Standing Waves on a String
When waves reflect and interfere, stable patterns form with nodes and antinodes. Standing waves are central to musical instruments and resonance.
Nodes and antinodes
Nodes are points that stay still; antinodes oscillate with maximum amplitude. The number of segments depends on frequency and boundary conditions.
Resonance and harmonics
At resonant frequencies, integer multiples of half-wavelengths fit on the string. Each harmonic adds another antinode between fixed ends.
See resonance in the lab
Drive a string at different frequencies in the simulation and watch standing patterns form with live wavelength and speed readings.
The Wave Equation
A wave's speed is set by the medium it travels through, like a string's tightness or the air it moves in — not by how fast you shake the source. Because the speed stays fixed, every cycle you send out must fit into that same speed, so a faster wiggle packs the cycles closer together into a shorter wavelength.
v = f · λ
- v — wave speed (m/s): how fast the wave pattern moves through the medium
- f — frequency (Hz): how many wave cycles pass a point each second
- λ — wavelength (m): the length of one full wave cycle
A wave travels along a string, completing 5 full cycles every second. Each cycle stretches out over 2 meters. How fast does the wave travel?
- f = 5 Hz
- λ = 2 m
- v = f · λ
- v = 5 Hz × 2 m
v = 10 m/s