Formulation of a Burgers’ equation for shock waves in a turbulent atmosphere

Abstract

The theoretical formulation of a Burgers’ equation accounting for the attenuation and dispersion of shock waves due to scattering in a turbulent medium is presented. The physical effect treated here is in addition to the effects of geometric spreading, large-scale inhomogeneity, nonlinearity, viscous and thermal dissipation, and molecular vibrational relaxation. Although all Burgers’ equations are inherently a one-way approximation, here the scattering effects are incorporated into the Burgers’ equation through a correction to the small-amplitude sound speed. Such a correction is obtained from the full wave equation using a perturbative technique, and physically arises from multiple scattering of the signal due to small-scale inhomogeneities. The sound speed correction is complex, and is dependent on the frequency and the variance and length scale of the fluctuations. The real part of the correction gives rise to the dispersion of the signal while the imaginary part leads to signal attenuation. The latter is manifested by the increase in the rise time of the shock waves. The physical effect presented here is integrated into an existing sonic boom propagation code with minimal additional computational time. Numerical results will be compared with experimental data.