78 Generation, Growth and Propagation of Waves 



Waves with a wave length smaller than A min are called capillary waves or, 

 after Kelvin, ripples. These theoretical results are in good agreement with 

 observations of Matthieson (1889) and Kaplan (1950). 



If in the case of two superposed currents, we take the capillarity into 

 consideration, then we have to add to the equation (IV. 10) to the right side 

 the term Tx/(q 1 + q 2 ) and we obtain for the wave velocity of such waves again 

 the equation (IV. 11), with the change that instead of c we now have c of 

 the equation (IV. 13). For quantities fulfilling the inequality 



M-u^Z&z^+ik^^ , (iv. 14) 



X Q X Q 2 Q X Q 2 



the velocity of propagation of the wave becomes imaginary, in other words 

 the slightest disturbances cause the formation of small waves, and the 

 amplitudes of these waves will increase gradually. The equation on the right- 

 hand side has a minimum for 



f(Qi-Q2) 



1/2 



(IV. 15) 



or for a wave length A = A min . The smallest difference in velocities where 

 there exists the possibility of wave formation is 



Qi+Qt 



K ^0,1 — ^0,2;" ^ ^ 



also: 



(U0.1- U , 2 f 2> 2 ^^gT( Ql -Q 2 )^ , (IV. 16) 



Q1Q2 



TJ TJ > 6l+Qi 



V(QiQd 



In case the media are water of 15°C and 35°/ 00 salinity and air where 

 Ql = 1026 and & = 0-00122 



U ,i - U 0>2 ;> 67 1 cm/sec , (IV. 1 7) 



whereas A = 1 -7 cm and c min = 23 cm/sec. The wind will only start generat- 

 ing waves when its velocity is about 6-7 m/sec, which corresponds to a force 

 on the Beaufort scale of 4. The first waves are of the order of the capillary 

 waves. This result is not in accordance with the observations and we can, 

 therefore, see at once that the capillarity is not as important an element in 

 the formation of waves as it was believed formerly. 

 {d) Frictional Influences 



The amplitude of waves in a wave train due to the internal friction would de- 

 crease gradually and the wave would finally disappear completely, were it not 

 for the fact that energy is fed continuously into the wave train. We have men- 

 tioned previously that the first capillary waves which come on to a motion- 



