454 Mr. stokes, ON THE THEORY OF OSCILLATORY WAVES. 



where A and B have to be determined. The conditions (32), (33) and (45) give 



DA + A^- B^= 0, 

 p (gD - mc'S') A + p^{g+ mc) A^ - p,(.g- mc^) fi, = 0, 

 (g + mc') (-'""■A, -(g- mc") e""'fi, = 0. 

 Eliminating A, A^ and B^ from these equations, and putting 



m 

 we find 



(pSS, + pDD) X^-p {SD^ + Sp) ^ + (jo - p) DD^ = 0. ... (46). 



The equilibrium of the fluid being supposed to be stable, we must have p^ < p. This being 

 the case, it is easy to prove that the two roots of (46) are real and positive. These two roots 

 correspond to two systems of waves of the same length, which are propagated with the same 

 velocity. 



In the limiting case in which " = eo, (46) becomes 



SS'X' - (SD^ + Sp) ^ + DD, = 0, 



the roots of which are — and — ^ , as they evidently ought to be, since in this case the motion of 



the under fluid will not be affected by that of the upper, and the upper fluid can be in motion 

 by itself. 



SD 4- S D 



When o — o one root of (46) vanishes, and the other becomes ——- — — ^ 

 ^' ^ ^ ' SS, + BD^ 



The former of these roots corresponds to the waves propagated at the common surface of the fluids, 



while the latter gives the velocity of propagation belonging to a single fluid having a depth equal 



to the sum of the depths of the two considered. 



D 



When the depth of the upper fluid is considered infinite, we must put — ' = 1 in (46). The 



two roots of the equation so transformed are I and -^—- — , the former corresponding to waves 



pA + pP 



propagated at the upper surface of the upper fluid, and the latter agreeing with Art. 15. 



When the depth of the under fluid is considered infinite, and that of the upper finite, we 



must put — = 1 in (46). The two roots will then become 1 and -^ — — — =-' . The value of the 

 S pS, + pP^ 



former root shows that whatever be the depth of the upper fluid, one of the two systems of 

 waves will always be propagated with the same velocity as waves of the same length at the sur- 

 face of a single fluid of infinite depth. This result is true even when the motion is in three 

 dimensions, and the form of the waves changes with the time, the waves being still supposed to 

 be such as could be excited in the fluids, supposed to have been previously at rest, by means of 

 forces applied at the upper surface. For the most general small motion of the fluids in this case 

 may be regarded as the resultant of an infinite number of systems of waves of the kind con- 

 sidered in this paper. It is remarkable that when the depth of the upper fluid is very great, the 

 root t^ = 1 is that which corresponds to the waves for which the upper fluid is disturbed, while 

 the under is sensibly at rest ; whereas, when the depth of the upper fluid is very small, it is the 

 other root which corresponds to those waves which are analogous to the waves which would 

 be propagated in the upper fluid if it rested on a rigid plane. 



