OCEAN TEMPERATURES 359 



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



of the time. 



If the flow is the periodic function of the time 



(62) 



the term 6' (equation 22, p. 344) will be the same as before, but 6" will 

 be given by equation (55) where a 4 and a 5 are retained. The integra- 

 tions can be readily performed with the aid of well known standard 

 forms and the result is 



TT 11 ( / ~Rn p-6 



JJ.-.II I / -DCt t/ 



) 



s) 1 



O 7 I * 



2afc }. si 

 J 



-2X] sin2at 



4 2a 2 a 5 ] cos 2a^ ( 63 ) 



The solution of equation (57) when a 4 and a s are retained, found 

 by Lagrange's method, is 



(64) 

 where f 1 [ ] is an arbitrary function of 



( t 4 COS at -| 5 Sin at } 77 



L\ a- a / #,J 



This solution can be verified by substitution in equation (57), and 

 can be readily changed into the more suitable form 



fc(z Z ) fc(a cos at a s sin oO r- / \ 



fl'" = e - #1 o " / ( f 4 COS a# + - 5 Sin atf ) 



L V a a / 



(65) 



