370 MISCELLANEOUS STUDIES 



and, if the solution is to be a periodic function of the time having 



the period where 04 is positive 

 i 



I ^(a l 2 b l z ) w l a 1 = (86) 



(2a 1 ^ w 1 )b 1 = a 1 (87) 



Solving equation (86) for a l and equation (87) for b t we have 



/! ;+: v Wl " ^7* "\ /QQN 



i= jri 



and 



7. - a l 



Since the temperature and the amplitude of the temperature 

 decrease as the depth increases, the exponent a^y and hence a l must 

 be negative (y is positive in the direction from the upper surface 

 downward). Therefore only the negative sign is admissible before the 

 radical in equation (88) and 



is definitely determined by given values of u\, ft 2 and b l . Solving 

 equation (87) for a a gives 



therefore because of equation (90) 01L ai must be negative, or & t 



- 



must be negative since a x is assumed to be positive. From equations 

 (90) and (91) we have 



1 =6 1 X+ VV (92) 



and 



where only the plus sign is admissible since &^ is necessarily positive. 

 Substituting this value of fe^ in equation (90) gives 



a _ i 

 V 



