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
Wavelength— cells
Harmonic—
Frequency0.90
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.
How It Works
One equation, a reflection that may or may not flip, and standing waves at the right 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 the whole string with a stable leapfrog scheme, and disturbances travel at a speed set by the tension and mass, v = √(T/μ).
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 string can move, nothing needs cancelling, and the pulse returns upright. Toggle the end and send a pulse to watch the two cases side by side.
3
Standing waves
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 — the resonances, or harmonics, marked by the dots at each node.
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.