PLANE WAVES OF SOUND 185 



In the case of a pure shearing motion (17), the formula (6) 

 takes the shape 



w^ = /*V ......................... (8) 



In plane waves of sound we have e 2 = 0, e 3 = 0, and therefore 

 from (1) and (4) 



p l ^-p-\-^fJi l = -p -KS + ^fJL / l .......... (9) 



Moreover, in the notation of 59, 



2- *- ............... <'> 



The equation of motion, viz. 



o^- 9 ^ (11) 



*!*- ..................... l " J 



therefore becomes 



* 



To obtain a solution appropriate to the case of free waves 

 we put 



f=Pcosfcp, ..................... (13) 



where P is a function of t, to be determined. We find that 

 (12) will be satisfied, provided 



<"> 



This has the form of 11 (3), and the solution is therefore 



P = ta-<<' T cos(7tf-He), ............... (15) 



provided r = 3/2i/& 2 , n? = & 2 c 2 - 1/r 2 ............. (16) 



In all cases of interest cr is a considerable multiple of the 

 wave-length (X = 2?r/A?), so that n = kc, practically, the friction 

 having as usual no appreciable effect on the period. Thus 



f = Ce~ tir cos (kct + e) . cos kx .......... (17) 



This represents a system of standing waves with fixed nodes 

 and loops. There is a similar solution in which cos kx is 

 replaced by sin&#, and by superposition of the two we can 

 construct a progressive wave-system 



% = Ce-*l r wak(ct-x) ................ (18) 



Putting i/ = -132 for the case of air, we find T = '288X 2 , the 

 units being the second and the centimetre. 



