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## Homework Statement

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]

Show that the wave is damped in time. 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]. Find the phase velocity for this case.

## Homework 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]

## The Attempt at a Solution

I substitued

**ii**into

**i**to obtain the expression: [tex]\psi (x,t) = Ae^{-\beta t}e^{i(\alpha t - kx)}[/tex]. Therefore, the factor of [tex]e^{-\beta t}[/tex] represents the damping of the wave in time. There is no damping in the position of the wave.

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.

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