Essay: Existence of Electromagnetic-Hydrodynamic Waves

Context and aim

Alfven addresses whether a conducting fluid permeated by a magnetic field can support wave motion that is neither an ordinary sound wave nor an electromagnetic light wave. The essay establishes a new class of disturbances that couple Maxwell’s equations to fluid dynamics, showing that magnetic stresses can act as an elastic agent in a highly conducting medium. These hydromagnetic disturbances, later called Alfven waves, provide a mechanism for transmitting forces and energy along magnetic field lines in space and laboratory plasmas.

Physical setting and assumptions

The analysis considers a homogeneous, perfectly conducting fluid threaded by a uniform magnetic field. Conductivity is taken sufficiently large that the field is “frozen” into the fluid, so electric fields in the fluid rest frame vanish to leading order and magnetic diffusion is negligible. Small, linear perturbations are imposed on the equilibrium state, with viscosity and resistive dissipation neglected. Under these conditions, the momentum equation of fluid dynamics is coupled to the magnetic force density, and the induction equation ties magnetic perturbations to the fluid motion.

Derivation and core result

By linearizing the coupled equations about a uniform field B0 and density ρ, Alfven demonstrates the existence of transverse waves whose restoring force is magnetic tension. The disturbances propagate strictly along the background field with a dispersion relation ω = k∥ v_A, where v_A = B0 / sqrt(4πρ) in cgs units (v_A = B0 / sqrt(μ0 ρ) in SI). The velocity and magnetic perturbations are mutually perpendicular and also perpendicular to the direction of propagation; density and pressure remain essentially unchanged, marking these waves as non-compressive. Because the restoring force is geometric tension along field lines, propagation is highly anisotropic: no propagation occurs across the field in the ideal limit.

Character and distinctions

Alfven distinguishes these waves from ordinary sound waves, which rely on compressibility and pressure gradients, and from electromagnetic waves in vacuum, which travel at the speed of light and do not require matter. The phase speed v_A depends on the magnetic-field strength and mass density, not on temperature or compressibility, and is typically far below light speed but can exceed sound speeds in magnetized, rarefied media. With ideal conductivity, the waves are essentially dispersionless and weakly damped; energy is carried along the field by coupled magnetic and kinetic fluctuations rather than by radiation.

Implications for geophysics and astrophysics

The existence of such waves implies that magnetic structures behave elastically, transmitting stresses and perturbations over macroscopic distances. Alfven points to applications in the Earth’s ionosphere and magnetosphere, where geomagnetic disturbances may propagate along magnetic field lines, and to solar and stellar contexts, where forces can be communicated through magnetized plasma without significant compression. The mechanism offers a pathway for energy and momentum transport in sunspots, prominences, and interplanetary space, setting the stage for a magnetically mediated dynamics of cosmic plasmas.

Observational and experimental prospects

Alfven notes that the predicted speeds and anisotropy make detection plausible both in terrestrial laboratories using conducting liquids under strong magnetic fields and in geophysical observations of magnetic pulsations. The key signatures include transverse polarization, linkage between velocity and magnetic fluctuations, and propagation constrained to the direction of the ambient field. Finite conductivity and viscosity would introduce attenuation, but for sufficiently high conductivity the damping length remains large.

Significance

The essay establishes the theoretical foundation for magnetohydrodynamic wave motion and identifies a specific, robust mode, now universally known as the Alfven wave. By showing that magnetic tension endows conducting fluids with an effective elasticity, it opens a unified view of electromagnetic and fluid phenomena and provides essential tools for interpreting space and astrophysical plasmas.