The propagation of Alfvén waves along a uniform horizontal field in a highly conducting incompressible fluid, subject to the convective forces produced by a uniform vertical temperature gradient, is treated in a Boussinesq approximation. It is shown that there are exact solutions with large amplitude but restricted form. Their restricted form means that an arbitrary disturbing force produces other motions as well as Alfvén waves. An arbitrary initial disturbance of small amplitude produces waves whose state of polarization varies along the direction of propagation. For large amplitudes, however, any mixtures of polarization states causes scattering into new modes.