Cylindrically symmetric waves in the magnetohydrodynamic approximation

The equations of motion of an ideally conducting medium, in the magnetohydrodynamic approximation, are solved exactly under the hypotheses that (i) one component of the magnetic field is constant everywhere, (ii) the magnetic and hydrodynamic pressures are in equilibrium, and (iii) the solution is invariant under a continuous group of transformations which preserves the symmetry of the uniform magnetic field. It is seen that the invariants of the group of transformations form the basis for a parametric description of the full solution that describes the propagation of cylindrically symmetric magneto-sonic waves in the direction of the uniform field. A number of general features of the motion are deduced, and an integral expression is given for the amplitude of the component of velocity, perpendicular to the uniform magnetic field, which undergoes potential well oscillations.