176 

 or, if 



Lord Rayleigh on the 



iv = Wei^ a % (10) 



dt 2 



d 2 W 

 dz 2 



(11) 



= 0. 



(12) 



the ordinary equation of sound in a uniform medium. Waves 

 of the kind contemplated are therefore propagated without 

 change of type except for the effect of the exponential factor 

 in (10), indicating the increase of motion as the waves pass 

 upwards. This increase is necessary in order that the same 

 amount of energy may be conveyed in spite of the growing 

 attenuation of the medium. In fact w 2 cr must retain its value, 

 as the waves pass on. 



If w vary as e int , the original equation (9) becomes 



d 2 w g dw 

 dz 2 a 2 dz 



Let m l} m 2 be the roots of 



m 2 — —„ m-\- -s =0. 

 cr a 1 



so that 



£±vV-W) 

 m ~ 2a 2 ..... (Id) 



then the solution of (12) is 



w=Ae m > z + Be m * z , (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 



m — 



g + 2nai 

 2d 2 



and 



(15) 



A wave propagated upwards is thus 



w = e ig*la* cos n (t—z/a), .... (16) 



and there is nothing of the nature of reflexion from the upper 

 atmosphere. 



A stationary wave would be of type 



TiZ 



iv = e* gz/a2 cos nt sin—, 



w being supposed to vanish with 



z. 



■ ■ • (17) 



According to (17), the 



