Waves with Negative Energy. Linked Waves

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.