20 



WAVE ACOUSTICS 



Equation (47a) can also be shown to hold good for 

 the wave moving in the positive direction if the 

 initial conditions are for the space interval < x < Xo 

 at the time t = O.li the wave is moving in the nega- 

 tive X direction, it can easily be shown by an argu- 

 ment similar to the above that the pressure and par- 

 ticle velocity are related by 



c 

 — P 



K 



.JL. 



PqC 



(47b) 



For the particular case of a plane harmonic wave, 

 we have from equations (47a) and (47b) 



a 



u = + — cos 27r/(f — e). 

 PoC 



The following interesting result stems directly from 

 equations (47). If the particles in a plane wave are 

 moving in the direction of wave propagation, they 

 are in a region of positive excess pressure; if they are 

 moving in a direction opposite to the route of the 

 wave, they are in a region of negative excess pres- 

 sure; and if the particles are not moving, they are in 

 a region of zero excess pressure. Also, we can argue 

 from equations (47) that if the initial conditions ful- 

 fill neither 



nor 



u{x,0) = — pix,0) 

 PoC 



u(x,0) = p(xfi)> 



PoC 



then waves are propagated in both a positive and a 

 negative direction from the initial source of pressure 

 disturbance. 



Spherical Waves 



At great distances from the source a small section 

 of a spherical wave approximates a plane wave. For 

 this reason, many of the foregoing results can be re- 

 written in a form valid for spherical disturbances far 

 from the source. Since the mathematical proofs, 

 though straightforward, are rather cumbersome they 

 will not be reproduced here. 



For a general spherical wave far from the source, 

 the following relation exists between the excess pres- 

 sure and particle velocity : 



c 1 



U = ±-p = + p; 



K PoC 



in analogy with equations (47). For a spherical har- 

 monic wave it will be remembered that the maximum 

 pressure change at the distance r from the source is 



given by a/r. Thus, for the case of a spherical har- 

 monic wave, 



la 



Wmax = (48) 



Pocr 



The relations (47) are not necessarily true for the 

 general solution of the wave equation (27). 



2.4.2 Acoustic Energy and Sound 

 Intensity 



The vibration of the particles of a fluid disturbed 

 by wave propagation is a process which involves both 

 kinetic and potential energy. The energy of vibration 

 of the sound source is propagated through the fluid 

 along with the sound wave. In a perfect fluid where 

 frictional heat losses are zero, the energy content of 

 the wave is unchanged as the wave travels. The en- 

 ergy passes from one region to another, "activating" 

 the region through which the wave is passing. Thus, 

 there are two quantities of interest. One is the energy 

 found at any location as a function of time; the other 

 is the rate at which energy is transported from one 

 region to another as a function of time. In the follow- 

 ing sections, both of these quantities are expressed in 

 terms of the wave parameters we have introduced. 



The kinetic energy possessed by a volume element 

 V, whose volume was Vo at equilibrium, and whose 

 speed is u, is given by 



Kinetic energy oi v = ^poVoU^. (49) 



The potential energy possessed by the volume ele- 

 ment V is the work which was done on it to change its 

 volume from vo to v. This work can be calculated as 

 follows: By equation (9), the relative change in 

 volume produced by an infinitesimal alteration of 

 condensation from a to a + dais just —da. The total 

 volume change caused by this infinitesimal alteration 

 of 0- is, to a first approximation, —Vgdcr. The work 

 done during this infinitesimal alteration is merely the 

 pressure times the small volume change, that is, 

 pvoda, which, because of equation (18), equals 

 KdVoda. The total amount of work done on the volume 

 element as its volume changes from Vo to v can be 

 obtained by integrating this infinitesimal amount of 

 work between a condensation of zero and condensa- 

 tion of a. 



Potential energy of t; = Vok j 

 ~ 2(c 



iwoKc^ 



(50) 



