350lect04 - 4 Time harmonic sources and retarded potentials...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
4 Time harmonic sources and retarded poten- tials The solution of forced wave equation 2 V - μ o ± o 2 V ∂t 2 = - ρ ± o for scalar potential V is most conveniently obtained in the frequency domain: Consider a time-harmonic forcing function ρ and a time-harmonic re- sponse V expressed as ρ ( r ,t )= Re { ˜ ρ ( r ) e jωt } and V ( r ,t )= Re { ˜ V ( r ) e jωt } in terms of phasors ˜ ρ ( r ) and ˜ V ( r ) . Then, the above wave equation transforms — upon replacing ∂t by —intophasorformas 2 ˜ V + μ o ± o ω 2 ˜ V = - ˜ ρ ± o . For ω =0 the above equation reduces to Poisson’s equation, which we know has, with an impulse forcing ˜ ρ ( r )= δ ( r ) , an impulse response solution ˜ V ( r )= 1 4 π± o | r | 1 4 π± o r 1
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
where r ≡| r | denotes the distance of the observing point r from the impulse point located at the origin. Note that this impulse response 1 /r is symmetric with respect to the origin just like the impulse input δ ( r ) . Note that by substituting the source function δ ( r ) and response function 1 4 π± o r back into the Poisson’s equation we obtain an equality 2 ± 1 | r | ² = - 4 πδ ( r ) , which is a useful vector iden- tity. We now postulate and subsequently prove that for ω 0 ,theimpu l sere - sponse solution of the forced wave equation — i.e., with forcing function ˜ ρ ( r )= δ ( r ) —is ˜ V ( r )= e - jk | r | 4 π± o | r | with k ω μ o ± o = ω c . Proof: For ˜ ρ ( r )= δ ( r ) the source of the forced wave equation (for an arbitrary ω )is symmetric with respect to the origin ,imp ly ingthatthecorre- sponding solution ˜
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 09/27/2011 for the course ECE 450 taught by Professor Staff during the Fall '08 term at University of Illinois, Urbana Champaign.

Page1 / 8

350lect04 - 4 Time harmonic sources and retarded potentials...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online