234-235] ' FORM OF PROFILE. 421 



must inevitably suffer a continual change of form as it advances, the changes 

 being the more rapid the greater the elevation above the undisturbed level. 

 The investigation referred to postulates, however, a length so great that the 

 vertical acceleration may be neglected, with the result that the horizontal 

 velocity is sensibly uniform from top to bottom (Art. 169). The numerical 

 table above given shews, on the other hand, that the longer the 'solitary 

 wave ' is, the lower it is. In other words, the more nearly it approaches to 

 the character of a 'long' wave, in the sense of Art. 169, the more easily is 

 the change of type averted by a slight adjustment of the particle- velocities *. 



The motion at the outskirts of the solitary wave can be represented by a 

 very simple formula f. Considering a progressive wave travelling in the 

 direction of ^-positive, and taking the origin in the bottom of the canal, at a 

 point in the front part of the wave, we assume 



(j) = Ae~ m(x ~ ct) cosmy ........................... (xiii). 



This satisfies v 2 $=0, and the surface-condition 



will also be satisfied for y = h, provided 



, tan mh 



This will be found to agree approximately with Lord Rayleigh's investigation 

 if we put m = b~ 1 . 



235. The theory of waves of permanent type has been brought 

 into relation with general dynamical principles by von HelmholtzJ. 



If in the equations of motion of a 'gyrostatic' system, Art. 

 139 (14), we put 



* Stokes, "On the Highest Wave of Uniform Propagation," Proc. Camb. Phil. 

 Soc. t t. iv., p. 361 (1883). 



For another method of investigation see M c Cowan " On the Solitary Wave," 

 Phil. Mag., July 1891; and "On the Highest Wave of Permanent Type," Phil. 

 Mag., Oct. 1894. The latter paper gives an approximate determination of the 

 extreme form of the wave, when the crest has a sharp angle of 120. The limiting 

 value for the ratio ajh is found to be '78. 



t Kindly communicated by Sir George Stokes. 



J " Die Energie der Wogen und des Windes," Berl. Monatsber., July 17, 1890; 

 Wied. Ann., t. xli., p. 641. 



