Damping of Gravity Waves by Surface Films 



ro(dT/dr)o = -AT(0)(7o/7,) (21a) 



^ T(ro) - T(0) (7o«7i) (21b) 



(dT/dy), = AT(O)(R0)-i7i(7o + 7i)"^ (22a) 



^ (A/Wy,)T(0) (yo«yi)' (22b) 



where A is a dimensionless constant (A = 0.18 for fatty acids), y^ is that value 

 of 7o ^t which Fq attains half its saturation {y^ ^ co) value, R is the gas constant, 

 and 6 is the absolute temperature. The results (Eqs. (21a,b)) are not, of course, 

 applicable to insoluble surfactants. 



DISSIPATION 



Let Eg be the total energy of the oscillation on the hypothesis e = o, and 

 let a be the logarithmic decrement of the damped oscillation as e -» o. Invoking 

 the hypothesis of small oscillations, which implies that the energy must be 

 quadratic in the amplitude, we infer that the mean energy must decay like 



<E> = Eg exp(-aat/7T) , (23) 



where < > implies an average over one cycle of the oscillation, say t = (0. 2n/cr). 

 Introducing the Rayleigh dissipation function 



D = - i-dE/dt , (24) 



equating its mean value to the corresponding mean value implied by Eq. (23), 

 and anticipating that a. = 0(e), we obtain 



a= (27T/crEo) <D> [l + 0(e)] . (25) 



The total energy of the undamped oscillation is equal to twice the mean 

 kinetic energy and is given by 



o{\\<y<p>^d\ (26a) 



^plk'^ tanh kh , (26b) 



where 



I = ||(v0)2ds - k^JJ^^ds. (27) 



The two forms of i are equivalent by Green's theorem. 



The viscous dissipation in the boundary layer on S, say Ej , is equal to that 

 for a flat plate oscillating with the velocity v and is given by (21b) 



579 



