Damping of Gravity Waves by Surface Films 



implication that is a slowly varying function of time for free oscillations. We 

 then account for the time dependence of only indirectly, by introducing a dissi- 

 pation function (see the section on Dissipation). The consequent errors in the 

 kinematical description are 0(e). 



The tangential component of the velocity in the boundary layers at S and w 

 can be calculated by analogy with the flow over a flat plate that moves with the 

 tangential velocity 



V = V0O - u , (9) 



where V0q denotes the tangential component of V0 at the boundary, and u denotes 

 the actual tangential velocity at the boundary (u= on w). Evaluating v^g at z = 

 from Eq. (7) and remarking that the velocity in the surface film must be parallel 

 to V0O, say u = (l-C)V0oj ^^ obtain 



u = (l-C)V0e^"* , (10a) 



V = CV0e'^* (z= 0) , (10b) 



where c is a constant that is determined (in next section) by the equality between 

 the viscous shearing stress in the liquid and the tangential stress in the surface 

 film. Following Stokes (3), who remarked that "the effect of [viscosity] may be 

 calculated with a very close degree of approximation by regarding each element 

 of the [boundary] as an element of an infinite plane oscillating with the same 

 linear velocity," we obtain (cf. Landau and Lifshitz (21a)) 



r = -p{ivcr)^''^v (11a) 



= -Cp(ii^a)»/2v^ei'^t (lib) 



as the boundary- layer approximation to the viscous shearing stress 



SURFACE FILM 



We neglect surface viscosity (a more detailed investigation reveals that 

 surface viscosity is likely to be significant only for very short capillary waves) 

 and assume that the surface tension is uniquely specified by the superficial 

 concentration of the surface film, say T = T(r). The shearing stress associated 

 with small variations of r about the equilibrium concentration, say r^ , then is 

 given by 



T = vT % (dT/dr)o vcr-To) . (12) 



The calculation of r - r^ for an insoluble film follows directly from the 

 continuity equation 



3r/3t + rv-u = . (13) 



577 



