Topological Aspects of Wave Propagation
Carlos Valero · arXiv (Cornell University) · 2014
In the context of wave propagation on a manifold X, the characteristic functions are real valued functions on cotangent bundle of X that specify the allowable phase velocities of the waves. For certain classes of differential operators (e.g Maxwell's Equations) the associated characteristic functions have singularities. These singularities account for phenomena like conical refraction and the transformation of longitudinal waves into transversal ones (or viceversa). For a specific class of differential operators on surface, we prove that the singularities of the characteristic functions can be accounted from purely topological considerations. We also prove that there is a natural way to desingularsize the characteristic functions, and observe that this fact and Morse Theory establishes a specific connection between singularities and critical points of these functions. The relation between characteristic functions and differential operators is obtained through what is known as the symbol of the operator. We establish a connection between these symbols and holomorphic vector fields, which will provide us with symbols whose characteristic functions have interesting singularity sets.