StandingWave

A standing wave is a vibration pattern in which the spatial distribution of displacement does not propagate along the medium. Instead, certain positions, called nodes, remain stationary, while other positions, called antinodes, reach maximum displacement. Standing waves arise when two waves of the same frequency and amplitude travel in opposite directions and interfere, or when a wave is reflected at a boundary, producing a persistent pattern of interference.

The temporal variation at each point is harmonic with the same frequency, even though the spatial pattern remains fixed. A common mathematical description for a standing wave on a string is obtained by the superposition of two counter-propagating waves, yielding a form such as f(x,t)=2A cos(kx) cos(ωt) or f(x,t)=2A sin(kx) cos(ωt), depending on boundary conditions. Nodes occur where the spatial factor is zero, and antinodes occur where it is maximal. Energy in a standing wave oscillates locally between potential and kinetic forms, but there is typically little net energy transport along the medium.

Standing waves occur in a variety of physical systems. On a string fixed at both ends, fixed boundaries support discrete standing-wave modes with wavelengths λn = 2L/n and frequencies fn = n v/(2L), where v is the wave speed. In air columns, standing waves form in pipes and depend on whether ends are open or closed. Standing waves also arise in electromagnetic cavities and laser resonators, where they define resonant frequencies and mode structures.