---
title: "Resonance Tube & Sound Speed Interactive Lab | Sandbox Physics"
description: "Why is the loudest resonance shifted from the tube’s geometric length?"
canonical_url: "https://sandboxphysics.com/labs/resonance-tube/"
language: "en"
page_type: "lab"
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  zh-CN: "https://sandboxphysics.com/zh/labs/resonance-tube/"
last_modified: "2026-10-28"
---

IN07 · Air beyond the open rim

# Resonance Tube & Sound Speed

**Why is the loudest resonance shifted from the tube’s geometric length?** Air just outside an open rim also moves, so the wave reflects as if the tube extended a little farther. Resonance therefore depends on an effective length. Comparing measured resonances at several geometric lengths separates sound speed from this end correction.

Slide the end plate, move the microphone and compare open and closed tubes. Save sweeps at several lengths to infer sound speed and end correction.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

Model assumptions

One-dimensional linear dry-air acoustics with a weak, non-obstructing microphone. Thin unflanged ends use a constant 0.6133-radius correction only within a dimensionless mouth-size parameter of at most 0.3. A chosen conservative guard excludes lengths below four radii; it is not an accuracy specification. A virtual extension and passive resistance approximate end inertia and aggregate loss; this is not full radiation CFD. Temperature sets the PASCO approximate reference speed. Humidity, mean flow, viscothermal dispersion, nonlinear motion and probe loading are omitted. Particle displacement and tube width are enlarged. Complex pickup voltage, ruler/counter indications, calibration, temperature, seed and version are frozen with each scan. Sound speed and end correction need distinct measured lengths; one fixed length only identifies their ratio. Fits use raw peak positions and do not claim confidence intervals.

Simulated microphone observations

## Find peaks, then compare lengths

Inspect saved acquisition

Resolved peaks in this scan $4$

Compatible saved scans $3$

Sound speed from records $344\,\mathrm{m/s}$

Correction per open end $6.13\,\mathrm{mm}$

Peaks searched in the saved voltage samples

| Peak | Frequency / length | Pressure amplitude | Assigned mode |
| --- | --- | --- | --- |

Calibration and inference limits

$$
L=\frac{q_n}{f_n}c-N_o\delta
$$

Different geometric lengths separate the slope, sound speed, from the intercept, end correction. Many orders at one fixed length only determine speed divided by effective length. Frequency scans identify the fundamental within a broad speed interval; length scans reject ambiguous or missing order sequences. The displayed fit is a deterministic estimate, not a confidence interval.

Acoustic model and primary sources

One-dimensional linear acoustics, weak non-obstructing probe and a thin unflanged mouth. The low-frequency correction is 0.6133 radii per open end, used only while the dimensionless mouth-size parameter is at most 0.3. A virtual extension and a passive dimensionless resistance of 0.025 represent end inertia and aggregate loss. They are a teaching termination model, not a complete radiation calculation. A chosen conservative guard requires a length of at least four radii to avoid very short, wide cavities; this is not an accuracy specification. The left effective open end is pressure-release. Dry-air sound speed uses the PASCO temperature approximation. Viscothermal dispersion, humidity, mean flow, nonlinear sound and probe loading are omitted. Particle displacement and tube width are enlarged; all particles use one displacement gain within each frame.

$$
ka=2\pi fa/c\leq0.3,\qquad L/a\geq4
$$

- [PASCO WA-9612, 012-03541F (3/95), pp.1–6,20–21](https://cdn.pasco.com/product_document/Resonance-Tube-Manual-WA-9612.pdf)
- [Levine & Schwinger, Physical Review 73, 383 (1948)](https://journals.aps.org/pr/abstract/10.1103/PhysRev.73.383)

Physics tutorial

## The reflection reaches beyond the rim

**Background** A speaker pushes air along the tube. Reflections can reinforce the motion, but the pressure pickup and moving air have different nodes.

**Why it matters** A small shift at the open rim can bias a sound-speed measurement even when the resonances look sharp.

Start with the essentials

- **Focus question**: Why is the loudest resonance shifted from the tube’s geometric length?
- **One-sentence intuition**: The open end carries extra air inertia; changing tube length separates that extra length from sound speed.

### Core mathematical model

#### One closed end selects odd quarter waves

$$
f_n=\frac{(2n-1)c}{4(L+\delta)}
$$

The rigid end fixes air displacement; the effective open end approximately fixes pressure. Each extra mode adds half a wavelength.

#### Two open ends add two corrections

$$
f_n=\frac{nc}{2(L+2\delta)}
$$

Both thin unflanged mouths contribute the same low-frequency extension in this model.

#### Pressure follows compression

$$
p=-\rho c^2\frac{\partial\xi}{\partial x},\qquad u=\frac{\partial\xi}{\partial t}
$$

Pressure responds to the spatial compression of particle displacement. Particle velocity is the time derivative, so pressure and velocity have a different time phase from pressure and displacement.

#### Fit slope and intercept from measured peaks

$$
L=\frac{q_n}{f_n}c-N_o\delta
$$

The known boundary chooses the quarter-wave or half-wave factor. Distinct recorded lengths identify a slope and an intercept.

### Common difficulties

#### A pressure node is not still air

**Typical misconception** A quiet microphone marks a place where nothing moves.

**Better mental model** The microphone senses pressure. At a pressure node, air can move most strongly while its compression stays small.

### Run the experiment

1. 01

   #### Move the actual end

   Drag the silver end plate slowly, keeping drive frequency fixed.

   **What to observe:** The pressure colours and particle motion brighten near resonances. Adjacent resonant lengths differ by half a wavelength.

2. 02

   #### Find a quiet pickup

   Choose A quiet microphone, then drag the green probe away from that point.

   **What to observe:** A pressure node gives little microphone voltage even though nearby air moves strongly.

3. 03

   #### Save a frequency and length sweep

   Use Save frequency sweep, then Save length sweep.

   **What to observe:** The saved curves contain complex pickup voltage and calibrated axes. Changing the live apparatus leaves those samples unchanged.

4. 04

   #### Separate the two unknowns

   Choose Measure three lengths after changing temperature or boundary.

   **What to observe:** The three setups supply different lengths at the same temperature and radius. Repeated orders at one fixed length cannot supply the missing geometric information.

5. 05

   #### Check the pickup and scales

   Change Pickup gain factor or Ruler zero offset, then Calibrate pickup & scales before another acquisition.

   **What to observe:** Later scans use the new saved reference observations. Earlier scans retain their own calibration.

Physical conclusion

Keep the raw pressure response, find its peaks and compare genuinely different lengths.

## Related experiments and articles

How physics instruments measure · Reading path

### How physics instruments measure

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **Why the air beyond a pipe changes its resonance**](https://sandboxphysics.com/stories/instruments/resonance-tube/)

Motion & dynamics · Starter path

## From a falling ball to an orbit

Start by measuring gravity and splitting motion into components. Then add contact forces and collisions, and follow the same laws all the way to an orbit.

[Start here → **Free Fall & Measuring Gravity** Capture a few drops and fit the camera positions to estimate gravitational acceleration. Then add air damping and watch the fit drift from the truth. Observe · 14 min](https://sandboxphysics.com/labs/free-fall/)

1. [1 Free Fall & Measuring Gravity](https://sandboxphysics.com/labs/free-fall/)
2. [2 Projectile Flight Designer](https://sandboxphysics.com/labs/projectile-flight/)
3. [3 Inclined Plane & Friction](https://sandboxphysics.com/labs/inclined-plane/)
4. [4 One-Dimensional Collision Bench](https://sandboxphysics.com/labs/collision-bench/)
5. [5 Mass–Spring Oscillator](https://sandboxphysics.com/labs/mass-spring/)
6. [6 How Orbits Take Shape](https://sandboxphysics.com/labs/orbit/)

[View the full path →](https://sandboxphysics.com/topics/mechanics/#starter-path)