Distortion of sonic bangs by atmospheric turbulence

Abstract

Recorded pressure signatures of supersonic aircraft often show intense, spiky perturbations superimposed on a basicN-shaped pattern. A first-order scattering theory, incorporating both inertial and thermal interactions, is developed to explain the spikes. Scattering from a weak shock is studied first. The solution of the scattering equation is derived as a sum of three terms: a phase shift corresponding to the singularity found by Lighthill; a small local compression or rarefaction; a surface integral over a paraboloid of dependence, whose focus is the observation point and whose directrix is the shock. The solution is found to degenerate at the shock into the result given by ray acoustics, and the surface integral is identified with the scattered waves that make up the spikes. The solution is generalized for arbitrary wave-forms by means of a superposition integral. Eddies in the Kolmogorov inertial subrange are found to be the main source of spikes, and Kolmogorov's similarity theory is used to show that, for almost all timestafter a sonic-bang shock passes an observation point, the mean-square pressure perturbation equals$(\Delta p)^2 (t_c/t)^{\frac{7}{6}}$, where Δpis the pressure jump across the shock andtcis a critical time predicted in terms of meteorological conditions. For an idealized model of the atmospheric boundary layer,tcis calculated to be about 1 ms, a figure consistent with the qualitative data currently available. The mean-square pressure perturbation just behind the shock itself is found to be finite but enormous, according to first-order scattering theory. It is conjectured that a second-order theory might explain the shock thickening that actually occurs.