THE PATTERN STAYS. THE PARTICLES MOVE.
Standing waves
Find the nodes. Follow the particles. Compare their phase.
A string with fixed ends
N · displacement nodeA · displacement antinode● Particle A● Particle B
Animation is slowed to half a cycle per second. Time is shown in periods T; the frequency below is the physical resonant frequency.
Phase difference between A and B
Air column · textbook views
Compare the first three resonances. The highlighted row is the selected mode; click a row to select it. If a higher mode is selected, it appears in an extra row.
Solid and dashed curves show two opposite extremes of longitudinal displacement. They are not paths of particles moving vertically. N = displacement node; A = displacement antinode.
Each vertical line follows a thin layer of air moving along the tube. Closely spaced lines show compression; widely spaced lines show rarefaction. These are oscillating layers of the medium, rather than travelling wavefronts. The nodes of particle displacement are the antinodes of pressure variation.
This view follows the selected mode and the same pause and time controls above. At a quarter-period, layers pass through equilibrium and the spacing becomes uniform.
Connect motion to the graphs
Displacement vs position
A snapshot of the whole medium. Dashed lines show the maximum positive and negative displacement at each position.
Displacement vs time · A and B
Compare the timing of the peaks and troughs. Different amplitudes can still have the same phase.
Same frequency. Different amplitudes.
String fixed at both ends
Both ends are displacement nodes. All harmonics are possible. Between adjacent nodes, particles oscillate in phase; neighbouring loops oscillate in antiphase.
Standing sound waves
An open end is a displacement antinode. A closed end is a displacement node. Open–open columns allow all harmonics; open–closed columns allow only odd harmonics.
A particle at a node has zero oscillation amplitude, so its oscillation phase is undefined. Every particle passes through equilibrium during a cycle: being at zero displacement at one instant does not make it a node.
In air, particles move along the column, not up and down. The displacement–position curve is a graph of that longitudinal displacement; it is not the shape of a bent air column. A pressure node occurs at a displacement antinode, and vice versa.
Ideal model and sources
The medium follows a single undamped resonant normal mode. The string is uniform and fixed at both ends. Air columns use ideal boundaries with no end correction. Wave speed and length set the frequency; changing the amplitude does not change it. Dots represent small elements of the medium, not individual gas molecules with thermal motion. Movement is magnified for visibility. In air, magnification is limited to prevent the displayed particles from crossing each other; it is reported above the scene. A phase difference is 0 or π according to the signs of the mode shape, independently of the current instant.
OpenStax: standing waves on strings · OpenStax: standing sound waves
Explore standing waves and harmonics
Compare standing waves on a string fixed at both ends, an air column open at both ends, and an air column closed at one end. Select a resonant mode, adjust length, wave speed and amplitude, and observe transverse or longitudinal particle motion with displacement nodes and antinodes.
Select two particles to compare their phase and displacement-time graphs. Explore textbook air-column patterns and moving compression bands, including why an open-closed tube supports only odd harmonics. This free interactive resource supports IB, A Level and AP Physics learning.