224 DYNAMICAL THEOBY OF SOUND 



the time-factor e int being understood. Thus a train of simple- 

 harmonic waves travelling in the direction of ^-positive is 

 represented by 



When we proceed to calculations of energy it is of course 

 necessary to revert to real forms. Thus, taking the real part 

 of (6), we have 



<I> = A cos k(ct x). '. ................. (7) 



The mean energy per unit volume, as given by 70 (7), (8), 

 is ^pk*A 2 , and the mean energy transmitted per unit time, 

 per unit area of the wave-front, is 



%pk 2 cA 2 , or $pn*/c.A* ................ (8) 



We may call this the " energy-flux " in the wave-system (7). 



The equation of symmetrical spherical waves, 71 (6), now 

 takes the form 



^ + *W)-0, .............. ....(9) 



and the solution is 



r<t> = Ae~ ikr + Be ikr , .... .............. (10) 



or r<f> G cos kr + D sin AT, ............ (11) 



the time-factor being understood as before. The two terms 

 in (10) correspond to waves diverging from, or converging to, 

 the origin, respectively. In particular, the diverging waves 

 due to a source Ae ikct at the origin are represented by 



(12) 



or, in real form, <t> = ~ A oosnff -] ............ (13) 



\ c) 



This is of course a particular case of 73 (1). 



The maintenance of such a source in an unlimited medium 

 requires a certain expenditure of energy. The work done per 

 unit time at the surface of a sphere of radius r, on the fluid 

 outside, is the product of the pressure, the area, and the 

 outward velocity, or 



. (14) 



