## Calculations of wave propagation through statistical random media, with and without a waveguide

Optics Express, Vol. 10, Issue 16, pp. 777-804 (2002)

http://dx.doi.org/10.1364/OE.10.000777

Acrobat PDF (3186 KB)

### Abstract

Simulations of laser-beam propagation through atmospheric turbulence and acoustic pulse propagation though ocean internal waves in the presence of a transverse waveguide are described. Both these problems are amenable to the parabolic wave equation for propagation in the forward direction. In the optical case, the question treated is the irradiance variance and spatial spectrum. In the ocean case, pulse propagation over long ranges is investigated. Determining the travel time of a pulse requires expanding the numerical simulation in a broadband, multifrequency calculation that takes even more time. Much effort has been expended in approximating the propagation by rays, so that the trajectory of the energy propagating from a given source to a given receiver can be tracked, and coherence calculations can be ray-based, using semi-classical formulas. This paper reviews the comparisons of several analytic ray-based approximations with numerical parabolic-equation simulations to determine their accuracies.

© 2002 Optical Society of America

## 1 Introduction

### Atmospheric Optics

1. A. Consortini, F. Cochetti, J. H. Churnside, and R. J. Hill, “Inner-scale effect on intensity variance measured for weak to strong atmospheric scintillation,” J. Opt. Soc. Am. A **10**, 2354–2362 (1993). [CrossRef]

2. S. M. Flatté, G. Y. Wang, and J. Martin, “Irradiance variance of optical waves through atmospheric turbulence by numerical simulation and comparison with experiment,” J. Opt. Soc. Am. A **10**, 2363–2370 (1993). [CrossRef]

3. R. Hill and S. Clifford, “Modified spectrum of atmospheric temperature fluctuations and its applications to optical propagation,” J. Opt. Soc. Am. **68**, 892–899 (1978). [CrossRef]

4. S. Flatté and J. Gerber, “Irradiance variance behavior for plane- and spherical-wave optical propagation through strong turbulence,” J. Opt. Soc. Am. A **17**, 1092–1097 (2000). [CrossRef]

5. R. Dashen and G.-Y. Wang, “Intensity fluctuation for waves behind a phase screen: a new asymptotic scheme,” J. Opt. Soc. Am. **A10**, 1219–1225 (1993). [CrossRef]

6. R. Dashen, G.-Y. Wang, S. M. Flatté, and C. Bracher, “Moments of intensity and log intensity: new asymptotic results for waves in power-law media,” J. Opt. Soc. Am. **A10**, 1233–1242 (1993). [CrossRef]

7. R. Dashen, G. Wang, S. M. Flatté, and C. Bracher, “Moments of intensity and log-intensity: New asymptotic results for waves in power-law media,” J. Opt. Soc. Am. A **10**, 1233–1242 (1993). [CrossRef]

9. L. Andrews, R. Phillips, C. Hopen, and M. Al-Habash, “Theory of optical scintillation,” J. Opt. Soc. Am. A **16**, 1417–1429 (1999). [CrossRef]

### Ocean Acoustics

11. S. Flatté, “Wave propagation through random media: Contributions from ocean acoustics,” Proc. of the IEEE **71**, 1267–1294 (1983). [CrossRef]

12. T. Duda, S. M. Flatté, J. Colosi, B. Cornuelle, J. Hildebrand, W. Hodgkiss Jr., P. Worcester, B. Howe, J. Mercer, and R. Spindel, “Measured wavefront fluctuations in 1000-km pulse propagation in the Pacific Ocean,” J. Acoust. Soc. Am. **92**, 939–955 (1992). [CrossRef]

13. J. Colosi, S. M. Flatté, and C. Bracher, “Internal-wave effects on 1000-km oceanic acoustic pulse propagation: Simulation and comparison with experiment,” J. Acoust. Soc. Am. **96**, 452–468 (1994). [CrossRef]

12. T. Duda, S. M. Flatté, J. Colosi, B. Cornuelle, J. Hildebrand, W. Hodgkiss Jr., P. Worcester, B. Howe, J. Mercer, and R. Spindel, “Measured wavefront fluctuations in 1000-km pulse propagation in the Pacific Ocean,” J. Acoust. Soc. Am. **92**, 939–955 (1992). [CrossRef]

*μ*light, this would correspond to a fiber of diameter 400

*μ*and length 70 mm.

11. S. Flatté, “Wave propagation through random media: Contributions from ocean acoustics,” Proc. of the IEEE **71**, 1267–1294 (1983). [CrossRef]

14. S. Reynolds, S. Flatté, R. Dashen, B. Buehler, and P. Maciejewski, “AFAR measurements of acoustic mutual coherence functions of time and frequency,” J. Acoust. Soc. Am. **77**, 1723–31 (1985). [CrossRef]

15. R. Dashen, S. Flatté, and S. Reynolds, “Path-integral treatment of acoustic mutual coherence functions for rays in a sound channel,” J. Acoust. Soc. Am. **77**, 1716–22 (1985). [CrossRef]

16. S. Flatté, S. Reynolds, R. Dashen, B. Buehler, and P. Maciejewski, “AFAR measurements of intensity and intensity moments,” J. Acoust. Soc. Am. **82**, 973–980 (1987). [CrossRef]

17. S. Flatté, S. Reynolds, and R. Dashen, “Path-integral treatment of intensity behavior for rays in a sound channel,” J. Acoust. Soc. Am. **82**, 967–972 (1987). [CrossRef]

### Rays through random media

19. J. Simmen, S. M. Flatté, and G.-Y. Wang, “Wavefront folding, chaos, and diffraction for sound propagation through ocean internal waves,” J. Acoust. Soc. Am. **102**, 239–255 (1997). [CrossRef]

## 2 The Parabolic Equation

*ψ*is the wave function,

*U*

_{0}is the deterministic (average) deviation of the sound speed from a constant. (For optical propagation through turbulence

*U*

_{0}= 0, while for acoustic propagation through the ocean,

*U*

_{0}represents the waveguide.) The quantity

*μ*(

*x*,

*y*,

*z*) is the fractional sound-speed fluctuation from zero due to the statistically fluctuating medium. Any PE equation has the critical advantage that it can be advanced through a series of range steps (in

*x*) without recourse to costly iterative or relaxation methods.[10, 20, 21

21. F. Tappert, “The parabolic approximation method,” In *Wave Propagation and Underwater Acoustics*, J. Keller and J. Papadakis, eds., pp. 224–287 (Springer-Verlag, 1977). [CrossRef]

11. S. Flatté, “Wave propagation through random media: Contributions from ocean acoustics,” Proc. of the IEEE **71**, 1267–1294 (1983). [CrossRef]

*z*and

*y*are used. In the acoustic propagation in the ocean case, only the vertical coordinate

*z*is used, because internal waves are so anisotropic. However, the ocean case is broadband, requiring of the order of 1024 frequencies, so the required computer times are similar.

## 3 Optical Propagation through Atmospheric Turbulence

*k*is the wavenumber of the propagating wave, and

*L*is the range from the source to the final screen. In the case of a spherical wave:

*n*

_{s}of size

*N*

_{s}. Also, the Fresnel length

*R*

_{f}will be necessary. Atmospheric turbulence has two parameters: its strength

*l*

_{0}. If the fluctuations are weak, the inner scale is negligible, so that it does not appear in the previous equations, but in the strong-fluctuation regime it is crucial. How to do an accurate simulation in the face of all these parameters is explained (with references) in Flatté and Gerber.[4

4. S. Flatté and J. Gerber, “Irradiance variance behavior for plane- and spherical-wave optical propagation through strong turbulence,” J. Opt. Soc. Am. A **17**, 1092–1097 (2000). [CrossRef]

### 3.1 Spatial Spectra

### 3.2 Irradiance Variances

4. S. Flatté and J. Gerber, “Irradiance variance behavior for plane- and spherical-wave optical propagation through strong turbulence,” J. Opt. Soc. Am. A **17**, 1092–1097 (2000). [CrossRef]

*a*and

*b*are different for each initial condition and inner scale. The results of these fits are shown in Tables 1 and 2. For finite inner scale, all the slopes for a given initial condition are close to being the same. Therefore lines with the average slope have been plotted over each data set corresponding to a given inner scale.

24. R. L. Fante, “Inner-scale size effect on the scintillations of light in the turbulent atmosphere,” J. Opt. Soc. Am. **73**, 277–281 (1983). [CrossRef]

25. R. Frehlich, “Intensity covariance of a point source in a random medium with a Kolmogorov spectrum and an inner scale of turbulence,” J. Opt. Soc. Am. **A4**, 360–366 (1987). [CrossRef]

*a*have been made by Andrews et al.[8, 9

9. L. Andrews, R. Phillips, C. Hopen, and M. Al-Habash, “Theory of optical scintillation,” J. Opt. Soc. Am. A **16**, 1417–1429 (1999). [CrossRef]

26. L. Andrews, M. Al-Habash, C. Hopen, and R. Phillips, “Theory of optical scintillation: Gaussian-beam wave model,” Waves Random Media **11**, 271–291 (2001). [CrossRef]

### 3.3 Empirical Power-Law Formulae

## 4 Acoustical Propagation through Ocean Internal Waves

### 4.1 Introduction

28. S. Flatté and G. Rovner, “Calculations of internal-wave-induced fluctuations in ocean-acoustic propagation,” J. Acoust. Soc. Am. **108**, 526–534 (2000). [CrossRef] [PubMed]

21. F. Tappert, “The parabolic approximation method,” In *Wave Propagation and Underwater Acoustics*, J. Keller and J. Papadakis, eds., pp. 224–287 (Springer-Verlag, 1977). [CrossRef]

30. S. Flatté and F. Tappert, “Calculation of the effect of internal waves on oceanic sound transmission,” J. Acoust. Soc. Am. **58**, 1151–1159 (1975). [CrossRef]

15. R. Dashen, S. Flatté, and S. Reynolds, “Path-integral treatment of acoustic mutual coherence functions for rays in a sound channel,” J. Acoust. Soc. Am. **77**, 1716–22 (1985). [CrossRef]

*τ*), the internal-wave-induced travel-time bias (

*τ*

_{1}), the induced spread of the acoustic pulse (

*τ*

_{0}), and the length scale characterizing the coherence of the acoustic signal at different depths (

*z*

_{0}). An important motivation for an investigation into the accuracy of these approximations is their computational efficiency and the relative simplicity of subsequent analysis. Calculations using both techniques were performed with two different sound-speed profiles – a Munk Canonical profile and a profile from the Slice89 experiment[12

12. T. Duda, S. M. Flatté, J. Colosi, B. Cornuelle, J. Hildebrand, W. Hodgkiss Jr., P. Worcester, B. Howe, J. Mercer, and R. Spindel, “Measured wavefront fluctuations in 1000-km pulse propagation in the Pacific Ocean,” J. Acoust. Soc. Am. **92**, 939–955 (1992). [CrossRef]

### 4.2 Ocean Model

31. W. H. Munk, “Sound channel in an exponentially stratified ocean, with application to SOFAR,” J. Acoust. Soc. Am. **55**, 220–226 (1974). [CrossRef]

**92**, 939–955 (1992). [CrossRef]

*U*

_{0}(

*z*) represents the dependence of the background profile,

*c*(

*z*) =

**c**

_{0}[1+

*U*

_{0}(

*z*)], on depth and

*μ*(

*x*,

*t*) contains the effect of internal waves on the speed. A useful definition of

*c*

_{0}is the average of

*c*(

*z*) over depth. Ocean internal waves are not affected by pressure. For adiabatic vertical displacements, the value of

*μ*can be expressed,[10]

*ζ*(

*x*,

*t*) is the magnitude of the displacement and [

*∂*

_{z}

*U*

_{0}(

*z*)]

_{p}is the fractional potential gradient of the sound-speed profile. (“potential” means with the pressure effect removed)

*ζ*several parameters are needed, as well as the profile of buoyancy frequency

*N*(

*z*). (See Figure 8.) (The buoyance frequency is a measure of the density gradient that allows internal waves.)

*j*is the vertical mode number,

*k*is the magnitude of the horizontal wave number (the spectrum is horizontally isotropic),

*E*= 6.3 ∙ 10

^{-5}(for the reference internal-wave energy referred to as the Garrett-Munk energy level),

*N*

_{0}= 3 cycles per hour and

*j*

_{*}= 3 are empirical constants,

*M*=

*j*

^{2}+

^{-1}.[32, 33

33. C. Garrett and W. Munk, “Space-time scales of internal waves: a progress report,” J. Geophys. Res. **80**, 291–297 (1975). [CrossRef]

34. S. Flatté and R. Esswein, “Calculation of the phase-structure function density from oceanic internal waves,” J. Acoust. Soc. Am. **70**, 1387–96 (1981). [CrossRef]

35. J. Colosi and M. Brown, “Efficient numerical simulation of stochastic internal-wave-induced sound-speed perturbation fields,” J. Acoust. Soc. Am. **103**, 2232–2235 (1998). [CrossRef]

*μ*

^{2}〉; the correlation length parallel to a given ray direction, called

*L*

_{p}; and a measure of the transverse (which means vertical) inverse correlation length squared, designated by {

*L*

_{p}has been shown to be sufficiently accurate:[28

28. S. Flatté and G. Rovner, “Calculations of internal-wave-induced fluctuations in ocean-acoustic propagation,” J. Acoust. Soc. Am. **108**, 526–534 (2000). [CrossRef] [PubMed]

*O*(100) m in the vertical.

*l*

_{υ}depends on experimental constants like

*j*

_{*}and ∫

*N*(

*z*)

*dz*.[34

34. S. Flatté and R. Esswein, “Calculation of the phase-structure function density from oceanic internal waves,” J. Acoust. Soc. Am. **70**, 1387–96 (1981). [CrossRef]

*l*

_{o}≈ 1000 m.

### 4.3 Parabolic-Equation Simulations

*z*

_{b}= 5118.75 m, was 2048. In order to implement the reflecting surface boundary condition, an image ocean was constructed above the surface for a total of 4096 vertical points.

**92**, 939–955 (1992). [CrossRef]

13. J. Colosi, S. M. Flatté, and C. Bracher, “Internal-wave effects on 1000-km oceanic acoustic pulse propagation: Simulation and comparison with experiment,” J. Acoust. Soc. Am. **96**, 452–468 (1994). [CrossRef]

### 4.4 Integral-Approximation Techniques

*ID*, that counts the number of turning points from source to receiver.) The weighting functions that enter into the integrals are not range independent, nor even periodic in range. Nevertheless, the nature of the weighting functions is such that great gains in computational speed are achievable compared with straightforward integration.[28

28. S. Flatté and G. Rovner, “Calculations of internal-wave-induced fluctuations in ocean-acoustic propagation,” J. Acoust. Soc. Am. **108**, 526–534 (2000). [CrossRef] [PubMed]

*z*

_{0}is the characteristic depth scale that will be calculated using an integral along the ray trajectory. Similarly, for changes in frequency,

*σ*is the acoustic frequency,

*τ*

^{2}gives the variance of travel time,

*τ*

_{1}is the internal-wave bias (the average difference in travel time caused by the presence of the speed disturbances), and

*τ*

_{0}describes the spreading of an intensity peak in time due to internal waves.[37

37. S. Flatté and R. Stoughton, “Predictions of internal-wave effects on ocean acoustic coherence, travel-time variance, and intensity moments for very long-range propagation,” J. Acoust. Soc. Am. **84**, 1414–1424 (1988). [CrossRef]

15. R. Dashen, S. Flatté, and S. Reynolds, “Path-integral treatment of acoustic mutual coherence functions for rays in a sound channel,” J. Acoust. Soc. Am. **77**, 1716–22 (1985). [CrossRef]

*τ*

^{2}, is calculated from the properties of internal waves in the following way,[10]

*μ*

^{2}(

*z*)〉, is a profile in depth. The quantity

*L*

_{p}(

*θ*,

*z*) expresses the correlation length of the internal waves along the ray direction.

*τ*

^{2}is the simplest of the integral approximations.

*z*

_{ray}(

*x*), and a nearby ray,

*z*(

*x*), this separation is,

**108**, 526–534 (2000). [CrossRef] [PubMed]

*τ*

_{1}, is the average change in travel time caused by the presence of sound-speed fluctuations. It is calculated according to,

*τ*

_{0}. This quantity characterizes the additional width in time of the intensity peak for a timefront ID segment, at the ray’s arrival depth, caused by the presence of internal-wave fluctuations. The pulse spreading is given by,

*τ*

_{0}is also one contribution to the estimate of the coherent bandwidth.

**108**, 526–534 (2000). [CrossRef] [PubMed]

### 4.5 Comparison of Results from Integral Expressions and the Parabolic Equation

*τ*,

*τ*

_{1},

*τ*

_{0}, and

*z*

_{0}) that have been calculated from integral approximations have been compared to corresponding results from simulations using the parabolic equation: The results obtained at 450 Hz did not differ qualitatively from those obtained at 250 Hz.

#### 4.5.1 Root-Mean-Square Travel-Time Variability: τ

*τ*as a function of range, averaged over ID, are displayed in Figure 13. The values for individual timefront ID segments at a range of 1000 km are shown in Figure 14.

**92**, 939–955 (1992). [CrossRef]

*τ*increases with internal-wave strength and also varies somewhat with the choice of sound-speed profile and timefront ID number. The statistical uncertainty in the PE values was approximately 10% for the cases employing fifty realizations.

*τ*analytic values were within statistical uncertainty of the 1000-km simulation results. Canonical

*τ*analytic values were within the statistical error of the simulation results in a few cases, at low timefront ID numbers and internal-wave strengths. In other cases they overestimated by between 20% and 60%.

#### 4.5.2 Internal-Wave-Induced Travel-Time Bias: τ_{1}

#### 4.5.3 Depth-Coherence Length: z_{0}

*z*

_{0}determined from Equation 15 averaged over ID. The parabolic-equation results are shown averaged over all ID segments in each case. The coherence functions predicted by integral methods decrease over substantially shorter length scales than the parabolic-simulation results in all cases

13. J. Colosi, S. M. Flatté, and C. Bracher, “Internal-wave effects on 1000-km oceanic acoustic pulse propagation: Simulation and comparison with experiment,” J. Acoust. Soc. Am. **96**, 452–468 (1994). [CrossRef]

#### 4.5.4 Internal-Wave-Induced Pulse Spreading: τ_{0}

*τ*

_{0}, is based on a comparison of the mean pulse shapes, 〈

*I*(

*t*–

*T*

_{i,ID})〉, from parabolic simulations with internal waves to those without internal waves. The construction of these mean pulses depends on the use of multifrequency techniques. Pulses are pictured in Figures 17 and 18 for each speed profile, an acoustic frequency of 250 Hz, and an internal-wave strength of 1 GM at a range of 1000 km. The pulse-spread approximations, calculated according to Equation 17 and averaged over each timefront ID, are included as numbers within each box.

_{0}parameter, the predicted value greatly exceeds the spreading seen in the PE results, often by an order of magnitude. An example of this situation occurs in the Slice89 profile for ID-37.

39. J. Colosi*et al*., “Comparisons of measured and predicted acoustic fluctuations for a 3250-km propagation experiment in the eastern North Pacific Ocean,” J. Acoust. Soc. Am. **105**, 3202–3218 (1999). [CrossRef]

40. J. Colosi, F. Tappert, and M. Dzieciuch, “Further analysis of intensity fluctuations from a 3252-km acoustic propagation experiment in the eastern North Pacific,” J. Acoust. Soc. Am. **110**, 163–169 (2001). [CrossRef]

*τ*

_{0}approximation was too large by more than an order of magnitude. It was also reported that the effects of internal waves on the mean pulse shape are more accurately characterized as increases in the mean intensity at times away from the maximum, rather than as spreading of the central peaks themselves.

## 5 Geometrical-Optics Rays: Energy Diffusion and Semi-classical Interference

19. J. Simmen, S. M. Flatté, and G.-Y. Wang, “Wavefront folding, chaos, and diffraction for sound propagation through ocean internal waves,” J. Acoust. Soc. Am. **102**, 239–255 (1997). [CrossRef]

19. J. Simmen, S. M. Flatté, and G.-Y. Wang, “Wavefront folding, chaos, and diffraction for sound propagation through ocean internal waves,” J. Acoust. Soc. Am. **102**, 239–255 (1997). [CrossRef]

## 6 Summary and Conclusion

*l*

_{0}< 8 mm). In the strong-focussing regime (near the variance peak) the simulations agree very well with experiment. In the strong fluctuation regime (where logarithmic scaling is observed) the simulations show scaling, but the scaling slopes do not agree at all with the analytic predictions from first order; it is known that third order is needed.

*l*

_{0}at which they should be valid, and our simulations are in the region of expected validity.[8, 9

9. L. Andrews, R. Phillips, C. Hopen, and M. Al-Habash, “Theory of optical scintillation,” J. Opt. Soc. Am. A **16**, 1417–1429 (1999). [CrossRef]

*τ*is found to be successful, to which we attribute the correctness of the basic assumptions that go into that analytical integral expression. The quantities

*τ*

_{1},

*τ*

_{0}, and

*z*

_{0}from the analytic integral expressions do not correspond to the PE results over the entire propagation range. Speculations have been made as to the reasons for this failure.[18]

*τ*

_{1}and

*τ*

_{0}) make the assumption that internal-wave effects at separation distances of order a Fresnel length are accurate. But a Fresnel length at these small frequencies is of order several hundred meters. It seems that the expressions for the medium fluctuations fail at those large separations.

*τ*

_{0}.

14. S. Reynolds, S. Flatté, R. Dashen, B. Buehler, and P. Maciejewski, “AFAR measurements of acoustic mutual coherence functions of time and frequency,” J. Acoust. Soc. Am. **77**, 1723–31 (1985). [CrossRef]

*O*(100) km, the analytic integrals appear not to be useful, with the single exception of the travel-time variance, which does not depend either on the Fresnel length or the internal-wave-induced vertical curvature of sound speed.

*τ*as a fundamental parameter in tomographic measurements of internal-wave strength.

**96**, 452–468 (1994). [CrossRef]

40. J. Colosi, F. Tappert, and M. Dzieciuch, “Further analysis of intensity fluctuations from a 3252-km acoustic propagation experiment in the eastern North Pacific,” J. Acoust. Soc. Am. **110**, 163–169 (2001). [CrossRef]

## 7 Acknowledgments

## References and links

1. | A. Consortini, F. Cochetti, J. H. Churnside, and R. J. Hill, “Inner-scale effect on intensity variance measured for weak to strong atmospheric scintillation,” J. Opt. Soc. Am. A |

2. | S. M. Flatté, G. Y. Wang, and J. Martin, “Irradiance variance of optical waves through atmospheric turbulence by numerical simulation and comparison with experiment,” J. Opt. Soc. Am. A |

3. | R. Hill and S. Clifford, “Modified spectrum of atmospheric temperature fluctuations and its applications to optical propagation,” J. Opt. Soc. Am. |

4. | S. Flatté and J. Gerber, “Irradiance variance behavior for plane- and spherical-wave optical propagation through strong turbulence,” J. Opt. Soc. Am. A |

5. | R. Dashen and G.-Y. Wang, “Intensity fluctuation for waves behind a phase screen: a new asymptotic scheme,” J. Opt. Soc. Am. |

6. | R. Dashen, G.-Y. Wang, S. M. Flatté, and C. Bracher, “Moments of intensity and log intensity: new asymptotic results for waves in power-law media,” J. Opt. Soc. Am. |

7. | R. Dashen, G. Wang, S. M. Flatté, and C. Bracher, “Moments of intensity and log-intensity: New asymptotic results for waves in power-law media,” J. Opt. Soc. Am. A |

8. | L. Andrews and R. Phillips, |

9. | L. Andrews, R. Phillips, C. Hopen, and M. Al-Habash, “Theory of optical scintillation,” J. Opt. Soc. Am. A |

10. | S. Flatté, R. Dashen, W. Munk, K. Watson, and F. Zachariasen, |

11. | S. Flatté, “Wave propagation through random media: Contributions from ocean acoustics,” Proc. of the IEEE |

12. | T. Duda, S. M. Flatté, J. Colosi, B. Cornuelle, J. Hildebrand, W. Hodgkiss Jr., P. Worcester, B. Howe, J. Mercer, and R. Spindel, “Measured wavefront fluctuations in 1000-km pulse propagation in the Pacific Ocean,” J. Acoust. Soc. Am. |

13. | J. Colosi, S. M. Flatté, and C. Bracher, “Internal-wave effects on 1000-km oceanic acoustic pulse propagation: Simulation and comparison with experiment,” J. Acoust. Soc. Am. |

14. | S. Reynolds, S. Flatté, R. Dashen, B. Buehler, and P. Maciejewski, “AFAR measurements of acoustic mutual coherence functions of time and frequency,” J. Acoust. Soc. Am. |

15. | R. Dashen, S. Flatté, and S. Reynolds, “Path-integral treatment of acoustic mutual coherence functions for rays in a sound channel,” J. Acoust. Soc. Am. |

16. | S. Flatté, S. Reynolds, R. Dashen, B. Buehler, and P. Maciejewski, “AFAR measurements of intensity and intensity moments,” J. Acoust. Soc. Am. |

17. | S. Flatté, S. Reynolds, and R. Dashen, “Path-integral treatment of intensity behavior for rays in a sound channel,” J. Acoust. Soc. Am. |

18. | S. Flatté and M. Vera, “Comparison between ocean-acoustic fluctuations in parabolic-equation simulations and estimates from integral approximations,” J. Acoust. Soc. Am. in review (2002). |

19. | J. Simmen, S. M. Flatté, and G.-Y. Wang, “Wavefront folding, chaos, and diffraction for sound propagation through ocean internal waves,” J. Acoust. Soc. Am. |

20. | F. Jensen, W. Kuperman, M. Porter, and H. Schmidt, |

21. | F. Tappert, “The parabolic approximation method,” In |

22. | S. M. Flatté, C. Bracher, and G.-Y. Wang, “Probability-density functions of irradiance for waves in atmospheric turbulence calculated by numerical simulation,” JOSA |

23. | K. Gochelashvili and V. Shishov, “Saturated fluctuations in the laser radiation intensity in a turbulent medium,” Sov. Phys. JETP |

24. | R. L. Fante, “Inner-scale size effect on the scintillations of light in the turbulent atmosphere,” J. Opt. Soc. Am. |

25. | R. Frehlich, “Intensity covariance of a point source in a random medium with a Kolmogorov spectrum and an inner scale of turbulence,” J. Opt. Soc. Am. |

26. | L. Andrews, M. Al-Habash, C. Hopen, and R. Phillips, “Theory of optical scintillation: Gaussian-beam wave model,” Waves Random Media |

27. | R. Dashen, G.Y. Wang, Stanley M. Flatté, and C. Bracher, “Moments of intensity and log-intensity: new asymptotic results from waves in power-law random media,” J. Opt. Soc. Am. A |

28. | S. Flatté and G. Rovner, “Calculations of internal-wave-induced fluctuations in ocean-acoustic propagation,” J. Acoust. Soc. Am. |

29. | R. Hardin and F. Tappert, “Applications of the Split-Step Fourier Method to the Numerical Solution of Nonlinear and Variable Coefficient Wave Equations,” Society for Industrial and Applied Mathematics Review |

30. | S. Flatté and F. Tappert, “Calculation of the effect of internal waves on oceanic sound transmission,” J. Acoust. Soc. Am. |

31. | W. H. Munk, “Sound channel in an exponentially stratified ocean, with application to SOFAR,” J. Acoust. Soc. Am. |

32. | C. Garrett and W. Munk, “Space-time scales of ocean internal waves,” Geophys. Fluid Dyn. |

33. | C. Garrett and W. Munk, “Space-time scales of internal waves: a progress report,” J. Geophys. Res. |

34. | S. Flatté and R. Esswein, “Calculation of the phase-structure function density from oceanic internal waves,” J. Acoust. Soc. Am. |

35. | J. Colosi and M. Brown, “Efficient numerical simulation of stochastic internal-wave-induced sound-speed perturbation fields,” J. Acoust. Soc. Am. |

36. | S. Flatté and G. Rovner, “Path-integral expressions for fluctuations in acoustic transmission in the ocean waveguide,” In |

37. | S. Flatté and R. Stoughton, “Predictions of internal-wave effects on ocean acoustic coherence, travel-time variance, and intensity moments for very long-range propagation,” J. Acoust. Soc. Am. |

38. | J. Colosi, Ph.D. thesis, University of California at Santa Cruz, 1993. |

39. | J. Colosi |

40. | J. Colosi, F. Tappert, and M. Dzieciuch, “Further analysis of intensity fluctuations from a 3252-km acoustic propagation experiment in the eastern North Pacific,” J. Acoust. Soc. Am. |

**OCIS Codes**

(010.1290) Atmospheric and oceanic optics : Atmospheric optics

(030.6600) Coherence and statistical optics : Statistical optics

**ToC Category:**

Focus Issue: Rays in wave theory

**History**

Original Manuscript: June 11, 2002

Revised Manuscript: July 18, 2002

Published: August 12, 2002

**Citation**

Stanley Flatte, "Calculations of wave propagation through statistical random media, with and without a waveguide," Opt. Express **10**, 777-804 (2002)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-10-16-777

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