PAPER BY LORD RAYLEIGH. 291 



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

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



dp dw_gw_ 



dt + dz a 2 ~^ (i) 



or by elimination of p, 



a 2 ' dt 2 dz 2 , a 2 dz 

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



dp_ dw. 



a dz~ St' (8) 



1 d 2 w d 2 w q dw ,_, 



(9) 







\dz~2a z J 4a 4 w ' 



and in this the latter term may be neglected when the variation of w 

 with respect to z is not too slow. If X be of the nature of the wave- 



(llll 74) 



length, — is comparable with — ; and the simplification is justifiable 



when a 2 is large in comparison with gX, that is when the velocity of 

 sound is great in comparison with that of gravity- waves (as upon water) 

 of wave length A. The equation then becomes 



d2w - a *( d - ^Yw 

 S#- a \Jz 2tf) W ' 



or, if 



w --= We* , (10) 



d 2 W_ 2 d 2 W. (U) 



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 <r must 

 retain its value, as the waves pass on. 



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



d?jv_][dw nhc^Q ( 12 ) 



d 2 z a 2 dz a 2 



Let Mi, m 2 be the roots of 



m 2 — ^ m + -, j = 0, 

 a 2 a 2 



