300-302] EFFECT OF VISCOSITY ON WATER-WAVES. 545 



whence a = a. Q e~^ m ........................ (10). 



The modulus of decay/ r, is therefore given by r= I/2v&, or, 



in terms of the wave-length (X), 



T = X 2 /87T 2 Z/ (11)*. 



In the case of water, this gives 



r = -71 2 X 2 seconds, 



if X be expressed in centimetres. It follows that capillary waves 

 are very rapidly extinguished by viscosity; whilst for a wave 

 length of one metre r would be about 2 hours. 



The above method rests on the assumption that or is moderately large, 

 where o-( = kc] denotes the speed. In mobile fluids such as water this 

 condition is fulfilled for all but excessively minute wave-lengths. 



The method referred to fails for another reason when the depth is less than 

 (say) half the wave-length. Owing to the practically infinite resistance to 

 slipping at the bottom, the dissipation can no longer be calculated as if the 

 motion were irrotational. 



302. The direct calculation of the effect of viscosity on 

 water waves can be conducted as follows. 



If the axis of y be drawn vertically upwards, and if we assume 

 that the motion is confined to the two dimensions x, y, we have 



du 1 dp 



dt p dx 



dv _ I dp 



dt p dy 



du dv 



.,, 



Wlth 



These are satisfied by 

 dd&amp;gt; 



- 



~r-&amp;gt; v = -~^- + -j ............ (3), 



dy dy dx 



=-&amp;lt; ........................ w, 



* Stokes, 1. c. ante p. 518. (Through an oversight in the calculation the value 

 obtained for T was too small by one-half.) 



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