438 Lord Rayleigh: Further Calculations 



By definition the momentum of the wave, whose length 

 may be supposed to be limited, is per unit of cross-section 



\pudx, (13) 



the integration extending over the whole length of the wave. 

 If we introduce the value of u given in (11), we get 



and the question to be examined is the sign of (14). For 

 brevity Ave may write unity in place of p , and we suppose 

 that the wave is such that its mean density is equal to that 



of the undisturbed fluid, so that §pdx-=l, where I is the 



length of the wave. If I be divided into n equal parts, then 

 when n is great enough the integral may be represented by 

 the sum 



r 7+1 y+l v+i 1 I 



*\j>l 2 +P2 2 +/>3 2 +... -/>l-p 2 -...|-, (15) 



in which all the p's are positive. Now it is a proposition m 

 Algebra that 



y+i y+i y+l 



Pi 2 +P2 2 +... > / />!+?»+.. A 2 



n \ n 1 



when i(7 + l) is negative, or positive and greater than 

 unity ; but that the reverse holds when -Hy+1) is positive 

 and less than unity. Of course the inequality becomes an 

 equality when all the n quantities are equal. In the present 

 application the sum of the p's is w, and under the adiabatic 

 law (3), y and i(y + l) are positive. Hence (15) is positive 

 or negative according as ^ (y + 1) is greater or less than 

 unity, viz., according as 7 is greater or less than unity. In 

 either case the momentum represented by (13) is positive? 

 and the conclusion is not limited to the supposition of small 

 disturbances. 



In like manner if the law of pressure be that expressed 

 in (6), we get from (12) 



from which we deduce almost exactly as before that the 

 momentum (13) is positive if/3 (being positive) is less than 1 



