Implemented phase dispersion in retarded time

[enc08]

Hi,

I am trying to implement phase dispersion in a retarded time frame.

$$c_{phase}(ω) = c_{0} + c'(ω)$$

where $$c'(ω)$$ is a small deviation from the reference phase speed $$c_{0}$$.

In the frequency domain, the propagation term appears as an exponent:

$$e^{-(\alpha + iω/c_{phase}(ω))z}$$

where z is distance. I can re-write this as

$$e^{-\alpha z}e^{-izωc_{0}^{-1} c_{0}/(c_{0} + c'(ω))}$$

Now this where I am confused...In my wave equation, I am using retarded time, so there is no $$ω/c_{0}$$term. However, it seems I can't implement dispersion due to c'(ω) as it's coupled (multiplied) with a $$ω/c_{0}$$ term.

Any input is appreciated.

 

[AndyW]

Hello,

Thank you for reaching out with your question. It seems like you are working on a very interesting problem involving phase dispersion in a retarded time frame. I can see how you are trying to implement the phase speed equation in the frequency domain and how the propagation term appears as an exponent.

One approach you can take is to first solve the wave equation without the dispersion term, and then introduce the dispersion term afterwards. This can be done by using a perturbation method, where you treat c'(ω) as a small parameter. This way, you can still use the retarded time frame, and the ω/c_{0} term will not be an issue.

Another option is to use a different time frame, such as the advanced time frame, where the ω/c_{0} term does not appear. However, this may require some modifications to your wave equation.

I hope this helps and gives you some ideas on how to proceed. It would also be helpful to know more about your specific problem and the equations you are using, so I can provide more specific advice. Good luck with your research!