Miles 



Linearizing Eq. (13) with respect to perturbations about r= r^, and u = 0, we 

 obtain ., 



r-To = -(ia)-' r.v-u. (14) 



Substituting Eq. (14) into Eq. (12) and invoking Eq. (10a) and Eq. (8), we obtain 



T = -(icr)-' ro(dT/dr)o vv-u (15a) 



= (i-c)(ic7)-i ro(dT/dr)ok2v>/'ei^* . (15b) 



Equating Eq. (15b) and Eq. (lib) and solving for c, we place the result in the 

 form 



C = ^/(^- 1+i), (16) 



where 



^^ -p-^ ro(dT/dr)o(2/v)i/2k2a-3/2. (17) 



We extend this last result to soluble surfactants on the hypothesis that the 

 relaxation time for equilibrium between the material in the film and the dis- 

 solved material in the underlying liquid is negligible compared with l-n/cr (there 

 can be little departure from this assumption for gravity waves; cf . the relaxa- 

 tion times given by Davies and Vose (9)). We then can relate the changes in 

 surface concentration to the corresponding changes in bulk concentration, say 

 7- 7o, by the linearized equilibrium equation 



r-r^ ^ (dr/d7)o(7-7o), (18) 



where y satisfies the diffusion equation* 



By/Bt = DB^y/Bz^ , (19) 



and the right-hand side of Eq. (13) is replaced by the transfer rate DCBy/Bz)^; D 

 is the bulk diffusion coefficient. Solving this set of equations for y - y^ and 

 r - Tq, we obtain results that differ from Eq. (14) and Eq. (15) only by an addi- 

 tional factor of [1 + 1/2 (1 - i )t7]- ^ on the right-hand side of each, where 



T) -- (2D/cr)l/2(dr/d7)-/. (20) 



The corresponding generalization of Eq. (16) is 



C = ^/[^- 1+ 1(1 + 7,)] . (21) 



The quantities rp(dT/dr)Q and (dT/dy),, must be determined either experi- 

 mentally or from surf ace -chemistry considerations. Invoking the semiempirical 

 equations of Szyszkowski and Langmuir for soluble surfactants (18a), we find that 



♦ This is a boundary -layer approximation (V^7 % 3'^7/3z^). Typically, D << v, so 

 that the diffusion boundary layer lies inside of the viscous boundary layer. 



578 



