Phase singularities in isotropic random waves

Abstract

The singularities of complex scalar waves are their zeros; these are dislocation lines in space, or points in the plane. For waves in space, and waves in the plane (propagating in two dimensions, or sections of waves propagating in three), we calculate some statistics associated with dislocations for isotropically random Gaussian ensembles, that is, superpositions of plane waves equidistributed in direction but with random phases. The statistics are: mean length of dislocation line per unit volume, and the associated mean density of dislocation points in the plane; eccentricity of the ellipse describing the anisotropic squeezing of phase lines close to dislocation cores; distribution of curvature of dislocation lines in space; distribution of transverse speeds of moving dislocations; and position correlations of pairs of dislocations in the plane, with and without their strength (topological charge) ±1. The statistics depend on the frequency spectrum of the waves. We derive results for general spectra, and specialize to monochromatic waves in space and the plane, and black–body radiation.