A-4 



2 . Solution to Homogeneous Wave Equation 



Let cp be a solution to the homogeneous scalar wave equation (A-1) every- 

 where in some volume V, bounded by a surface S. Let if be the solution (A- 9) of 

 the inhomogeneous equation (A- 10). In summary: 



V'^' cp + k cp = 

 V^ A + k^ * = - 



ik, 



^ / <\ L. 1 e r - r 

 (r - r ) where v 



4n 1 r - r'l 

 Multiply the first of these equations by i|(, the second by cp, and subtract: 



\|; V^ cp - cp V^ ilr = cp 6 (r - r') 



Integrate both sides over the volume V. The left side can be changed to a surface 

 integral by Green's theorem. The right side simply selects cp at the point r', so 

 we obtain: 



cp( 



H 



Hr, r') ^^^-cp(r)^ ^ (r. r') 

 — — On — on — 



s •- 



dS 



(A- 14) 



This is the Helmholtz formula, which gives the value of the wave function inside 

 a closed surface in terms of a distribution of simple and dipole sources on the 

 surface . 



3. Solution to the Inhomogeneous Wave Equation 



If cp is a solution of the inhomogeneous scalar wave equation 



v^ cp + k^ cp = f (r) (A-15) 



we can again use the same Green's function i|/ (r, r') which satisfies 



v^ ili+k^ ^ = 6 (!-£') (A-9) 



to obtain a solution. Again we multiply (A-15) by ^, (A- 9) by cp, subtract, integrate 

 over a volume V, and apply Green's theorem. This yields: 



cp(r')=r 

 S 



^E.r^)^--iL^h^<^--^-> 



J 



dS - dVf(r) ^ (r, r') 



(A- 16) 



Thus we see that cp consists of a solution to the homogeneous wave equation plus a 

 particular solution to the inhomogeneous equation. This particular solution has 

 the form of a volume distribution of simple sources. 



Arthur Sl.littlpJnt. 



S-7001-0307 



