Higher-Order Statistical Moments of Predicted Sonic Boom Waveforms Through Turbulence

Sonic booms generated by supersonic aircraft are affected by turbulence in the atmospheric boundary layer through which they propagate. Turbulence effects lead to random variability of the sonic boom waveforms measured on the ground, complicating the prediction of such waveforms. As an initial effort to predict the waveform variability, the solution of the coherent or mean sonic boom waveform has previously been formulated by the author. The current paper extends the formulation to derive an expression of the second-order statistical moment necessary to calculate the variance of the spectral amplitudes. Since the derivation uses a full wave equation, results using the derived expression will be compared with those obtained using a parabolic approximation. In order to fully quantify the uncertainty of our predictions, the higher order statistical moments are also formulated and are shown to be approximately zero. Consequently, the probability density function (pdf) of the spectral amplitudes is predicted to be Gaussian. The formulation is further extended to determine the pdf of the loudness of sonic booms and quantify the uncertainties associated with the loudness prediction. Results from the formulation are compared with available flight test data.