Solving Wave Equation / Imaginary Numbers

[piano.lisa]

Homework Statement

Consider the simplified wave function: \psi (x,t) = Ae^{i(\omega t - kx)}
Assume that \omega and \nu are complex quantities and that k is real:
\omega = \alpha + i\beta
\nu = u + i\omega
Use the fact that k^2 = \frac{\omega^2}{\nu^2} to obtain expressions for \alpha and \beta in terms of u and \omega.

Homework Equations

i \psi (x,t) = Ae^{i(\omega t - kx)}
ii \omega = \alpha + i\beta
iii \nu = u + i\omega
iv k^2 = \frac{\omega^2}{\nu^2}

The Attempt at a Solution

I cannot seem to find expressions for \alpha and \beta in terms of u and \omega. I have tried rearranging the given equations in many such ways, but have not come up with any conclusive result.

Any suggestions are greatly appreciated. Thank you.

 

[HallsofIvy]

You are given that k^2= \frac{\omega^2}{\nu^2} and that k is real so k² is positive real, so we can write k= \frac{\omega}{\nu} or k= -\frac{\omega}{\nu}
With \omega= \alpha+ i\beta and \nu= u+ iw. Then
k= \frac{\alpha+ i\beta}{u+ iw} or k= -\frac{\alpha+ i\beta}{u+ iw}. Simplify the fraction on the right by multiplying both numerator and denominator by u- iw and use the fact that k is a real number so the imaginary part must be 0.