The general solution to the wave equation

Question

Consider the wave equation, \begin{equation} \nabla^{2}\textbf{A}(\textbf{r},t)-\frac{1}{v^{2}}\frac{\partial^{2} \textbf{A}(\textbf{r},t)}{\partial t^{2}}=0\label{eq:WaveEQN}. \end{equation} It is relatively straight forward to see that $$\textbf{A}(\textbf{r},t)=e^{i(\textbf{k}\cdot\textbf{r}-\omega t)}$$ is a solution if $|\textbf{k}|=\omega/v$. I also know that any integrable continuous vector function $\textbf{B}(\textbf{r},t)$ can be written in the form \begin{equation} \textbf{B}(\textbf{r},t)=\int \textbf{B}(\textbf{k},\omega)e^{-i(\textbf{k}\cdot\textbf{r}-\omega t)}d\textbf{k}d\omega. \end{equation} where the integral is over all possible values of $\textbf{k}$ and $\omega$. I think this means that any solution to the wave equation can be written as, \begin{equation} \textbf{A}(\textbf{r},t)=\int_{|\textbf{k}|=\omega/v} \textbf{A}(\textbf{k},\omega)e^{-i(\textbf{k}\cdot\textbf{r}-\omega t)}d\textbf{k}d\omega, \end{equation} where the condition that $|\textbf{k}|=\omega/v$ ensures that the above expression is a solution to the wave equation. I'm not 100% sure that this is the case though. Just because $|\textbf{k}|=\omega/v$ ensures that the above expression is a solution to the wave equation doesn't necessarily prove that all solutions can be written in this form. Does anyone have any ideas?