In order to determine the velocity with which energy is transported, one needs to first determine the appropriate for (18) mulation of energy and energy flow in a space-time disper sive medium. For the purely propagating wave modes in a This proves the equality of the group velocity and energy plasma-like medium, one can show that the average (in space velocity, Ug=ve, for purely propagating waves in a linear time)energy density is given by generally dispersive, and nondissipative"electric" medium =B+5E.,人,E like a plasma. Note that, in the above proof, no specific (8) model of the linear, loss-free, dispersive dynamics had to be where the first term is clearly the average magnetic energy sive dynamics of a loss-free medium Two remarks are in order. First, in relation to the assump. the electric field and in all of the collective ""mechanical" tion of a nondissipative medium, the Kramers-Kronig rela- fields. One can also show that the average energy flow den- tions for a dispersive medium require that the permittivity sity is given by tensor, K(kr, ar), have both a Hermitian and an anti- Hermitian part. The relative magnitudes of these parts can (=ReEx日 ) o, E ak E however, vary from region to region in(k,, o)space (9) Weakly damped waves [lo (k,)<]o, (k,)I], for which(18) holds, exist in regions of (k,, w,) where the anti-Hermitian ting)energy flow density and the second term is the average part of k is small compared to its Hermitian part so that collective"mechanical"energy flow density. Using(8)and a,(kr)is essentially determined by (13). Second, group ve- (9), one can define an energy flow velocity for a natural wave locity(as its name is intended to remind us) applies to the velocity of a group of waves-a wave packet-and by(18) his must also be true for energy velocity in a loss-free, dis- = (10) pensive medium. Allowing for the wave fields [exp(ik,F iw,] to have amplitudes that vary slowly in space and where sk=(5)lo, (, and wk=(w)lo, tk. are the average wave time(slowly compared to the fast scales of, respectively, k, energy flow density and wave energy density, respectively and or), their velocity is also found to be given by(14). In The proof that(7) and(10) are equal to each other pro- addition, averaged (on the fast scales of either k, or o) ceeds as follows. Consider Maxwell's equations(2)and (3), energy and energy flow densities are again found to be given for k=kr, a=Or, and k(k,o)=K,(kr, or) by(8)and(9), respectively, and one can show that(17)and (18)also hold for such wave fields with slow space-time K XE=orFoH (11) amplitude modulations. A detailed proof of the above,in- cluding the account of weak dissipation, is given by Bers k,×H=-0n∈0Kh(k,o)E (12)(Ref.8) Several concluding remarks are also in order. The above As followed from(2)and (3), these entail the dispersion derivation carries through for a weakly inhomogeneous and/or weakly time-varying medium as long as geometrical optics is applicable to describe the wave propagation. The det K, K, -K27+K,(k,, or)=D,(K,, o,)=0,(13) proof of (18)can also be carried out for a weakly dissipative or weakly unstable medium. However, in a linearly unstable iving w (k), and thence the group velocity, medium [i.e, in which for some kr, o(kr)=o(k, +io(kr)has o kr)>0], the group velocity direction for Po.(-dDh/akr)o U (dDb1o)a、(k (14) purely propagating wave modes having w(k,)=wr(k,),for some other kr, may not be the same as the direction of signal In addition, consider the variation of(11) and(12)with re pect to kr and K M. Awati and T Howes, Am J Phys, 64(11), 1353(1996) (Sk,)XE+k,x(SE)=(So,)uoH+o, lo( SH),(15) K T. MeDonald, C S Helrich, R.J. Mathar, S. Wong, and D Styer, Am (ok,)×H+k×(6H=-E0Kb)·E-o∈0Kh Well-defined wave propagation in a dispersive medium is understood to take place in a regime of weak dissipation where the modes are character- (16) 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 Dot-multiplying(15) by H,(16) by -E, the complex considering these concepts, one can idealize the situation by considering conjugate of (I )by-(8H), the complex conjugate of( 2) Coe medium to be"nondissiraiNe s a. 30 (len for tempore as we do in the following variant electrodynamic ge wave energy and by(SE), and adding these equations, one obtains mentum and their flows in linear med sive media by S. M. Rytov, JETP 17, 930(1947), and ger (17) spatially and temporally dispersive media by M. E. Gertsenshte 680(1954). Independently, a simpler formulation of average from which one immediately finds and energy flow in linear media with spatial and temporal dispersion was 483 Am J Phys., VoL 68, No 5, May 2000 Notes and discussionsIn order to determine the velocity with which energy is transported, one needs to first determine the appropriate for￾mulation of energy and energy flow in a space–time disper￾sive medium. For the purely propagating wave modes in a plasma-like medium, one can show that the average ~in space or time! energy density is given by6 ^w&5 m0 4 uHW u 21 e 0 4 EW *• ]~vrKJh! ]vr •EW , ~8! where the first term is clearly the average magnetic energy density and the second term is the average energy density in the electric field and in all of the collective ‘‘mechanical’’ fields. One can also show that the average energy flow den￾sity is given by6 ^sW&5ReS 1 2 EW 3HW *D 2 e 0 4 vrEW *• ]KJh ]kWr •EW , ~9! where the first term is the average electromagnetic ~Poyn￾ting! energy flow density and the second term is the average collective ‘‘mechanical’’ energy flow density. Using ~8! and ~9!, one can define an energy flow velocity for a natural wave vr(kWr): vW e5 sW k wk , ~10! where sW k5^sW&uvr(k W r) and wk5^w&uvr(k W r) are the average wave energy flow density and wave energy density, respectively. The proof that ~7! and ~10! are equal to each other pro￾ceeds as follows. Consider Maxwell’s equations ~2! and ~3!, for kW 5kWr , v5vr , and KJ(kW,v)5KJh(kWr ,vr), kWr3EW 5vrm0HW , ~11! kWr3HW 52vre 0KJh~kWr ,vr!•EW . ~12! As followed from ~2! and ~3!, these entail the dispersion relation detF kWrkWr2kr 2JI 1 vr 2 c2 J Kh~kWr ,vr!G [Dh~kWr ,vr!50, ~13! giving vr(kWr), and thence the group velocity, vW g5]vr ]kr 5 ~2]Dh /]kWr!vr~k W r! ~]Dh /]vr!vr~k W r! . ~14! In addition, consider the variation of ~11! and ~12! with re￾spect to kWr and vr : ~dkWr!3EW 1kWr3~dEW !5~dvr!m0HW 1vrm0~dHW !, ~15! ~dkWr!3HW 1kWr3~dHW !52e 0d~vrKJh!•EW 2vre 0KJh •~dEW !. ~16! Dot-multiplying ~15! by HW *, ~16! by 2EW *, the complex conjugate of ~11! by 2(dHW ), the complex conjugate of ~12! by (dEW ), and adding these equations, one obtains ~dkWr!•^sW&5~dvr!^w& ~17! from which one immediately finds ]vr ]kWr 5 sW k wk . ~18! This proves the equality of the group velocity and energy velocity, vW g5vW e , for purely propagating waves in a linear, generally dispersive, and nondissipative ‘‘electric’’ medium, like a plasma. Note that, in the above proof, no specific model of the linear, loss-free, dispersive dynamics had to be specified; the result ~18! is thus valid for any linear, disper￾sive dynamics of a loss-free medium. Two remarks are in order. First, in relation to the assump￾tion of a nondissipative medium, the Kramers–Kro¨nig rela￾tions for a dispersive medium require that the permittivity tensor, KJ(kWr ,vr), have both a Hermitian and an anti￾Hermitian part. The relative magnitudes of these parts can, however, vary from region to region in (kWr ,vr) space. Weakly damped waves @uvi(kr)u!uvr(kr)u#, for which ~18! holds, exist in regions of (kWr ,vr) where the anti-Hermitian part of KJ is small compared to its Hermitian part so that vr(kWr) is essentially determined by ~13!. Second, group ve￾locity ~as its name is intended to remind us! applies to the velocity of a group of waves—a wave packet—and by ~18! this must also be true for energy velocity in a loss-free, dis￾persive medium.7 Allowing for the wave fields @exp(ikWr•rW 2ivrt)# to have amplitudes that vary slowly in space and time ~slowly compared to the fast scales of, respectively, kWr and vr!, their velocity is also found to be given by ~14!. In addition, averaged ~on the fast scales of either kWr or vr! energy and energy flow densities are again found to be given by ~8! and ~9!, respectively, and one can show that ~17! and ~18! also hold for such wave fields with slow space–time amplitude modulations. A detailed proof of the above, in￾cluding the account of weak dissipation, is given by Bers ~Ref. 8!. Several concluding remarks are also in order. The above derivation carries through for a weakly inhomogeneous and/or weakly time-varying medium as long as geometrical optics is applicable to describe the wave propagation.8 The proof of ~18! can also be carried out for a weakly dissipative or weakly unstable medium.8 However, in a linearly unstable medium @i.e., in which for some kWr , v(kWr)5vr(kWr) 1ivi(kWr) has vi(kWr).0#, the group velocity direction for purely propagating wave modes having v(kWr)5vr(kWr), for some other kWr , may not be the same as the direction of signal propagation.9 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 character￾ized 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 mo￾mentum and their flows in linear media were given for temporally disper￾sive 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 483 Am. J. Phys., Vol. 68, No. 5, May 2000 Notes and Discussions 483