542 VISCOSITY. [CHAP, xi 



addition of fixed parallel disks at a short distance above and below greatly 

 increases the effect of viscosity. 



The free modes of motion are expressed by (iii), with the conditions that 

 u = for y=Q and y = h. This gives .4=0 and mh=8Tr, where s is integral. 

 The corresponding moduli of decay are then given by r=l/vm 2 . 



300. As a further example, let us take the case of a force 

 X=fcos(at + ) ..................... (1), 



acting uniformly on an infinite mass of water of uniform depth h. 

 The equation (1) of Art. 298 is now replaced by 



du d*u 



-T: = *> -j-; -+X ........................ ^2). 



dt d* 



If the origin be taken in the bottom, the boundary -conditions 

 are u = for y = 0, and dujdy = for y = h ; this latter condition 

 expressing the absence of tangential force on the free surface. 

 Replacing (1) by 



we find 



if = (<r/2i/)*, as before. 



When fth is large, the expression in { } reduces practically to 

 its first term for all points of the fluid whose height above the 

 bottom exceeds a moderate multiple of ft~ l . Hence, taking the 

 real part, 



u = sin (<rt -f e) ........................ (5). 



(7 



This shews that the bulk of the fluid, with the exception of a 

 stratum at the bottom, oscillates exactly like a free particle, the 

 effect of viscosity being insensible. For points near the bottom 

 the formula (4) becomes 



(7 



or, on rejecting the imaginary part, 



u =- sin(<7* + e) --er*san (at - 

 cr (7 



(6), 



