How to use it
Choose a system at the top: a string fixed at both ends, a string with one free end, a pipe open at both ends or a pipe closed at one end. It vibrates in slow motion. The dashed outline is the envelope, the furthest the wave ever reaches. Points marked N are nodes, which never move; points marked A are antinodes, where the swing is largest.
Harmonics shows one mode at a time. Mix adds several modes with amplitudes you set. Two waves draws the two travelling waves behind a standing wave. Resonance drives the system at any frequency and plots its response. Press Play sound to hear the mode, the mix or the driving tone.
Controls
- Mode buttons 1–8 (or keys 1–8) pick the mode. In Mix the keys switch a mode on or off; in Resonance they jump to that natural frequency.
- Freeze (key F) stops the picture; Animation speed makes the slow motion slower or faster. A readout says how many times slower than reality it runs.
- Play sound (key S) and Volume. Esc stops the sound and a running sweep.
- Parameters: length, tension and mass per metre for a string; length and air temperature for a pipe, plus a switch between air displacement and pressure.
- Mix: one slider per mode from −1 to 1 (a negative value flips the mode), plus presets.
- Resonance: a driving-frequency slider (or drag on the curve), Sweep and Damping.
Note names assume A4 = 440 Hz; 100 cents make one semitone.
How it works
A wave that reaches the end of a string or pipe is reflected and runs back through the incoming wave. Two equal waves travelling in opposite directions add up to
The sum does not travel: every point swings in place by an amount set by sin(kx). Where that is zero there is a node, and neighbouring nodes are half a wavelength apart. Only wavelengths that fit the ends survive. A fixed end of a string is a node and a free end an antinode, so:
one end free: fk = ((2k−1)/4L) √(T/μ)
L is the length, T the tension in newtons and μ the mass per metre in kilograms; √(T/μ) is the wave speed. A harmonic is a whole multiple of the lowest frequency. With one free end only the odd ones exist, so mode k is harmonic 2k−1.
Air columns follow the same pattern with the speed of sound, about v = 331.3 + 0.606 × (temperature in °C) metres per second:
closed at one end: fk = (2k−1)v/4L
At an open end the air moves freely while the pressure stays at the outside value. At a closed end the air cannot move and the pressure swings the most. So displacement nodes are pressure antinodes, and the two views swap N and A.
The Resonance view is a simplified model, not a full simulation. Each of the eight modes is a separate damped oscillator; all are driven equally hard and lose energy at the same rate. The response of mode n at driving frequency f is
Q is the quality factor of the lowest mode. Every peak reaches Q at f = fn and is f1/Q wide at 71% of its height. The curve plots the square root of the sum of the eight squared responses. Real damping varies with frequency, and how strongly a mode responds depends on where you push.
Things to try
- Switch between the two strings: freeing one end drops the lowest note by an octave and removes the even harmonics.
- Multiply the tension by four: the frequency doubles and the note rises one octave.
- Warm a pipe from 0 °C to 30 °C: the pitch climbs by about 3 cents per degree.
- In Two waves, freeze the picture at different moments: at every node the two travelling waves are equal and opposite.
- In Mix, choose Triangle on a string fixed at both ends. That is roughly a string pulled aside in the middle and let go; watch the corners run to the ends and back.
- In Resonance, sweep the closed pipe: peaks sit at 1, 3, 5… times the lowest frequency and the response stays low at the even multiples. Then raise the damping.
Frequently asked questions
What is a standing wave?
A vibration pattern that stays in place: fixed nodes with loops swinging between them. It forms when a wave and its reflection overlap and the wavelength fits the string or pipe.
Why does a closed pipe sound lower than an open one of the same length?
Its lowest mode holds a quarter of a wavelength instead of half. The wavelength is twice as long, so the frequency is half: one octave lower.
Why does the sound sometimes differ from the readout?
Tones below 40 Hz are hard to hear and very high ones are unpleasant, so the page shifts the sound by whole octaves and leaves out partials above 8,000 Hz. It says so under the buttons; the readouts always show the true values.
Are real instruments this exact?
No. Stiff strings have overtones slightly sharper than whole multiples, an open pipe end acts as if the pipe were longer by roughly 0.6 of its radius, and the speed-of-sound formula is an approximation for dry air.