424 Drs. Korteweg and de Yries on the 



equation for stationary waves, and it is easily shown that the 

 well-known equation 



7? = /isech 2 «#A /_ 

 ' V 4a- 



of the solitary wave is included as a particular case in the 

 general solution of this equation. But, in referring to this 

 kind of wave, we have to notice the result that, taking 

 capillarity into account, a negative wave will become the sta- 

 tionary one, when the depth of the liquid is small enough. 



On proceeding then to the general solution, a new type of 

 long stationary wave is detected, the shape of the surface 

 being determined by the equation 



, = W.*^(n.od.M-^). 



We propose to attach to this type of wave the name of 

 cnoidal waves (in analogy with sinusoidal waves). For k = 

 they become identical with the solitary wave. For large 

 values of k they bear more and more resemblance to sinusoidal 

 waves, though their general aspect differs in this respect, that 

 their elevations are narrower than their hollows ; at least 

 when the liquid is not too shallow, in which latter case this 

 peculiar feature is reversed by the influence of capillarity. 



For very large values of k these cnoidal waves coincide 

 with the train of oscillatory waves of unchanging shape dis- 

 covered by Stokes"*, which therefore in the theory of long 

 waves t constitutes a particular case of the cnoidal form. 

 Indeed the equation \ obtained by Stokes, when written in 

 our notation, becomes 



, =Acos ______ C0S _. 



but, as Sir Gr. Stokes remarks, in order that the method of 



X z h 

 approximation adopted by him may be legitimate, -™- must 



be a small fraction. Now, when capillarity is neglected, the 

 wave-length X of our cnoidal waves is equal to 



* Transactions of the Cambridge Phil. Soc. vol. viii. (1847), reprinted 

 in Stokes, Math, and Phys. Papers, vol. i. p. 197. 



t Stokes' solution is more general in so far as it applies also to those 

 cases wherein the depth of the liquid is moderate or large in respect 

 to the wave-length. 



\ Stokes, Math, and Phys. Papers, vol. i. p. 210. 



