Parametric Amplification of Alfvén Waves

A calculation describing the parametric amplification of Alfvén waves in a uniform, perfectly conducting, magnetohydrodynamic fluid is reported. The calculation has been motivated by a recent experiment of Lehane and Paoloni. A magnetically supported uniform column of the magnetohydrodynamic fluid is imagined to be subjected to an externally imposed, sinusoidally varying, magnetic field which can be idealized as spatially uniform. It is shown that a certain class of Alfvén waves which have an axial wavelength long compared with the radius of the column can be parametrically amplified in the case where their frequency and the modulating frequency satisfy certain integer quotient relations. The mathematical behavior is determined by Mathieu equations.