231] FINITE WAVES OF PERMANENT TYPE. 411 



The figure shews the wave-profile, as given by (3), in the case 

 of ka = J, or a/\ = '0796. 



The approximately trochoidal form gives an outline which is 

 sharper near the crests, and flatter in the troughs, than in the case 

 of the simple-harmonic waves of infinitely small amplitude investi- 

 gated in Art. 218, and these features become accentuated as the 

 amplitude is increased. If the trochoidal form were exact, instead 

 of merely approximate, the limiting form would have cusps at the 

 crests, as in the case of Gerstner's waves to be considered presently. 

 In the actual problem, which is one of irrotational motion, the 

 extreme form has been shewn by Stokes*, in a very simple manner, 

 to have sharp angles of 120. 



The problem being still treated as one of steady motion, the motion near 

 the angle will be given by the formulae of Art. 63 ; viz. if we introduce polar 

 coordinates r, 6 with the crest as origin, and the initial line of 6 drawn 

 vertically downwards, we have 



(i), 



with the condition that >//=() when 0=a (say), so that ma=^n. This 

 formula leads to 



q=mr m ~ l .................................... (ii), 



where q is the resultant fluid velocity. But since the velocity vanishes at the 

 crest, its value at a neighbouring point of the free surface will be given by 



q 2 =2grcosa ................................. (iii), 



as in Art. 25 (2). Comparing (ii) and (iii), we see that we must have w=f , 

 and therefore a 



In the case of progressive waves advancing over still water, the particles 

 at the crests, when these have their extreme forms, are moving forwards with 

 exactly the velocity of the wave. 



Another point of interest in connection with these waves of permanent 

 type is that they possess, relatively to the undisturbed water, a certain 



* Math, and Phys. Papers, t. i., p. 227. 



t The wave-profile has been investigated and traced, for the neighbourhood of 

 the crest, by Michell, " The Highest Waves in Water," Phil. Mag., Nov. 1893. He 

 finds that the extreme height is -142 X, and that the wave-velocity is greater than in 

 the case of infinitely small height in the ratio of 1*2 to 1. 



