14. Cnoidal wave trains behind bores 



However, Mr. Brooke Benjamin and I thought it desirable to reexamine this 

 theory on a more accurate basis [2], partly because* the ratio a\ 2 /h 2 3 is too great for 

 sinusoidal-wave theory to be applicable (it is about 10) and so the long waves behind 

 the bore should really be treated as cnoidal waves. (Another reason was that although 

 the 1.15 in (12) is close to the mean of the observations of many workers, it has been 

 common to find on occasion values either much larger or much smaller.) To under- 

 stand the subject better we re-worked the theory of cnoidal waves as follows. The 

 momentum flow per unit span (divided by the density) is in general 



S = / ( - + iA dy (13) 



and for irrotational flow this can be calculated, for given volume flow Q per unit span 

 and total head R ~ Vi{iC- + v 2 ) + gy, as 



S = R v - Vm 2 + l A— f i - )W 2 + ° (-) ) ( 14 ) 



Thus, taking in frequency dispersion to the first approximation only, we have a differ- 

 ential equation 



m* ( — +m l - 2#i?» + 2>St7- Q2 = (15) 



\dx) 

 as the equation for all stationary long waves. 



The character of the solutions is easily seen by thinking of x as the time, and the 

 equation as that of a particle of mass % Q 2 , velocity drj/dx, potential energy equal to 

 the cubic, and zero total energy. The particle would oscillate if the cubic has two 

 zeros between which it is negative as in Fig. 8, curve B; this behavior oscillatory in x 



g7) 3 -2R-9 z +2S-9-Q 



Figure 8. Different possible positions of the graph of the cubic occurring in equation 115) 

 * Note that our a is twice the a of reference 2. 



27 



