Uniform formulation of the problem of distortion of sonic booms or bangs by atmospheric turbulence

Abstract

A now-classic paper relating to wave propagation through turbulence is S. C. Crow's “Distortion of sonic bangs by atmospheric turbulence” [J. Fluid Mech. 37, 529–563 (1969)]. Crow treats the linear acoustics problem of an (initially) step discontinuity in pressure (weak shock} propagating through a nominally uniform atmosphere with weak turbulence superimposed, uses a first-order perturbation theory to determine the turbulence-induced corrections to the waveform, and obtains some simple predictions concerning the variance at times somewhat later than shock onset, which have subsequently been well confirmed by field data. The singular nature of Crow's solution near the nominal time of shock onset, however, has always been somewhat disquieting to the present author, so a formulation is here given that avoids this singularity using a version of the method of strained coordinates. The recently derived approximate wave equation [Pierce, J. Acoust. Soc. Am. (in press)] for wave propagation through a fluid with inhomogeneous unsteady flow is used as a starting point. The solution for the acoustic pressure is taken in the form of a rippled wavefront (Δp)(1 + ψ)H(1 − c−1x + φ), where φ is a function of position, while ψ is a function of position and time. Here, H(t) is the Heaviside unit step function and Δp is the nominal pressure jump at the start of the boom. Linear perturbation equations governing the functions φ and ψ are developed and solved by approximate methods. [Work supported by NASA Langley Research Center and by the William E. Leonhard endowment to Pennsylvania State University.]