Laplace and Fourier Transform of a PDE

[bvic4]

Homework Statement

In this problem I'm trying to derive an explicit solution for Langmuir waves in a plasma. In part (a) of the problem I derived the wave equation

(\partial_t_t+\omega_e^2-3v_e^2\partial_x_x) E(x,t) = 0

This matches the solution in the book so I believe it's correct.

The initial conditions are
E(x,t=0)=f(x)
\partial_tE(x,t=0)=0
(The second i.c. should be the time derivative, but I can't figure out how to write it correctly)

For this part of the problem I am supposed to solve this equation explicitly. The problem statement says we should use the method of Laplace and Fourier transforms to do this. Generally, for this class, we assume the solution is a plane wave (E(x,t)=exp[-iwt+ikx]), but we don't get to do that here.

Homework Equations

I don't have a lot of experience using Fourier transforms so the equations that I think I should use are:

Fourier Transform:

f(x)=\frac{1}{2\pi}\int_{-\infty}^{\infty}f(\alpha)e^{-i\alpha x}d\alpha
f(\alpha)=\int_{-\infty}^{\infty}f(t)e^{i\alpha x}dx

The Attempt at a Solution

I'm doing this problem by taking the Laplace Transform with respect to time and the Fourier transform with respect to x, which I think is right, but I'm not 100% sure about.

First I took the Fourier Transform (using tables) which gave me:

\delta_t_tE(\alpha,t) + \omega_e^2 E(\alpha,t) + 3 v_e^2 \alpha^2 E(\alpha,t) = 0

Then I took the Laplace Transform, which after some simplification, gave me:

E(\alpha,s) = s f(\alpha)/(s^2 + \omega_e^2 + 3 v_e^2 \alpha^2)

I used contour integration to get the inverse Laplace transform:

E(\alpha,t) = f(\alpha) (e^{i \sqrt{\omega_e^2 + 3 v_e^2 \alpha^2} t} + e^{-i \sqrt{\omega_e^2 + 3 v_e^2 \alpha^2} t})

Now I need to take the inverse Fourier transform and I think I'm done. The problem is I'm not sure how to do this. The only explicit way I know to do this is using contour integration, which isn't useful here because there aren't any poles. I'm not sure how I can use the tables because of that alpha inside of the square root. Any ideas?

Thanks,

Brian

 

[bvic4]

I realized that the last equation I wrote can be written as

E(\alpha,t) = f(\alpha) (2*\cos{\sqrt{\omega_e^2 + 3 v_e^2 \alpha^2} t})

Is that helpful? Does anyone know how to take the inverse Fourier transform?

Thanks,

Brian