The Spread of a Noise Field in a Dispersive Medium
© Leon Cohen. 2010
Received: 31 January 2010
Accepted: 17 May 2010
Published: 17 June 2010
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.
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.
5. Wigner Spectrum Approach
This expresses the Wigner spectrum at an arbitrary time given the spectrum at time zero.
which evaluates to (33).
This paper is supported by the Office of Naval Research.
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