Generation, Growth and Propagation of Waves 11 



waves have a velocity of propagation ±c, which is given by the equation 



*-*-T£*& v *- vif ' (,V11) 



One can easily recognize that the values from c in (IV. 1 1) becomes imaginary, 

 in other words that the wave motion becomes unstable, when 



(U x -U 2 f> 69 "~ Q - 2 



x QiQ2 



As the right member of the equation decreases unlimitedly with A, this would 

 mean that, in the absence of all other factors, the smallest wind will be capable 

 of generating waves of extremely small wave lengths. 

 (c) Influence of the Surface Tension 



The wave lengths of the waves which are generated first (capillary waves) 

 are extremely small. Therefore it is to be expected that the surface tension 

 has to be considered if the condition of pressure on the boundary surface 

 is to be fulfilled. If both media are at rest, the velocity of propagation of 

 the waves on the boundary surface is now given by (IV. 13) if we designate 

 the capillary coefficient, or surface tension by T 



rf-l/feSCfc + J*-). (IV .l3) 



r \* Q1+Q2 Q1+Q2I 



If the values of X are sufficiently large, the first term of this equation becomes 

 large relative to the second term, in other words the decisive factor in a wave 

 motion is the gravity. However, if a is very small, then the second term is 

 more important, and the wave motion is conditioned mainly by the surface 

 tension. 



Although the wave length decreased continuously from 00 to zero, the 

 wave velocity tends towards a minimum and then increases again. If we 

 put TJQi — o., == T', this minimum value c^ ia is given by 



Ql ~ Qi 2 ten* 



.Q1+Q2 



and the corresponding wave length is 



•Vi- 



''min — ■*- 



We can compute that for A > 3A min the influence of the surface tension on 

 the wave velocity will not exceed 5 %, and the influence of the gravity de- 

 creases to a like degree if I < p min . For water-air at 20° C, Tis approximately 

 74 (cm/sec -2 ), whiles = 981 (cm/sec -2 ) A min = 172 cm and c' miQ = 23T5 cm/sec. 



