Vector fields for monochromatic waves with general dispersion: vortex and stagnation singularities

Abstract Three vector fields are associated with complex scalar fields representing monochromatic waves: the local wavevector (canonical momentum), parallel to the phase velocity along which wavefronts move; the local group velocity, generalising the particle velocity in Hamiltonian mechanics; and the divergenceless local current. The streamline patterns of each field display typical singularities: phase vortices, where the underlying wave vanishes, and stagnation points, where any of the vectors vanish. These features are studied for waves in the plane for general dispersion relations. The stagnation singularities typically occur in different places for the three fields, and have different topological indices. The different wave geometries are illustrated with numerical calculations.