372 MISCELLANEOUS STUDIES 



W 2 



where h n 2 = 5-5 - z is small. If the velocity w t equals 0, 



a n =b n = ~\~n anc * e Q uat i n (98) reduces to the well known 



form 



n = o i . . . 

 \ I JG*.\~ (. / 1 L \ / 



4 



(103) 



A solution independent of the time results, if in equation (83) we 



W 



make & = and a= , or b = a = Q. The result is 



e = Ce^ + D = C e>-y + D ; (104) 



where C and D are arbitrary constants. This expression for can be 

 added to the right-hand member of equation (98), giving the more 

 general solution 



(105) 



Solution for the case in which the vertical flow is a periodic 

 function of the time. 



If the vertical velocity has the value w^=w 1 [1 + rcos a] the 

 general equation (80) becomes 



= 2 w [I- 1 J " 66 



Let 



(108) 



(109) 



and substitute in equation (106), the result is 



df(t] 2fl2 ^ _ 

 dt 



from which 



a 



where c is an arbitrary constant. 



