Chapter.1.Appendix.Greens.function

# Chapter.1.Appendix.Greens.function - APPENDIX to Chapter 1...

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APPENDIX to Chapter 1: Green’s Function Solution to the Wave Equation Given 2 1 c 2 2 t 2 ( r , t )=− 4 F ( r , t ) and 2 1 c 2 2 t 2 G ( r , r ; t , t 4  ( r r ) ( ct ct ) one can write the following, integrating over all space and from t =−∞ to t =∞ : ∫∫∫∫ ( r , t ) 2 1 c 2 2 t 2 G ( r , r ; t , t ) d 3 x d ( ct )− G ( r , r ; t , t ) 2 1 c 2 2 t 2 ( r , t ) d 3 x d ( ct ) = ( r , t )(− 4 ) ( r r ) ( ct ct ) d 3 x d ( ct G ( r , r ; t , t )[− 4 F ( r , t )] d 3 x d ( ct ) =− 4 ( r , t ) − G ( r , r ; t , t )[− 4 F ( r , t )] d 3 x d ( ct ) A1 Note that r and t must be inside the volume and time integration limits. This expression can also be written in terms of ”surface” integrals as follows: d ( ct ) ∫∫∫ [ ( r , t )∇ G ( r , r ; t , t ) − G ( r , r ; t , t )∇ ( r , t )] d 3 x ct [ ( r , t ) ct G ( r , r ; t , t ) − G ( r , r ; t , t ) ct ( r , t )] d ( ct ) d 3 x = d ( ct ) ∫∫ r →∞ [ ( r , t )∇ G ( r , r ; t , t ) − G ( r , r ; t , t )∇ ( r , t )] r ̂ dS

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Chapter.1.Appendix.Greens.function - APPENDIX to Chapter 1...

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