Educational model: dry air, ideal tube, typical values. Motion is slowed about 100 times so the standing wave can be seen; wavelengths and lengths are to scale.
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Lab & science
Drag to orbit, wheel / pinch to zoom. Arrow keys nudge the tube by 1 mm (Shift = 1 cm) when the page has focus. F = fork, R = reset.
Graphs (same as the panel)
Setup
Tube inner radius (sets the end correction)
1.5 cm 2.35 cm 3.5 cm
Presets
Classroom 20 °C, 512 Hz Freezing morning −5 °C Hot room 38 °C Low fork 384 Hz High fork 1024 Hz Wide tube 3.5 cm
Show predicted resonances on the graph Jump to resonance 1 Jump to resonance 2
Camera
Bench Fork and mouth Water surface
Graphics (High on every device; Low = smaller shadows, lower pixel ratio)
High Low
Live readout (model values)
Experiments
1. Slide the tube like a student: the program finds the resonances on the loudness curve with a 1 mm reading error and computes v = 2 f (L2 − L1 ) and the end correction. 2. See how many millimetres the first resonance moves when the room warms by 10 K.
Measure v (with reading error) Warm the room by 10 K
Missions
Mission 5: how fast does sound travel in air at 20 °C, in m/s (nearest whole number)?
Check
The science
What you see The fork pushes the air at the open mouth back and forth. The sound runs down the tube, bounces off the water surface (a rigid, closed end) and comes back. When the round trip fits the wave so that the reflected wave reinforces the next push, a standing wave builds up: the air at the water cannot move (a displacement node, where the pressure swings most) and the air at the mouth moves freely (an antinode). That happens when the column is about a quarter wavelength long, or three, five quarters… The loudness curve shows the peaks, and the dots show how the air parcels are shaken (greatly exaggerated and slowed down).
Equations
Speed of sound v = 331.3 sqrt(1 + T/273.15) m/s ( = sqrt(gamma R T), gamma = 1.4, R = 287 J/kg/K )
Wavelength lambda = v / f
End correction e = 0.6133 r (open, unflanged end, small k r)
Resonant lengths L_n = (2n - 1) lambda / 4 - e , n = 1, 2, 3 ...
Spacing of peaks L_(n+1) - L_n = lambda / 2 -> v = 2 f (L_2 - L_1)
Damped column p_water / p_mouth = 1 / cosh( (alpha + i k) (L + e) ), k = 2 pi f / v
Wall losses alpha_wall = (1 + (gamma - 1)/sqrt(Pr)) sqrt(omega nu / 2) / (r v) , Pr = 0.71 Assumptions & honest simplifications
Plane waves in a straight rigid tube, dry air, small amplitude (linear acoustics). Higher tube modes are ignored (valid when f is below about 0.29 v / r, about 4 kHz here).
The mouth is modelled by the textbook end correction 0.6133 r and a small radiation-loss term; the fork is treated as a fixed-pressure source at the mouth. Real end corrections depend a little on frequency.
A constant extra loss of 0.3 Np/m stands for everything not modelled (fork coupling, water-surface absorption, leaks). It is illustrative: it is tuned so the resonance is about 1.5 cm wide, as it is by ear in a real class. With only wall losses the peaks would be needle-sharp.
The air temperature is uniform in the tube; humidity and CO2 are ignored (humid air is about 0.3 % faster at 20 °C). Water level, tube and jar sizes are drawn to scale; the motion is slowed about 100 times and the displacement of the air dots is exaggerated. Independent numerical checks
Also in node: node check.js. The time-domain check solves the 1-D wave equation on a staggered grid with a sinusoidal source and rigid wall, a different method from the frequency-domain formula the graph uses.
Sources Textbook closed-open pipe acoustics (Kinsler, Frey, Coppens & Sanders, Fundamentals of Acoustics); end correction 0.6133 r from Levine & Schwinger (1948); boundary-layer attenuation from Kirchhoff’s theory (see Rossing, Springer Handbook of Acoustics). Typical values; spot-check the numbers before classroom use.