332 On Plane and Spherical Sound- Waves, 



actual motion may be written 



whence by subtraction 



^ = _^_ W _||t_BW_ . (24) 



Again, 



expanding this and substituting in (24) we get 



*vW V 3*' d?' 'dr 2 r 2 r )] r ' ^ 0) 

 Now differentiate (22) with respect to t and we have, remem- 

 bering (23), 



v o4>ojr _ I dp "d$p 

 ^ <y or " pdp ' ~dt ' 



In this equation we have to substitute the value of ~d$p/ot 

 from (25), and if we then put j^=^rr, and perform the 

 necessary reductions, we finally obtain as the differential 

 equation satisfied by yjrr, 



where 



a 2 =the variable -j- (in the steady motion) , 



~d<b constant 



UEEL ^~= 2 — 5 



dr /or* ' 



TT lB(wp) 2?i^a 2u 2u "da 



p d?' a "or~~ r a or' 



or\aJ p or 



