Difference between odd and even dimensional wave equations

$$ u(t,x_{1},x_{2},x_{3},\dots), \ \ \ -1 < x_{i} < 1 $$

Wave equation.

$$ \frac{\partial^2 u}{\partial t^2} = \sum_{i=1}^{N} \frac{\partial^2 u}{\partial x_{i}^2} $$

Initial conditions.

$$ u(t=0,x_{1},x_{2},x_{3},\dots) = \frac{\partial u(t=0,x_{1},x_{2},x_{3},\dots)}{\partial t} = \exp \left\lbrace - 40 \sum_{i=0}^{N}x_{i}^2 \right\rbrace $$

Boundary Conditions: Dirichlet 0.

visualization 1D, 2D, 3D, 4D wave equation.

comparison.