Alfvén simple waves

<jats:title>Abstract</jats:title><jats:p>Multi-dimensional Alfvén simple waves in magnetohydrodynamics (MHD) are investigated using Boillat's formalism. For simple wave solutions, all physical variables (the gas density, pressure, fluid velocity, entropy, and magnetic field induction in the MHD case) depend on a single phase function ϕ, which is a function of the space and time variables. The simple wave ansatz requires that the wave normal and the normal speed of the wave front depend only on the phase function ϕ. This leads to an implicit equation for the phase function and a generalization of the concept of a plane wave. We obtain examples of Alfvén simple waves, based on the right eigenvector solutions for the Alfvén mode. The Alfvén mode solutions have six integrals, namely that the entropy, density, magnetic pressure, and the group velocity (the sum of the Alfvén and fluid velocity) are constant throughout the wave. The eigenequations require that the rate of change of the magnetic induction <jats:bold>B</jats:bold> with ϕ throughout the wave is perpendicular to both the wave normal <jats:bold>n</jats:bold> and <jats:bold>B</jats:bold>. Methods to construct simple wave solutions based on specifying either a solution ansatz for <jats:bold>n</jats:bold>(ϕ) or <jats:bold>B</jats:bold>(ϕ) are developed.</jats:p>