544 VISCOSITY. [CHAP. XT 



When f$h is small the real part of (4) gives 



............... (8), 



the velocity being in the same phase with the force, and varying 

 inversely as v. 



301. To find the effect of viscosity on free waves on deep 

 water we may make use of the Dissipation -Function of Art. 287, 

 in any of the forms there given, the simplest for our purpose 

 being 



since, by Art. 279, the dissipation may, under a certain restriction, 

 be calculated as if the motion were irrotational. 



To put the calculation in a form which shall apply at once to 

 the case where capillary as well as gravitational forces are taken 

 into account, we recall that, corresponding to the surface-elevation 



7j = a sin k (x ct) (2), 



we have <f> = ace k v C os k (x ct) (3), 



since this makes diy/dt = d$/dy for y 0. Hence 



and the dissipation is, by (1), 



2/*& 3 c 2 a 2 (5), 



per unit area of the surface. The kinetic energy, 



-j7&& (6), 



has a mean value ^pktfa? per unit area. The total energy, being 

 double of this, is 



.............................. (7). 



Hence, equating the rate of decay of the energy to the dissipa- 

 tion, we have 



.................. (8), 



or 



