01 / THE BASICSFrequency, amplitude & phase
Frequency counts cycles per second, measured in hertz (Hz). A 2 Hz wave completes two cycles each second. Its period—the time for one cycle—is 0.5 seconds.
Amplitude controls the distance from the center line to a wave’s peak. Here it uses arbitrary units: doubling amplitude doubles the signal’s height.
Phase shifts a wave within its cycle. A full turn is 360°; 180° is half a cycle. In this studio, positive phase advances the wave to the left. A 90° advance is one quarter of a cycle.
Sine wave: y(t) = A sin(2πft + φ)A = amplitude · f = frequency · t = time
φ = phase in radians (degrees × π / 180)
02 / THE SHAPESFour ways to repeat
Sine: a smooth oscillation at one frequency. It is the simplest building block for understanding more complex signals.
Square: switches between positive and negative amplitude, spending half a cycle at each level.
Triangle: rises and falls along straight slopes. Our triangle starts at its negative peak.
Sawtooth: rises steadily from negative toward positive amplitude, then jumps back down to start again.
Try “Four shapes” to compare equal-frequency signals. Their frequency matches, but their shape—and harmonic content—differs.
03 / SUPERPOSITIONAdding signals, moment by moment
The combined waveform adds the values of all enabled waves at each instant. Peaks can reinforce one another; positive and negative values can cancel.
Two equal-frequency sine waves with equal amplitude and phase produce a sine wave with twice the amplitude. Shift one by 180° and their sum becomes zero everywhere.
Try “Phase cancellation,” then disable one wave or change its phase. Cancellation depends on the waves matching in frequency and amplitude.
The combined plot automatically adjusts its amplitude scale. Read its axis numbers when comparing heights; the separate lanes use a shared amplitude scale. In “All together,” every trace shares one amplitude scale; the thicker white line is the combined sum. Click a graph to inspect its contributions.
04 / EXPERIMENTSHarmonics & slow beats
Harmonics are frequencies at whole-number multiples of a fundamental. “Odd harmonics” combines 1, 3 and 5 Hz sine waves with amplitudes of 1, 0.33 and 0.20. These approximate the first three terms of a square-wave series.
Adding more odd harmonics with amplitudes proportional to 1/n makes the transitions sharper. A finite sum retains ripples near the transitions.
Beats appear when two sine waves have nearby frequencies. “Slow beats” mixes 2 and 2.3 Hz: the envelope swells and fades about once every 3.33 seconds, at the 0.3 Hz frequency difference.
Choose “Slow beats” and set Window to 8 s to see several swells. Speed changes how quickly you explore time; it does not change a wave’s frequency setting.