Boundary conditions and time domain electromagnetic waves

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Discussion Overview

The discussion revolves around the boundary conditions and behavior of electromagnetic waves at the interface between a lossy dielectric medium and a lossless dielectric medium. Participants explore the implications of these conditions in both time and frequency domains, examining the continuity of the tangential electric field and the decay of wave amplitudes in lossy media.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant suggests that the decay term in the lossy medium should depend on distance (z) rather than time (t), questioning the initial formulation.
  • Another participant explains that in a lossy medium without sources, the wave amplitude must decay over time due to energy loss, complicating the application of boundary conditions.
  • A different viewpoint emphasizes that the wave equation for lossy media can yield complex resonant frequencies, indicating damped waves, which contrasts with the behavior expected in lossless media.
  • Several participants discuss the mathematical formulation of the wave equation, with some correcting each other on the appropriate equations and boundary conditions.
  • One participant raises the issue of charge density at the boundary, suggesting that it could allow for discontinuities in the electric field, while another insists on a source-free assumption that excludes mobile charges.
  • A participant expresses confusion about the continuity of the tangential electric field at the boundary, noting that the decaying field in the lossy medium cannot equal the non-decaying field in the lossless medium.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the correct application of boundary conditions between lossy and lossless media. Multiple competing views and interpretations of the mathematical formulations and physical implications remain unresolved.

Contextual Notes

Limitations include assumptions about the presence of sources, the treatment of charge density at the boundary, and the applicability of different mathematical models in the context of lossy versus lossless media.

  • #31
ashade said:
I need the field distributions. A transmission line model won't fit because I'm looking for TE and TM modes on a coaxial cable, semi-infinite, with 2 dielectrics: one lossy and the other lossless. And I need distributions of field because I need to find the best way to place some receivers for a very very specific application.

PS: I didnt say in any moment I want a dumped wave on lossless medium. You said so.

Then what is the difficulty that you are having? Why do you keep complaining that the lossless medium will not allow a damped solution?
 
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  • #32
Born2bwire said:
Then what is the difficulty that you are having? Why do you keep complaining that the lossless medium will not allow a damped solution?

Because dumped and undumped solutions cannot equal for all t. Therefore, continuous tangential electric field is not guarateed in the boundary of both media.
 
  • #33
ashade said:
Because dumped and undumped solutions cannot equal for all t. Therefore, continuous tangential electric field is not guarateed in the boundary of both media.

They are equal because one wave is the source for the other. As I explained previously, a wave that transitions from a lossy to a lossless region has already been attenuated by its passage through the lossy region. A wave that goes from lossless to lossy will become attenuated as it travels into the lossy region. But the wave isn't going to decay until it is traveling in the lossy media. The causality of the problem and the fact that you are working with waves what you are forgetting here. Think of what happens if you were to send a wave pulse down your lossless section of your waveguide and it hits the the lossy dielectric section.

In addition, the analysis of the boundary conditions that I gave in post #19 demonstrates the general case that you are discussing using a plane wave solution. If you convert it from the time-harmonic form to the time domain form the wave solutions match the same conditions that you stipulated in your OP. That is, the left-hand side is dependent upon a sinusoidal function demonstrating the spatial and time phase dependence multiplied by what is equivalently an exponentially decaying function in time (by virtue of the constant dispersion curve relating the wave number with frequency and the phase velocity of the wave). The right hand side will be just a sinusoidal function with a spatial and time dependence.
 

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