Ill -3 



In the case of harmonic time dependence, e~^ '*''-, we may compute the rate of 

 energy flow (i.e. , work done) across unit surface of a coordinate plane by aver- 

 aging the above over one cycle in time: 



2TT/UJ 



h 2n 



dt Re p Re u. 







If we carry out the integration over time, using (III-5), and express u in terms 

 of p by means of (III-2a), we find (using an asterisk to denote complex conjugate): 



I. = Reip(x)uMx) = 2^Imp* (x)^)P (x) (III-6) 



The simplest solution of (III-4) is the plane wave. For a plane wave 

 propagating with wave number vector k (i.e. , propagating in the direction of k 



with wave number k = | k | = — ), we may write: 



~ Co 



/ \ i k • X - iu)t 



p (x,t) = p e - - (III- 7a) 



These are, for different values of k, all the different possible "separable" solu- 

 tions in Cartesian coordinates. The energy flux in the direction of propagation 



3 



(i.e. , k) is seen from (III- 6) to be -r-^ . 



- 2 Po Co 



If we ask for the separable (i.e. , product form) solutions in spherical 

 coordinates, we obtain another very useful set of solutions. Specifically, let r 

 be the radial coordinate (r = | x | ), 9 be the polar angle, and the azimuthal 

 angle. Normally, we deal with problems whose solution is independent of 0, in 

 that we treat scattering of waves incident along the polar axis on objects with com- 

 plete symmetry. We may then take as a complete set of wave functions: ^^^ 



p (x,t)=P (cose)j (kr)e"^'"^ 

 m - m -"m 



p (x,t) = P (cos 9) n (kr) e"^"^*^ 

 m - m m 



m = 0,1, ..., " (III-7b) 



1. P. M. Morse, "Vibration and Sound." 



artbur KlLitth.linc. 



S-7001-0307 



