A-1 



APPENDIX A 



SOME SOLUTIONS OF THE WAVE EQUATION 



We shall summarize here some well known facts about the solutions of 

 the scalar wave equation: 



V^ cp + r cp = (A-l) 



solution, (A-l) becomes: 



-iuot 

 The time dependence e is suppressed. If we wish a spherically symmetric 



J^ 7- r^r- P+k%=0 (A-2) 



r dr S r 



ikr -ikr 



e e 



It is readily seen that both - and - satisfy (A-2), and if the suppressed 



-iuut '"' ^ 



time dependence is e , the outgoing wave is given by: 



ikr 

 cp = ^ (A-3) 



At the origin, this wave function blows up, and V^ cp is ill-defined. In fact, we 

 must resort to the definition of the divergence as a limiting ratio of flux to volume 

 to evaluate V cp at the origin. Given any vector function v(r), the divergence is 

 usually defined as: 



div V = V -v = lim — i v • n dcr (A- 4) 



T-o "^ 



o 



where: 



T = small volume element containing the point at which div v is to be computed 



o = surface enclosing t 



n = local normal to o 



Hence, for V^ cp, we substitute v = grad cp = Vcp in (A- 4) and obtain: 



V^cp = lim ^ f 1-^ da (A-5) 



„ T J dn 



T-»o "^ 



artbur B.lLittUMt. 



S-7001-0307 



