436 Drs. Korteweg and de Vries on the 



If, now, we take account of the fact that, as may easily 

 be inferred from (31), the wave-surface becomes steeper in 

 proportion as p is increased, we are then justified in saying 

 that a solitary wave which is steeper than the stationary one, 

 correspondiug to the same height, becomes less steep in front 

 and steeper behind, but that its behaviour is exactly opposite 

 when it is less steep than the stationary one. 



Cnoldal Waves. — Applying formula (12) to the cnoidal 

 wave, 



r) = h en* p;e, (34) 



<fc 2 I 3(4o-My-/0 en . 7 Aij^(7iu^ 



— h)snpx . cn/>^ . dnjuw. „ . . (35) 

 Supposing then 



2[«-o-/(2-4M 2 )] = 3(4crM 2 /; 2 -/0, 

 we have 



-J- = +j— (4c7M 2 /> 2 — 7i) sn 3 ^ . en px . dn j>#. (36) 



Here fig. 3 shows the change of form calculated for the 

 case A-4(rM 2 /9 2 >0. 



Fig. 3. 



When h— 4crM 2 /? 2 =0, the waves are stationary in accord- 

 ance with (20), whilst for A — 4o-M 2 p 2 <0 they become steeper 

 behind; and this last result, since p is inversely proportional 

 to the wave-length, may be stated by saying that cnoidal 

 waves become less steep in front and steeper behind when, 

 for a given modulus and a given height, their length is smaller 

 than the one required for the stationary wave of this modulus 

 and height. 



In proportion as M is taken smaller the cnoidal waves 

 more and more resemble sinusoidal waves. They would take 

 the sinusoidal form for M = 0, but then an infinitely small wave- 

 length would be required for the stationary case. For this 

 reason sinusoidal waves may always be considered as pnoidal 

 waves whose length is too large to be stationary, that is, they 

 are always becoming steeper in front. 



