A-3 



The function given in (A- 9) is known as the Green's function and permits solution 

 of both the homogeneous and inhomogeneous wave equations through the use of 

 Green's theorem. We shall, therefore, discuss briefly: 



1. Green's theorem 



2 . the general solution to the homogeneous wave equation 



3. the general solution to the inhomogeneous wave equation. 



1. Green's Theorem 



It follows from the definition of the divergence that, for any volume V 

 bounded by a surface S with local normal n, the volume integral of the divergence 

 of a vector function v is related to the outward flux by: 



fdivv dV = r V . n dS (A-11) 



V 



Suppose the vector function is given in terms of two arbitrary scalar functions 

 cp and t according to: 



V = cpV \|; - i|; V cp (A- 12) 



Application of (A-11) then yields: 



I V . (cpv^ - ,if V(p)dv = r 



(cp V ^ - ,|f V (p) dV = (cp V ^ - \|; V cp) -n dS 

 V S 



Both sides may be simplified to yield Green's theorem: 



("(^ov^ t - ,1, v^ cp)dv = r(cp|| ,^^ 



V s 



i^-P-) dS (A-13) 



artliur M.lLittU.linc. 



S-7001-0307 



