The dynamics of a highly conducting fluid stream coupled by means of a transverse electric or parallel magnetic field to a conducting elastic medium is examined in detail. From the dispersion relation and the Bers-Briggs criterion, the stability of the infinite length system is described. Eigen-frequencies and eigenfunctions are computed for the finite length system and good agreement is obtained with an electrohydrodynamic experiment. A physical explanation is given for the overstabilities observed. The close analogy between a limiting case of the field-coupled system and an electron beam coupled to a traveling wave structure is discussed.