Geometrical hydromagnetics

Abstract

The propagation of small disturbances in an infinitely conducting, perfect, compressible medium is studied from the standpoint of geometrical hydromagnetics. The medium need not be homogeneous. A wavelike time-harmonic disturbance is assumed given on an arbitrary reference surface. Then the various phase fronts, slow, Alfvén, and fast, that may be thought to evolve out of this surface are obtained as solutions of first-order partial differential equations. These eikonal-like equations are solved, as in optics, by means of rays, which are shown to satisfy Hamilton's equations and Fermat's principle. The hydromagnetic analog of Huygens' phase front construction is described. Next, the variation of the disturbance strengths along the rays is determined in terms of the disturbance strengths on the reference surface, to complete the geometrical solution. A noteworthy feature of this solution is that its component modes of propagation, slow, Alfvén, and fast, are uncoupled as long as the medium is free of diffracting objects and surfaces of discontinuity. As an illustration of the general theory, the complete geometric solution for the propagation of Alfvén disturbances in a dipole field is described with the aid of explicit formulas. Fast wave propagation in various types of inhomogeneous magnetic fields is also briefly discussed. Finally, the problem of reflection and refraction of slow, Alfvén, and fast disturbances at (possibly curved) surfaces of discontinuity is treated. The hydromagnetic analogs of Snell's laws are derived, and their relation to Huygens' wavelet construction is elucidated. It is shown that the introduction of an appropriate set of direction angles for the unperturbed magnetic field -on each side of the discontinuity) (1) reveals the symmetries of ‘Snell's laws’ (2) leads immediately to a simplification of V. C. A. Ferraro's laws relating to the reflection and refraction of Alfvén waves, and (3) makes possible a simple graphical method for determining the angles and speeds at which the various reflected and refracted waves emerge.