Evolution of statistical properties for a nonlinearly propagating sinusoid

Abstract

The nonlinear propagation of a pure sinusoid is considered using time domain statistics. The probability density function, standard deviation, skewness, kurtosis, and crest factor are computed for both the amplitude and amplitude time derivatives as a function of distance. The amplitude statistics vary only in the postshock realm, while the amplitude derivative statistics vary rapidly in the preshock realm. The statistical analysis also suggests that the sawtooth onset distance can be considered to be earlier than previously realized.

Skewness and shock formation in laboratory-scale supersonic jet data

“…Data with a greater bandwidth allow for steeper shock content, which results in greater maximum derivative values and a corresponding greater skewness. Reexamination of the results of Shepherd et al 16 for the evolution of derivative skewness for a nonlinearly propagating initial sinusoid while using different sampling rates has revealed sensitivity of the results to the relative data discretization near the shock formation distance. Further quantitative differences are still being explored.…”

“…However, skewness of the time derivative, which increases significantly as waveforms steepen and acoustic shocks form, was proposed as a useful measure by McInerny. 15 Shepherd et al 16 recently calculated the statistics in the preshock region for the canonical nonlinear example, a planar, initially sinusoidal wave. They found a large increase in derivative skewness near the shock formation distance.…”

Near-field shock formation in noise propagation from a high-power jet aircraft

“…However, Shepherd et al 22 have studied the nonlinear evolution of an initial sine wave in the preshock region while calculating the skewness of the time derivative. They show that the skewness increases exponentially from 0 to values greater than 10 as the shock formation distance is approached.…”

Evolution of the average steepening factor for nonlinearly propagating waves

“…Rudenko and Chirkin 17 and Webster and Blackstock 18 showed that the probability density function of the waveform remains stationary until shocks form, which suggests that useful measures might be based on the temporal rates of change of the pressure. The statistics (i.e., skewness and/or kurtosis) of the waveform time derivative have been used to characterize the nonlinearity for initial sinusoids, 19 noise in a plane-wave tube, 20 and jet 10,[21][22][23][24][25][26] and rocket 27,28 noise. Baars and Tinney 11 have recently investigated a shock detection algorithm in the context of supersonic jet noise propagation for such metrics as number of shocks per unit time.…”