Moving source illusion in geometric optics of anisotropic media

Abstract

We discuss an effect related to wave propagation in an inhomogeneous medium: the center of curvature of the wave front arriving at some point (x, y, z) proves to be moving at a velocity whose dependence on the radius of curvature is given by a quadratic trinomial. The coefficients of the trinomial are completely determined by the corresponding Riemannian metric at the observation point, the front orientation at the observation point, and the geometric parameters of the front (geodesic torsion) at the observation point. The resulting dependence is similar to the well-known Hubble law; however, in this case, the motion is not that of the source but of the center of curvature, which plays the role of a hypothetical source for an observer assuming that the ambient medium (or the perturbation propagation velocity) is homogeneous. The result is generalized to the case of an arbitrary medium in which the front equation t = ψ(x) is specified by a solution of the partial differential equation H(ψ, x,∇ψ) = 0 of the general form.