548 VISCOSITY. [CHAP, xi 



If we put a = + kAjcr, the equation (9) of the free surface 

 becomes, approximately, on taking the real part, 



< n = ae-* m $m(kxat) .................. (21). 



The wave-velocity is o-jk, or (g/k + Tk)*, as in Art. 246, and 

 the law of decay is that investigated independently in the last Art. 



To examine more closely the character of the motion, as affected by 

 viscosity, we may calculate the angular velocity (o>) at any point of the fluid. 

 This is given by 



dv du 



2a>=-= --- =- 



dx dy 



Now, from (7) and (18), we have, approximately, 



ra = (l^)/3, where /3 = (o-/2i/)*. 

 With the same notation as before, we find 



>=+<rkae- Zvk '' t+ Msm{kx(<rt+py)} ................. (ii). 



This diminishes rapidly from the surface downwards, in accordance with 

 the analogy pointed out in Art. 286. Owing to the oscillatory character of 

 the motion, the sign of the vortex-motion which is being diffused inwards 

 from the surface is continually being reversed, so that beyond a stratum of 

 thickness comparable with 2?r//3 the effect is insensible, just as the fluctuations 

 of temperature at the earth's surface cease to have any influence at a depth 

 of a few yards. 



In the case of a very viscous fluid, such as treacle or pitch, 6 

 may be large even when the wave-length is considerable. The 

 admissible roots of (16) are then both real. One of them is 

 evidently nearly equal to 20, and continuing the approximation 

 we find 



whence, neglecting capillarity, we have, by (15), 



a = -g/2kv ........................ (22). 



The remaining real root is T090, nearly, which gives 



(23). 



The former root is the more important. It represents a slow 

 creeping of the fluid towards a state of equilibrium with a horizontal 

 surface ; the rate of recovery depending on the relation between 



