15.1 Standing Waves

A standing wave appears when two waves with the same wavelength, frequency, and amplitude travel in opposite directions through the same medium. At each point in space, displacement is the algebraic sum of the two contributions. Because the waves repeatedly reinforce and cancel at fixed locations, the pattern shape changes in time but key spatial points remain fixed.

This product form is the core standing-wave fingerprint. The cosine term controls temporal oscillation, while the sine term controls location-dependent amplitude. At some positions, the sine factor is always zero and the medium never moves. At others, the sine factor reaches magnitude one and oscillation amplitude is maximal.

Nodes are fixed points of permanent zero displacement from destructive interference. Antinodes are points of largest oscillation amplitude from constructive interference. Node-to-node spacing is lambda/2, while node-to-nearest-antinode spacing is lambda/4. These spacing rules are the geometric backbone for all later string and pipe formulas.

Every point in a standing wave oscillates at the same frequency, but not with the same amplitude. This is often counterintuitive if you learned waves only as moving crests. In a standing pattern, the crest locations are not objects traveling along the medium; they are the moments when antinode locations reach extreme displacement.

A traveling wave transports phase and net energy in one direction through the medium. A standing wave is different: it is built from two traveling components that carry equal energy in opposite directions, so net energy transfer through the standing pattern is zero. Local energy still oscillates between kinetic and potential forms, but there is no sustained directional flux.