The Spread of a Noise Field in a Dispersive Medium
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Research Article The Spread of a Noise Field in a Dispersive Medium Leon Cohen City University of New York, 695 Park Avenue, New York, NY 10021, USA Correspondence should be addressed to Leon Cohen, [email protected] Received 31 January 2010; Accepted 17 May 2010 Academic Editor: Aydin Akan Copyright © 2010 Leon Cohen. 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. 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.
1. Introduction In many situations noise is induced by a pulse due to the scattering of the pulse from many sources. To take a specific example, consider a pulse that hits a school of fish; each fish scatters the wave and the acoustic field seen at an arbitrary point is the sum of the waves received from each fish; the sum of which is noise like. Another example is the creation of a set of bubbles by a propeller in a finite region in space. When each bubble explodes it produces a wave and the acoustic pressure seen is the sum of the waves produced by each bubble. Of course this is a simplified view since there could be many other effects such as multiple scattering. Now as time evolves the noise field is propagating and can be changing in a significant way if the medium has dispersion. Our aim here is to investigate how a noise field which is composed of a group of pulses behave as it is propagating and in particular we want to investigate the spreading of the field as a function of time. Suppose we consider a space-time signal composed of the sum of elementary signal, un , ψ(x, t) = A
N
an un (x, t; εn ),
(1)
n=1
where un is a deterministic function and an and εn are random parameters. We have put in an overall A for normalization convenience. The production of noise by expressions like (1) are sometimes called FOM models of noise production [1–7]. Here, we consider the case where the only random variables are the means of each of the
elementary signals and where all un are the same. Hence we write N
ψ(x, t) = A
an u(x − xn , t),
(2)
n=1
where xn are the means of the elementary signals, assuming that the mean of u(x, t) is zero. The approach we take is the following. At time t = 0 we form an ensemble of signals ψ(x, 0) = A
N
an u(x − xn , 0).
(3)
n=1
We then evolve u(x − xn , 0) into u(x − xn , t) in a medium with dispersion giving (2) and calculate the appropriate ensemble averaged moments of ψ(x, t). For the elementary signal we define the moments in the standard
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