My graduate student, Alex Carr, defended his dissertation on sonic boom through the turbulent atmosphere.

Title: SONIC BOOM PROPAGATION IN THE TURBULENT ATMOSPHERIC BOUNDARY LAYER

Abstract:

Sonic boom waveforms measured at ground level exhibit variability due to scattering and diffraction caused by the presence of turbulence in the atmospheric boundary layer (ABL). Sonic boom propagation in the ABL is considered in the context of a partially one-way equation for a finite amplitude pressure perturbation that incorporates turbulence effects. This equation is shown to simplify to several well known equations in nonlinear acoustics, if certain assumptions are made. A wide-angle parabolic approximation is applied to the heterogeneous terms of the governing equation. The forward solution along the propagation direction is computed with a split-step method. The solution at each propagation plane is the composition of solutions to subproblems that account for each physical effect: nonlinear distortion, diffraction, atmospheric absorption, and the effects of mean flow and turbulence. Simulation results for benchmark acoustic problems are compared to the corresponding analytical solutions to validate the code. Turbulence is synthesized in the computational domain with Fourier synthesis techniques that have been used previously for sound propagation simulations. The turbulent kinetic energy spectrum is approximated by a von Ka ́rma ́n model. Simulations of traditional N-wave and shaped booms are performed through homogeneous turbulence, as well as inhomogeneous turbulence that is representative of daytime conditions in the atmospheric boundary layer. A length scale is proposed to non-dimensionalize the propagation distance and collapse the probability density functions of the caustic locations obtained from the N-wave simulations. For the simulations performed through homogeneous turbulence, the average loudness levels for both waveforms are shown to decrease along the propagation direction due to turbulence effects. The standard deviation of the loudness metrics increases linearly for small non-dimensional distances. The maximum standard deviation of each loudness metric considered for both waveforms fell between 2 to 4 dB. For the simulations performed through inhomogeneous turbulence, the effects of ABL height and convection level on a traditional N-wave and shaped waveform are considered. An empirical modification to the scaling length is proposed to account for the varying ABL height altering the turbulence integral length scales in the mixed layer region. Results for the loudness metrics are shown to follow a normal distribution for both waveforms when the non-dimensional distance is less than 2, beyond which the observations become increasingly skewed to the right of a normal distribution. Results indicate that the loudness metric distributions of both waveforms are likely to be normally distributed in the region undertrack of the flight path for weak to strong convection levels in the ABL. Additional simulations are conducted of sonic boom decay into the shadow zone region. Results indicate that loudness levels of both waveforms in the shadow zone region are sensitive to the turbulence levels in the ABL. As the turbulence level increases, the average of each loudness metric increases.

Carr, Defense, 2022.