192 WAVES IN LIQUIDS. [CHAP. VII. 



From (28) we find 



d.Jcc T 



Q-Ikh 



The ratio of this velocity of propagation to that of the waves in 

 creases as kh diminishes, being ^ when the depth is infinite, and 

 unity when it is small compared with the wave-length*. 



160. Rankine *|* has by a synthetic process arrived at the 

 following exact equations, expressing a possible form of wave- 

 motion when the depth of the fluid is infinite, viz. : 



1 -I 



x = a +j e~ A 5 sin k (a + ct) \ 



1 (34). 



= ) 4 - 



kb cos k (a + ct} 



J 



k 



Here x t y are the co-ordinates at the time t of the particle defined 

 by the parameters a, b, whilst k, c are absolute constants. The 

 axis of x is horizontal, that of y vertical and downwards. 



We find at once 



d(a, I 



so that the Lagrangian equation of continuity [see Art. 17] 

 is satisfied. Also, substituting from (34) in the Lagrangian 

 equations of motion, we have 



( -r V] = kc 2 e~ kb sin k (a + ct), 

 da \p J 



A ( + F) - kc* e^ b cos k (a + ct) - kc* e~ 2kb , 

 whence, since F= gy, 



P = a\b + j e ~ ** cos k (a -f- ct) j- - c z e~ kb cos k (a + ct} 4- Jc 2 e~ n b . 

 P ( k 



Now at the free surface p must be independent of t, so that 



g = 7,v 2 , 



* Stokes, Smith s Prize Examination, 1876. See also Prof. O. Beynolds, On 

 Waves, Nature, Vol. xvi., p. 343; and Lord Bayleigh, Theory of Sound, Art. 191. 

 and Proc. Lond. Math, Sor. Nov. 8, 1877. 



f Phil. Tram. 1803. 



