PLANE WAVES OF SOUND 161 



x + f -t- &B + Sf , and its thickness is therefore changed from 8x 

 to &c -f 8f , or (1 + 9f/9#) 8#, and the dilatation is accordingly 



- ' A - *= ...................... <*> 



Hence, in the case of infinitely small disturbances, we have, 



by(3)> 



In forming the equation of motion we assume that the 

 pressure varies with the density according to some definite law. 

 We have then, for small values of s, 



p=p Q + KS, ........................ (6) 



where K is a coefficient of cubic elasticity. t Considering the 

 acceleration of momentum of unit area of a stratum originally 

 bounded by the planes x and x -f &c, we have 



where 8p represents the excess of pressure on the anterior face. 

 Hence, by (5) and (6), 



**-*** (7) 



dP'^da?' 



where c = V(*/?o) ......................... (8) 



The solution of (7) is as in 23, 43 



%=f(ct-x) + F(ct + x), ............... (9) 



and represents two systems of waves travelling in opposite 

 directions with the velocity c*. 



If we assume, as Newton -f- did, that, the expansions and 

 contractions of a gas, as a sound-wave passes, take place 

 isothermally, i.e. without variation of temperature, the relation 

 between p and p is given by Boyle's law, viz. p/p Q = p/p = 1 4- s, 

 whence K=p , as already proved. This makes 



Now for air at C. we may put, as corresponding values, 

 p = 76 x 13-60 x 981, p 9 = '00129, 



* The analytical theory of plane waves of sound is due to Euler (1747) and 

 Lagrange (1759). 



t The investigation is given in Prop. 48 of the second book of the Principia 

 (1726). 



L. 11 



