What is the exact difference between wavenumber and propagation constant in an electromagnetic wave propagating in a medium such as a transmission line, cause i am a bit confused. Does it have to do with loss in the medium?


See "Mathematical descriptions of opacity", Wikipedia.

The propagation constant has a real and an imaginary part. One of those is equal to the angular wavenumber, the other is proportional to the absorption coefficient.

Which is which (which is the real part and which is the imaginary part) depends on what definition you're using for the term "propagation constant". There is more than one definition in common use.


Here is the consensus for microwave engineering. Other fields of science may vary.

$\newcommand{\j}{{\rm{j}}}$$\newcommand{\e}[1]{\,{\rm{e}}^{#1}}$ It was shown by Hertz that an arbitrary electromagnetic field in a source free homogeneous linear isotropic medium can be defined in terms of a single vector potential $\vec{\Pi}$. Assuming $\e{\,\j\omega t}$ time dependency, a wave in the Hertz vector potential field can be written as: $$\vec{\Pi}(x) = \vec{\Pi}(0) \e{-\gamma x}$$

The propagation constant $\gamma$ is a complex quantity: $$\gamma = \alpha + \j\beta$$

where $\alpha$ is the attenuation constant, and $\beta$ is the phase constant.

However, since the attenuation in an air medium is negligible, it is customary to write the wave equation solely in function of a complex phase constant $\beta$: $$\vec{\Pi}(x) = \vec{\Pi}(0) \e{-\j\beta x}$$

where $\beta = \beta' -\j \beta''$, such that $\gamma \equiv \j \beta = \j (\beta' -\j \beta'') = \beta'' + \j \beta' \Rightarrow \beta'' \equiv \alpha$.

The free space angular wave number $k_0$ is defined as: $$k_0 \equiv \frac{\omega}{c_0} = \frac{2\pi}{\lambda_0}$$

The unit is $\frac{\text{rad}}{\text{m}}$

Only for TEM waves: $$\beta = k_0 = \frac{2\pi}{\lambda_0}$$

Whereas for TE and TM waves, separation of variables in a Helmholtz equation results in a transcendental dispersion function that needs to be solved involving the free space wave number $k_0$ and a transverse wave number $\tau$.

In such cases, $$\tau^2 = -\left(\gamma^2 + {k_0}^2\right) = \beta^2 - {k_0}^2$$

$$\Rightarrow \beta = \sqrt{{k_0}^2 + \tau^2}$$

Some worked out examples for EM surface waves can be found here.


The attenuation constant is specifically the imaginary part of the wave number (ki), while the wave number in dissipative media is the real part of the wave number (kr). Let me know what you think, because I just started studying this.


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