Fundamental Wave Functions in an Unbounded Magneto-Hydrodynamic Field. I. General Theory

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Abstract
This is the first of two papers dealing with a systematic study of the linearized, unbounded medium problems in magneto-hydrodynamics of incompressible and compressible fluids. Part I deals with the fundamental equations which are set up quite generally for an ideal, homogeneous and isotropic, conducting fluid devoid of viscosity and expansive friction, subject only to the initial assumption that the externally applied field of magnetic induction be constant and uniform. The energy and momentum balance in a magneto-hydrodynamic field is verified with the aid of the exact fundamental equations and the conservation laws of energy and momentum, for a rigid volume fixed in the (stationary) observer's inertial frame of reference, are displayed in differential and in integral form. By successive eliminations there is obtained a single partial differential equation in the particle velocity from which the unwanted second-order terms are merely dropped in a linearized small amplitude theory, a process which is fully justified by considering the special case of infinite conductivity, zero displacement current, and incompressible fluids. Also, assuming that a particular solution of the linearized magneto-hydrodynamic wave equation has been obtained, it is shown how to compute quite generally, from the linearized Maxwellian set, the accompanying electromagnetic field vectors expressed in terms of the assumed velocity field. These computations are carried out for plane homogeneous waves and for time-harmonic cylindrical waves. The actual determination of particular wave functions appropriate for incompressible and compressible fluids, together with the computation of the corresponding wave numbers, is reserved for the sequel to this paper, Part II.

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