Physics · Waves

Standing Waves

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Wrap-up

What you learned

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Read next: Resonance in Air ColumnsSame node rules in pipes

Key takeaway

Nodes are fixed cancellation points, while nearby parts of the string can still oscillate strongly.
Antinodes mark the largest local envelope; one instant of zero displacement does not make an antinode a node.
Mode number sets the fitted wavelength, node spacing, and frequency for a string with fixed ends.

Common misconception

A standing wave is not a frozen string. The mode pattern is fixed in space, but most points still move up and down in time.

The pattern in space stays fixed, but most points between the nodes still move up and down in time.

Keep the wave story going

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Open vs closed tubesSound

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.

Resonant responseOscillations

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Watch kinetic and potential energy trade places in simple harmonic motion while the total stays fixed by amplitude and spring stiffness.

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Reference and support

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Equation snapshotMode fit and local motion · Allowed wavelengthShow 3 equations

Use the fitted wavelength to count nodes, then use the displacement equation to decide what one probe point does in time.

Mode fit and local motion
$y_{n} (x, t) = A sin (\frac{nπ x}{L}) cos (ω_{n} t)$
The sine term sets how the amplitude varies along the string, and the cosine term makes each allowed point oscillate in time.
Allowed wavelength
$λ_{n} = \frac{2 L}{n}$
Only wavelengths that fit an integer number of half-wavelengths between the fixed ends can form a standing wave.
Allowed frequency
$f_{n} = \frac{n v}{2 L}$
Once the wave speed and string length are fixed, each harmonic has one matching frequency.

Why it behaves this way

Explanation

At first a standing wave can look frozen, but the string is not motionless. It forms when two equal waves travel in opposite directions on the same string and keep interfering. At some points the cancellation is permanent, so those points stay at zero displacement and become nodes. Halfway between them, reinforcement is strongest, so those points become antinodes with the largest oscillation.

This lab keeps the stage, mode-shape graph, and probe trace tied to the same harmonic. The graph shows where the string can have zero motion or maximum motion, the stage shows that pattern across the whole string, and the probe trace shows what one chosen point does in time. When you change the mode number, you are not just changing the picture. You are choosing a new allowed wavelength and frequency for a string with fixed ends.

Key ideas

01Nodes are fixed positions where destructive interference persists. The wave has not vanished from the whole string.

02Antinodes are the centers of the loops, where the local oscillation is largest and the time trace reaches its biggest amplitude.

03Higher harmonics fit more half-wavelength segments into the same length, so node spacing shrinks and frequency rises.

Worked examples

Live harmonic checks

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Frozen walkthrough

Step through the frozen example

Frozen walkthrough

These examples use the current harmonic, string length, probe position, and inspected time from the live bench, so each calculation refers to the same string you are watching.

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Example 1 of 2

Frozen valuesUsing frozen parameters

For the current mode number and string length, what wavelength and oscillation frequency are allowed on this fixed string?

n

Mode number

L

String length

1.6 m

v

Wave speed

1.2 m/s

1. Start from the fixed-end 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 current harmonic and string length

λ_{n} = \frac{2 ( 1.6 )}{2}

and

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

3. Calculate the allowed wavelength and frequency

That gives

λ_{n} = 1.6 m

and

f_{n} = 0.75 Hz

, with node spacing

0.8 m

Current allowed values

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.

Quick test

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Accessibility

Open the text-first descriptions when you need the simulation and graph translated into words.

The simulation shows a horizontal string fixed at both ends with a movable probe marker and a ruler underneath. The string oscillates in one selected harmonic, and optional overlays can mark nodes, antinodes, and the two traveling-wave components that combine to make the standing pattern.

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

Graph summary

The mode-shape graph plots the spatial factor of the standing wave against position on the string. Zero crossings correspond to nodes and peaks correspond to antinodes.

The probe-motion graph plots the chosen point's displacement in time together with its local envelope, so a node produces a flat trace while an antinode reaches the largest swing.

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Waves

Standing Waves appears later in this track, so it is cleaner to start from the beginning first.

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