551 



length, A', than does the linear growth rate, ^^ 

 (A'). The long-wave linear growth rate, Q-, is 

 also shown for comparison; it is much smaller. Of 

 course the interaction growth rate also involves 

 (k'C') of the short waves which would be typically 

 0.01 but the division by c-c' would somewhat offset 

 the effect of small slope. One calculation for an 

 upstream travelling long wave verified that a was 

 negative and energy was removed from the long wave 

 by interaction with the short wave. We have not 

 carried the calculations further to date. 



Some idea of the wavelengths, involved in any 

 practical application of these ideas can be seen 

 from Figure 4 which shows the group velocity and 

 phase velocities for gravity-capillary waves. The 

 requirement for strong coupling is c = c^. We 

 further note that waves shorter than say 0.3 cm 

 are unlikely to be important in a viscous fluid. 

 Thus short waves in the range 0.3 cm could interact 

 with a 20 cm long wave in the manner we have dis- 

 cussed but waves longer than 20 cm would be unlikely 

 to be affected. 



Although the effects of surface drift are not 

 yet included in our calculations, the range of 

 affected long waves can be somewhat broadened by 

 considering surface drift. Drift velocities are 

 typically 5% of the wind velocity; this is the same 

 order as the friction velocity which we have taken 

 as uT = 0.05 Ua,. If we assume that a surface layer 

 will advect the short waves [Valuenzuela (1976)] 

 but leave the phase velocity of the long waves 

 unaffected (Valenzuela' s calculations did not extend 

 to long waves) , we can consider a broader range of 

 interaction possibilities, as sketched in Figure 4. 

 For a group velocity augmented by a surface current 

 of 30 cm/sec, interactions between a long wave of 

 about 50 cm and waves longer than 0.3 cm become 

 possible and a 20 cm wave may interact with waves 

 of order 1.4 cm. 



Experimental data in the range of wave lengths 

 and friction velocities of interest for the inter- 

 actions we have investigated here was presented by 

 Plant and Wright (1977) . Some of their results are 

 reproduced in Figure 5, showing the temporal growth 

 rate vs. wave number for several values of friction 

 velocity. Of particular interest is that while the 

 short-wave growth rate is accurately predicted by 

 linear theory, there is a departure of theory and 



S 3.0 - 



I 



cf 



2.0- 



1.0- 



10 



20 30 50 

 X ~ cm 



100 



10 



100 

 C ~ cm/sec 



FIGURE 4. Group velocity and phase velocity for 

 gravity-capillary waves. 



100 



k~ cm 



FIGURE 3. Coupling coefficients for long-wave and 

 short-wave interaction; linear temporal growth rate 

 Q. . u =30 cm/sec. 



FIGURE 5. Measured temporal growth rates for various 



u cm/sec; from Plant and Wright (1977) . 

 T 



