Miles 



predictions of Levich (we recall that Levich did not establish, at least in any 

 systematic way, regimes of validity for his various approximations). 



Benjamin and Ursell (2), Case and Parkinson (5), and Keulegan (6) all at- 

 tributed the observed discrepancies between a and a^^, to capillary hysteresis at 

 the contact line of the meniscus, but none of them attempted a direct calculation 

 of this effect. Although Benjamin and Ursell appear to have been the first to 

 recognize the significance of capillary hysteresis for surface-wave damping, the 

 basic phenomenon is well known in the literature of surface chemistry (see, e.g., 

 Schwartz et al. (17)), and Adam (18) refers to A. A. Milne (19) for an everyday 

 observation. 



Keulegan (6) established that damping in a hydrophobic basin may be sub- 

 stantially larger than in a geometrically similar, hydrophilic basin. Subse- 

 quently, Keulegan and Brockman (20) carried out a series of measurements 

 aimed at isolating the effect and concluded, at least tentatively, that both the 

 advance and the recession of a meniscus are opposed by constant forces that 

 depend only on the material properties of the three-phase interface. This con- 

 clusion agrees with that generally accepted in the literature of surface chemistry. 



We have reexamined the results of Case and Parkinson (5) and Keulegan (6) 

 and regard it as probable that both surface contamination and capillary hystere- 

 sis were significant in their experiments. We emphasize, however, that these 

 two effects are not likely to be entirely independent, for both the modulus of 

 compression for the surface film and capillary hysteresis at the meniscus de- 

 pend on the contamination of the free surface through mechanisms that are far 

 from being fully understood. 



FLUID MOTION 



We consider a cylindrical container filled to a mean depth h with a slightly 

 viscous liquid of density p and designate the upper surface of the liquid by s and 

 the wetted boundary of the container by w. The velocity field associated with a 

 wave motion on s will be essentially irrotational outside of viscous boundary 

 layers at s and w and can be derived from a velocity potential, say q = v^. 

 Choosing cylindrical coordinates, with z measured vertically downward from 

 the mean position of s, we pose this potential in the form (Lamb (4c)) 



0(x,y,z,t) = (//(x.y) sech kh cosh [k(z-h)] e^°^ , (<) 



where ^ is real and satisfies the two-dimensional Helmholtz equation 



v20+k20 = . (8) 



The usual conventions hold in respect to complex, time-dependent variables, 

 such as <t>. 



The wave number k is determined (as a member of an infinite, discrete set 

 of eigenvalues) by the requirement that the normal derivative of r vanish on w. 

 The angular frequency, t, is given by Eq. (3). We regard c as real, with the 



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