banner image
No data available.
Please log in to see this content.
You have no subscription access to this content.
No metrics data to plot.
The attempt to load metrics for this article has failed.
The attempt to plot a graph for these metrics has failed.
Validity of the Markov approximation in ocean acousticsa)
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)].
Rent this article for
View: Figures


Image of FIG. 1.
FIG. 1.

(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 .

Image of FIG. 2.
FIG. 2.

(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.

Image of FIG. 3.
FIG. 3.

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.

Image of FIG. 4.
FIG. 4.

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 .

Image of FIG. 5.
FIG. 5.

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.

Image of FIG. 6.
FIG. 6.

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.

Image of FIG. 7.
FIG. 7.

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 .

Image of FIG. 8.
FIG. 8.

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.

Image of FIG. 9.
FIG. 9.

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.


Article metrics loading...


Full text loading...

This is a required field
Please enter a valid email address
752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Validity of the Markov approximation in ocean acousticsa)