Sketch of the considered acoustic scenario. A plane wave propagates in the positive direction of the x-axis (arrow direction) through atmospheric turbulence for positive x values. The turbulent volume (shaded) is considered infinite along the z-axis and positive x-axis.
Range-frequency diagram. The dashed lines and equations show the boundaries between the various fluctuations regimes. The solid lines represent the simulation sets considered in this study.
(Top) Wind amplitude (color plot in m/s) and direction (arrows) generated from the RFG model, and (bottom) amplitude (normalized at 1 for X = 0 m) of a 300 Hz plane wave propagated with the FDTD model in positive x direction through this realization of wind turbulence.
Normalized amplitude of the mean pressure as function of range for a harmonic source of 50 Hz (left), 300 Hz (middle), and 600 Hz (right). The solid line represents the results obtained from the FDTD model and the dashed line represents the theoretical exponential decay. In the 50 Hz set, is calculated from Eq. (4) , whereas in the 300 Hz and 600 Hz sets, it is calculated from Eq. (8) .
Log-amplitude variance with range for a harmonic source of 50 Hz (top), 300 Hz (middle), and 600 Hz (bottom). The solid line represents the results obtained from the FDTD model. The dashed lines are the limiting values for the saturation regime (horizontal line) and Tatarski's (1961) expression in the weak fluctuations regime. Triangles are the MPS simulation results, circles in the 50 Hz set are Brownlee's (1973) theoretical results.
Transverse coherence as a function of transverse separation for different propagation ranges for a harmonic source of 600 Hz. The solid lines represent the results obtained from the FDTD model and the dashed lines give Tatarski's (1961) theoretical expression in the weak fluctuations regime. The propagation ranges are, from top to bottom, 30 m, 60 m, 100 m, and 300 m.
Joint probability density of the complex pressure for various propagation ranges and sets (given in brackets in top-right corner on each plot) obtained from the FDTD model. Colormap is linear between zero (in white) and the maximum value (in black). The dashed circle is the unit circle and the solid lines are contours of iso-probability: 90% of the sample lie within the outer border and 50% within the inner border.
Probability density functions of the normalized amplitude (top) and phase (bottom) for the 300 Hz set at different propagation ranges (7 m, 21 m, 49 m, 105 m, and 301 m) obtained from the FDTD model. The PDFs for the shorter propagation range overflow the figure. The theoretical PDF in the saturation regime is shown with squares.
Probability density functions of the normalized amplitude for the 50 Hz set at five propagation ranges (126 m, 294 m, 360 m, 966 m, and 1302 m). The solid lines are from the FDTD model and the dashed lines from Brownlee's (1973) theory. At these propagation ranges, is, respectively, 0.92, 2.15, 4.60, 7.05, and 9.50.
The two parameters of the distribution fitting the simulated PDF of the intensity for the 300 Hz set. The white squares give and the black circles give .
Mean and variances of the main acoustic parameters in the saturated fluctuations regime. is the Euler constant.
Numerical details for each set of simulations.
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