235] DYNAMICAL CONDITION FOR STATIONARY WAVES. 423 



spending to the ignored coordinate may be taken to be the flux 

 per unit breadth of the channel, and the constant momentum of 

 the circulation may be replaced by the cyclic constant K. The 

 coordinates q lt q 9 , ... of the general theory are now represented by 

 the value of the surface-elevation (77) considered as a function of 

 the longitudinal space -coordinate x. The corresponding com- 

 ponents of extraneous force are represented by arbitrary pressures 

 applied to the surface. 



If I denote the whole length of the circuit, then considering 

 unit breadth of the canal we have 



(5). 



o 

 where 77 is subject to the condition 



/: 



.(6). 



If we could with the same ease obtain a general expression for 

 the kinetic energy of the steady motion corresponding to any 

 prescribed form of the surface, the minimum condition in either of 

 the forms above given would, by the usual processes of the Calculus 

 of Variations, lead to a determination of the possible forms, if any, 

 of stationary waves*. 



Practically, this is not feasible, except by methods of successive 

 approximation, but we may illustrate the question by reproducing, 

 on the basis of the present theory, the results already obtained 

 for ' long ' waves of infinitely small amplitude. 



If h be the depth of the canal, the velocity in any section when the surface 

 is maintained at rest, with arbitrary elevation 77, is x/(A + 7), where x is the 

 flux. Hence, for the cyclic constant, 



* For some general considerations bearing on the problem of stationary waves 

 on the common surface of two currents reference may be made to von Helmholtz' 

 paper. This also contains, at the end, some speculations, based on calculations of 

 energy and momentum, as to the length of the waves which would be excited 

 in the first instance by a wind of given velocity. These appear to involve the 

 assumption that the waves will necessarily be of permanent type, since it is only on 

 some such hypothesis that we get a determinate value for the momentum of a train 

 of waves of small amplitude. 



