Electromagnetic wave equation

The electromagnetic wave equation is a partial differential equation governing the propagation of electric and magnetic fields through space or a material medium. It follows from Maxwell’s equations and predicts wave speed from the medium’s electromagnetic properties.

Maxwell's equations: Four equations relating electric and magnetic fields to charges, currents, and their changes. Taking curls of the field equations yields wave equations when sources vanish.
Partial differential equation: An equation involving an unknown function and its derivatives with respect to multiple independent variables. The electromagnetic wave equation relates field derivatives across space and time.
Plane wave: An ideal wave whose fields have constant phase across parallel planes. Plane-wave solutions make propagation direction, speed, and polarization explicit.
Inhomogeneous wave equation: A wave equation with a nonzero source term that drives the field. Charges and currents supply sources absent from the source-free form.
Wave equation: A partial differential equation describing disturbances that propagate at a finite speed. The electromagnetic equation is a vector-field instance of this broader mathematical form.
Electric field: A vector field describing electric force per unit charge at each point in space and time. One wave equation governs the electric-field components.
Waveguide: A structure that confines and directs electromagnetic waves along a path. Boundary conditions turn the wave equation into discrete guided modes.
Helmholtz equation: A spatial differential equation obtained from a wave equation for a single-frequency field. Assuming sinusoidal time dependence converts the time-domain equation into this form.
Speed of light: The speed at which electromagnetic radiation propagates in vacuum, exactly 299,792,458 meters per second. The vacuum wave equation identifies this speed from electric and magnetic constants.
Magnetic field: A vector field describing magnetic influence on moving charges and magnetic materials. A companion wave equation governs the magnetic-field components.

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