Published in IET Signal Processing Received on 13th May 2009 Revised on 25th November 2009 doi: 10.1049/iet-spr.2009.0132 Special Issue on Time-Frequency Approach to Radar Detection, Imaging, and Classification ISSN 1751-9675 Time –frequency approach to radar, sonar and seismic wave propagation with dispersion and attenuation L. Cohen City University of New York, 695 Park Avenue, New York, NY 10021, USA E-mail: leon.cohen@hunter.cuny.edu Abstract: In many physical situations that involve wave propagation there is dispersion and attenuation. Examples include sonar in shallow water, underground radar, seismic wave propagation, fibre optics, among many others. The author shows that phase-space methods are particularly suited to study propagation with dispersion since in such situations the velocity of propagation is frequency dependent. Depending on the situation the phase space may be time–frequency or position–wavenumber. The author derive explicit expressions for the Wigner distribution for both cases and how it evolves with time are derived. The application to the propagation of noise fields is also discussed. 1 Introduction Wave propagation in matter and waveguides often exhibits dispersion and attenuation and the fundamental characteristic of such propagation is that different frequencies travel at different speeds and that attenuation is frequency dependent. Examples of situations where dispersion occurs are underwater acoustics, particularly in shallow water, seismic wave propagation, underground penetration radar and propagation in fibres. Since such dispersive propagation is frequency dependent, time – frequency analysis should be an effective method [1, 2]. In this article we show how phase space methods can be effectively used to study pulse propagation. As we will see the phase space can be time – frequency or position – wavenumber and which one is chosen depends on the physical situation. We present results for both cases. In addition, often one wants to study the propagation of noise in a dispersive medium. For example, suppose noise is produced at a spatial point and one asks for the statistics of the noise at different spatial points and for different times. We show that phase-space methods are particularly effective. 2 Dispersion and attenuation We now review how dispersion and attenuation arise from linear wave equations. Consider the general differential equation with constant coefficients: X N n¼0 a n @ n u @t n ¼ X M n¼0 b n @ n u @x n (1) Substituting e ikxivt into (1) gives an algebraic equation [3–7] X N n¼0 a n (iv) n ¼ X M n¼0 b n (ik) n (2) One solves for v as a function of k or k as a function of v v(k) ¼ v R (k) þ iv I (k) (3) k(v) ¼ k R (v) þ ik I (v) (4) where we have explicitly written the solutions in terms of real and imaginary parts. Furthermore, it is generally the case that there is more than one solution to (2) and each solution is called a mode. If the physical situation is such that we know u(x, t 0 ) at time t 0 then the solution for a given mode is [3–7] u(x, t ) ¼ 1 ffiffiffiffiffiffi 2p p ð S(k; t 0 )e ikxiv(k)(t t 0 ) dk (5) where S(k, t 0 ) is obtained from the initial pulse, u(x, t 0 ), by IET Signal Process., 2010, Vol. 4, Iss. 4, pp. 421–427 421 doi: 10.1049/iet-spr.2009.0132 & The Institution of Engineering and Technology 2010 www.ietdl.org