Why does the propagation constant in the Helmholtz equation include j?

[baby_1]

Homework Statement

This is my own question: Why do we intentionally include the imaginary unit jj in the propagation constant after taking its square root?

Relevant Equations

Helmholtz equation

Here is my question:
$$
\nabla^2 \bar{E} + \omega^2 \mu \left( 1 - j \frac{\sigma}{\omega} \right) \bar{E} = 0
$$

where

$$
\gamma = \alpha + j \beta
$$

$$
\gamma^2 = \omega^2 \mu \left( 1 - j \frac{\sigma}{\omega} \right)
$$

$$
\gamma = j \sqrt{ \omega^2 \mu \left( 1 - j \frac{\sigma}{\omega} \right) }
$$

why is there a factor of j in the expression for γ?

 

[TSny]

$$
\gamma^2 = \omega^2 \mu \left( 1 - j \frac{\sigma}{\omega} \right)
$$

$$
\gamma = j \sqrt{ \omega^2 \mu \left( 1 - j \frac{\sigma}{\omega} \right) }
$$

why is there a factor of j in the expression for γ?

Yes, the $j$ outside the square root in the second equation above looks like a mistake. If you square this equation, you do not get the first equation.

Where did you see the second equation?

 

[baby_1]

Dear TSny,

Thank you for your help.
In many electromagnetics resources, I have noticed that the propagation constant is often represented with a "j," but I am having trouble understanding why this is the case. I have attached one of the references for your review.

 

[baby_1]

I realized my mistake: in the main equation, the sign of the propagation constant should be negative. Thank you once again for your help!