292 THE MECHANICS OF THE EARTH'S ATMOSPHERE. 



so that • 



w== ^^72 '-> ( 13 ) 



then the solution of (12) is 



w = Ae m i z + Be™** , (14) 



A and B denoting arbitrary constants in which the factor e int may be 

 supposed to be included. 



The case already considered corresponds to the neglect of g 2 in the 

 radical of (13), so that 



_#±2 nai 



and 



W 



'"■- 2a> 



111 



in ( t+ ^j, |L* l ( | -i) (15) 



we a '• -Ae x a '+Be 

 A wave propagated upwards is thus 



w=e coswf t ) . . . . (16) 



and there is nothing of the nature of reflection from the upper atmos- 

 phere. 

 A stationery wave would be of type 



w=e cos w£ sin — (17) 



a 



w being supposed to vanish with z. According to (17), the energy of 

 vibration is the same in every wave length, uot diminishing with ele- 

 vation. The viscosity of the rarefied air in the upper regions would 

 suffice to put a stop to such a motion, which can not therefore be taken 

 to represent anything that could actually happen. 



When 2 na<.g, the values of m from (13) are real, and are both posi- 

 tive. We will suppose that m, is greater than m 2 . If w vanish with z, 

 we luwe from (14) as the expression of the stationary vibration 



e —e Ji (18) 



which shows that w is of one sign throughout. Again by (8) 



a 2 p=n sin nt \ e _ 1_ > (10) 



( »»i w? 2 ) 



Hence J*, proportional to w, is of one sign throughout; p itself is 



negative for small values of 2, and positive for large values, vanishing 

 once when 



e <-»-*' = «i (20 ) 



m 2 



