1II-2 



Equations III-l through III-3 govern the wave motion. A little manipulation of 

 these shows that p, p and u all satisfy the ordinary wave equation. For ex- 

 ample, we may obtain the wave equation for p by taking the partial derivative 

 of (III-l) with respect to t, and then substituting for p from (III-3) and for 

 Po ua ^ from (III-2). This yields: 



;:^ p - p = (III-4) 



Co .tt ,JJ 



We shall usually start with (III- 4) as the basic equation of motion, and use (III- 2) 

 and (III- 3) to derive u and p from p. 



The time dependence of the acoustic variables will almost always be 

 harmonic. We shall therefore introduce state variables which are functions of 

 position X only, according to: 



p (x,t) = p (x) e'''^^ p (x,t) = p (x) e"''*'^ u (x,t) = u (x)"''^^ (III-5) 



It should be noted that the physical values of p, p and u are given by the real 

 parts of (III-5). The resulting space -dependent state variables satisfy the anal- 

 ogous equations of motion: 



-iujp + Po Ujj =0 (III- la) 



-iuupo Uj + p j = (III-2a) 



p = c|p (III-3a) 



— T- p + p =k^p + p . = (where k = — = wave number) (III-4a) 

 Co 'JJ 'JJ ^° 



A word should be said about the flow of energy associated with the wave 



iVO] 



ordinate plane* in a time interval dt is**: 



motion of p, p and u. The amount of work done across unit area of the j*" co 



6W. 

 J 



Re p(x,t) 



Re u.(x. 



dt 



*The coordinate plane with normal in the x; direction. 

 **Re preceding a complex number means "Real Part of; Im is "Imaginary Part of. 



:arthur ai.HittkJnir. 



S-7001-0307 



