520 Internal Waves 



c\ =^tanhx(/j + /z'), 



X 



2 g Q-Q 



x QCO\h.xh J r q' coih.xh' 



(XVI. 10) 



The first type of wave is identical with the ordinary surface waves which travel 

 with a velocity c l5 on a layer of water of the thickness {h + ti). With this type 

 of wave the vertical component of the displacements of the water particles 

 decreases gradually with increasing depth from the surface. The surface has 

 the largest amplitude, the boundary surface a smaller one and the phase of 

 the wave motion at the surface is the same as at the boundary surface. This 

 "external" wave, therefore, is in no way different from the ordinary wave 

 at the surface of homogeneous water-masses. 



The second type of wave progressing with a velocity of c 2 has its largest 

 amplitude at the boundary surface and it decreases rapidly upwards and 

 downwards. Consequently, this is an internal wave. In the upper layer close 

 to the free surface there is a certain depth where the vertical motions vanish. 

 Consequently, the internal boundary surface and the free surface oscillate 

 with opposite phases, but the amplitude at the free surface is so small that 

 for all practical purposes it can be disregarded. The ratio between the am- 

 plitude at the free surface and that at the boundary surface is given by 



(XVI. 11) 



cosh xh'—(g/c 2 x 2 ) sin xli 



In practice, the depth of the lower layer is large compared to the wave 

 length. Then we can put tanhx/z = 1 and the roots of (XVI. 9), and (XVI. 10) 

 become 



and 



c\= S - 



X 



cl = i g , £; (XVI. 12) 



Qcothxh +@ x 



the ratio between the amplitudes at the free surface and at the boundary 

 surface becomes 



-i^-e-*. (XVI. 13> 



Q 



When q differs slightly from q' this is always a small quantity. 



If, on the contrary, the thickness of the upper layer is small compared 

 to the wave length, one can replace coth^/i' by \/xh' and one gets in first 

 approximation : 



A=£ and ct = ?— Q -gh', (XVI. 14). 



X Q 



