today I've got a problem with understanding what is meant by the term "traveling wave solution" of a PDE in a more general sense.
To be more precise, I've got a following Poisson equation
\begin{align}\Delta u &= \delta (r,\theta,z-ct) \ &in \ \Omega \subset \mathbb{R}^3\\ u&=0 \ &on \ \partial\Omega\end{align}
where $\delta$ is the delta-distribution, $\Omega$ unbounded domain and the PDE is given in cylindrical coordniates.
Now it is said that there exists a travelling wave solution $u$ satisfying the PDE. But what does that mean? I know the condition that there exists a function $w$ und $c$ with $u(x,t)=w(x-ct)$, but $x-ct$ makes no sense with $x\in \mathbb{R}^3$ and in addition I can't find literature dealing with a definition of traveling waves in higher dimensions.
Does someone know how to read that or knows relevant literature?