Nonlinear Dispersion Relation with Imaginary Part

[rmjmu507]

Homework Statement

I've determined the dispersion relation for a particular traveling wave and have found that it contains both a real and an imaginary part. So, I let $k=\alpha+i\beta$ and solved for $\alpha$ and $\beta$

I found that there are $\pm$ signs in the solutions for both $\alpha$ and $\beta$.

Considering the two different cases for both $\alpha$ and $\beta$, I find:

For a: if +, then $\alpha=\frac{\omega}{c}$. If -, $\alpha=0$

For b: if +, then $\beta=\frac{\omega}{c*\sqrt2}$. If -, $\beta=\frac{\omega}{c*\sqrt2}+\sqrt2$

How am I supposed to decide which of the roots (either positive or negative) to use for each?

Homework Equations

The Attempt at a Solution

 

[Nate810]

When you have an equation with both a positive and a negative solution, the way to decide which one to use is to consider the physical situation or system you are studying. Depending on the particular details of the system, one root may be more appropriate than the other. For example, in this case, it appears that \alpha is related to the speed of the wave (given by \frac{\omega}{c}), so the + sign would make more sense in this case. For \beta, it appears that it is related to the attenuation of the wave, so the - sign would make more sense here.