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.