Wave packets, rays, and the role of real group velocity in absorbing media

Autor: Elazar Sonnenschein, Dan Censor, I. Rutkevich
Rok vydání: 1998
Předmět:
Zdroj: Physical Review E. 57:1005-1016
ISSN: 1095-3787
1063-651X
DOI: 10.1103/physreve.57.1005
Popis: In an absorbing medium, where the vector W5]v/]k usually is complex for real values of the wave vector k, the group velocity W may become real for some complex values of k. The role of real group velocity in the propagation of one-dimensional wave packets in homogeneous absorbing media is examined. Applying the saddle point method to an analysis of the asymptotic behavior of the Gaussian wave packets shows that for absorbing media, at large times and distances, the real group velocity appears as a local characteristic of any small section of a wave packet. For each section we can find the complex values of the local wave number and the local frequency defining a real group velocity. Thus, the real group velocity concepts in absorbing media do not have to be based on the signals having real wave vectors or real frequencies. The analysis of the exact solution for a Gaussian wave packet in a medium with a complex law of dispersion describing whistler waves in a collisional plasma is performed. It is shown that at all times the initial carrier wave number exists as a real part of the local complex wave number at some point of the Gaussian envelope and this point moves with a constant real group velocity. For large times the local wave group with the initial carrier wave number can be found far away from the envelope center. @S1063-651X~98!05601-3# The concepts of wave packet~s !~ WP!, group velocity, and ray tracing come up in many areas of physics—quantum mechanics, optics, plasma physics, fluid mechanics, solid state physics, geophysics, and astrophysics. The group velocity concept seems to have been first introduced by Rayleigh @1# for the transverse sound waves propagating in thin elastic rods. Since then this concept was applied to studying WP and signals based on various kinds of waves in dispersive, nonabsorbing media. The theory of WP in dispersive media without absorption of the wave energy has been amply discussed in the literature @2,3#. As is well known, any WP in a homogeneous medium is constructed by the continuous superposition ~integration! of the elementary plane waves, sinusoidal in both space and time, with neighboring values of the wave vector k and the frequency v. Usually such a superposition is presented in the form of the integral of the function A(k)exp@i(kix2vt)# in the k space. The dispersion equation D(v,k)50 characterizes the properties of the medium with respect to wave propagation. The spatial Fourier transform of the WP at t50 is the function A(k) concentrated in some vicinity of the carrier wave vector kc . The spatial maximum of the WP envelope propagates with the group velocity W5]v/]k calculated at k5kc . The concept of rays in the theory of WP propagation in nonabsorbing media appears in studies of the asymptotic behavior of the WP for large values of t and uxu. A powerful tool that leads to the asymptotic formulas for the WP solutions is the saddle point ~SP! method for the Fourier integrals. Whitham @3# shows that for large times each small section of the WP can be characterized by the instantaneous values of the local wave vector k (x,t) and the local frequency v (x,t)5v@k (x,t)#, where the dependence v 5v(k) is determined by the dispersion equation D(v,k) 50. These local characteristics of the WP remain constant along the straight-line trajectories x5X(t) termed the rays. The value of the local wave vector k at the point x5xm(t) corresponding to the spatial maximum of the WP envelope equals kc . The description of the rays associated with the WP propagation admits the Hamiltonian formalism: the vectors X(t) and k determine the position of the dynamic system in the configuration ~coordinate! space and in representation ~momentum! space, respectively, while the frequency
Databáze: OpenAIRE
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