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

Wolfgang Quapp, Josep Maria Bofill

Published 2013-09-12Version 1

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'.