524 Internal Waves 



Its two roots are 



c x =V[g(h + h')], 



(XVI. 17) 



Jq- 



q § h + h'' 



For the first of these waves the velocity of progress c x is identical with that 

 of ordinary long waves at the surface of a water-mass having the total depth 

 of the two layers. This is the external wave, which will not be discussed heie. 

 The value of c 2 is the velocity of progress of the internal wave. If the thickness 

 of the bottom layer h is large compared to h', this equation is reduced to the 

 second equation of (XVI. 14). The amplitude of the internal wave has its 

 maximum at the boundary surface and decreases upwards and downwards. 

 At the free surface it is, according to (XVI. 16), 



y =-Z^-, (XVI. 18) 



Q 



if Z is the amplitude at the boundary surface. The negative sign indicates 

 that at the surface the phase is opposite to the phase at the boundary surface. 

 If one assumes q—q' = 10~ 3 , which is already a large value corresponding 

 to a t = 24-5 and a' t = 23-5, then if Z = 10 m, r) Q becomes of the order of 

 magnitude of 1 cm. It is so small that it can be neglected. At the bottom 

 there is no vertical displacement of the water particles, so that here the in- 

 ternal wave must also vanish. We can assume as it is allowed in a first ap- 

 proximation — that the amplitude of the internal wave increases linearly from 

 the free surface to the boundary surface and decreases linearly from the 

 boundary surface to the bottom. The variation per unit length in the upper 

 layer is then Z/h', in the bottom layer —Zjh. We can compute the amplitude of 

 the horizontal velocities of the water particles from the vertical amplitudes 

 with the equation 



(XVI. 19) 



and one obtains for the two layers 



U' = c 2 ^ and U = -c 2 ? (XVI. 20) 



n n 



The amplitudes of the horizontal velocities are constant within each layer; 

 however, they change their sign at the boundary surface; consequently, the 

 velocities have opposite directions on the two sides of the boundary surface. 

 As we have already stated above (p. 518), with internal waves there is always 

 a vortex-sheet at the boundary surface. As, for reasons of continuity, U'h' must 

 equal Uh, these amplitudes of the velocity are inversely proportional to the 

 thickness of the two layers and one obtains 



