Interactive Wave Generator

This is a playground for seeing where a sine wave comes from and what happens when you add waves together. A point moving steadily around a circle traces a sine wave: its height at each moment is the wave. Add a second or third wave and watch them stack into a new shape, view the spinning point as a 3D spiral, and turn on sound to hear them.

How to Use It

1. Three waves are already running (frequencies 1x, 2x, and 3x). Drag any wave’s frequency, amplitude, and phase sliders and watch the circles and the curve respond.
2. Press + Add wave to layer another one, or remove waves you do not want. Each wave gets its own colour, its own circle, and its own animated point. The bold curve is the sum.
3. Use Pause, Speed, and Sound as you like. Sound starts only when you turn it on.

Where the Wave Comes From

A point going round a circle of radius A at a steady rate has a height that rises and falls as a sine wave. If the point completes f turns per second and starts at angle φ, its height at time t is:

$$y(t) = A \sin(2\pi f t + \varphi)$$

That is exactly what the spinning dot draws: amplitude A is the circle’s radius, frequency f is how fast it spins, and phase φ is where it started. This is the honest, exact version of “simple harmonic motion” - the projection of steady circular motion.

Adding Waves (Superposition)

When two or more waves overlap, the combined wave is just their values added at every instant:

$$y(t) = A_1 \sin(2\pi f_1 t + \varphi_1) + A_2 \sin(2\pi f_2 t + \varphi_2) + \cdots$$

Small changes in frequency or phase can produce very different combined shapes - this is the idea behind everything from music chords to noise cancellation.

The 3D Spiral

Take the spinning point and stretch it out along a time axis and the circle becomes a helix (a spiral). Its shadow on the back wall is the flat wave - projecting the helix onto that wall drops the depth and leaves a pure sine curve.

The tool above draws one spiral per wave you add, so you can watch several helices at once.

Three Waves, One Point

Instead of stacking the waves, you can let each one drive a different direction in space: wave 1 moves a point left and right (X), wave 2 moves it up and down (Y), and wave 3 moves it toward and away from you (Z). Plotting that single point over time traces a 3D Lissajous figure:

• Two waves give a flat Lissajous curve - the looping shapes seen on oscilloscopes.
• Three waves give a 3D Lissajous knot.

When the frequencies are simple whole-number ratios (like 1:2:3) the curve closes into a steady repeating loop; off-ratios drift and slowly fill a box. This view uses the first three waves.

Orbital Resonance: The Same Idea in Space

That whole-number-ratio rule is not just a math curiosity - it shapes the solar system. When two moons or planets orbit so that their periods form a small whole-number ratio (one finishes two orbits for every one of the other, or three for every two), they keep lining up in the same places. Those repeated tugs in the same spots add up, and the orbits can settle into a stable, repeating pattern. This is orbital resonance.

In resonance - 2:1. The inner body goes round exactly twice for each outer orbit, so they keep lining up in the same spots. The meeting points stay put.

Out of resonance - about 1.9:1. Just off a whole-number ratio, so each meeting happens a little further around than the last. The meeting points smear into a ring.

The two systems above differ only in their period ratio. On the left the inner body orbits exactly twice for each outer orbit, so the bodies meet at the same points over and over - the meeting marks stay put. On the right the ratio is just off a whole number, so each meeting lands a little further around than the last, and the marks smear into a full ring. A whole-number ratio holds the pattern; an off-ratio lets it drift.

The clearest real example is Jupiter’s three large inner moons. Io, Europa, and Ganymede are locked in a 1:2:4 chain: in the time Ganymede orbits once, Europa orbits twice and Io four times. The repeated line-ups keep all three held together - a setup called the Laplace resonance.

Io 1.77 days 4 orbits

Europa 3.55 days 2 orbits

Ganymede 7.15 days 1 orbit

Period ratio 1 : 2 : 4 - in the time Ganymede orbits once, Europa orbits twice and Io four times.

Pluto and Neptune show it on a bigger scale: Neptune completes three orbits for every two of Pluto’s - a 3:2 resonance. Their paths cross, yet the resonance means the two are never close at the crossing.

It is the same math as the curves above. Those same moons - 1x, 2x, and 4x - are just three sine waves, and because the frequencies are whole-number multiples their sum repeats cleanly:

Set the generator’s three waves to frequencies 1x, 2x, and 4x and the 3D Lissajous figure closes into one steady loop - a 1:2:4 resonance you can watch. Nudge a frequency off a whole number and the loop never closes, exactly like the drifting orbits.

Related Tools

• The circle behind it all: the circle calculator.
• Another step-by-step visual: square roots, step by step.
• The right-triangle identity behind sine and cosine: the Pythagorean theorem calculator.