^{1}, Purnima Ratilal

^{2}and Nicholas C. Makris

^{3}

### Abstract

The mean and variance of the acoustic field forward propagated through a stratified ocean waveguide containing three-dimensional (3-D) random internal waves is modeled using an analytic normal mode formulation. The formulation accounts for the accumulated effects of *multiple forward scattering*. These lead to redistribution of both coherent and incoherent modal energies, including attenuation and dispersion. The inhomogeneous medium’s scatter function density is modeled using the Rayleigh-Born approximation to Green’s theorem to account for random fluctuations in both density and compressibility caused by internal waves. The generalized waveguideextinction theorem is applied to determine attenuation due to scattering from internal wave inhomogeneities. Simulations for typical continental-shelf environments show that when internal wave height exceeds the acoustic wavelength, the acoustic field becomes so randomized that the expected total intensity is dominated by the field variance beyond moderate ranges. This leads to an effectively saturated field that decays monotonically and no longer exhibits the periodic range-dependent modal interference structure present in nonrandom waveguides. Three-dimensional scatteringeffects can become important when the Fresnel width approaches and exceeds the cross-range coherence length of the internal wave field. Density fluctuations caused by internal waves are found to noticably affect acoustic transmission in certain Arctic environments.

I. INTRODUCTION

II. FORMULATION IN A TWO-LAYER WATER COLUMN

A. Statistical description of internal waves

1. Joint spatial probability density of internal wave displacement

2. Linear internal wave field as a stationary random process

B. Scatter function of an internal wave inhomogeneity

1. Mean and correlation function of the scatter function density of internal wave inhomogeneities

2. Isotropic internal waves

C. Statistical moments of the forward propagated field

D. Modal attenuation from generalized waveguideextinction theorem

III. COMPUTING 2-D SPATIAL REALIZATIONS OF A RANDOM INTERNAL WAVE FIELD

IV. ILLUSTRATIVE EXAMPLES

A. Mid-latitude Atlantic continental shelf environment

B. Effect of internal wave density fluctuations on acoustic transmission in an Arctic environment

C. Comparison of 3-D analytic model with 2-D Monte-Carlo simulations

V. CONCLUSION

### Key Topics

- Internal waves
- 130.0
- Wave attenuation
- 37.0
- Coherence
- 29.0
- Acoustical effects
- 21.0
- Acoustic waveguides
- 20.0

## Figures

Geometry of mid-latitude Atlantic continental shelf and Arctic environments with two-layer water column of total depth , and upper layer depth of . The bottom sediment half-space is composed of sand. The internal wave disturbances have coherence length scales and in the and directions, respectively, and are measured with positive height measured downward from the interface between the upper and lower water layers.

Geometry of mid-latitude Atlantic continental shelf and Arctic environments with two-layer water column of total depth , and upper layer depth of . The bottom sediment half-space is composed of sand. The internal wave disturbances have coherence length scales and in the and directions, respectively, and are measured with positive height measured downward from the interface between the upper and lower water layers.

(a) Normalized spectrum of internal wave field over an observation period of approximately 10 min with minimum wave number . The spectrum is computed using Eq. (22) with a dependence at high frequencies. (b) Correlation function of the isotropic internal wave field with a coherence length of 100 m. The correlation function was obtained from the inverse Fourier transform of the internal wave spectrum plotted in (a).

(a) Normalized spectrum of internal wave field over an observation period of approximately 10 min with minimum wave number . The spectrum is computed using Eq. (22) with a dependence at high frequencies. (b) Correlation function of the isotropic internal wave field with a coherence length of 100 m. The correlation function was obtained from the inverse Fourier transform of the internal wave spectrum plotted in (a).

(a) Fresnel half-width for receiver ranges and . The Fresnel width is approximately equal to , where is the source-receiver separation and is the range from source to inhomogeneity. (b) The maximum Fresnel width as a function of source-receiver separation .

(a) Fresnel half-width for receiver ranges and . The Fresnel width is approximately equal to , where is the source-receiver separation and is the range from source to inhomogeneity. (b) The maximum Fresnel width as a function of source-receiver separation .

Acoustic field intensity at 415 Hz as a function of range and depth in the mid-latitude Atlantic continental shelf waveguide of Fig. 1, when there are no internal waves present so that the waveguide is undisturbed. The boundary between the warm and cool water is at the depth of 30 m from the water surface in this static waveguide. The source is at 50 m depth with source level 0 dB *re* at 1 m. The acoustic intensity exhibits range- and depth-dependent variations due to coherent interference between waveguide modes.

Acoustic field intensity at 415 Hz as a function of range and depth in the mid-latitude Atlantic continental shelf waveguide of Fig. 1, when there are no internal waves present so that the waveguide is undisturbed. The boundary between the warm and cool water is at the depth of 30 m from the water surface in this static waveguide. The source is at 50 m depth with source level 0 dB *re* at 1 m. The acoustic intensity exhibits range- and depth-dependent variations due to coherent interference between waveguide modes.

Intensities of the (a) mean or coherent field, (b) variance or incoherent field, and (c) the total field at 415 Hz as functions of range and depth in the mid-latitude Atlantic continental shelf waveguide of Fig. 1 when there is a random internal wave field present in the waveguide. The internal wave disturbances have a height standard deviation of and coherence lengths of . The source is at 50 m depth with source level 0 dB *re* at 1 m. This medium is only slightly random and the total intensity in (c) is dominated by the coherent intensity out to 50 km range and exhibits range- and depth-dependent variations due to coherent interference between waveguide modes, similar to the static waveguide example in Fig. 4. Figure 5(d) shows the acoustic intensity as a function of range at a single receiver depth of 50 m for the fields shown in (a)–(c). For a comparison, the acoustic intensity of the static waveguide is also plotted.

Intensities of the (a) mean or coherent field, (b) variance or incoherent field, and (c) the total field at 415 Hz as functions of range and depth in the mid-latitude Atlantic continental shelf waveguide of Fig. 1 when there is a random internal wave field present in the waveguide. The internal wave disturbances have a height standard deviation of and coherence lengths of . The source is at 50 m depth with source level 0 dB *re* at 1 m. This medium is only slightly random and the total intensity in (c) is dominated by the coherent intensity out to 50 km range and exhibits range- and depth-dependent variations due to coherent interference between waveguide modes, similar to the static waveguide example in Fig. 4. Figure 5(d) shows the acoustic intensity as a function of range at a single receiver depth of 50 m for the fields shown in (a)–(c). For a comparison, the acoustic intensity of the static waveguide is also plotted.

Similar to Fig. 5, but for a waveguide with an internal wave height standard deviation of . This medium is highly random and the total intensity in (c) is dominated by the variance or incoherent intensity beyond the 11 km range. The total acoustic intensity decays monotonically as a function of range at sufficiently long ranges since the field is now completely incoherent and the waveguide loses the coherent range- and depth-dependent variations due to modal interference.

Similar to Fig. 5, but for a waveguide with an internal wave height standard deviation of . This medium is highly random and the total intensity in (c) is dominated by the variance or incoherent intensity beyond the 11 km range. The total acoustic intensity decays monotonically as a function of range at sufficiently long ranges since the field is now completely incoherent and the waveguide loses the coherent range- and depth-dependent variations due to modal interference.

Contributions of the waveguide modes to the depth-integrated total intensity of the forward field at (a) the source location, (b) 1 km, (c) 10 km, and (d) 50 km ranges from the source for a source strength of 0 dB *re* at 1 m. All values are absolute except those in (a), which are normalized by the maximum modal contribution.

Contributions of the waveguide modes to the depth-integrated total intensity of the forward field at (a) the source location, (b) 1 km, (c) 10 km, and (d) 50 km ranges from the source for a source strength of 0 dB *re* at 1 m. All values are absolute except those in (a), which are normalized by the maximum modal contribution.

Acoustic intensity at a single receiver depth of 50 m in the presence of an internal wave field with coherence lengths and height standard deviations of (a) and (b) .

Acoustic intensity at a single receiver depth of 50 m in the presence of an internal wave field with coherence lengths and height standard deviations of (a) and (b) .

Effect of (a) including and (b) neglecting internal wave density fluctuations on acoustic transmission in an Arctic waveguide with geometry described in Fig. 1. The internal wave field has coherence lengths of and a height standard deviation of . The acoustic intensity is plotted as a function of range for the source and receiver at 50 m depth.

Effect of (a) including and (b) neglecting internal wave density fluctuations on acoustic transmission in an Arctic waveguide with geometry described in Fig. 1. The internal wave field has coherence lengths of and a height standard deviation of . The acoustic intensity is plotted as a function of range for the source and receiver at 50 m depth.

Comparison of intensities from 2-D Monte-Carlo simulations and 3-D analytical model at the single receiver depth of 50 m in the presence of an internal wave field with height standard deviation of . A total of 1000 simulations were made using the parabolic equation to compute the 2-D Monte Carlo field statistics. (a) Coherent field comparison, (b) incoherent field comparison, (c) total field comparison, (d) only the 2-D Monte-Carlo simulated acoustic intensities of the coherent, incoherent, and total fields used in (a)–(c).

Comparison of intensities from 2-D Monte-Carlo simulations and 3-D analytical model at the single receiver depth of 50 m in the presence of an internal wave field with height standard deviation of . A total of 1000 simulations were made using the parabolic equation to compute the 2-D Monte Carlo field statistics. (a) Coherent field comparison, (b) incoherent field comparison, (c) total field comparison, (d) only the 2-D Monte-Carlo simulated acoustic intensities of the coherent, incoherent, and total fields used in (a)–(c).

Similar to Fig. 10, but for a waveguide with an internal wave height standard deviation of .

Similar to Fig. 10, but for a waveguide with an internal wave height standard deviation of .

Total depth-integrated intensities for the waveguide used in Fig. 4. The static case with no internal waves in the medium is compared to the 3-D analytical model and 2-D Monte-Carlo simulations with internal wave height standard deviations of and . The attenuation or power loss due to scattering is most significant in the 3-D analytical model for the highly random waveguide.

Total depth-integrated intensities for the waveguide used in Fig. 4. The static case with no internal waves in the medium is compared to the 3-D analytical model and 2-D Monte-Carlo simulations with internal wave height standard deviations of and . The attenuation or power loss due to scattering is most significant in the 3-D analytical model for the highly random waveguide.

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