# Solution of wave equation

1. Jul 30, 2011

### mahblah

1. The problem statement, all variables and given/known data

The wave equation is
$\nabla^2 \mathbf{A}(\mathbf{r},t) = \frac{1}{c^2} \frac{\partial^2 \mathbf{A}(\mathbf{r},t)}{\partial t^2}$

I want to get a solution for the vector potential A.

2. Relevant equations
we can use the Fourier transformation

$\mathbf{A}(\mathbf{k},\omega) = \int{d\mathbf{r}}\int{dt \mathbf{A}(\mathbf{r},t) \exp{[-i(\mathbf{k \cdot r} - \omega t)]}}$

$\mathbf{A}(\mathbf{r},t) = \int{\frac{d\mathbf{k}}{(2\pi)^3}} \int{\frac{d\omega}{2 \pi} \mathbf{A}(\mathbf{k},\omega) \exp{[-i(\mathbf{k \cdot r} - \omega t)]}}$

to get

$\left( \mathbf{k}^2 - \frac{\omega^2}{c^2}\right) \mathbf{A}(\mathbf{k},\omega) =0$

(so $\omega = c k$ )

3. The attempt at a solution

Now the the solution is (formally)

$\mathbf{A}(\mathbf{r},t) = \sum_{\lambda =1,2} \int{\frac{d\mathbf{k}}{(2\pi)^3}} \int{\frac{d\omega}{2 \pi} A_\lambda(\mathbf{k},\omega) \hat{\epsilon}_\lambda(k) \cos{(\mathbf{k \cdot r} - \omega t + \varphi_\omega)} \delta(\omega -ck)}$

but i don't understand well why we have 2 polarization vector and form where they come...

thanks all,
MahBlah