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OscillationsIntermediateStarter track

Concept module

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.

Interactive lab

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

Time

0.00 s / 4.67 sLiveThe mode-shape graph stays position-based while the time rail inspects the live string motion.

0.00 s4.67 s

Standing Waves

Fixed ends, one live probe, and optional traveling-wave overlays keep nodes, antinodes, and interference on the same compact stage.

Live setup

Graphs

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

Mode shape

Shows the signed spatial shape of the current harmonic, so adjacent loops flip sign across each node even though the node positions stay fixed.

position on string (m): 0 to 2.4signed amplitude scale (m): -2 to 2

Mode shape

Hover or scrub to link the graph back to the stage.position on string (m) / signed amplitude scale (m)

Controls

Adjust the physical parameters and watch the motion respond.

Antinode amplitude

A

1.1 m

Controls how large the antinode swing can be.

String length

L

1.6 m

Changes the fixed-end distance and therefore the allowed harmonic spacing.

Mode number

n

2 harmonic

Selects which harmonic fits on the string.

Probe position

x_{p}

1 m

Moves the live measurement point along the string. Values past the end clamp to the current string length.

More tools

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

Show

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 mode-shape graph moves the live probe to the matching string position instead of inventing a second spatial state.

Try this

Hover near a node, then hover near an antinode, and compare how the probe trace changes.

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.

A

Antinode amplitude

1.1 m

Sets the largest possible standing-wave displacement, so antinodes swing farther while nodes still stay fixed.

Graph: Mode shapeGraph: Probe motionOverlay: Antinode guidesOverlay: Counter-traveling waves

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 standing-wave pattern that the current stage, graph, or overlay is already showing.

Graph readingPrompt 1 of 2

Graph: Mode shape

Notice that hovering the mode-shape graph moves the live probe to the matching string position instead of inventing a second spatial state.

Try this

Hover near a node, then hover near an antinode, and compare how the probe trace changes.

Why it matters

The graph and stage are two views of the same standing-wave mode shape.

Graph: Mode shapeGraph: Probe motionOverlay: Node guidesOverlay: Antinode guides

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

Node guides

Marks the fixed points that never move in the current mode.

What to notice

Nodes stay on the center line even while neighboring parts of the string cross above and below it.

Why it matters

It keeps destructive interference visible instead of making nodes look like a decorative marker.

Control: Mode numberControl: String lengthControl: Probe positionGraph: Mode shapeGraph: Probe motionEquation

n

Equation

L

Equation

x_{p}

Challenge mode

Use the mode shape, overlays, and inspect-time rail to turn standing-wave patterns into compact probe-placement tasks.

0/2 solved

ConditionStretch

1 of 5 checks

Center zero crossing

From the fundamental mode, keep the probe at the center antinode and pause right as that antinode crosses through zero displacement.

Inspect timeGraph-linkedGuided start2 hints

Suggested start

Keep the probe at the center antinode, then pause at a zero crossing.

Pending

Open the probe-motion graph.

Mode shape

Matched

Keep the antinode overlay visible.

Pending

Pause into inspect mode.

live

Pending

Keep the probe envelope between

0.95

and

1.1 m

0.78 m

Pending

Bring instantaneous probe displacement close to zero.

-0.78 m

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

At t = 0 s, the standing wave is in the 2nd harmonic on a 1.6 m string. The allowed wavelength is 1.6 m and the probe at x = 1 m is at an in-between point. Its instantaneous displacement is -0.78 m while its oscillation envelope there is 0.78 m.

Equation detailsDeeper interpretation, notes, and worked variable context.

Standing-wave displacement

The spatial mode shape and the shared time oscillation multiply together to give the local displacement.

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

The sine factor fixes the node pattern in space.

The cosine factor keeps almost every allowed point oscillating in time.

A

Antinode amplitude 1.1 m

n

Mode number 2 harmonic

L

String length 1.6 m

x_{p}

Probe position 1 m

Allowed wavelength

The nth harmonic fits n half-wavelength segments onto the fixed-length string.

λ_{n} = \frac{2 L}{n}

n

Mode number 2 harmonic

L

String length 1.6 m

Allowed frequency

Once wave speed and string length are set, each harmonic has a matching oscillation frequency.

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

n

Mode number 2 harmonic

L

String length 1.6 m

Node positions

Nodes sit at fixed fractions of the string where destructive interference keeps the displacement at zero.

x_{m} = \frac{m L}{n}

n

Mode number 2 harmonic

L

String length 1.6 m

x_{p}

Probe position 1 m

Progress

Not startedMastery: NewLocal-first

Start exploring and Open Model Lab will keep this concept's progress on this browser first. Challenge mode has 2 compact tasks ready. No finished quick test, solved challenge, or completion mark is saved yet.

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 8 of 90 / 9 complete

Waves

Earlier steps still set up Standing Waves.

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

Previous step: Wave Interference.

Start from track beginning

Short explanation

What the system is doing

A standing wave is what you get when equal waves travel in opposite directions and keep interfering on the same bounded system. Some points cancel all the time, so they become nodes. Other points reinforce most strongly, so they become antinodes.

This lab keeps one string, one live probe, and one authoritative oscillation state in view. The mode-shape graph, the moving string, and the probe trace all come from the same harmonic, which is why mode number, nodes, antinodes, interference, and local oscillation stay tied together.

Key ideas

01Nodes are fixed points created by persistent destructive interference, not places where the wave has disappeared from the whole string.

02Antinodes are the points of largest oscillation, so the local time trace there reaches the full standing-wave amplitude.

03Higher harmonics fit more half-wavelength segments onto the same string, which adds nodes and raises the oscillation frequency.

Live harmonic checks

Solve the exact state on screen.

These examples read the current harmonic, string length, probe position, and inspected time directly from the live state so the algebra stays attached to the same string you are watching.

Live valuesFollowing current parameters

For the current mode number and string length, what wavelength and oscillation frequency does this standing wave require?

n

Mode number

L

String length

1.6 m

v

Wave speed

1.2 m/s

1. Start from the allowed-mode relations

For a string fixed at both ends, use

λ_{n} = \frac{2 L}{n}

and

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

2. Substitute the live harmonic

λ_{n} = \frac{2 ( 1.6 )}{2}

and

f_{n} = \frac{( 2 ) ( 1.2 )}{2 ( 1.6 )}

3. Compute the allowed oscillation

That gives

λ_{n} = 1.6 m

and

f_{n} = 0.75 Hz

, with node spacing

0.8 m

Current harmonic requirements

λ_{n} = 1.6 m, f_{n} = 0.75 Hz

Raising the mode number adds another half-wavelength segment to the same string, which creates one more loop and one more interior node.

Node checkpoint

You want one more interior node, but you are not allowed to change the fixed-end boundary condition. What is the most direct change you can make?

Prediction prompt

Decide whether you should move the probe, raise the mode number, or just wait longer in time.

Check your reasoning

Raise the mode number to the next harmonic.

Moving the probe only changes where you inspect the string, and waiting changes only the time within the same oscillation. The number of interior nodes is set by the harmonic itself, so you must change the mode number.

Common misconception

A standing wave is frozen in place, so nothing is really oscillating once the pattern appears.

The spatial pattern is fixed, but almost every point between the nodes still oscillates up and down in time.

What stays fixed are the node positions and the overall mode shape, not the instantaneous displacement of the string.

Quick test

Misconception check

Question 1 of 4

Use the stage, graph, and interference idea together. These checks ask what must be happening on the string, not what one formula looks like by itself.

A student says, "A node is just a place where the whole wave vanished, so nothing interesting is happening 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 a fixed string stretched horizontally with one movable probe marker and a ruler underneath. The string oscillates in one selected harmonic while optional overlays can mark node positions, antinode positions, and the two traveling-wave components that interfere to make the standing pattern.

Changing amplitude, string length, mode number, or probe position immediately updates the same stage, mode-shape graph, and probe-motion graph so the standing-wave state stays synchronized.

Graph summary

The mode-shape graph plots signed standing-wave amplitude scale against position on the string, so zero crossings correspond to nodes and peaks correspond to antinodes.

The probe-motion graph plots the selected point's displacement in time together with its local envelope, which makes node points flatten while antinode points reach the full standing-wave amplitude.

Keep the wave story 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.

Open vs closed tubesOscillations

Standing Waves

See each variable before you move it.

Node guides

Center zero crossing

Waves

What the system is doing

Solve the exact state on screen.

For the current mode number and string length, what wavelength and oscillation frequency does this standing wave require?

A student says, "A node is just a place where the whole wave vanished, so nothing interesting is happening there." What is the best correction?

Keep moving through the concept map.

Resonance in Air Columns / Open and Closed Pipes

Damping / Resonance

Oscillation Energy