Damping of Gravity Waves by Surface Films 



first satisfactory explanations appear to have been given by Reynolds (11) and 

 Aitken (12), each of whom invoked the variations of surface tension with wave 

 motion; Aitken also carried out laboratory experiments. Lamb provided the 

 necessary mathematical analysis in support of the Reynolds -Aitken model in the 

 second (1895) edition of his "hydrodynamics," but abbreviated the analysis to that 

 for an inextensible film in the sixth edition (4b). Subsequent analyses, dealing 

 with the physicochemical as well as the hydrodynamical problem, have been 

 given by Levich (8,13), Dorrestein (14), Goodrich (15), and Tempel and Riet (16). 



Levich considered both insoluble and soluble surfactants, but appears to 

 have assumed (as, implicitly, did Lamb) that damping would increase monotoni- 

 cally with concentration to the limiting value a^i). This tacit assumption led 

 him to overlook the fact that his general formulation for an insoluble film yields 

 the maximum value a^ = 2ag ■' at an intermediate concentration, a result estab- 

 lished independently by Dorrestein. Levich considered various limiting cases 

 of his general formulation, but his approximations do not appear to be entirely 

 systematic and, in any event, do not give a clear picture of the interrelations 

 among such parameters as frequency, surface modulus of compression, and 

 solubility. Dorrestein considered only insoluble surfactants, but included sur- 

 face viscosity in his analysis (Levich included surface diffusion of the surfac- 

 tant, which is essentially similar to surface viscosity in its effect on damping). 

 Tempel and Riet attributed differences among the results of Levich, Dorrestein, 

 and Goodrich to "the use of an incorrect boundary condition for the tangential 

 stress." In fact, Tempel and Riet's formulation appears to agree with those of 

 Levich and Dorrestein when appropriate comparisons are made; detailed com- 

 parisons of their analytical results are not possible, as Tempel and Riet used 

 an electronic computer to solve the algebraic equations implied by their bound- 

 ary conditions. 



None of the formulations of Levich, Dorrestein, Goodrich, and Tempel and 

 Riet is directly applicable to gravity waves in closed basins. In the following 

 three sections we present a derivation of ag for either soluble or insoluble 

 films that, by virtue of its direct appeal to boundary -layer approximations, is 

 not only simpler than its antecedents, but also is applicable to closed basins. 



Davies and Vose (9) have made careful measurements of the damping of 

 150-cps capillary waves by various types of surface films. Their experiments 

 were carried out in a "draught-free darkroom"; their apparatus was "cleaned 

 with hot chromic acid followed by washings with syrupy phosphoric acid, hot 

 tap water and distilled water" before each run. In their experiments on clean 

 water, "the surface was cleaned prior to the observations by spreading ignited 

 talc on the surface and then sucking off the talc and any contamination through 

 a fine capillary," a procedure that was repeated between observations. They 

 monitored surface tension continuously. They found that a did approximate a^°^ 

 for clean water and also confirmed Dorrestein' s prediction that a increases 

 from a^°^ to approximately 2a^^' and then decreases to the limiting value a^^^ 

 with increasing concentration of an insoluble surfactant. They also obtained 

 qualitative confirmation of Dorrestein' s calculations for the effect of surface 

 viscosity. They found that ag was typically smaller for soluble surfactants pro- 

 vided that the relaxation frequency for solution was above that of the capillary 

 waves; however, their results were not in complete agreement with the 



575 



