The general theory of magneto-hydrodynamic waves in an ideal conducting fluid embedded in a uniform field of magnetic induction, and the application of the theory to the systematic analysis of the various modes of propagation in incompressible and compressible fluids have been presented by the author in two earlier papers[1, 2]. In these papers, however, no effort was made to include the thermodynamics of the situation, which amounts to the tacit assumption that the fluid is of zero heat conductivity. In this case the resulting modes are of two kinds: isothermal (v-modes) and adiabatic (p-modes).
In this paper we first establish the conservation laws of momentum and energy for a (macroscopic) compressible fluid with finite viscosity and finite thermal and electrical conductivities, which is embedded in a uniform field of magnetic induction, and we then derive quite generally the exact (nonlinearized) equation governing the distribution of temperature in such a fluid. Next, making use of the linearized magneto-hydrodynamic wave equation in the fluid velocity, combined with the resulting heat diffusion equation and with the equation of state of the fluid, and applying the mathematical techniques developed earlier, we obtain a higher order partial differential equation in the fluid temperature from which ensue all the temperature modes.
In particular, we examine in detail the behavior of plane homogeneous waves, and it is shown that a compressible fluid with the indicated properties sustains altogether six different modes, two of which are pure shear modes, devoid of density, pressure, and hence temperature fluctuations (v-modes), while the remaining four are shear-compression waves accompanied necessarily by density, pressure, and temperature fluctuations (p-modes). The two shear modes, which are isothermal, comprise a slightly attenuated Alfvén wave, and a highly attenuated viscous mode, sometimes referred to as a vorticity mode. The four shear-compression modes have in general very complex properties, but in the low frequency and low heat conductivity case they are easily identified as (1) a modified (adiabatic) sound wave slightly attenuated; (2) a slightly attenuated modified Alfvén p-wave; (3) a highly attenuated viscous wave; and (4) a highly attenuated thermal wave governed in the main by the thermal properties of the medium.