303-304] CALMING EFFECT OF OIL ON WAVES. 553 



drop of oil thrown on water is gradually drawn out into a thin film. 

 If the film be sufficiently thin, say not more than two millionths of 

 a millimetre in thickness, the tension is increased when the thick 

 ness is reduced by stretching, and conversely. It is evident at 

 once from the figure on p. 374 that in oscillatory waves any 

 portion of the surface is alternately contracted and extended, 

 according as it is above or below the mean level. The consequent 

 variations in tension produce an alternating tangential drag on the 

 water, with a consequent increase in the rate of dissipation of 

 energy. 



The preceding formulae enable us to submit this explanation, to a certain 

 extent, to the test of calculation. 



Assuming that the surface tension varies by an amount proportional to 

 the extension, we may denote it by 



where T is the tension in the undisturbed state, is the horizontal displace 

 ment of a surface particle, and / is a numerical coefficient. 



The internal motion of the water is given by the same formulae as in 

 Art. 302. The surface-conditions are obtained by resolving normally and 

 tangentially the forces acting on an element of the superficial film. We thus 

 find, in the case of free waves, 



^__ 

 ~ J 



p dx~ da 



where T =T l /p. In the derivation of the first of these equations a term of 

 the second order has been neglected. 



Since the time-factor is e at , we have ^=u/a, whence, substituting from 

 Art. 302 (8), (9), (11), we find, as the expression of the surface-conditions (ii), 



(a 2 + 2^ 2 a +gk+ T k*} A - i (gk + T & + 2i//bw a ) (7= 0, 



i(2 v k*a+fT f t?) A + (a z + 2vk*a+fT k*m) (7=0 

 If we write 



the elimination of the ratio A : C between the above equations gives 



+ &amp;lt;r 2 -4,/ 2 X +fj o- 2 ja 2 + (l --} o4 = ...... (v). 



