Extension of Crow’s paraboloid of dependence to spherical waves

Abstract

Crow1 discussed the distortion of sonic booms in the presence of atmospheric turbulence. He showed that the scattering from a weak shock is given by a surface integral over a paraboloid of dependence. The paraboloid of dependence describes the two-dimensional surface that contains the scatterers that contribute to the scattered sound at a point behind the shock at a given time. In Crow’s analysis the sonic boom is treated as a plane wave. Lipkens et al.2,3 used electrical sparks that propagate through turbulence created by a plane jet to simulate sonic boom propagation through a turbulent atmosphere. Experiments were done with plane and spherical waves. An extension of Crow’s scattering surface for spherical wave propagation is presented. For spherical waves the two-dimensional scattering surface is an ellipsoid of dependence.