206 WAVES IN AIR. [CHAP. vm. 



General Equations of Sound Waves. 



169. We proceed to the general case of propagation of .air 

 waves. We neglect, as before, the squares of small quantities, so 

 that the dynamical equation is 



Also, writing p = p Q (l + s) in the general equation of continuity, 

 (8) of Art. 8, we have, to the same order of approximation 



ds d&quot;(j) d c&amp;gt; - d d&amp;gt; _ . /on\ 



dt dx z dij* dz* 



The elimination of 5 between (10) and (22) gives 



d&quot; 

 or, with our former notation, 



.=cW&amp;gt; (23). 



170. Let us suppose a sphere of radius r described about any 

 point (a?, y, z) as centre. Multiplying both sides of (23) bydxdyds, 

 and integrating throughout the volume of the sphere, we find 



-r^ 1 1 1 (/&amp;gt; dxdydz = c 2 1 II V 2&amp;lt; ^ dxdydz = c 2 1 1 -^ dS, 



or, writing dS = r 2 d^, so that d-& denotes an elementary solid 

 angle, 



Let us differentiate both sides of this equation with respect to r. 

 The left-hand side gives 



a?//*** 



so that if we write 



&amp;lt; = ^ 1 1 (/&amp;gt;ckr = mean value of &amp;lt; over the sphere, 



