Damping of Gravity Waves by Surface Films 

 Cj. - Cj -> l). Referring to Eq. (21), we obtain 



which is plotted vs ^ in Fig. 1 for 77 = 0, 1, 2. 



(^- 1)' + (i + vf 



(34) 



Cr-Ci 



Fig. 1 - The surface-damping parameter 

 C^ - C^ as given by Eq. (34) 



The parameter f vanishes like 



as cT -. (i.e., ^oca^/^ fQj. iQj^g gravity 



gravity waves) a- ^^^ as ex -» 00 (very short capillary waves) and is typically 

 smaller for gravity waves than for capillary waves. The parameter v varies 

 monotonically like o-- ^^ g^j^^ therefore is more important for gravity waves. 

 This is consistent both with the observations of Davies and Vose (9), who found 

 that bulk solubility of the film material has only a slight effect on the damping 

 of capillary waves in the absence of relaxation effects (i.e., in the frequency 

 range for which Eq. (18) is valid), and with the empirical (and ancient) observa- 

 tion that soluble oils are less effective in the damping of gravity waves than are 

 insoluble oils. We remark that <E2> is proportional to -q and therefore vanishes 

 for an insoluble film; more generally. 



<E2>/<Ei> = (C^-Ci- Id 2)/|c| 



V^. 



(35) 



The maximum value of C^- Cj is 2 and is attained for <f = 2 and 77= 0. Dr. T. B. 

 Benjamin (private communication) has pointed out that the film velocity u is in 

 quadrature with the irrotational velocity v^g at this point, in consequence of 

 which |v| ^ = 2 Ivi/'l ^, thereby doubling the dissipation rate for an inextensible 

 film, for which u = and | v| ^ = Ivs/^l ^. 



WIND- GENERATED WAVES 



We now consider the effect of an inextensible surface film on the generation 

 of deep-water gravity waves by wind. That this effect is substantial is a matter 



581 



