# The Spread of a Noise Field in a Dispersive Medium

- Leon Cohen
^{1}Email author

**2010**:484753

https://doi.org/10.1155/2010/484753

© Leon Cohen. 2010

**Received: **31 January 2010

**Accepted: **17 May 2010

**Published: **17 June 2010

## Abstract

We discuss the production of induced noise by a pulse and the propagation of the noise in a dispersive medium. We present a simple model where the noise is the sum of pulses and where the mean of each pulse is random. We obtain explicit expressions for the standard deviation of the spatial noise as a function of time. We also formulate the problem in terms of a time-frequency phase space approach and in particular we use the Wigner distribution to define the spatial/spatial-frequency distribution.

## Keywords

## 1. Introduction

We then evolve into in a medium with dispersion giving (2) and calculate the appropriate ensemble averaged moments of

## 2. Pulse Propagation

### 2.1. An Exactly Solvable Case

## 3. Spread of the Wave Group

## 4. The Spread in Wave Number

Thus we see that the ensemble of the wave number moments are the moments of the individual pulse. This is the case because our model deals with random spatial translations only.

### 4.1. Example

## 5. Wigner Spectrum Approach

This expresses the Wigner spectrum at an arbitrary time given the spectrum at time zero.

which evaluates to (33).

## 6. Conclusion

where is inserted for normalization purposes. The combination of and allows us to make both the means and standard deviation of each pulse random variable.

First we point out that one can consdier the case for the ensomble as one particle having an initial mean given and an initial standard deviation

Hence for large times the spread, is a linear function of time.

## Declarations

### Acknowledgment

This paper is supported by the Office of Naval Research.

## Authors’ Affiliations

## References

- Faure P: Theoretical model of reverberation noise.
*Journal of the Acoustical Society of America*1964, 36: 259-266. 10.1121/1.1918943MathSciNetView ArticleGoogle Scholar - Ol'shveskii VV:
*Characteristics of Sea Reverberation*. Consultants Bureau, New York, NY, USA; 1967.Google Scholar - Middleton D: A statistical theory of reverberation and similar first-order scattered fields, part I.
*IEEE Transactions on Information Theory*1967, 13: 372-392.MathSciNetView ArticleMATHGoogle Scholar - Middleton D: A statistical theory of reverberation and similar first-order scattered fields, part II.
*IEEE Transactions on Information Theory*1967, 13: 393-414.MathSciNetView ArticleGoogle Scholar - Middleton D: A statistical theory of reverberation and similar first-order scattered fields, part III.
*IEEE Transactions on Information Theory*1972, 18: 35-67. 10.1109/TIT.1972.1054754MathSciNetView ArticleMATHGoogle Scholar - Middleton D: A statistical theory of reverberation and similar first-order scattered fields, part IV.
*IEEE Transactions on Information Theory*1972, 18: 68-90. 10.1109/TIT.1972.1054755MathSciNetView ArticleMATHGoogle Scholar - Cohen L: The history of noise.
*IEEE Signal Processing Magazine*2005, 22(6):20-45.View ArticleGoogle Scholar - Jackson JD:
*Classical Electrodynamics*. John Wiley & Sons, New York, NY, USA; 1992.MATHGoogle Scholar - Lighthill J:
*Waves in Fluids*. Cambridge University Press, Cambridge, UK; 1978.MATHGoogle Scholar - Morse PH, Ingard KU:
*Theoretical Acoustics*. McGraw-Hill, New York, NY, USA; 1968.Google Scholar - Tolstoy I, Clay CS:
*Ocean Acoustics: Theory and Experiment in Underwater Sound*. Acoustical Society of America; 1966.Google Scholar - Whitham GB:
*Linear and Nonlinear Waves*. John Wiley & Sons, New York, NY, USA; 1974.MATHGoogle Scholar - Graff K:
*Wave Motion in Elastic Solids*. Oxford University Press, Oxford, UK; 1975.MATHGoogle Scholar - Cohen L: Why do wave packets sometimes contract?
*Journal of Modern Optics*2002, 49(14-15):2365-2382. 10.1080/0950034021000011248MathSciNetView ArticleMATHGoogle Scholar - Cohen L, Loughlin P, Okopal G: Exact and approximate moments of a propagating pulse.
*Journal of Modern Optics*2008, 55(19-20):3349-3358. 10.1080/09500340802428280View ArticleMATHGoogle Scholar - Mark WD: Spectral analysis of the convolution and filtering of non-stationary stochastic processes.
*Journal of Sound and Vibration*1970, 11(1):19-63. 10.1016/S0022-460X(70)80106-7MathSciNetView ArticleMATHGoogle Scholar - Martin W: Time-frequency analysis of random signals.
*Proceedings of the International Conference on Acoustics, Speech and Signal Processing (ICASSP '82), 1982, Paris, France*1325-1328.View ArticleGoogle Scholar - Martin W, Flandrin P: Wigner-ville spectral analysis of nonstationary processes.
*IEEE Transactions on Acoustics, Speech, and Signal Processing*1985, 33(6):1461-1470. 10.1109/TASSP.1985.1164760View ArticleGoogle Scholar - Cohen L:
*Time-Frequency Analysis*. Prentice-Hall, Upper Saddle River, NJ, USA; 1995.Google Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.