- Research Article
- Open Access

# 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

- Ensemble Average
- Random Parameter
- Elementary Signal
- Dispersive Medium
- Spatial Spectrum

## 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

In the above we have assumed that and are independent.

where is given by (7).

## 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

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## 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.