Finite Amplitude Effects on Hydromagnetic Waves

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Abstract
Hydromagnetic waves traveling along an axial magnetic field in a cylindrical plasma bounded by a conductor have a frequency ω determined by the exciter. The nonlinear terms appearing in the equation of motion and in Ohm's law, usually neglected in wave theory, will induce a small amplitude wave of frequency 2ω accompanying the basic wave. In addition, the boundary conditions at the exciter will produce further double-frequency waves, but moving at speeds different from the basic wave. In this paper the amplitudes of these double-frequency waves are calculated for all the components of the magnetic field and for the plasma density. The existence of certain critical frequencies, at which resonance between the basic and double frequency waves occurs, is established. In the limit as Ω ≡ ω/ωci tends to zero, the amplitudes of the magnetic double-frequency waves are shown to depend on Ω/ε, where ε−1 is the damping length of the wave, whereas the corresponding density wave amplitude is found to be independent of both ε and Ω.

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