Can we prove that plane waves form a complete basis of solutions to the wave equation?

Question

For the 3D wave equation, \begin{equation} \frac{\partial^{2} \textbf{A}(\textbf{r},t)}{\partial t^{2}}-\frac{1}{v^{2}}\nabla^{2}\textbf{A}(\textbf{r},t)=0, \end{equation} can we prove that plane waves, \begin{equation} \textbf{A}(\textbf{r},t)=e^{-i(\textbf{k}\cdot\textbf{r}-\omega t)}, \end{equation} form a complete basis of solutions to this equation?