﻿352 Dr. J. McCowan on the Highest 



wave for such depth ; for the height to which waves can 

 attain without breaking must evidently increase with their 

 length, and the solitary wave may be regarded as the limiting 

 type to which each individual wave, reckoned from trough to 

 trough, in a permanent train of finite waves approaches as 

 the wave-length indefinitely increases. In fact this paper 

 and the former, " On the Solitary Wave," may be regarded 

 as giving a very close approximation to the form and motion 

 of the individual waves in a train of finite waves if the wave- 

 length is even so small a multiple of the depth as ten or 

 twelve. 



It will thus be convenient to follow to some extent the 

 methods and notation of the paper " On the Solitary Wave/' 

 and references to it will be briefly indicated by an S prefixed. 



Consider, then, a solitary wave propagated with uniform 

 velocity U along the direction in which x increases in an 

 endless straight channel of uniform rectangular cross section, 

 the axis of x being taken along the bottom and that of z 

 vertically upwards. 



Let the motion be regarded as reduced to steady motion by 

 having superposed on it a velocity equal and opposite to the 

 velocity of propagation of the wave, and take x = and z—c 

 as the coordinates of the crest. Let u and w be the horizontal 

 and vertical components respectively of the resultant velocity 

 q in the steady motion at x, z, of which, further, <j> is the 

 velocity potential and yfr the current function. 



We shall now, referring to the " General Theory of the 

 Wave," (S. § 1), seek to determine a form of the relation 

 between ^-{-icj), or, as we shall here find more convenient, 

 u + ow and z + lx corresponding to S. (1) and (2), but only 

 containing so many disposable constants as will suffice for 

 the degree of accuracy at present desired. Noting that 

 for the limiting form the velocity at the crest must vanish, 

 and remembering Sir George Stokes's expression* for the 

 leading term in the velocity near the crest of a wave at 

 the breaking-limit, we shall assume (compare S. (6)): — 



u + lw=— U{1— fk 2 sec 2 lm(z + ix)} v 1— k 2 sec 2 ^m (z + lx), (1) 



where 



k— cos ^mc ; (2) 



as a form conveniently integrable with respect to z + ix so as 

 to give yfr + 1$ in finite terms if so desired. 



It should be noted that all the conditions required to be 



* "On the Theory of Oscillatory Waves," Appendix B. ' Collected 

 Papers,' vol. i. 



