Solving Wave Equation / Imaginary Numbers

1. The problem statement, all variables and given/known data
Consider the simplified wave function: [tex]\psi (x,t) = Ae^{i(\omega t - kx)}[/tex]
Assume that [tex]\omega[/tex] and [tex]\nu[/tex] are complex quantities and that k is real:
[tex]\omega = \alpha + i\beta[/tex]
[tex]\nu = u + i\omega[/tex]
Use the fact that [tex] k^2 = \frac{\omega^2}{\nu^2}[/tex] to obtain expressions for [tex]\alpha[/tex] and [tex]\beta[/tex] in terms of [tex]u[/tex] and [tex]\omega[/tex].


2. Relevant equations
i [tex]\psi (x,t) = Ae^{i(\omega t - kx)}[/tex]
ii [tex]\omega = \alpha + i\beta[/tex]
iii [tex]\nu = u + i\omega[/tex]
iv [tex] k^2 = \frac{\omega^2}{\nu^2}[/tex]

3. The attempt at a solution
I cannot seem to find expressions for [tex]\alpha[/tex] and [tex]\beta[/tex] in terms of [tex]u[/tex] and [tex]\omega[/tex]. 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.
 
Last edited:

HallsofIvy

Science Advisor
41,626
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You are given that [tex]k^2= \frac{\omega^2}{\nu^2}[/tex] and that k is real so k2 is positive real, so we can write [itex]k= \frac{\omega}{\nu}[/itex] or [itex]k= -\frac{\omega}{\nu}[/itex]
With [itex]\omega= \alpha+ i\beta[/itex] and [itex]\nu= u+ iw[/itex]. Then
[itex]k= \frac{\alpha+ i\beta}{u+ iw}[/itex] or [itex]k= -\frac{\alpha+ i\beta}{u+ iw}[/itex]. Simplify the fraction on the right by multiplying both numerator and denominator by [itex]u- iw[/itex] and use the fact that k is a real number so the imaginary part must be 0.
 

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