Turbulence-Driven Losses in Sonic Booms Using Multiple Scattering Theory

Accurate prediction of sonic-boom propagation through the turbulent atmospheric boundary layer is essential for the certification and public acceptance of next-generation low-boom aircraft such as NASA's X–59. Flight test data show that turbulence produces mean loudness reduction that grow with both turbulence strength and propagation distance, effects not fully captured by current sonic boom prediction tools such as PCBoom. Models based on a nonlinear parabolic equation combine the Burgers equation for nonlinear absorption with the linear parabolic equation for turbulence effects. The one-way parabolic formulation conserves mean energy density as it cannot represent turbulence-induced backscattering losses. To overcome this limitation, the multiple scattering theory (MST) has been used that employs the three-dimensional wave equation and explicitly accounts for scattering from turbulent refractive-index fluctuations. The MST framework reveals two effective propagation modes whose interference produces spectral features observed in flight test data. Extension of the analysis to the Bethe–Salpeter equation yields an expression for the mean energy density that includes both coherent modes and turbulence-induced modal coupling, enabling the prediction of energy losses absent from parabolic models. The expression establishes a physics-based approach for sonic boom prediction and supports the development of next-generation propagation tools for low-boom aircraft.