190 Sir W. Thomson on the Rectilineal Motion of 



and, for brevity, p now denotes, instead of as before the pres- 

 sure, the pressure 4-^cosIy. 



We still suppose v to be a function of y and t determined 

 by (4) and (6). Thus (1) and (7), (8), (9) are four equa- 

 tions which, with proper initial and boundary conditions, 

 determine the four unknown quantities u, v, w, p ; in terms 

 of a-, y, z, t. 



29. It is convenient to eliminate u and w ; by taking 



-j-j -j-i -j- of (7), (8), (9), and adding. Thus we find, in 

 *rtJof(l), 2 (|_^J = _ VV .... (1]) . 



This and (8) are two equations for the determination of v 

 and p. Eliminating p between them, we find 



^-\df- C )Tx + ^-l<b*-f)]-^=^v . . (12), 



a single equation which, with proper initial and boundary 

 conditions, determines the one unknown, v. When v is thus 

 found, (8), (7), (9) determine p, u, andw. 



30. An interesting and practically important case is pre- 

 sented by supposing one or both of the bounding planes to be 

 kept oscillating in its own plane ; that is, F and § °f (6) to 

 be periodic functions of t. For example, take 



F = acosotf, $ = (13) 



The corresponding periodic solution of (4) is 



v=a w =- ->v^ eos r~ y v» ■ • (l4) - 



e 2 M — e 2fi. 



In connexion with this case there is no particular interest 

 in supposing a current to be maintained by gravity; and we 

 shall therefore take c=0, which reduces (7), (8), (9), (11), 

 (12), to d u fa d v fa 



cti +V dx + dy V = ^ U -£ ■ ■ ■ W> 



dv dv 2 dp na , 



dt +V dlc =^ V ~dy • • • < 16 >> 



dw dio . dp ,.,_. 



_ +v _ =^V%-| • • . (17), 



~dv dv ,_- '_. 



2 Tyd-c =-^P < 18 >> 



dV*v d*vdv dV*v 



-dT-dfT x +v ^r=^ v < 19 > ; 



in all of which v is the function of (y, t) expressed by (14). 



