SECT. 5J RIPPLES 725 



By numerical integration of (8), wo find 



^2+-^2 = ^[i„ (3.32/,„i/rjr-i) + |(72(27r/„,f7)-2+ . . .]. (9) 



In Fig. 3 we exhibit equation (9) after arbitrarily setting /,„ — 2 c/s as an 

 estimate of the high-frequency cutoff induced by slicks. The measurements 

 (circles) show rough agreement (cf. Burling, 1959). 



The fact that the mean square sloj^e of a clean water surface is scarcely 

 larger than that of oil-smoothed \\ater surfaces when the wind speed is less 

 than 4 m sec~i implies then that ripples make an inappreciable contribution 

 at these low wind speeds. 



The increasing deviation bet^^'een the observed mean square slope and the 

 calculated gravity contribution above 4 m sec"i implies that ripples become 

 increasingly important. The laboratory measurements show that ripples start 

 growing above 5 m sec~i. The measured values of Sx'^, since they include 

 effects of gravity waves, are overestimates (by a factor of about 2) for the up 

 and down wind components of mean square slope due to ripples alone ; these in 

 turn are underestimated (by a factor of about 0.8) for the sum of both com- 

 ponents of mean square slope due to ripples. The increase in mean square slope 

 appears to be at about the correct rate to sujiply the difference between the 

 gravity contribution and the total mean square slojDe. This gives an indirect 

 indication that the laboratory spectra are approximately correct for ripples 

 even though they exaggerate the importance of short gravity waves. 



2. Effect of Slicks 



According to Reynold's classical theory (Lamb, 1945, Art. 351) the effect of 

 an inextensible surface film is to damp waves by creating a thin layer of intense 

 friction at the surface. The modulus of decay for waves of length A and fre- 

 quency / is 



r = rr-^<^u-^2Xf-l. (10) 



in terms of the kinematic viscosity of water, v. For comparison, the modulus of 

 decay for waves on a clean surface is X"I{8tt'^p). 



Contamination of water surfaces by surface-active compounds results in a 

 somewhat extensible film which can withstand forces of the order of a few 

 dynes per centimeter, and the foregoing expression will be a lower limit. 

 Natural slicks seem to be formed by monomolecular layers of organic material 

 which have an effective strength of only a few dynes per cm. The change of 

 surface tension from a clean surface is frequently negligible. Artificial slicks 

 composed of fish oil and kerosene have a higher tensile strength, which is 

 adequate to maintain an unbroken film in strong winds. They reduce the 

 surface tension appreciably. For ripples of frequency /m (13.5 c/s), equations 

 (2) and (10) yield t = 0.83 sec for a surface contaminated with a natural slick 

 and an even smaller value for an oil slick. Higher frequency ripples are damped 

 out still more rapidly. A slick-covered water surface of a few meters in size is, 



