494 WAVES OF EXPANSION. [CHAP. X 



For plane waves travelling horizontally, the equation (4) takes the form 



The waves are therefore propagated unchanged with velocity c, as we should 

 expect, since on the present hypothesis of uniform equilibrium-temperature 

 the wave-velocity is independent of the altitude*. 



273. We may next consider the large-scale oscillations of an 

 atmosphere of uniform temperature covering a globe at rest. 



If we introduce angular coordinates 0, w as in Art. 190, and 

 denote by u t v the velocities along and perpendicular to the 

 meridian, the equations (4) of Art. 271 give 



du c 2 d , dv c 2 d , 



s 5fc-ft s--iiE&quot;=&amp;lt;*-i&amp;gt; (1)&amp;gt; 



where a is the radius. If we assume that the vertical motion (w) 

 is zero, the equation of continuity, Art. 271 (6), becomes 



ds 1 (d(usm6) dv 



dt a sin 6 ( dd o 



The equations (1) and (2) shew that u, v, s may be regarded as 

 independent of the altitude. The formulae are in fact the same as in 

 Art. 190, except that s takes the place of %/h, and c 2 of gh. Since, 

 in our present notation, we have c 2 = #H, it appears that the free 

 and the forced oscillations will follow exactly the same laws as 

 those of a liquid of uniform depth H covering the same globe. 



Thus for the free oscillations we shall have 



Q n ( 4- \ \ ( *\\ 



where S n is a surface-harmonic of integral order n, and 



(4). 



As a numerical example, putting c=2 80 x 10 4 , 2vra = 4 x 10 9 [c. s.], 

 we find, in the cases n l,n = 2, periods of 28*1 and 16 2 hours, 

 respectively. 



* The substance of this Art. is from a paper by Lord Kayleigh, u On Vibrations 

 of an Atmosphere,&quot; Phil. Mag., Feb. 1890. For a discussion of the effects of 

 upward variation of temperature on propagation of sound-waves, see the same 

 author s Theory of Sound, Art. 288. 



