22 



WAVE ACOUSTICS 



We can calculate F by means of equation (58). 

 Since a sphere of radius r has the area 4irr^, the total 

 energy crossing such a spherical surface per unit of 

 time is merely the intensity times this area : 



2ira? 

 Rate at which energy crosses sphere = (60) 



Because of the assumption of conservation of sound 

 energy, equation (60) must be equal to the amount of 

 energy radiated by the source per unit of time. 

 Dividing equation (60) by 47r gives 



F = 



2poC 



(61) 



General Sound Waves 



We now examine the transport of acoustic energy 

 for the case of a general solution of the wave equa- 

 tion (27). 



In the general case, it is useful to start with an 

 equation of continuity for energy flow analogous to 

 the exact equation of continuity (2) for mass flow. 

 It will be recalled that equation (2) followed directly 

 from the law of conservation of mass. The law of con- 

 tinuity for energy flow will follow from the law of con- 

 servation of energy in exactly the same fashion. For 

 the mass density p, the energy density which may be 

 denoted by Z is substituted. Also, for the instanta- 

 neous flow of matter with components u,:,Uy,Uz, the 

 instantaneous flow of energy is substituted. The 

 components of the instantaneous energy flow past 

 normal unit area may be denoted by Ei,Ey,E^. The 

 equation of the continuity for energy flow becomes, 

 in analogy with equation (2) , 



dZ 

 dt 



L dx 



+ 





dE, 

 dz. 



(62) 



Equation (62) is the mathematical expression of the 

 assertion that the energy flow through a closed sur- 

 face is equal to the decrease of energy inside this sur- 

 face. A rather complicated argument must be used to 

 calculate the components of energy flow E^,Ey,E^. 

 Equations (21) are rewritten by using p = kc, as 



(63) 



Also, from equations (4) and (18), we have the rela- 

 tion 



1 dp VdUx dUy du, 



K dt L 5a: dy dz 



]• 



(64) 



Multiplying the first equation of (63) by Ux, the 

 second by Uy, the third by m^, and equation (64) by 

 p, and adding them all up, we obtain 



-I 



dtl 



l^(U^ + Uy + Uj + -J 



[3(pux) djpuy) d{pu, 



n- 



(65) 



L dx ' dy ' dz 



Because of equation (52), we see that the left-hand 

 sides of equations (65) and (62) are equal. Hence the 

 right-hand sides are also equal, and we must have 



Ex = pux] Ey = puy] E, = pu^. (66) 



The instantaneous energy flow E is the resultant of 

 its three components Ex,Ey,Ez and is numerically 

 equal to V^^ + El + El. Thus, we have the general 

 result that 



E = pu. (67) 



According to equation (66), this energy flow is always 

 along the direction of the particle velocity. 



The intensity /, which was defined as the time 

 average of E, is therefore always given by the fol- 

 lowing formula : 



/ = ^. (68) 



2.4.3 Complex Representation of 

 Harmonic Vibrations 



The complex number e'"° is defined by the equation 



e™ = cos w + i sin w 



where v' = —1. The one-dimensional harmonic vi- 

 bration 



d = a cos 27r/i 



can therefore be regarded as the real part of ae*^*^^'. 

 Similarly, the vibration 



d = acos 2vf(t - e) (69) 



can be rewritten as 



d = real part of ae'''^"-'^ 



The latter relation can be expressed in the following 

 less cumbersome form 



D 



ae 



,t27r/((-e) 



(70) 



if the conventions are adopted that the actual phys- 

 ical displacement is the real part of the complex dis- 



