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Resonance in Air Columns / Open and Closed Pipes

Compare open and closed pipe boundary conditions on one compact air column so standing-wave shapes, missing even harmonics, probe motion, and pressure cues stay tied to the same resonance state.

Interactive lab

Keep the stage, graph, and immediate control feedback in one working view.

Time

0.00 s / 0.14 sLiveThe shape and harmonic-ladder graphs stay parameter-based while the time rail inspects one live tube state honestly.

0.00 s0.14 s

Resonance in Air Columns / Open and Closed Pipes

One bounded tube keeps parcel motion, pressure standing waves, boundary conditions, and the allowed harmonic ladder on the same acoustics bench.

Live setup

Resonance state

pipe: open-open
mode: 2nd resonance
harmonic: 2nd harmonic
lambda: 1.2 m
frequency: 28.33 Hz
probe x: 0.6 m
parcel shift: -0.12 m
pressure cue: 0

Open-open tubes allow every integer harmonic, so each resonance order steps to the next whole-number harmonic.
Probe motion envelope = 0.12 m at a antinode point.
Pressure envelope = 1.47e-17 m and swaps roles with the parcel-motion pattern.

Graphs

Switch graph views without breaking the live stage and time link.

Displacement shape

Shows the signed parcel-motion shape of the current resonance, so open ends, closed walls, and interior nodes stay tied to one live tube.

position in tube (m): 0 to 1.8signed parcel-shift scale (m): -0.5 to 0.5

Displacement shape

Hover or scrub to link the graph back to the stage.position in tube (m) / signed parcel-shift scale (m)

Controls

Adjust the physical parameters and watch the motion respond.

Tube length

L

1.2 m

Changes the resonating air-column length and therefore the allowed wavelengths.

Close the left end

BC

Switch between open-open and closed-open boundary conditions on the same tube.

Resonance order

m

2 mode

Selects which allowed resonance pattern fits the current boundary condition.

Probe position

x_{p}

0.6 m

Moves the live parcel probe along the tube. Values past the mouth clamp to the current tube length.

More tools

Secondary controls, alternate presets, and less-used toggles stay nearby without crowding the main bench.

Show

Reference amplitude

A

0.12 m

Scales the displacement and pressure cue size without changing the allowed resonance family.

More presets

Presets

Predict -> manipulate -> observe

Keep the active prompt next to the controls so each change has an immediate visible consequence.

Graph readingPrompt 1 of 2

Notice that hovering the displacement-shape graph moves the live probe to the same tube position instead of inventing a second pipe state.

Try this

Hover near the closed wall, then hover near the open mouth, and compare the probe-motion graph.

Equation map

See each variable before you move it.

Select a symbol to highlight the matching control and the graph or overlay it most directly changes.

L

Tube length

1.2 m

Changes how much room the standing pattern has, so every allowed wavelength and frequency shifts with it.

Graph: Displacement shapeGraph: Allowed harmonicsGraph: Probe motionOverlay: Boundary rulesOverlay: Parcel nodes and antinodesOverlay: Pressure complements

Equations in play

Choose an equation to sync the active symbol, control highlight, and related graph mapping.

More tools

Detailed noticing prompts, guided overlays, and challenge tasks stay available without taking over the main bench.

Hide

What to notice

Use one cue at a time. The best prompt should point you at a real resonance feature that the current tube, graph, or overlay is already showing.

Graph readingPrompt 1 of 2

Graph: Displacement shape

Notice that hovering the displacement-shape graph moves the live probe to the same tube position instead of inventing a second pipe state.

Try this

Hover near the closed wall, then hover near the open mouth, and compare the probe-motion graph.

Why it matters

The graph and the tube are two views of the same standing-wave pattern.

Control: Probe positionGraph: Displacement shapeGraph: Probe motion

Guided overlays

Focus one overlay at a time to see what it represents and what to notice in the live motion.

2 visible

Overlay focus

Boundary rules

Marks how the current tube ends constrain parcel motion and pressure variation.

What to notice

Every open end lines up with large parcel motion but weak pressure variation, while a closed wall reverses that pairing.

Why it matters

It keeps the resonance family tied to a physical boundary condition instead of turning open and closed pipes into memorized formulas.

Control: Close the left endControl: Tube lengthControl: Resonance orderGraph: Displacement shapeGraph: Allowed harmonicsEquation

BC

Equation

L

Equation

m

Challenge mode

Use the tube, overlays, and graphs to turn open-vs-closed boundary rules into compact resonance tasks.

0/2 solved

ConditionCore

2 of 5 checks

Closed-end stillness

Starting from the closed-pipe third harmonic, move the probe onto the closed wall so parcel motion nearly disappears while the pressure cue stays strong.

Graph-linkedGuided start2 hints

Suggested start

Use the boundary and pressure overlays together while you move the probe.

Matched

Open the displacement-shape graph.

Displacement shape

Matched

Keep the boundary overlay visible.

Pending

Turn on the pressure guides.

Off

Pending

Make the parcel-motion envelope smaller than

0.015 m

0.12 m

Pending

Keep the pressure envelope above

0.09

1.47e-17

The checklist updates from the live simulation state, active graph, overlays, inspect time, and compare setup.

At t = 0 s, the air column is open at both ends. The selected mode is the 2nd resonance and corresponds to the 2nd harmonic. The allowed wavelength is 1.2 m and the frequency is 28.33 Hz. At x = 0.6 m, the probe is near a displacement antinode and a pressure node.

Equation detailsDeeper interpretation, notes, and worked variable context.

Open-open parcel displacement

Both ends are displacement antinodes, so the parcel-motion shape uses the cosine pattern across the full tube.

y_{n} (x, t) = A cos (\frac{nπ x}{L}) cos (ω_{n} t)

Open ends let parcels move most freely.

Pressure variation is smallest at those open ends.

L

Tube length 1.2 m

BC

Boundary type Off

m

Resonance order 2 mode

x_{p}

Probe position 0.6 m

A

Reference amplitude 0.12 m

Closed-open parcel displacement

A closed wall forces a displacement node, so only odd harmonic multiples fit between the still wall and the open end.

y_{m} (x, t) = A sin (\frac{( 2 m - 1 ) π x}{2 L}) cos (ω_{m} t)

L

Tube length 1.2 m

BC

Boundary type Off

m

Resonance order 2 mode

x_{p}

Probe position 0.6 m

A

Reference amplitude 0.12 m

Open-open resonance ladder

Open-open tubes allow every integer harmonic because both ends follow the same antinode boundary rule.

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

L

Tube length 1.2 m

BC

Boundary type Off

m

Resonance order 2 mode

Closed-open resonance ladder

Closed-open tubes keep only odd harmonic multiples because the wall and the mouth obey different boundary conditions.

λ_{m} = \frac{4 L}{2 m - 1}, f_{m} = \frac{( 2 m - 1 ) v}{4 L}

L

Tube length 1.2 m

BC

Boundary type Off

m

Resonance order 2 mode

Progress

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Let the live model runChange one real controlOpen What to notice

Try this setup

Copy the live bench state and reopen this concept with the same controls, graph, overlays, and compare context.

Stable links

Starter track

Step 9 of 90 / 9 complete

Waves

Earlier steps still set up Resonance in Air Columns / Open and Closed Pipes.

1. Simple Harmonic Motion2. Wave Speed and Wavelength3. Sound Waves and Longitudinal Motion4. Pitch, Frequency, and Loudness / Intensity+5 more steps

Previous step: Standing Waves.

Start from track beginning

Starter track

Step 5 of 50 / 5 complete

Sound and Acoustics

Earlier steps still set up Resonance in Air Columns / Open and Closed Pipes.

1. Sound Waves and Longitudinal Motion2. Pitch, Frequency, and Loudness / Intensity3. Beats4. Doppler Effect+1 more steps

Previous step: Doppler Effect.

Start from track beginning

Short explanation

What the system is doing

Air-column resonance is the sound-wave version of a standing-wave constraint. The tube length and the boundary conditions decide which standing-wave patterns can survive, so the allowed wavelengths and frequencies are not arbitrary.

This bench keeps one tube, one live probe parcel, and one authoritative resonance state in view. The tube picture, the displacement-shape graph, the harmonic ladder, and the probe trace all read from that same state, so open ends, closed ends, nodes, antinodes, and odd-only closed-pipe harmonics stay tied together.

Key ideas

01Open ends behave like displacement antinodes and pressure nodes, while a closed end behaves like a displacement node and a pressure antinode.

02Open-open tubes allow every integer harmonic, but closed-open tubes skip the even harmonics because one end must stay still while the other end breathes.

03Changing tube length or boundary type changes the allowed wavelength first, and the resonance frequency follows from the same sound speed.

Live pipe checks

Solve the exact state on screen.

These examples read the current tube length, boundary condition, resonance order, and probe state directly from the live air column so the algebra stays attached to the same resonant pattern you are watching.

Live valuesFollowing current parameters

For the current tube length and boundary condition, what wavelength and resonance frequency are allowed?

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 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

Here the selected mode is the 2nd resonance and the 2nd harmonic, so

λ = \frac{2 ( 1.2 )}{2}

with

L = 1.2 m

and

v = 34 m/s

3. Compute the allowed resonance

That gives

λ = 1.2 m

and

f = 28.33 Hz

, with all integer harmonics for this boundary.

Current resonance requirement

λ = 1.2 m, f = 28.33 Hz

Open-open tubes allow every integer harmonic, so each higher resonance adds one more half-wavelength segment across the same tube.

Boundary checkpoint

You want the same tube length but a lower fundamental resonance without changing the length. What is the most direct change?

Prediction prompt

Decide whether you should close one end, raise the resonance order, or move the probe.

Check your reasoning

Close one end of the tube.

Closing one end changes the boundary condition from a half-wave fundamental to a quarter-wave fundamental. That doubles the allowed wavelength from

2 L

4 L

and halves the fundamental frequency, while moving the probe only changes what you inspect and raising the resonance order moves to a higher mode instead.

Common misconception

An open end should be a node because the air can move out of the tube there.

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

What becomes small at an open end is the pressure variation, which is why pressure nodes and displacement antinodes trade places.

Quick test

Misconception check

Question 1 of 4

Use the tube, graph, and boundary ideas together. These checks ask what the resonance pattern means physically, not which formula you memorized.

A student says, "An open end must be a node because the air can just escape there." What is the best correction?

Choose one answer to reveal feedback, then test the idea in the live system if a guided example is available.

Accessible description

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 immediately updates the same tube view, displacement-shape graph, harmonic ladder, and probe-motion graph so the resonance state stays synchronized.

Graph summary

The displacement-shape graph plots signed parcel-motion scale against position in the tube, so zero crossings correspond to 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 graph plots the allowed resonance frequencies for the current tube and makes the missing even harmonics in a closed-open pipe visible.

Keep the acoustics path going

Keep moving through the concept map.

These suggestions come from the concept registry, so the reason label reflects either curated guidance or the fallback progression logic.

Driven responseResonance

Resonance in Air Columns / Open and Closed Pipes

See each variable before you move it.

Boundary rules

Closed-end stillness

Waves

Sound and Acoustics

What the system is doing

Solve the exact state on screen.

For the current tube length and boundary condition, what wavelength and resonance frequency are allowed?

A student says, "An open end must be a node because the air can just escape there." What is the best correction?

Keep moving through the concept map.

Damping / Resonance

Pitch, Frequency, and Loudness / Intensity

Standing Waves