NOTES AND DISCUSSIONS Note on group velocity and energy propagation Abraham Bers Department of Electrical Engineering Computer Science and Plasma Science Fusion Center, (Received 12 January 1999; accepted 10 June 1999) A general proof is given for the equality between group velocity and energy velocity for linear wave propagation in a homogeneous medium with arbitrary spatial and temporal dispersion. C 2000 American Association of Physics Teachers. t In the series of physics"Questions"in the American function. The Fourier-Laplace transform of this convolution urnal of Physics, K. M. Awati and T. Howes ask for a relationship is expressed by the conductivity tensor function general proof of the fact that in linear dispersive wave propa- of wave vector k and frequency w as gation, energy propagates at the group velocity. Several an- swers to this question have appeared recently, but a general J(k, o)=o(k, o).E(k,o) nondissipative medium, such a general proof is indeed The other linear response functions have similar interpreta- media. It can be readily argued that this type of proof can be =o(k, o)(-iwEo): K(k,)=I+X(k, o); E(k, w) carried out for any other(than electrodynamic)description of Eok(k, o). The Fourier-Laplace transform of Maxwells dynamics of a)medium. In an electrodynamic formulation, equations for the self-consistent electromagnetic fields are a stable, nondissipative, linear (or, more generally, linearized favor of the electromagnetic fields; one can easily visualize kXE=OLoH (2) doing the opposite. For exposing the simplest electrod- namic proof, I will focus on a nonmagnetic medium uoH) which is arbitrarily (spatially and temporally k×=-0e0k(k,ω)·E spersive-a plasma. The generalization to gnetic media is straightforward Taking kx(2)and using(3) to eliminate H, one find he continuum electrodynamics of a plasma-like mediu omogeneous set of equations for E Is described by Maxwells equations for the electromagnetic fields e andh. wherein the collective " mechanic D(ko).E=0 namics of the medium as a function of E and H are ex- where the dispersion tensor D is pressed by electric current and electric charge densities (, P). The latter, expressed as functions of E and H, are D(k, o)=kk-k21+-K(k,o) what one can call the electrodynamic response functions of the medium. For a plasma-like medium, since B=HoH, For nontrivial solutions of (4), Faradays equation provides a way of eliminating H in favor det[ d(k,o)]=D(k, w)=0 (6) of E, so that the response functions are only functions of E In addition, since J and p are related by the continuity equa- which is the dispersion relation giving, e.g., o(k). These are tion, the mechanical dynamics, regardless of the particular the natural modes of the system, with fields whose space model chosen for describing the dynamics, can be expressed time dependence exp[i(k-r-of)] is constrained by the disper by a single"electrical"response(or"influence")function, sion relation(6). Natural modes that are purely propagating e.g., the conductivity tensor o, or the susceptibility tensor x, waves are those for which solutions of (6)entail real k=k or the permittivity tensor K, or the dielectric tensor E-all of and real o=@r, i.e., o(k). The group velocity for such these being related to each other waves is given by In a homogeneous medium of the type we are considering the most general linear response function is one which ex presses both spatial and temporal dispersion through a con- volution integral in both space and time. Thus, for example, the current density J(, 1), at a point location F and time t, In a dissipation-free medium, the permittivity tensor is Her- depends upon the time-history and location-neighborhood mitian for real k and real a, K(kr, w)=Kh. In a linearly (consistent with causality and relativity)of the electric field stable and dissipation-free medium, the direction of signal E(r, n through the space-time conductivity tensor influence propagation is given by the direction of u Am J Phys. 68(5), May 2000 c 2000 American Association of Physics TeachersNOTES AND DISCUSSIONS Note on group velocity and energy propagation Abraham Bers Department of Electrical Engineering & Computer Science and Plasma Science & Fusion Center, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 ~Received 12 January 1999; accepted 10 June 1999! A general proof is given for the equality between group velocity and energy velocity for linear wave propagation in a homogeneous medium with arbitrary spatial and temporal dispersion. © 2000 American Association of Physics Teachers. In the series of physics ‘‘Questions’’ in the American Journal of Physics, K. M. Awati and T. Howes ask for a general proof of the fact that in linear dispersive wave propa￾gation, energy propagates at the group velocity.1 Several an￾swers to this question have appeared recently,2 but a general proof has not been provided by any of them. For a stable and nondissipative3 medium, such a general proof is indeed available from classical, continuum electrodynamics of media.4 It can be readily argued that this type of proof can be carried out for any other ~than electrodynamic! description of a stable, nondissipative, linear ~or, more generally, linearized dynamics of a! medium. In an electrodynamic formulation, one chooses to eliminate the ‘‘mechanical’’ field variables in favor of the electromagnetic fields; one can easily visualize doing the opposite. For exposing the simplest electrody￾namic proof, I will focus on a nonmagnetic medium (BW 5m0HW ) which is arbitrarily ~spatially and temporally! dispersive—a plasma. The generalization to linear waves in magnetic media is straightforward. The continuum electrodynamics of a plasma-like medium is described by Maxwell’s equations for the electromagnetic fields EW and HW , wherein the collective ‘‘mechanical’’ dy￾namics of the medium as a function of EW and HW are ex￾pressed by electric current and electric charge densities (JW,r). The latter, expressed as functions of EW and HW , are what one can call the electrodynamic response functions of the medium. For a plasma-like medium, since BW 5m0HW , Faraday’s equation provides a way of eliminating HW in favor of EW , so that the response functions are only functions of EW . In addition, since JW and r are related by the continuity equa￾tion, the mechanical dynamics, regardless of the particular model chosen for describing the dynamics, can be expressed by a single ‘‘electrical’’ response ~or ‘‘influence’’! function, e.g., the conductivity tensor sJ, or the susceptibility tensor Jx, or the permittivity tensor KJ, or the dielectric tensor Je—all of these being related to each other. In a homogeneous medium of the type we are considering, the most general linear response function is one which ex￾presses both spatial and temporal dispersion through a con￾volution integral in both space and time. Thus, for example, the current density JW(rW,t), at a point location rW and time t, depends upon the time-history and location-neighborhood ~consistent with causality and relativity! of the electric field EW (rW,t) through the space–time conductivity tensor influence function. The Fourier–Laplace transform of this convolution relationship is expressed by the conductivity tensor function of wave vector kW and frequency v as JW~kW,v!5sJ~kW,v!•EW ~kW,v!. ~1! The other linear response functions have similar interpreta￾tions, and are simply related to each other: Jx(kW,v) 5sJ(kW,v)/(2ive 0); KJ(kW,v)5JI 1Jx(kW,v); Je (kW,v) 5e 0KJ(kW,v). The Fourier–Laplace transform of Maxwell’s equations for the self-consistent electromagnetic fields are then kW 3EW 5vm0HW ~2! and kW 3HW 52ve 0KJ~kW,v!•EW . ~3! Taking kW 3 ~2! and using ~3! to eliminate HW , one finds the homogeneous set of equations for EW : DJ ~kW,v!•EW 50, ~4! where the dispersion tensor DJ is DJ ~kW,v!5kW kW 2k2JI 1 v2 c2 J K~kW,v!. ~5! For nontrivial solutions of ~4!, det@DJ ~kW,v!#[D~kW,v!50, ~6! which is the dispersion relation giving, e.g., v(kW). These are the natural modes of the system, with fields whose space– time dependence exp@i(kW•rW2vt)# is constrained by the disper￾sion relation ~6!. Natural modes that are purely propagating waves are those for which solutions of ~6! entail real kW 5kWr and real v5vr , i.e., vr(kWr). The group velocity for such waves is given by vW g5]vr ]kWr . ~7! In a dissipation-free medium, the permittivity tensor is Her￾mitian for real kW and real v, KJ(kWr ,vr)5KJh . In a linearly stable and dissipation-free medium, the direction of signal propagation is given by the direction of vW g . 5 482 Am. J. Phys. 68 ~5!, May 2000 © 2000 American Association of Physics Teachers 482