^{a)}

^{a)}Portions of this work were presented in “Some validity issues in the theory and modeling of WPRM,” by Terry Ewart and Frank Henyey at the 143rd Meeting of the Acoustical Society of America [J. Acoust. Soc. Am. 111, 2351 (2002)].

^{1,c)}and Terry E. Ewart

^{1,d)}

### Abstract

Moment equations and path integrals for wave propagation in random media have been applied to many ocean acoustics problems. Both these techniques make use of the Markov approximation. The expansion parameter, which must be less than one for the Markov approximation to be valid, is the subject of this paper. There is a standard parameter (the Kubo number) which various authors have shown to be sufficient. Fourth moment equations have been successfully used to predict the experimentally measured frequency spectrum of intensity in the mid-ocean acoustic transmission experiment (MATE). Yet, in spite of this success, the Kubo number is greater than 1 for the measured index of refraction variability for MATE, arriving at a contradiction. Here, that contradiction is resolved by showing that the Kubo parameter is far too pessimistic for the ocean case. Using the methodology of van Kampen, another parameter is found which appears to be both necessary and sufficient, and is much smaller than the Kubo number when phase fluctuations are dominated by large scales in the medium. This parameter is shown to be small for the experimental regime of MATE, justifying the applications of the moment equations to that experiment.

The authors thank Barry Uscinski for conversations and Darrell Jackson for suggestions on improving the paper. This work was supported by the Office of Naval Research.

I. INTRODUCTION

II. EXPERIMENTAL VALUES

III. THE EFFECT OF LARGE VERTICAL SCALES IN THE OCEAN

IV. DERIVATION OF THE NEW EXPANSION PARAMETER

V. EVALUATION OF THE NEW PARAMETER FOR MATE

VI. DISCUSSION AND CONCLUSIONS

### Key Topics

- Internal waves
- 18.0
- Integral equations
- 14.0
- Correlation functions
- 12.0
- Speed of sound
- 12.0
- Oceans
- 11.0

## Figures

(Color online) Local bathymetry from the precision depth recorder, upper path and lower path eigenrays traced using the mean sound speed profile, and potential density contours taken with an autonomous vehicle, SPURV, depth cycling over the lower ray. The grayscale are equally spaced, with a total range of .

(Color online) Local bathymetry from the precision depth recorder, upper path and lower path eigenrays traced using the mean sound speed profile, and potential density contours taken with an autonomous vehicle, SPURV, depth cycling over the lower ray. The grayscale are equally spaced, with a total range of .

(Color online) Spectra of the moored displacement and travel time measured during MATE (normalized to integral one). The fit of the model to the moored spectrum and its prediction for the travel time spectrum are shown.

(Color online) Spectra of the moored displacement and travel time measured during MATE (normalized to integral one). The fit of the model to the moored spectrum and its prediction for the travel time spectrum are shown.

Intensity for the four frequencies of MATE taken during the last 66 hours of the experiment with equal time records of the lower path and upper path results.

Intensity for the four frequencies of MATE taken during the last 66 hours of the experiment with equal time records of the lower path and upper path results.

Intensity spectra for the four frequencies of MATE with the moment theory predictions of ^{ Macaskill and Ewart (1996) } superimposed. The inertial frequency is , and the buoyancy frequency is .

Intensity spectra for the four frequencies of MATE with the moment theory predictions of ^{ Macaskill and Ewart (1996) } superimposed. The inertial frequency is , and the buoyancy frequency is .

Expansion parameter, , according to the literature for 2, 4, and computed using the MATE lower path oceanographic data. Since these values are above 1, the Markov approximation is called into question.

Expansion parameter, , according to the literature for 2, 4, and computed using the MATE lower path oceanographic data. Since these values are above 1, the Markov approximation is called into question.

A diagrammatic representation of Eq. (29) . The horizontal represents range, while the vertical does not have a similar meaning. The solid line represents the factor for propagation of the acoustic field, for mode . The dots indicate interaction factors , and the dashed line represents that these interactions are to be correlated. Not indicated is the integration over the range variable with , nor is the factor in front of the , integral, that relates the unscattered field at to that at , indicated. In Eq. (29) , the correlation function is expressed in terms of the spectrum.

A diagrammatic representation of Eq. (29) . The horizontal represents range, while the vertical does not have a similar meaning. The solid line represents the factor for propagation of the acoustic field, for mode . The dots indicate interaction factors , and the dashed line represents that these interactions are to be correlated. Not indicated is the integration over the range variable with , nor is the factor in front of the , integral, that relates the unscattered field at to that at , indicated. In Eq. (29) , the correlation function is expressed in terms of the spectrum.

The three ways of pairing four interactions. The fourth moment of the internal wave field can be expressed in terms of correlated pairs because of the assumption that it comprises a zero mean Gaussian process. These diagrams can be interpreted as parts of the contribution from the medium fourth moment, using the same interpretation as in Fig. 6 . Integrations over , , are understood with .

The three ways of pairing four interactions. The fourth moment of the internal wave field can be expressed in terms of correlated pairs because of the assumption that it comprises a zero mean Gaussian process. These diagrams can be interpreted as parts of the contribution from the medium fourth moment, using the same interpretation as in Fig. 6 . Integrations over , , are understood with .

A diagrammatic representation of how the three contributions of Fig. 7 occur in the solution, Eq. (39) , of Eq. (18) with Eq. (29) (Fig. 6 ) for . No interactions occur between a correlated pair. The correction to Eq. (29) is the difference between Figs. 7 and Fig. 8 . The contribution is correctly given, whereas and are different in Figs. 7 and 8 . The diagonal lines represent “backward” propagation. For example, in Fig. 8(b) the factor is . van Kampen gives a purely algebraic method of constructing the correction, so the diagrams should be taken as only interpretational, not as necessary for obtaining the expressions.

A diagrammatic representation of how the three contributions of Fig. 7 occur in the solution, Eq. (39) , of Eq. (18) with Eq. (29) (Fig. 6 ) for . No interactions occur between a correlated pair. The correction to Eq. (29) is the difference between Figs. 7 and Fig. 8 . The contribution is correctly given, whereas and are different in Figs. 7 and 8 . The diagonal lines represent “backward” propagation. For example, in Fig. 8(b) the factor is . van Kampen gives a purely algebraic method of constructing the correction, so the diagrams should be taken as only interpretational, not as necessary for obtaining the expressions.

Estimate of the modified expansion parameter computed from the MATE oceanographic data for the lower path. These values are below 1, so the Markov approximation is valid. The vanishing at the center of the path is not to be understood as the Markov approximation being better there. Rather, the fourth order happens to have a factor multiplying the expansion parameter that is small there.

Estimate of the modified expansion parameter computed from the MATE oceanographic data for the lower path. These values are below 1, so the Markov approximation is valid. The vanishing at the center of the path is not to be understood as the Markov approximation being better there. Rather, the fourth order happens to have a factor multiplying the expansion parameter that is small there.

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