Statistical Theory of Atmospheric Turbulence Effects on Sonic-Boom Rise Times

The measured sonic-boom rise times at ground level are typically of the order of 1–10 msec, which is two to three orders of magnitude larger than what would be predicted on the basis of a planar shock propagating in a homogeneous atmosphere. A tentative explanation of how such anomalous rise times are caused by atmospheric turbulence is given in terms of the Keller-Friedlander geometrical acoustics theory of weak shock propagation in an inhomogeneous medium. It is suggested that the shockfront initially develops ripples that subsequently are transformed into folds in the front when the shock passes vertices of caustics. This wavefront-folding mechanism may produce sonic-boom signatures that have a fine structure consisting of many small pressure jumps (microshocks), each jump corresponding to a segment of the folded wavefront. This fine structure may, however, be smeared out by viscosity. To assess the mechanism's effectiveness quantitatively, a stochastic model of an initially sharp shock propagating through the earth's turbulent boundary layer is derived by use of an adiabatic perturbation to the Green's-function solution for waves in a homogeneous medium subject to specified normal fluid velocity at a boundary. The model depends on turbulence statistics through three parameters, one of which is the characteristic time tc estimated previously by Crow as 0.7 msec. An analysis based on the model substantiates the supposition that typical waveforms are composed of many very small discrete microshocks. An approximate derivation gives an expression for the ensemble average of the early portion of ground-level signatures. The corresponding rise time is found to be of the order of (2 to 3)tc, which is in reasonable agreement with the data. Nonlinear effects, while not necessarily negligible, would appear to be insufficient to nullify the mechanism. However, because of the many approximations employed in the analysis, the conclusion that wavefront folding is the primary cause of anomalous rise times remains tentative.