Oscillations and Waves pp 195-212 | Cite as

# Waves with Negative Energy. Linked Waves

Chapter

## Abstract

In the preceding chapter we discovered that the energy density and the energy flow density of the slow space-charge wave in an electron beam are negative (see (9.31) and (9.32)). At first sight this seems to contradict some general principles. For example, some energy must be spent in disturbing the electromagnetic wave packet in a medium with dissipation, and hence when energy ceases being “pumped” from the outside, the dissipation existing in the dispersing system (even if it is small) must transform all the energy
(Chapter 9) into heat. Since entropy must increase, the heat must be released and not absorbed, and hence we obtain [1]
However, this is only true for equilibrium media. Equation (10.1) need not be fulfilled in nonequilibrium media; and it is in such media that disturbances and wave propagation may occur with negative energy. The physical meaning of this will be made clear below.

$$
\left\langle {W(t)} \right\rangle = \frac{1}{{16\pi }}\left[ {\frac{{d\left( {\omega \varepsilon } \right)}}{{d\omega }}\left\langle {{{\left| E \right|}^2}} \right\rangle + \frac{{d\left( {\omega \mu } \right)}}{{d\omega }}\left\langle {{{\left| H \right|}^2}} \right\rangle } \right]
$$

$$
< W\left( t \right) > {\text{ }} > {\text{ }}0,d\left( {\omega \varepsilon } \right)/d\omega ){\text{ }} > {\text{ }}0,d\left( {\omega \mu } \right)/d\omega > {\text{ }}0.
$$

(10.1)

## Keywords

Transmission Line Space Charge Slow Wave Negative Energy Positive Energy
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Preview

Unable to display preview. Download preview PDF.

## References

- 1.L. D. Landau and E.M. Lifshits, Electrodynamics of Continuous Media (2nd ed) [in Russian], Fizmatgiz, Moscow (1982).Google Scholar
- 2.M.I. Rabinovich and A.L. Fabrikant, “Nonlinear Waves in Nonequilibrial Media,” Izv. Vuzov: Radiofiz.,
**19**, 721–766 (1976).Google Scholar - 3.P.A. Sturrock, “In what sense do slow waves carry negative energy?,” J. Appl. Phys.,
**31**, 2052–2056 (1976).MathSciNetCrossRefGoogle Scholar - 4.J.R. Pierce, Almost All About Waves [Russian translation], Mir, Moscow (1976), Chs.3–4.Google Scholar
- 5.M.V. Nezlin, “Waves with negative energy and anomalous Doppler effects,” UFN,
**120**, 481–495 (1976).CrossRefGoogle Scholar - 6.B.B. Kadomtsev, Group Phenomena in Plasmas [in Russian], Nauka, Moscow (1976), pp.88–90.Google Scholar
- 7.J. Weiland and H. Wilhelmsson, Coherent Nonlinear Interaction of Waves in Plasmas [Russian translation], Energoizdat, Moscow (1981), Ch.8.Google Scholar
- 8.V.N. Shevchik, “Fundamentals of microwave electronics,” Sovetskii Radio, Moscow (1959), pp.87–92.Google Scholar
- 9.W.H. Louisell and J.R. Pierce, “Power flow in electron beam devices,” Proc. IRE,
**43**, 435–427 (1955).CrossRefGoogle Scholar - 10.C.K. Birdsall, G. R. Brewer, and A. V. Haeff, “The resistive wall amplifier,” Proc. IRE,
**41**, 865–874 (1953).CrossRefGoogle Scholar - 11.V.N. Tsytobich, Nonlinear Effects in Plasmas [in Russian], Nauka, Moscow (1967).Google Scholar
- 12.W.H. Louisell, Coupled Mode and Parametric Electronics [Russian translation], Izd. Ino. Lit., Moscow (1963).Google Scholar
- 13.V.N. Shevchik and D.I. Trubetskov, Analytical Calculation Methods in Microwave Electronics [in Russian], Sovetskii Radio, Moscow (1970).Google Scholar
- 14.C.C. Cutler, “Mechanical travelling wave oscillator,” Bell Lab. Record, 134–138 (1954).Google Scholar
- 15.V.L. Ginzburg, “On the radiation of an electron travelling near a dielectric,” Dokl. Akad. Nauk SSSR,
**6**, 145 (1947).Google Scholar - 16.V.L. Ginzburg, “On the radiation of a microwave and its absorption in air,” Izv. Akad. Nauk SSSR, Fiz,
**11**, 165 (1947).Google Scholar - 17.V.L. Ginzburg, “Various topics on radiation theory for superlight travel in a medium,” UVN,
**69**, 537 (1959).Google Scholar - 18.R.J. Briggs, “Twin-Beam Instability” in: A. Simon, W.B. Thompson (eds,) Advances in Plasma Physics (Wiley Interscience 1969) [Russian translation], Mir, Moscow (1974), pp.132–171.Google Scholar
- 19.T.B. Benjamin, “The threefold classification of unstable disturbances in flexible surfaces bounding inviscid flows,” J. Fluid Mech.,
**16**, No.3, 436–450 (1963).MATHCrossRefGoogle Scholar - 20.L.A. Ostrovskii, Yu.A. Stepanyants, and L.Sh. Tsimring, “Interactions between inner waves and flows and turbulence in the ocean” in: Nonlinear Waves. Self- Organization [in Russian], Nauka, Moscow (1963)..Google Scholar
- 21.L.J. Chu, “The kinetic power theorem” in: IRE Electron Devices Conference, Univ. of New Hampshire, (June 1951).Google Scholar
- 22.M. V. Nezlin, Dynamics of Beams in PLasmas [in Russian], Energoizdat, Moscow (1982).Google Scholar
- 23.G. Bekefi, Radiation Processes in Plasmas [Russian translation], Mir, Moscow (1971).Google Scholar
- 24.N.M Frank, “Einstein and optics,” UFN,
**129**, 694–703 (1979).CrossRefGoogle Scholar

## Copyright information

© Kluwer Academic Publishers 1989