256-258] ENERGY OF SOUND WAVES. 469 



258. The kinetic energy of a system of plane waves is given 



by 



T-faff&tadydM (1), 



where u is the velocity at the point (x, y, z] at time t. 



The calculation of the intrinsic energy requires a little care. 

 If v be the volume of unit mass, the work which this gives out in 

 expanding from its actual volume to the normal volume v is 



I pdv 



J V 



(2). 



Putting v = Vu/(l+s), p=po+KS, we find, for the intrinsic energy 

 (E) of unit mass 



^=boS + (i*-po)s 2 K (3), 



if we neglect terms of higher order. Hence, for the intrinsic 

 energy of the fluid which in the disturbed condition occupies any 

 given region, we have the expression 



W=jtfEpdxdydz = p 9 fffE (1 + s) dxdydz 



= S!5(p Q s + %KS&amp;gt;}dxdydz (4), 



since p v = I. If we consider a region so great that the con 

 densations and rarefactions balance, we have 



fjfsdxdydz = (5), 



and therefore W \ KJjjs^dxdydz (6). 



In a progressive plane wave we have cs = u, and therefore 

 T = W. The equality of the two kinds of energy, in this case, may 

 also be inferred from the more general line of argument given in 

 Art. 171. 



In the theory of Sound special interest attaches, of course, to 

 the case of simple-harmonic vibrations. If a be the amplitude 

 of a progressive wave of period 27r/cr, we may assume, in con 

 formity with Art. 257 (6), 



f = a cos (kx at + e) (7), 



where k = &amp;lt;r/c. The formulae (1) and (6) then give, for the energy 

 contained in a prismatic space of sectional area unity and length 

 \ (in the direction #), 



T+ W = ^p&amp;lt;rW\ (8), 



