# Waves with Negative Energy. Linked Waves

• M. I. Rabinovich
• D. I. Trubetskov
Part of the Mathematics and Its Applications (Soviet Series) book series (MASS, volume 50)

## 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
$$\left\langle {W(t)} \right\rangle = \frac{1}{{16\pi }}\left[ {\frac{{d\left( {\omega \varepsilon } \right)}}{{d\omega }}\left\langle {{{\left| E \right|}^2}} \right\rangle + \frac{{d\left( {\omega \mu } \right)}}{{d\omega }}\left\langle {{{\left| H \right|}^2}} \right\rangle } \right]$$
(Chapter 9) into heat. Since entropy must increase, the heat must be released and not absorbed, and hence we obtain [1]
$$< W\left( t \right) > {\text{ }} > {\text{ }}0,d\left( {\omega \varepsilon } \right)/d\omega ){\text{ }} > {\text{ }}0,d\left( {\omega \mu } \right)/d\omega > {\text{ }}0.$$
(10.1)
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.

## 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.

## References

1. 1.
L. D. Landau and E.M. Lifshits, Electrodynamics of Continuous Media (2nd ed) [in Russian], Fizmatgiz, Moscow (1982).Google Scholar
2. 2.
M.I. Rabinovich and A.L. Fabrikant, “Nonlinear Waves in Nonequilibrial Media,” Izv. Vuzov: Radiofiz., 19, 721–766 (1976).Google Scholar
3. 3.
P.A. Sturrock, “In what sense do slow waves carry negative energy?,” J. Appl. Phys., 31, 2052–2056 (1976).
4. 4.
J.R. Pierce, Almost All About Waves [Russian translation], Mir, Moscow (1976), Chs.3–4.Google Scholar
5. 5.
M.V. Nezlin, “Waves with negative energy and anomalous Doppler effects,” UFN, 120, 481–495 (1976).
6. 6.
B.B. Kadomtsev, Group Phenomena in Plasmas [in Russian], Nauka, Moscow (1976), pp.88–90.Google Scholar
7. 7.
J. Weiland and H. Wilhelmsson, Coherent Nonlinear Interaction of Waves in Plasmas [Russian translation], Energoizdat, Moscow (1981), Ch.8.Google Scholar
8. 8.
V.N. Shevchik, “Fundamentals of microwave electronics,” Sovetskii Radio, Moscow (1959), pp.87–92.Google Scholar
9. 9.
W.H. Louisell and J.R. Pierce, “Power flow in electron beam devices,” Proc. IRE, 43, 435–427 (1955).
10. 10.
C.K. Birdsall, G. R. Brewer, and A. V. Haeff, “The resistive wall amplifier,” Proc. IRE, 41, 865–874 (1953).
11. 11.
V.N. Tsytobich, Nonlinear Effects in Plasmas [in Russian], Nauka, Moscow (1967).Google Scholar
12. 12.
W.H. Louisell, Coupled Mode and Parametric Electronics [Russian translation], Izd. Ino. Lit., Moscow (1963).Google Scholar
13. 13.
V.N. Shevchik and D.I. Trubetskov, Analytical Calculation Methods in Microwave Electronics [in Russian], Sovetskii Radio, Moscow (1970).Google Scholar
14. 14.
C.C. Cutler, “Mechanical travelling wave oscillator,” Bell Lab. Record, 134–138 (1954).Google Scholar
15. 15.
V.L. Ginzburg, “On the radiation of an electron travelling near a dielectric,” Dokl. Akad. Nauk SSSR, 6, 145 (1947).Google Scholar
16. 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. 17.
V.L. Ginzburg, “Various topics on radiation theory for superlight travel in a medium,” UVN, 69, 537 (1959).Google Scholar
18. 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. 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).
20. 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. 21.
L.J. Chu, “The kinetic power theorem” in: IRE Electron Devices Conference, Univ. of New Hampshire, (June 1951).Google Scholar
22. 22.
M. V. Nezlin, Dynamics of Beams in PLasmas [in Russian], Energoizdat, Moscow (1982).Google Scholar
23. 23.
G. Bekefi, Radiation Processes in Plasmas [Russian translation], Mir, Moscow (1971).Google Scholar
24. 24.
N.M Frank, “Einstein and optics,” UFN, 129, 694–703 (1979).