Standing Waves & Harmonics
Guitar string harmonics. Builds on Traveling Waves.
A guitar string is fixed at both ends and vibrating in its 3rd harmonic — three beautiful half-wavelength loops are visible.
How many nodes (points of zero displacement) does the string have?
4 nodes! The 3rd harmonic has three loops (antinodes) separated by points that never move (nodes). But don’t forget — both fixed endpoints are also nodes! So that’s 2 interior nodes + 2 endpoints = 4 total. In the nth harmonic, there are always (n+1) nodes and n antinodes.
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Switch between harmonics 1, 2, 3, 4... and count the nodes (marked dots). Notice the endpoints are always nodes. Watch how two traveling waves combine to form the standing pattern.
fₙ = n · v/(2L) n = 1, 2, 3...
Only wavelengths that fit an integer number of half-wavelengths on the string are allowed. This quantization of frequencies is why strings produce musical notes, not noise!
Pressing a fret shortens L, raising the fundamental frequency. The 12th fret halves the string length, doubling the frequency — exactly one octave higher.
Lightly touching a violin string at 1/2, 1/3, or 1/4 of its length forces a node there, selecting that specific harmonic. The ethereal ‘harmonic’ tone rings out.
The 1940 collapse is often called "resonance," but it was aeroelastic flutter: wind pumped energy into a torsional standing-wave mode of the deck faster than damping could remove it — a self-excited oscillation, not a driven one. Same mode shapes, different physics.
Microwaves form standing waves inside the oven cavity. The nodes (cold spots) are why your food heats unevenly — and why the turntable rotates!
“Nature only allows certain frequencies on a bounded string. This quantization is everywhere — from guitar strings to electron orbitals.”
Standing Wave Lab
Create standing waves on a string. Adjust the driving frequency to find harmonics. Visualize how two opposing traveling waves combine to form nodes and antinodes.
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Start at the fundamental (n=1) and slowly increase the harmonic number. Count nodes at each step — verify it’s always (n+1). Toggle ‘Show traveling waves’ to see the two counter-propagating waves that combine to make the standing pattern. Try changing string length and tension to shift all the harmonic frequencies.
Standing waves are the superposition of two traveling waves moving in opposite directions. Only specific frequencies ‘fit’ the boundary conditions, creating the discrete harmonic series that underlies all of music.