Phase constant in wave propagations: what are its effects?

[Ionito]

In the electromagnetism theory, the phase factor or constant (usually BETA) in wave propagation for lossy medium has the unit rad/m.

I understood that it must be interpreted as the amount of phase shift that occurs as the wave travels one meter.

However, differently of the attenuation factor (usually ALFA), I cannot see examples relating the phase factor to the distance. In other words, we can see the signal attenuation as the form of 8.69*ALFA*d, where d is the distance between the sender and the receiver. However, this distance "d" is not used in conjunction with the phase factor BETA. Is it right? Is there any correlation between BETA and the signal attenuation? If not, what are the effects of having a HIGH and LOW BETA?

Can anyone provide me a complete example of the total attenuation (in dBs), given ALFA, BETA, frequency, and distance d, for a plane wave propagating in a lossy medium?

Thanks

 

[Born2bwire]

The wave number/vector is a complex quantity. The real part dictates the phase progression with regard to spatial displacement. This is your \beta. The imaginary part gives rise to an attenuating behavior with regards to space, this is your \alpha loss. They are both related with the position in space/distance traveled since they both come out of the phase term:
e^{i\mathbf{k}\cdot\mathbf{r}}
So the real part will dictate the phase progression and the effective wavelength of the wave while the imaginary part dictates the attenuation of the wave. Both of these are dependent on not only the physical properties of the medium of propagation but also the frequency of the wave as well since the wave number is defined as:
k = |\mathbf{k}|^{\frac{1}{2}} = \omega\sqrt{\epsilon\mu}

 

[Ionito]

Thank you for the answer. Besides the velocity, wavelength, and phase shift effects caused by different BETAs of lossy media, is it possible (and how) that these BETAs are also related to the attenuation behavior (a secondary factor in addition to the ALFA factor)?

 

[Born2bwire]

Thank you for the answer. Besides the velocity, wavelength, and phase shift effects caused by different BETAs of lossy media, is it possible (and how) that these BETAs are also related to the attenuation behavior (a secondary factor in addition to the ALFA factor)?

If you are talking about \beta and \alpha in a strictly mathematical sense, then no, there is no relation. However, they are derived from the same material properties and frequency of the wave by the fact that they come from the wave number, defined above. So the two are linked physicaly by the fact that they are directly determined by the material properties of your surrounding medium. If we had a dielectric of a given permittivity and permeability, the introduction of loss, without modifying its dielectric constant (real part of the permittivity), results in the shortening of the wavelength as the square root of the permittivity increases when we increase its magnitude.

So increasing the loss of the material, via the manipulation of the material's conductivity, will decrease the wavelength (increase \beta) and increase the attenuation factor (increase \alpha).