304 THE MECHANICS OF THE EARTH'S ATMOSPHERE. 



The notation and solution are as follows : 



d 2 F_ ( dF 

 dz 2 dz 



hF=( a n 2 - a 2 \ A 



h= ( *n 2 — ni 2 

 L g J 



F (z) ^B+K^+K^* 



A/IX,^ 2 



B=^ ( -n 2 — a 

 h\9 



^-~ a — A* 2 7, fr>— a 4- l a 2 



// 



In order to determine the constants of integration Ki and K 2 whose 

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

 ic=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 = - g — (K x ~k x e kiZ + K 2 lc 2 e k2Z — aA) cos (mx+nt) 

 an 



The bound ar y conditions give 



Rik 1 -\-K 2 h=cxA 



K l k i e klZ +K 1 k 2 e k * z =aA 



KJn = aA 

 K 2 h=(vA 



e k "- z — I 

 \-e^ 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 2 , then is 1c very small, aud Tc 2 nearly equal to a. Hence, K 2 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 K 2 e k *. With this limitation we put K 2 =0 and K 1 fc 1 =aA, and 

 obtain 



w=Al(e klZ — 1) cos (mx+nt) 



