All-lOI 



86 



u =^ 



u 



:V/: 



X = sx' , y = sy' T = 



cot, 6 



W 



r ' 



FT 5 



^ = #3-^' e = ^,H = £g!a,w,„v.,r = nr. 



In this notation and with the prescribed form for t^ and t: „ , 

 equation (7) becomes 



6 -1 9 V 3JJ 

 9t I a X' ~ ay' 



Y 



.ax' ax' "^ 9^12 ^ ax' ay' ax' ay' 



- 9U au ^_a^u _ av ay y_aiy 

 ax'ay' ax'ay' ay'ay' " Qy,2 



+ y 



ay + a V 

 .ax' ay' 



+ V =eA' 



9 V _ au " 

 .ax' ay' 



- (1 + a sin T)sin nsy' 



(8) 



The integrated, non-dlmensionalized continuity equation be- 

 comes 



ay , 9j ^ X an 

 ax' ay' " ^ 



aT 



(9) 



If we expand the velocities and the height, H, in a 

 series in 6 , then the solution can be looked upon as the sum 

 of a quasi-steady part plus a number of out-of-phase contribu- 

 tions. If 6 is small enough we may be justified in keeping 

 only the first two terms of such a series as a fairly accurate 

 representation of the complete series. 

 Hence, let 

 U = Uq + 6 Ui + 6 ^U2 + ... , V = Vq + 5 V]_ + 6 ^¥3 + ... 



H = Hq + 6 PL + 6 ^Hp + . . . 



