222-223] ARTIFICE OF STEADY MOTION. 385 



The condition for a free surface is therefore satisfied, provided 



, tanhM 



This determines the wave-length (2-7T/&) of possible stationary 

 undulations on a stream of given uniform depth h } and velocity c. 

 It is easily seen that the value of kh is real or imaginary 

 according as c is less or greater than (gh)^. 



If, on the other hand, we impress on everything the velocity 

 c parallel to #, we get progressive waves on still water, and (5) 

 is then the formula for the wave-velocity, as in Art. 218. 



When the ratio of the depth to the wave-length is sufficiently great, the 

 formulae (1) become 



=--* 



cons. - - 

 If we neglect F/3 2 , the latter equation may be written 



leading to = const. - gy - 1 (1 - 2*/3e* cos Tex + &&<?*} ............... (ii). 



/Y\ 



-= const. + (kc 2 g) y + fcc\lr ......................... (iii). 



Hence if c 2 = #/ ....................................... (iv), 



the pressure will be uniform not only at the upper surface, but along every 

 stream-line ^ = const.* This point is of some importance ; for it shews that 

 the solution expressed by (i) and (iv) can be extended to the case of any 

 number of liquids of different densities, arranged one over the other in 

 horizontal strata, provided the uppermost surface be free, and the total depth 

 infinite. And, since there is no limitation to the thinness of the strata, we 

 may even include the case of a heterogeneous liquid whose density varies 

 continuously with the depth. 



223. The method of the preceding Art. can be readily 

 adapted to a number of other problems. 



For example, to find the velocity of propagation of waves over 

 the common horizontal boundary of two masses of fluid which are 

 otherwise unlimited, we may assume 



cos Aw,) 



where the accent relates to the upper fluid. For these satisfy the 



* This conclusion, it must be noted, is limited to the case of infinite depth. 

 It was first remarked by Poisson, I.e. ante, p. 380. 



L. 25 



