Why the one-dimensional wave equation is generated as the current format

The one-dimensional wave equation is $frac{partial^2 u}{partial t^2} = v^2 frac{partial^2 u}{partial x^2}$. The higher dimensional wave equation is $frac{partial^2 u}{partial t^2} = v^2 nabla^2 u $. What is the reason for not any other format such as $frac{partial^2 u}{partial t^2} = (vec{v} cdot nabla)^2 u$? At least in the latter format the direction of the wave is part of the equation. Both are solutions for $u(vec{k} cdot vec{r} + vec{k} cdot vec{v} t)$, in which $vec{k}$ is the wave vector.