Standing waves
A string fixed at both ends vibrates in standing waves — patterns where certain points (nodes) remain perfectly still while others (antinodes) oscillate with maximum amplitude. Select a harmonic, toggle the component traveling waves, and watch the superposition principle in action.
λ = 2L / n fₙ = n · f₁ y(x,t) = 2A sin(nπx/L) cos(2πfₙt)
About this lab
A standing wave forms when two identical traveling waves move in opposite directions and superpose. On a string fixed at both ends, reflections at the boundaries create exactly this situation. The boundary conditions require that both ends are nodes (points of zero displacement), which constrains the allowed wavelengths to λ = 2L/n, where L is the string length and n is the harmonic number.
The fundamental (n = 1) has the longest wavelength (2L) and lowest frequency. Each higher harmonic has a shorter wavelength and proportionally higher frequency: fₙ = n · f₁. The nth harmonic has n + 1 nodes (including the fixed ends) and n antinodes.
Standing waves are not traveling — energy sloshes back and forth between kinetic and potential forms but does not propagate along the string. This is visible in the animation: the wave oscillates in place rather than moving left or right. Toggle the component waves to see the two traveling waves whose superposition creates the standing pattern.