[Math] Intuitive explanation of the difference between waves in odd and even dimensions

Question

Motivation: In odd dimensions, solutions to the wave equation: $$u_{tt}(x,t)=\nabla u(x,t), \qquad u_t(x,0)=0, \qquad u(x,0)=f(x)$$ where $t \geq 0$ and $x \in \mathbb{R}^n$, have the nice property that the value of $u(x,t)$ only depends on the values $f(y)$ with $|y-x|=t$. For even dimensions, the value $u(x,t)$ depend on all the values $f(y)$ with $|y-x|\leq t$. A consequence of this is that when you switch a light bulb on and then off (in 3D), there will be a light wave traveling with the speed on light and behind the wave, there will be total darkness. But when you throw a rock into a pond (with a 2D surface), there will be lots of waves traveling outwards from where the rock hit the water and, in theory, the water will never a still again.

Question: Can anyone give an intuitive explanation of this difference between odd and even dimensions?

Answer 1

I had this question sometime ago and was shown the explanation in Balazs 1954 paper "Wave propagation in even and odd dimensional spaces". The singularities of the integrands in the solution vary based on the dimension giving rise to this effect.

1. When n is even, the integrand has a pole in the complex plane, so the contour has to be deformed, and the result refers to only one time value -- that of the pole.
2. When n is odd, the integrand has a branch point, and the contour cannot encircle the singularity. Consequently the solution contains contributions from all times along the branch cut.