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 = and y = h. This gives .4=0 and mh=STT^ 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 + e) ..................... (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 



-J1 V T~o ~r ^ ........................ (^ ) 



at dy 2 



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 



X = f e irt+ e} ............................ (3), 



-, ifL coah(l+i)P(h-y)} .... , 



we find u = -^\l ---- \ ,., \ -\ 01 \ e } 



a- ( cosh(l +l)ph J 



if = (V/2j/)*, as before. 



When ph 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 /3~ l . Hence, taking the 



real part, 



f 

 u = J ~ sin (at + 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 



(6), 



or, on rejecting the imaginary part, 



u = i- sin (&amp;lt;rt + e) - e~^ sin (at - fiy + e) (7). 



(7 (T 



