The wave equation for level sets is not a Huygens' equation

TL;DR

This paper investigates the wave equation associated with level sets, demonstrating that it does not satisfy Huygens' principle in any dimension by analyzing geometric properties and initial value problems.

Contribution

It provides a detailed geometric analysis showing that the wave equation for level sets fails to meet the criteria of Huygens' principle, contrasting with classical wave equations.

Findings

Wave solutions have no wake in any dimension.

The wave equation for level sets does not satisfy the strong Huygens' principle.

Geometric analysis reveals the failure of Huygens' principle for this PDE.

Abstract

Any surface can be foliated into equipotential hypersurfaces of the level sets. A current result is that the contours are the progressing wave fronts of a certain hyperbolic partial differential equation, a wave equation. It is connected with the gradient lines, as well as with a corresponding eikonal equation. The level of a surface point, seen as an additional coordinate, plays the central role in this treatment. A wave solution can be a sharp front. Here the validity of the Huygens' principle (HP) is of interest: there is no wake of the wave solutions in every dimension, if a special Cauchy initial value problem is posed. Additionally, there is no distinction into odd or even dimensions. To compare this with Hadamard's 'minor premise' for a strong HP, we calculate differential geometric objects like Christoffel symbols, curvature tensors and geodesic lines, to test the validity of the strong HP. However, for the differential equation for level sets, the main criteria are not fulfilled for the strong HP in the sense of Hadamard's 'minor premise'.