Classification of phase singularities for complex scalar waves and their bifurcations

Abstract

Motivated by the importance and universal character of phase singularities which were clarified recently, we study the local structure of equi-phase loci near the dislocation locus of complex valued planar and spatial waves, from the viewpoint of singularity theory of differentiable mappings, initiated by Whitney and Thom. The classification of phase singularities is reduced to the classification of planar curves by radial transformations due to the theory of du Plessis, Gaffney and Wilson. Then fold singularities are classified into hyperbolic and elliptic singularities. We show that the elliptic singularities are never realized by any Helmholtz waves, while the hyperbolic singularities are realized in fact. Moreover, the classification and realizability of Whitney's cusp, as well as its bifurcation problem, are considered in order to explain the three point bifurcation of phase singularities. In this paper, we treat the dislocation of linear waves mainly, developing the basic and universal method, the method of jets and transversality, which is applicable also to nonlinear waves.