the left of the 



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