^{1,a)}, Sylvain Cheinet

^{1}, Daniel Juvé

^{2}and Philippe Blanc-Benon

^{2}

### Abstract

Sound propagation outdoors is strongly affected by atmospheric turbulence. Under strongly perturbed conditions or long propagation paths, the sound fluctuations reach their asymptotic behavior, e.g., the intensity variance progressively saturates. The present study evaluates the ability of a numerical propagation model based on the finite-difference time-domain solving of the linearized Euler equations in quantitatively reproducing the wave statistics under strong and saturated intensity fluctuations. It is the continuation of a previous study where weak intensity fluctuations were considered. The numerical propagation model is presented and tested with two-dimensional harmonic sound propagation over long paths and strong atmospheric perturbations. The results are compared to quantitative theoretical or numerical predictions available on the wave statistics, including the log-amplitude variance and the probability density functions of the complex acoustic pressure. The match is excellent for the evaluated source frequencies and all sound fluctuations strengths. Hence, this model captures these many aspects of strong atmospheric turbulence effects on sound propagation. Finally, the model results for the intensity probability density function are compared with a standard fit by a generalized gamma function.

The ISL Aerodynamics Department is acknowledged for providing access to their computational cluster. The authors thank the anonymous reviewers for their pertinent remarks and suggestions, which helped improve the paper.

I. INTRODUCTION

II. STATISTICAL BEHAVIOR

A. The diagram

B. Weak fluctuations regime

C. Strong fluctuations regime

D. Saturated fluctuations regime

III. THE NUMERICAL SIMULATIONS

A. Atmospheric turbulence

B. The FDTDmodel

C. Scenarii and numerical details

IV. RESULTS

A. First and second moments

B. Joint probability density and probability density function

V. DISCUSSION

VI. CONCLUSION

### Key Topics

- Finite difference time domain calculations
- 45.0
- Turbulent flows
- 43.0
- Turbulence simulations
- 29.0
- Acoustic waves
- 23.0
- Atmospheric turbulence
- 12.0

## Figures

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.

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.

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.

(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) .

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.

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.

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.

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 (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.

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 .

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 .

## Tables

Mean and variances of the main acoustic parameters in the saturated fluctuations regime. is the Euler constant.

Mean and variances of the main acoustic parameters in the saturated fluctuations regime. is the Euler constant.

Numerical details for each set of simulations.

Numerical details for each set of simulations.

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