TravelingWave

A traveling wave is a disturbance that propagates through a medium or vacuum, transporting energy and information from the source to distant locations while maintaining, in the simplest cases, a consistent shape. Unlike standing waves, traveling waves move in a definite direction.

Mathematically, a one-dimensional traveling wave can be expressed as a function of a single variable, such as u(x, t) = f(x − vt) for a wave moving in the positive x-direction, or u(x, t) = f(x + vt) for a wave moving in the negative x-direction, where v is the wave’s velocity. For monochromatic waves, the form often seen is u(x, t) = A cos(kx − ωt + φ), with phase velocity v = ω/k, angular frequency ω, and wavenumber k. The wavelength λ is related by λ = 2π/k and the frequency f is related by f = ω/2π, with v = fλ. In dispersive media, phase velocity and group velocity (the speed of energy or a wave packet) may differ.

Examples span mechanical, acoustic, and electromagnetic waves: light traveling through free space, sound in air, waves on a string, and water waves on a surface. In electrical engineering, signals on transmission lines are modeled as traveling waves, and boundary conditions determine forward or backward propagation.

Energy and momentum flow with traveling waves; the energy flux is tied to the square of the amplitude and, in non-dissipative media, travels with the wave. The superposition principle allows complex waveforms to be built from multiple traveling waves, leading to interference and Doppler effects when relative motion is involved.