hw5_p5406_s08 - HW 5 Phys5406 S08 due 2/28/08 1)We will use...

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Unformatted text preview: HW 5 Phys5406 S08 due 2/28/08 1)We will use the solution ¨( r ), obtained in class, inside the empty conducting box with ¨( x; y; c ) = ¨ and all other sides grounded, to get the Green function G D for all Dirichlet problems with this geometry. a) Obtain the completeness relation for expansion of any f ( x ) in a sine series for 0 < x < a , by obtaining the coe cients A j below, ( x x ) = X j [ A j sin j x=a sin j x =a ] : (1) b) You should now write the 3-d delta function in a sum of the product of sines. Recall for the work below that r 2 G D ( r ; r ) = 4 ( r r ). c) Now expand the sinh function below for z in the range (0,c) in a sine series, sinh jk z sinh jk c = X m [ A m sin m z=c ] ; (2) and obtain the coe cients A m . Note that in the above equation, the LHS doesn't vanish at z =c, but the RHS does vanish there. Thus the representation of a function in a Fourier series allows you to get accuracy arbitrarily close to the boundary, but perhaps not exactly at the boundary.at the boundary....
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This note was uploaded on 12/24/2011 for the course PHYS 5406 taught by Professor Blecher during the Spring '10 term at Virginia Tech.

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hw5_p5406_s08 - HW 5 Phys5406 S08 due 2/28/08 1)We will use...

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