W.Quapp, J.M.Bofill


 "The Wave Equation for Level Sets is not a HUYGENS' Equation"  


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. 
  [Bofill, Quapp, Caballero: 2012, JCTC 8 (1012) 4855].
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 some special wave solutions in every dimension, 
if a special Cauchy initial value problem is posed. 
There is no distinction into odd or even dimensions which usually
characterizes the validity of the HP.
To compare this with Hadamard's 'minor premise',
we calculate differential geometric objects like Christoffel symbols,
curvature tensors and geodesic lines, which we need to test the validity of 
the HP.  However, for this differential equation, the main criteria are not 
fulfilled for the strong HP in the sense of Hadamard's 'minor premise'. 

keywords:
Contours;  steepest ascent; wave equation;  progressing waves; Huygens' principle. 

AMS Subject Classification: 35A18, 35C07, 35L05