Vibrations of an Atmosphere. 175 



In considering the small vibrations, the component velo- 

 cities at any point are denoted by u, v, it', the original density 

 <j becomes (cr + ap), and the increment of pressure is Sp. On 

 neglecting the squares of small quantities the equation of 

 continuity is 



dp du dv , dw da ~ 

 dt dx dy dz dz 



or by (3), 



The dynamical equations are 



k + p + p + *Z-g»=0. ... (5) 



dt dx ay dz a* 



d8p du 

 dx dt' 



dSp dv 



-~ = —<T — . 



dy dt' 



dSp dw 

 _ gap-a-f 



or by (3), since 



8p = a 2 ap, 





2 dp du 

 dx dt ' 



dp _ dv 

 dy~ dt' 



2 dp dw 

 dz~ dt' 



(6) 



We will consider first the case of one dimension, where u, v 

 vanish, while p, w are functions of z and t only. From (5) 

 and (6), 



dp dw gw _ ( m 



Tt + dz"~~tf-"> () 



2 dp dw , Q . 



a I = -lTt' (8) 



or by elimination of p, 



1 d 2 w _ dhv g dw , Q . 



a 2 ~d? ~dz J ~a 2 dz~ [ > 



The right-hand member of (9) may be written 

 / d a \ 2 q 2 



and in this the latter term may be neglected when the varia- 

 tion of w with respect to z is not too slow. If \ be of the 

 nature of the wave-length, dw/dz is comparable with w/X ; 

 and the simplification is justifiable when a 2 is large in com- 

 parison with g\, that is when the velocity of sound is great in 

 comparison with that of gravity-waves (as upon water) of 

 wave-length X. The equation then becomes 



d 2 w „ / d q\ 2 



de= a {dz-ts) w; 



02 



