2

Both the wave equation and the transport equation are hyperbolic, but their solutions exhibit different behaviors in odd and even dimensions.

  1. Transport Equation: For the transport equation in one spatial dimension (where $x \in \mathbb{R}$): $$ u_t + c u_x = 0, $$ the solution is simply transported along the characteristic lines. In higher dimensions (where $\mathbf{x} \in \mathbb{R}^n$ for $n \geq 2$): $$ u_t + \mathbf{c} \cdot \nabla u = 0, $$ the solution still behaves as a simple transport, remaining largely unchanged regardless of whether $n$ is odd or even.

  2. Wave Equation: In contrast, the wave equation in one dimension (where $x \in \mathbb{R}$): $$ u_{tt} - c^2 u_{xx} = 0, $$ involves wave propagation in both directions.

In higher dimensions (where $\mathbf{x} \in \mathbb{R}^n$ for $n \geq 2$): $$ u_{tt} - c^2 \Delta u = 0, $$ the behavior of the solutions differs notably between odd and even dimensions.

In odd dimensions, wave solutions tend to exhibit more localized behavior and can have a richer structure due to the way they propagate through the medium.

In even dimensions, wave solutions often exhibit more uniform dispersion, leading to smoother wavefronts.

Why does the dimension affect the behavior of wave equation solutions, leading to different properties for odd and even dimensions, while the transport equation solutions remain unchanged?

Thanks!

Rosy
  • 1,087

1 Answers1

1

In $1$-D, the solution to an advection equation travels in one direction, and the solution to a wave equation travels in two directions. The key idea in generalizing this to higher dimensions is by noticing that when we say the wave equation solution travels in two directions, what we actually mean is that the wave equation solution travels in all directions. This is what generalizes to higher dimensions. Since the Laplacian $\nabla^2$ is a rotationally symmetric operator, it has no prefered direction of travel. In $1$-D, rotationally symmetric just means it is invariant to swapping $x\to-x$, but in $2$-D, we see can see waves genuinely traveling in all directions in a radial fashion.

As for the difference between even and odd dimensions as it applies to the wave equation, this is known as Huygens' principle and there is no intuitive explanation to my knowledge. This post contains a lot of good answers and links to detailed explanations.

whpowell96
  • 7,849