Our simplest solution to the wave equation becomes
with a complex wavenumber
where
and 
are the real and imaginary parts of the wavenumber 
, respectively.In a lossy medium, the equivalent permittivity is a complex quantity
Our simplest solution to the wave equation becomes
with a complex wavenumber
where and are the real and imaginary parts of the wavenumber , respectively.
In terms of and , the above electric field can be expressed as
and its real time-domain equivalent
which shows that the wave is attenuated as it propagates in the positive z-direction. The quantity can be viewed as the attenuation factor, and the remaining in the argument of the cosine is sometimes also called the propagation constant.
The skin depth (for a good conductor) or the penetration depth (for a poor conductor)
is a measure of how far a plane wave can penetrate into a lossy medium. Because the envelope of the electric field decays to 1/e of its original amplitude at one penetration depth (or, skin depth) into the lossy medium.
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Wave in a Good Conductor |
Copper (
) is a good example for a highly conductive material. At 1 MHz, the skin depth of copper is in the order of tens of microns.
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|
Wave in a Poor Conductor |
Distilled water (
) represents a good example for a poor conductor. At 1 MHz, the penetration depth of distilled water is in the order of kilometers.
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Evanescent Wave in Plasma |
The third animation shows the evanescent wave in an isotropic electron plasma when the wave frequency ( e.g., 1 MHz) is below the plasma frequency (e.g., 2 MHz). Or,
For this case, the wavenumber is purely imaginary:
and the wave is said to be evanescent. Our simplest solution becomes
in phasor form, and
in real time domain. However, one should note that such a plasma is a lossless medium.