Note on group velocity and energy propagation

American Journal of Physics

Volumn 68, Issue 5, 2000, Pages 482-484

  Bers, Abraham ^a  

^a MASSACHUSETTS INSTITUTE OF TECHNOLOGY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0034396164     PISSN: 00029505     EISSN: None     Source Type: Journal    
DOI: 10.1119/1.19471     Document Type: Article

References (12)

1. K. M. Awati and T. Howes, Am. J. Phys. 64 (11), 1353 (1996).
2. K. T. McDonald, C. S. Helrich, R. J. Mathar, S. Wong, and D. Styer, Am. J. Phys. 66 (8), 656-661 (1998).
3. Well-defined wave propagation in a dispersive medium is understood to take place in a regime of weak dissipation where the modes are characterized by frequencies and wave numbers that are essentially real. It is only in these regimes that the concepts of group velocity, time-, or space-averaged energies and their flows, and thus energy velocity, are well-defined. In thus considering these concepts, one can idealize the situation by considering the medium to be "nondissipative" as we do in the following.
4. Covariant electrodynamic formulations of average wave energy and momentum and their flows in linear media were given for temporally dispersive media by S. M. Rytov, JETP 17, 930 (1947), and generalized to spatially and temporally dispersive media by M. E. Gertsenshtein, ibid. 26, 680 (1954). Independently, a simpler formulation of average wave energy and energy flow in linear media with spatial and temporal dispersion was given by A. Bers in 1962; see W. P. Allis, S. J. Buchsbaum, and A. Bers, Waves in Anisotropic Plasmas (MIT, Cambridge, MA, 1963), Sec. 8.5. As shown in this last reference, the equality between group velocity and energy velocity follows simply from a variational form of Maxwell's equations for an arbitrarily dispersive medium; this general proof is the basis of this note.
5. Covariant electrodynamic formulations of average wave energy and momentum and their flows in linear media were given for temporally dispersive media by S. M. Rytov, JETP 17, 930 (1947), and generalized to spatially and temporally dispersive media by M. E. Gertsenshtein, ibid. 26, 680 (1954). Independently, a simpler formulation of average wave energy and energy flow in linear media with spatial and temporal dispersion was given by A. Bers in 1962; see W. P. Allis, S. J. Buchsbaum, and A. Bers, Waves in Anisotropic Plasmas (MIT, Cambridge, MA, 1963), Sec. 8.5. As shown in this last reference, the equality between group velocity and energy velocity follows simply from a variational form of Maxwell's equations for an arbitrarily dispersive medium; this general proof is the basis of this note.
6. Covariant electrodynamic formulations of average wave energy and momentum and their flows in linear media were given for temporally dispersive media by S. M. Rytov, JETP 17, 930 (1947), and generalized to spatially and temporally dispersive media by M. E. Gertsenshtein, ibid. 26, 680 (1954). Independently, a simpler formulation of average wave energy and energy flow in linear media with spatial and temporal dispersion was given by A. Bers in 1962; see W. P. Allis, S. J. Buchsbaum, and A. Bers, Waves in Anisotropic Plasmas (MIT, Cambridge, MA, 1963), Sec. 8.5. As shown in this last reference, the equality between group velocity and energy velocity follows simply from a variational form of Maxwell's equations for an arbitrarily dispersive medium; this general proof is the basis of this note.
7. See, for example, L. Brillouin, Wave Propagation and Group Velocity (Academic, New York, 1960), Chap. I.
8. r)], (8) and (9) were derived independently by several authors; see Ref. 4 above, and the treatment by Bers in Ref. 8 below.
9. g that exceed the speed of light and hence, in such regions, the group velocity no longer represents the velocity of energy flow; energy cannot travel at such speeds.
10. A. Bers, "Linear Waves and Instabilities," in Plasma Physics - Les Houches 1972, edited by C. DeWitt and J. Peyraud (Gordon and Breach, New York, London, Paris, 1975), Secs. II and VII; I. B. Bernstein, "Geometric Optics in Space- and Time-Varying Plasmas," Phys. Fluids 18, 320 (1975).
11. A. Bers, "Linear Waves and Instabilities," in Plasma Physics - Les Houches 1972, edited by C. DeWitt and J. Peyraud (Gordon and Breach, New York, London, Paris, 1975), Secs. II and VII; I. B. Bernstein, "Geometric Optics in Space- and Time-Varying Plasmas," Phys. Fluids 18, 320 (1975).
12. A. Bers, "Space-Time Evolution of Plasma Instabilities - Absolute and Convective" in Handbook of Plasma Physics, general editors, M. N. Rosenbluth and R. Z. Sagdeev, Vol. I Basic Plasma Physics, volume editors, A. A. Galeev and R. N. Sudan (North-Holland, Amsterdam, 1983), Chap. 3.2, Sec. 3.2.3.