228 Chapter 7. Waves in inhomogeneous plasmas and wave energy relations
so a negative energy wave hasω∂P/∂ω< 0 .If the dissipative termPiis the same for both
positive and negative waves, then for a given sign ofω,the critical difference between
positive and negative waves is due to the sign of∂P/∂ω.Equation (7.105) shows thatωi
will have opposite signs for positive and negative energy waves. Hence, dissipation tends
to drive negative energy waves unstable. Since there is usually some dissipation in any real
system, a negative energy wave will generally spontaneously develop if it isan allowable
mode and will grow at the expense of the free energy in the system (e.g., thefree energy in
the streaming particles).
7.8 Assignments
- Consider the problem of short wave radio transmission. Letxbe the horizontal di-
rection andzbe the vertical direction. A short wave radio antenna is designed in
such a way that it radiates most of the transmitter power into a specifiedkxandkz
at the antenna. This is determined essentially by the Fourier transformof the antenna
geometry.
(i) What is the frequency range of short wave radio communications?
(ii) For ionospheric parameters, and the majority of the short wave band should the
ionospheric plasma be considered magnetized or unmagnetized?
(iii)What is the appropriate dispersion for short wave radio waves (hint-it is very sim-
ple)
(iv) Using Snell’s law and geometric optics, sketch the trajectory of a radio wave
showing what happens at the ionosphere. - Using geometric optics discuss qualitatively with sketches how a low frequency wave
could act as a lens for a high frequency wave. - For the example of an electrostatic electron plasma wave [dispersionω^2 =ω^2 pe(1+
3 k^2 λ^2 de)]show that the generalized group velocity as defined in Eq.(7.89) gives the
same group velocity as found using the previous definition based on∂ω/∂k. - Calculate the wave energy density, wave energyflux, and group velocity for the elec-
trostatic wave that can exist when a beam of cold electrons having velocityv 0 streams
through a neutralizing background of infinitely massive ions. Discuss the signs of
these three quantities.