304 THE MECHANICS OF THE EARTH's ATMOSPHERE. 



The notation and solution are as follows : 



(FF AF 

 d. 



F dF , J T^ /a , , \ . 



z' dz \ (J J 



L g J 



In order to determine the constants of integration Ki and K?, whose 

 factors in the expression for € represent free vibrations we note that 

 rr=0 when z=0 and also when z has a very large value = Z which cor- 

 responds to a fictitious upper plane bounding the atmosphere. From 

 the second of equations (3) we obtain 



w = -^ (Eihe^''' +K-2he'''' —aA) cos {mx-\-Ht) 

 an 



The boundary conditions give 



Eil'i-{-K.]{2 = aA 



K,kre'''^+K2he'''^=aA 

 Kxki = a A —^ j^r, 



E2ky = CxA-r 



1-e 



.k,Z 



^k-iZ— ^k,Z 



If now, as in our example (where the wave length is the circumfer- 

 ence of the earth and the period is one day), h is very small compared 

 with a^, then is h very small, and A-j nearly equal to a. Hence, Eo will 

 be smaller in proportion as Z is larger. If we desire to apply the re- 

 sulting formula only to altitudes that are slight in comparison with Z, 

 then will E-ze^-- . With this limitation we put 112=0 and Eiki=aA, and 

 obtain 



w=A^(e'"^—l) cos (mx-i-7it) 

 n 



e=Af^t_'^A."e''^') sin (mx-\-nt) 

 V gh h ^ A-i y 



