Miles 



The contribution of viscous dissipation in the boundary layers on the walls 

 of a cylindrical basin to the logarithmic decrement has been calculated by 

 Ursell (1) (see Benjamin and Ursell (2) for the corrected result) and has the 

 form 



a^ = Ae + 0(e2) , (4) 



where A depends only on the geometrical shape of the container and the partic- 

 ular mode; see Eq. (32) and Eqs. (33). 



The logarithmic decrement for a clean, unbounded surface is given by 

 (Stokes (3) and Lamb (4a)) 



a^°^ = IrreK (5) 



It can be shown (cf. Ursell (1) and Case and Parkinson (5)) that this result also 

 is valid, within a factor of l + 0(e), for a bounded surface, but it then is negli- 

 gible compared with a^^,. (Case and Parkinson and others have retained a^°> in 

 the calculation of a for closed basins, but this is clearly inconsistent unless 

 terms of O(e^) and 0(e) are included in the dissipation in the boundary layer on 

 w and in the mean energy of the oscillation, respectively. 



Available measurements (Benjamin and Ursell (2), Case and Parkinson (5), 

 Keulegan (6), Van Dorn (7)) reveal that (the theoretical value of) a^ does not 

 provide an adequate approximation to a for gravity waves in laboratory-size 

 basins except under special and carefully controlled conditions and that the ob- 

 served values of a/a^ may be as large as three. 



Van Dorn (7) attributed the observed discrepancies to a surface film pro- 

 duced by spontaneous contamination. He found that "while the observed attenua- 

 tion agreed with that computed for the solid boundaries [a^] when the water was 

 fresh, the former tended to increase with time to some higher limiting value, 

 usually within an hour." He also reported that "the fully contaminated surfaces 

 exhibited no obvious visual manifestation" and that "the water always appear[ed] 

 to be in every respect as clear and fresh as when uncontaminated." He obtained 

 an adequate approximation to the limiting value of i simply by adding to a^ the 

 value of a calculated on the hypothesis of an inextensible film, namely (Lamb 

 (4b)) 



4''-T-^- (6) 



This agreement notwithstanding, we suggest that capillary hysteresis could have 

 been significant in Van Dorn's experiments, especially since some of his meas- 

 urements were made in Incite tanks (see following description of Keulegan's 

 experiments (6)). 



The observation that surfactants, such as oil on water, can lead to appre- 

 ciable damping of surface waves goes back to antiquity (Pliny the Elder, in the 

 first century A.D., is cited by Levich (8) and by Davies and Vose (9), and 

 Benjamin Franklin (10) attempted a phenomenological explanation in 1774). The 



574 



