Solution of inhomogeneous Helmholtz Equation

Solution of inhomogeneous Helmholtz Equation - NUS/ECE...

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NUS/ECE EE4101 __________________________________________________________ Hon Tat Hui Solution of Inhomogeneous Helmholtz Equation 1 Solution of the Inhomogeneous Helmholtz Equation Equation: ( ) ( ) ( ) 22 k μ ∇+ = AR JR (1) Soultion: () ' ' 4 jkR v e 'd v R π = ∫∫∫ J R (2) ( ) , , =field point, ', ', ' =source point ' r' r R θ φθ φ == =− RR
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NUS/ECE EE4101 __________________________________________________________ Hon Tat Hui Solution of Inhomogeneous Helmholtz Equation 2 Proof: Eq. (1) can be decomposed into 3 scalar inhomogeneous Helmholtz equations. For example, the x -component one is: ( ) ( ) ( ) 22 xx x Ak A J μ ∇+ = RR R (3) All the 3 scalar equations are of the same form. Hence it is sufficient to solve only one scalar equation. Consider the scalar inhomogeneous Helmholtz equation for a delta source () ' δ first. ( ) ( )( ) ' fk f = R R . (4)
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NUS/ECE EE4101 __________________________________________________________ Hon Tat Hui Solution of Inhomogeneous Helmholtz Equation 3 The solution for this equation can be shown to be: () 4 jkR e f R π = R . (5) First we move the origin of the coordinate system to the source point ' R . Then (with '0 = R ), ( ) ( ) ( ) 22 fk f δ ∇+ = RR R (6) and the solution is then: 44 jkR jkr ee f R r == R (7) where r = | R |
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This note was uploaded on 10/21/2011 for the course EE 4101 taught by Professor Yeotatsoon during the Spring '11 term at National University of Singapore.

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Solution of inhomogeneous Helmholtz Equation - NUS/ECE...

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