Turbulence‑Driven Loudness Loss in Sonic Booms: A Multiple Scattering Theory-Based Mean Energy Framework

This paper discusses how the existing nonlinear parabolic equation used to account for turbulence effects conserves the mean acoustic energy of sonic booms when averaged over an ensemble of turbulence realizations. Because the equation preserves the mean energy, it also preserves the mean loudness. In contrast, flight‑test measurements consistently show that the mean loudness of sonic booms decreases during propagation through atmospheric turbulence, with the reduction increasing with both propagation distance and turbulence strength. These observations indicate that atmospheric turbulence scatters acoustic energy in ways not captured by the parabolic‑equation framework. To address this discrepancy, the paper introduces a new method based on the multiple scattering theory for computing the mean energy spectral density that incorporates turbulence‑induced backscattering, providing a more complete description of energy loss. Using this formulation, the predicted mean energy spectral density and loudness level are compared directly with those obtained from sonic booms measured during flight testing. The comparison demonstrates the extent to which backscattering must be represented to achieve accurate predictions. Developing a predictive capability that captures these turbulence effects is essential for reliable loudness estimation of NASA’s X‑59 low‑boom flight demonstrator and for improving future sonic‑boom propagation models.