A systematic account is given of the derivation of the dispersion relation for helicon waves in a uniform cylindrical plasma bounded by a vacuum. By retaining finite resistivity in the equations, boundary conditions present no difficulties, since the wave magnetic field is continuous through the plasma-vacuum interface. Two unexpected results are found. First, the wave attenuation remains finite in the limit of vanishing resistivity. This is due to the energy dissipated at the interface by the surface currents required to match the plasma wave field to the vacuum wave field. Zero wave attenuation for zero resistivity is recovered if electron inertia is included. Secondly, it is found that waves with azimuthal numbers m of opposite sign propagate differently, but the sense of polarization at the axis of the cylinder is independent of the sign of m.The argument of the dispersion function is complex and numerical results were obtained using a computer. The method of programming is described, and results are given applicable to propagation in metals at low temperatures, or in a typical gas discharge plasma for the m = 0 and m = ± 1 modes.An example of the amplitude of the wave fields as a function of radius is given for the axisymmetric mode, and of amplitude and phase for the m = ± 1 modes.

The dispersion equation for helicon waves in a plasma-filled wave-guide is derived. The relation between the solution and the quasi-static modes of Trivelpiece & Gould is discussed. An approximate dispersion relation, appropriate for cases where Ωe≫ ν ≫ ω, is obtained allowing the real and imaginary parts of the propagation constant to be expressed very simply in a normalized form. Curves are presented for this approximation.

A new derivation of quasilinear theory for plasma oscillations is given, subject to the neglect of wave-wave interactions. The theory, which applies equally to damping and growing waves, requires that the individual wave resonances be narrow and the electric field spectrum sufficiently broad.

A numerical solution of the equations for unstable electrostatic plasma waves propagating in a uniform infinite plasma with a weak beam is performed. Space dependent variables are Fourier transformed and the computations are done first with only one and then with two harmonics included. In both cases the first harmonic electric field reaches a maximum amplitude and thereafter oscillates periodically, the period being different in the two cases. Thus, the time evolution of the field depends strongly on the presence of the second harmonic. The second harmonic electric field itself has negligible amplitude compared with the fundamental but the corresponding perturbed distribution functions are comparable in the vicinity of the wave resonance. In the frame of the background plasma the frequency of the first harmonic electric field is close to the plasma frequency ωp but the nonlinearly generated second harmonic frequency is 2ωp. Because of this it is found that the wave-wave interaction terms are important and resonant processes occur. The neglect of such terms in quasilinear theory is questioned.

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