76 Generation, Growth and Propagation of Waves 



not very satisfactory, because they do not agree with the observations. More 

 research work on this theory has been done by Burgers (1927, p. 333) and 



ROSENHEAD (1931). 



Lord Kelvin (1871) showed earlier that surface waves can be formed 

 in the absence of friction as a consequence of the dynamic pressure variations 

 previously discussed and, that this was the principal cause for the generation 

 of waves by the wind. Later on, Rayleigh has shown in a simple way how 

 a system of progressive waves may be maintained against dissipative forces 

 by a properly adjusted distribution of pressure over their slopes. This is 

 especially true when the pressure is largest on the wind side and smallest 

 on the lee side of the wave (Lamb, 1932). 



This shows that through an excess of pressure on the rear slopes of the 

 waves, combined with a tangential stress on the exposed wave crest when 

 the wind velocity is greater than the wave velocity, the wave crests will become 

 larger and that the energy lost by friction has been equalized by the work 

 accomplished by these forces on the surface. The following investigations, 

 made by different authors at the same time, have accentuated this concept 

 of wave formation. In order to gain a good insight into all the factors which 

 have to be considered we will discuss each one, its relation and its influence 

 on the formation of waves. Many equations will be derived later in the section 

 dealing with internal waves on boundary surfaces of the density (chap. XVI). 

 (b) Stability of Internal Waves in Moving Water Masses 



If we have two superposed homogeneous fluids with a density o t and o 2 

 then there is a possibility of internal waves on their boundary surface having 

 the character of short or deep water waves. Their wave velocity is given by 

 the equation 



IgX Q x —Q^ 



I 



In Qx-\-Q2 



(IV. 9) 



The wave velocity as expressed in equation (11.11) is reduced in 

 equation (IV. 9) with the square root of the difference in density of both 

 water layers. The equation (IV. 9) is in reality only valid when both layers 

 have an infinite depth, but if the water depth is several times a multiple of 

 the wave length (short waves), then the discrepancy with the exact formula 

 (see equation (XVI. 17) is extremely small. 



If both water masses have a motion parallel to their common boundary 

 layer and have the velocities U 1 and U 2 , the wave velocity of the internal 

 waves is 





8 Qi (h _ Q1Q2 ,tj _tj \ 2 



XQx+Qz {Q1+Q2) 2 



1/2 



(IV. 10) 



in which x = 2n/X. The first member of this equation can be considered the 

 average velocity of both currents and, in relation to this velocity, the internal 



