PAPER BY LORD RAYLEIGH. 291 



We will cousider first the case of oue diineusion, where u, v vanish, 

 while p, w are fimctious oi 3 aud t only. From ^5) and (6), 



<lpclw_gu^^ (7) 



dt dz a- ' ^ 



., dp die . ,0, 



a- -r = — -^ (8) 



dz dt ^ ' 



(9) 



or by elimination of p, 



1 d'w d'W (J dw 



a^' dp ~ dz- a- dz 



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



d g \2 q 



dz-^')"'-ta^'^^ 



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

 with respect to c^ is not too slow. If A. be of the nature of the wave- 

 length, -— is comparable with — ; aud the simplification is justifiable 



when d^ 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 



dt^ \dz 2d\ 



or, if 



w^ We""' , . (10) 



d^W ,dny ,,-.. 



y=^ W^' ^'^^ 



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^ff must 

 retain its value, as the wav^es pass on. 



If tv vary as e*"*, the original equation (9) becomes 



dho _ g die n^w _ ^ q2) 



d^z (^ dz d^ 



Let Wi, W2 be the roots of 



2 Q , ^*^ A 



m^ — \m-\-—^ = 0, 



