The simplest travelling wave there is — and a companion to our simple harmonic motion lab, where each point on the string is its own tiny oscillator. Send a single pulse and watch it race to the far end and bounce back; flip the end between fixed and free to see the reflection turn upside down or stay upright. Or drive the end steadily until the whole string locks into a shimmering standing wave. It all runs on your device.
Switch the end between fixed and free to see the reflection flip
Presets
ModeOscillate
End typeFixed
Wavelength— cells
Harmonic—
Interior nodes—
Frequency0.90
Resonance—
The first four harmonics of a string fixed at both ends. Each rose shape is that mode's up-and-down envelope; the hollow dots are its fixed nodes and the loops between them swing hardest at the antinodes. The highlighted row is the mode your current frequency and speed are driving — raise the frequency and the pattern climbs the ladder.
A fixed end flips the wave; a free end doesn't. At a fixed end the string can't move, so the only way to keep it pinned is for the reflected pulse to come back inverted — upside down. Let the end run free and the pulse bounces back the same way up. Drive the end steadily instead and, whenever a whole number of half-wavelengths fits the string, the outgoing and reflected waves lock into a standing wave: fixed nodes that never move and antinodes swinging between them. Wavelength, frequency and speed always obey v = f·λ.
Reading the simulation
What the flip at the end, the still points and the harmonic number are telling you.
1
The flip at a fixed vs a free end
Switch to Pulse mode and send a single hump down the string. Against a fixed end — the wall clamp on the right — the string can't move, so the pulse comes back inverted, upside down. Switch to a free end — the ring sliding on the rod — and the same pulse returns upright. It is the clearest one-shot way to see why the boundary decides the flip; try the "Pulse on a free end" preset.
2
How nodes and antinodes form
In Oscillate mode the driver's outgoing wave overlaps its own reflection. Where the two always cancel you get a node — a point that never moves, marked by a rose dot on the axis; halfway between each pair the string swings hardest at an antinode. The amplitude guides +A and -A mark the drive height, and at resonance the antinodes build well past it. The readout counts the interior nodes for you.
3
What the harmonic number means
The Harmonic readout, n, is how many half-wavelength loops fit along the string — n = 1 is the fundamental, n = 2 the second harmonic, and so on, each at a frequency that is a whole-number multiple of the first. When a whole number of loops fits exactly, the badge flips to Resonant and the string locks into a clean standing wave; the mode ladder below highlights that same n. The Fundamental, 2nd- and 3rd-harmonic presets step you straight through the first three.
How It Works
One equation, a reflection that may or may not flip, and standing waves at the resonant frequencies.
1
The wave equation
Each little segment of the string is pulled by its neighbours, so its acceleration is proportional to how sharply the string curves there — the one-dimensional wave equation. The simulator advances all 160 points with a Courant-stable leapfrog scheme, and disturbances travel at a speed set by the tension and mass, v = √(T/μ). The wave-speed slider scales that speed directly.
2
Reflection at the end
When a pulse reaches a fixed end it must leave that point pinned at zero, which forces the reflected pulse to invert. At a free end the last point is free to follow its neighbour, nothing needs cancelling, and the pulse returns upright. Toggle the end and send a pulse to compare the two cases.
3
Standing waves and harmonics
Drive the end continuously and the outgoing wave overlaps its own reflection. At most frequencies they partly cancel, but when a whole number of half-wavelengths fits the string they reinforce into a standing wave — a resonance, or harmonic. Wavelength, frequency and speed always obey v = f·λ, so the harmonics fall at evenly spaced frequencies, and the readout flags each one as it locks in.
What is a wave on a string?
A wave on a string is a disturbance that travels along a stretched string while the string itself only moves up and down. Flick one end and a pulse runs along the string at a speed set by the string's tension and mass; drive the end back and forth and you launch a continuous travelling wave. It is the simplest example of a mechanical wave.
What is a standing wave?
A standing wave forms when a wave reflects off the far end and overlaps the incoming wave. At special driving frequencies the two line up so that some points on the string — the nodes — never move, while the points between them — the antinodes — swing with maximum amplitude. The string appears to vibrate in fixed loops rather than carry the wave along.
Why does a wave flip over when it reflects off a fixed end?
At a fixed end the string cannot move, so the incoming pulse must be cancelled there at every instant. The only way for the total displacement to stay zero is for the reflected pulse to be inverted — flipped upside down. At a free end the string is free to move, nothing has to cancel, and the pulse reflects the same way up.
What determines the speed of a wave on a string?
The wave speed is v = √(T/μ), where T is the tension in the string and μ is its mass per unit length. Tightening the string speeds waves up; a heavier, thicker string slows them down. The speed does not depend on the wave's amplitude or frequency — those set the wavelength through λ = v/f.
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Classroom activity
A ready-to-assign lab using the simulation above. Free to use — nothing to install or sign up for, and no student data leaves the browser.
High school, College10–20 minInteractive simulation
Learning objectives
Relate wave speed, frequency, and wavelength.
Describe how tension and linear density affect wave speed.
Recognize standing waves and their nodes and antinodes.
Variables to change
Frequency
Amplitude
Tension
Damping
Procedure
Drive the string at a steady frequency and measure the wavelength.
Double the frequency and measure the wavelength again.
Increase the tension and observe the effect on wave speed and wavelength.
Tune the frequency until a stable standing wave (fixed nodes) appears.
Observations
Record frequency, wavelength, and computed wave speed (v = fλ) for each trial.
Questions
Write the relationship between wave speed, frequency, and wavelength.
When you doubled the frequency at fixed tension, what happened to the wavelength?
How does increasing tension change the wave speed?
What is a node? An antinode?
What conditions produce a standing wave on a fixed string?
Explanation
A wave transfers energy along the string at speed v = √(T/μ). Since v = fλ, raising the frequency shortens the wavelength. At resonant frequencies incident and reflected waves form standing waves with fixed nodes and antinodes.
Answer key (for instructors)
1. v = f·λ.
2. It halved, because v stays roughly constant so λ = v/f.
3. Wave speed increases; v = √(T/μ), so higher tension means faster waves.
4. A node is a point of zero displacement; an antinode is a point of maximum displacement in a standing wave.
5. When the driving frequency matches a resonant frequency so reflected waves reinforce, forming a stable pattern (L = n·λ/2).
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