518 Internal Waves 



Hence, according to equation (IT. 3), we must have 



-xC = xC' = iaa. (XVI. 3) 



The equations of motion (II. 2) are replaced by the equations 



p = d(p_ 

 q dt gZ ' 



p' d(p' 



7 = ~dT~ gZ 



(XVI. 4) 



The condition for continuity of pressure at the boundary surface (p = p' 

 for z = 0) gives 



q{iaC-ga) = q\iaC-ga) . (XVI. 5) 



Substituting the value of C and C from (XVI. 3) we have 



2 Q~ Q 



or with 



a 



= c , 



(XVI. 6) 



the velocity of propagation of progressive internal waves of the wave length 

 X = 2ti/h is given by 



cZ= g*e^£ (XVI7) 



In q-\- q 



The presence of the upper fluid has therefore the effect of reducing the 

 velocity of propagation of the internal waves in the ratio (q—q')I(q + q'). This 

 decrease in the velocity, has a twofold cause; the potential energy of a given 

 deformation of the common surface is decreased in the ratio 1 — q'/q, whilst 

 the interia is increased in the ratio 1 -f q'/q. 



In practice p— q' is of the order of magnitude 10~ 3 so that the decrease 

 of velocity of the internal waves c against the surface waves is about 45 times, 

 which is quite considerable. 



The waves discussed hitherto have the character of surface or "short waves" 

 because their wave length is small compared with the infinite thickness of the 

 two superposed layers.* 



Figure 213 shows, according to V. Bjerknes, the stream lines and orbits in 

 the two superposed layers for an internal wave travelling from left to right. 

 It should be noticed that there is a discontinuity of motion at the common 

 surface. The normal velocity —dcp/dz is of course continuous, but the tangential 

 velocity —d<pjdx changes a sign as we cross the surface; in other words, we 

 have a vortex-sheet. 



* Concerning the application of the equation (XVI. 7) to the air-water system see Chapter IV. p. 76. 



