Physics · Sound

Resonance in Air Columns / Open and Closed Pipes

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Wrap-up

What you learned

Recommended next: Open concept testCheck whether the core ideas are ready without leaving this concept.
Read next: Damping / ResonanceDriven response

Key takeaway

Open ends are displacement antinodes and pressure nodes; closed walls are displacement nodes and pressure antinodes.
Open-open pipes allow every integer harmonic, while closed-open pipes keep the odd harmonic family.
At the same length, closing one end lowers the fundamental because the allowed wavelength becomes longer.

Common misconception

Do not call an open end a displacement node just because air can leave; the free-moving mouth is where displacement can be largest.

For parcel displacement, an open end is where the air moves most freely, so it behaves like a displacement antinode.

Carry air-column resonance forward

Keep this idea moving

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Driven responseOscillations

Damping / Resonance

Explore how damping removes energy, how driving frequency changes amplitude, and why resonance becomes dramatic near the natural frequency.

Sound cues reviewSound

Pitch, Frequency, and Loudness / Intensity

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String-to-tube bridgeWaves

Standing Waves

Track fixed nodes, moving antinodes, and harmonic mode shapes on one live string while the same probe trace shows the underlying oscillation in time.

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Reference and support

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Equation snapshotPipe boundary snapshot · Closed-open resonance ladderShow 2 equations

Read the end condition first, then choose the wavelength family. Open-open tubes use half-wave counts; closed-open tubes use odd quarter-wave counts.

Pipe boundary snapshot
$λ_{n} = \frac{2 L}{n}, f_{n} = \frac{n v}{2 L}$
Because both ends follow the same antinode rule, an open-open tube allows the full sequence of integer harmonics.
Closed-open resonance ladder
$λ_{m} = \frac{4 L}{2 m - 1}, f_{m} = \frac{( 2 m - 1 ) v}{4 L}$
Because one end is closed and the other is open, the tube keeps only the odd harmonics.

Why it behaves this way

Explanation

An air column resonates only when the standing wave fits the tube and obeys the ends. An open end lets the air move most freely, while a closed wall forces the air there to stay still. Those boundary rules decide which wavelengths can survive, so the resonance frequencies are selected by the tube instead of chosen freely.

This bench shows that same resonance state in several linked views. The tube picture shows where the air moves most and least, the displacement-shape graph shows the standing-wave pattern along the tube, the harmonic ladder shows which frequencies are allowed, and the probe trace shows one parcel in time. Use them together to see why open-open tubes allow every harmonic but closed-open tubes keep only the odd ones.

Key ideas

01At an open end, the air can move most freely, so parcel displacement is largest there. At a closed end, parcel displacement must be zero.

02Open-open tubes allow every integer harmonic, but closed-open tubes allow only the odd harmonics because one end must be a node and the other an antinode.

03Changing tube length or boundary type changes the allowed wavelengths, and the resonance frequencies follow from the same sound speed.

Worked examples

Live resonance checks

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Frozen walkthrough

Step through the frozen example

Frozen walkthrough

These examples use the current tube length, boundary condition, resonance order, and probe position from the live bench, so the algebra stays attached to the same resonant pattern you are watching.

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Example 1 of 2

Frozen valuesUsing frozen parameters

For the current open-open tube with sound speed $v = 34 m/s$ , what wavelength and resonance frequency are allowed for the selected mode?

L

Tube length

1.2 m

m

Resonance order

n_{e f f}

Harmonic multiple

v

Sound speed

34 m/s

1. Start from the current boundary rule

For the current open-open tube, use

λ_{n} = \frac{2 L}{n}

and

f_{n} = \frac{n v}{2 L}

2. Substitute the live tube and selected mode

Here the selected mode is the fundamental resonance and the fundamental, so

λ = \frac{2 ( 1.2 )}{1}

with

L = 1.2 m

and

v = 34 m/s

3. Compute the allowed resonance

That gives

λ = 2.4 m

and

f = 14.17 Hz

, with all integer harmonics for this boundary.

Current allowed resonance

λ = 2.4 m, f = 14.17 Hz

An open-open tube lets both ends breathe as displacement antinodes, so the fundamental is a half-wave fit across the full length.

Quick test

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Accessibility

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The simulation shows one horizontal air column with a movable probe parcel inside the tube, a colored pressure ribbon, and a ruler underneath. The left end can switch between open and closed, the right end stays open, and optional overlays mark boundary rules, parcel-motion nodes and antinodes, and the complementary pressure pattern.

Changing tube length, boundary type, resonance order, probe position, or amplitude updates the same tube view, displacement-shape graph, harmonic ladder, and probe-motion graph together so the resonance state stays synchronized.

Graph summary

The displacement-shape graph plots parcel-motion scale against position in the tube, so zero crossings mark displacement nodes and the end behavior changes when the boundary condition changes.

The probe-motion graph plots one selected parcel in time together with its local envelope, while the harmonic ladder shows the allowed resonance frequencies and makes the missing even harmonics of a closed-open pipe visible.

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Starter track

Step 5 of 5

Sound and Acoustics

Resonance in Air Columns / Open and Closed Pipes appears later in this track, so it is cleaner to start from the beginning first.

Previous step: Doppler Effect

Also appears in Waves.

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