Miles 



1 ^''^ [( -> 



El = -P Y^ JJ^^S. (28) 



s 



To this we must add the dissipation associated with the motion in the film, 

 namely, 



If... 



Ej = JJ T.udS . (29) 



s 



Substituting r from Eq. (11a) into Eq. (29), adding the result to Eq. (28), and 

 averaging with respect to t , we obtain 



= \p (7^^)''' (C,-Ci)I, (30c) 



where Eq. (30c) follows from Eq. (30b) by virtue of Eqs. (10a,b) and Eq. (27). 

 Substituting Eq. (26b) and Eq. (30c) into Eq. (25) and invoking Eq. (1) and Eq. (2), 

 we obtain 



ag = -j77e(C^-C.) coth kh . (31) 



The contribution of the boundary layer on w to a, namely a^ can be derived by 

 replacing S by w in Eq. (28), substituting v = v^g from Eq. (7), and again invok- 

 ing Eq. (25); the result is 



I I 1 



i^ - 



e ]^ (J + K) + [2 - kh (J - K)] csch 2khl , (32) 



where 



H 



^ dL , 



(33a) 



-i[{ 



dL (33b) 



are essentially form factors. 



The parameter c^ - Cj is a measure of the damping that owes its existence 

 to the presence of the film; more precisely, it is the ratio of this damping to the 

 damping that would be produced by an inextensible film (<f = m , in which limit 



580 



