Magnetosonic Wave Speed Calculator

A magnetosonic wave is a compressional (pressure-carrying) wave in a magnetized plasma that combines the restoring effects of both magnetic tension/pressure and ordinary gas (thermal) pressure. Unlike the purely transverse Alfvén wave, magnetosonic waves compress and rarefy the plasma as they propagate, much like an ordinary sound wave but modified by the magnetic field.

v²_(fast,slow) = [(v_A²+v_s²) ± √((v_A²+v_s²)² − 4v_A²v_s²cos²θ)] / 2, the exact quadratic solution of the ideal MHD wave dispersion relation, where v_A is the Alfvén speed, v_s is the sound speed, and θ is the angle between the wave's propagation direction and the magnetic field.

The MHD wave equation for compressional waves is quadratic in the squared wave speed, giving two solutions at any given angle, a faster mode where magnetic and thermal pressure reinforce each other, and a slower mode where they partially work against each other. Both propagate simultaneously in a magnetized plasma, alongside the separate Alfvén mode.

At θ=90° (propagation exactly perpendicular to the magnetic field), the slow magnetosonic speed drops to exactly zero, leaving only the fast mode, which simplifies to v_fast = √(v_A² + v_s²). This is a clean, exact limiting case that this calculator reproduces precisely.

At θ=0° (propagation exactly along the magnetic field), the fast and slow speeds reduce to the larger and smaller of v_A and v_s respectively, with no mixing between the magnetic and thermal restoring forces. In this direction, the wave essentially decouples into a pure Alfvén-like mode and a pure sound-like mode.

Sound speed uses v_s = √(γP/ρ), the standard ideal-gas formula, where P is the plasma's total thermal pressure (matching the <a href="/science/plasma-physics/plasma-pressure-calculator/">Plasma Pressure Calculator</a>), ρ is the ion mass density, and γ = 5/3 is the adiabatic index for a monatomic ideal gas.

Fast magnetosonic waves are responsible for shock formation ahead of planetary magnetospheres (the bow shock) and in the solar wind, since they can propagate in any direction and steepen into shocks. Both wave types are important for understanding energy transport and wave heating throughout the heliosphere and other astrophysical magnetized plasmas.

No, the Alfvén wave is a separate, purely transverse mode that does not compress the plasma and has speed exactly v_A regardless of angle. Magnetosonic waves are compressional and their speed depends on both v_A and v_s, varying continuously with propagation angle. The related <a href="/science/plasma-physics/alfven-wave-speed-calculator/">Alfvén Wave Speed Calculator</a> computes that separate mode.

The magnetosonic dispersion relation depends only on cos²θ, so angles between 90° and 180° give identical results to their supplement between 0° and 90° (for example, 120° behaves the same as 60°). Restricting the input to 0-90° avoids redundant entries without losing any physical information.

Yes, through the mass density ρ = n·mᵢ that appears in both the Alfvén and sound speed terms, heavier ions (like deuterons compared to protons) give lower magnetosonic speeds at the same density, field, and pressure.