Inhomogeneous electromagnetic wave equation

[hunt_mat]

Hi,

I am looking at electron beam going through a plasma. I am modelling it using two regions, the electron beam and external to the electron beam. I am using the potential formulation of electrodynamics and I am modelling a rigid electron beam and assuming cylindrical symmetry for simplicity.

I come down (with my assumption for the current) to solving the following equation:
$$-\nabla^{2}\varphi +\frac{1}{c^{2}}\frac{\partial^{2}\varphi}{\partial t^{2}}+\mu_{0}\sigma\frac{\partial\varphi}{\partial t}=\frac{\rho}{\varepsilon_{0}}$$
The charge density is constant. I am not too sure how to go about solving this equation, would it be via Green's functions? Can someone provide me with a reference please.

Regards

Mat

 

[gabbagabbahey]

I haven't worked it out yet, but the first method that comes to mind is to Fourier transform your PDE from time domain to frequency domain, and then solve the resulting inhomogeneous Poisson's equation using separation of variables in cylindrical coordinates.

 

[hunt_mat]

Hi,

Thanks for the reply. The problem with Fourier transforms is that you have to take the Fourier transform of a constant which doesn't exist. With regards with the separation of variables, it won't work as rho is a constant. Separation of variables works for the region outside of the beam where rho=0 (I have worked that part out).

Mat

 

[hunt_mat]

Another problem with the Fourier transform is that it requires an infinite domain which this problem doesn't have.

 

[hunt_mat]

I have been doing some work on this and I have come to the conclusion that there no separable solutions for the case $$\rho =0$$. As I want to look at the problem of an electron beam propagating into the plasma, I thought about writing the solution as
$$\varphi =f(r,z-v_{0}t)$$
This will reduce the PDE down to one with two variables rather than three and this makes it more tangible for a solution to be written down.

Thought?

 

[samalkhaiat]

$$\alpha \frac{\partial\phi}{\partial t} + \beta \frac{\partial^{2} \phi}{\partial t^{2}} - \nabla^{2} \phi = \rho (r,t) \ \ (1)$$

This equation stands intermediate between one of wave propagation ($\beta \gg \alpha$) and one of diffusion ($\alpha \gg \beta$). It can be solved by the Green’s function befined by;

$$\alpha \frac{\partial G}{\partial t} + \beta \frac{\partial^{2} G}{\partial t^{2}} - \nabla^{2} G = \delta (r) \delta(t) \ \ (2)$$

Write

$$G(r,t) = \int d^{3}k d \omega \hat{G}(k,\omega) e^{i(\vec{k}.\vec{r} - \omega t )} \ \ (3)$$

From eq(2), you find

$$\hat{G}(k,\omega) = (1/2 \pi )^{4} \frac{1}{k^{2} - k_{0}^{2}} \ \ (4)$$

where

$$k_{0}^{2} = \beta \omega^{2} + i \alpha \omega$$

Eq(3) becomes

$$G(r,t) = (\frac{1}{2 \pi})^{4}\ \int d \omega e^{-i \omega t} \ \int d^{3}k \frac{e^{i\vec{k}.\vec{r}}}{k^{2} - k_{0}^{2}}$$

or, by doing the angular integration,

$$G(r,t) = \frac{1}{8 \pi^{3} r} \int d \omega e^{-i \omega t} \ \int_{-\infty}^{\infty} dk \frac{k \sin{kr}}{k^{2} - k_{0}^{2}}$$

The k-integral may be evaluated as a contour integral to give $\exp (ik_{0}r)$.
Thus

$$G(r,t) = \frac{1}{8\pi^{3}r} \int_{-\infty}^{+\infty} d \omega \ e^{i(k_{0}r - \omega t)}$$

I believe, this integral can be evaluated in terms of Bessel’s functions. Now, do you know how to write the field $\phi$ in terms of $G(r,t)$? In order to do that, you need to know the values of $\phi (r,t)$ and $\partial_{t}\phi (r,t)$ at t = 0, and the value of $\nabla \phi + a \phi$ on some specified surfaces.

sam

 

[hunt_mat]

Hi Sam,

Sorry for being rude and not saying thanks (thanks by the way) but I have been a little distracted. I know nothing about Greens function and I will have to learn about them before I can understand your answer.

I thought of more simple way of looking at things (I have convinced that there are no separable solutions to the problem) by looking for a traveling wave solution of the form $$\varphi =f(r,z-v_{0}t)$$ which would turn the PDE in three variables into one with two which can then be solved via Laplace transforms. in $$w=z-v_{0}t$$

With regard to the wave propagation question. Regardless of the relative sizes of alpha and beta, there is still wave propagation but the damping term may or may not have a large effect.

Mat