Sine waves, superposition, and the building blocks of Fourier analysis
Every periodic signal, from a violin note to a heartbeat, can be built out of one simple shape: the sine wave. Here is where it comes from and how it combines.
Watch the point go around the circle and follow its height across to the graph: steady circular motion traces out a sine wave. Change the speed to see the wave squeeze together.
A sine wave is simply the vertical projection of circular motion. As the point rotates around the circle, its height traces out a perfect sine wave.
Before diving into Fourier analysis, we need to understand its fundamental building block: the sine wave. Remarkably, any periodic function can be built from sine waves of different frequencies and amplitudes.
The deep connection between circles and waves is key to understanding Fourier analysis. Watch how uniform circular motion naturally generates a sine wave when we track the vertical position over time.
Move the frequency, amplitude, and phase sliders one at a time. Frequency packs in more cycles, amplitude makes the wave taller, and phase slides it sideways.
How many oscillations per unit
Height of the wave peaks
Horizontal shift
Every sine wave is fully described by three parameters.
Frequency (f): How many complete cycles occur per unit time. Higher frequency = faster oscillation.
Amplitude (A): The maximum height of the wave from the center line. Controls the "loudness" or intensity.
Phase (φ): The horizontal shift of the wave. Determines where the wave "starts" in its cycle.
Where:
A = amplitudef = frequency (cycles per second, or Hz)t = timeφ = phase (initial angle in radians)Switch waves on and off and tune each one's frequency, amplitude, and phase. The combined wave is just their sum, point by point.
Add waves together to see how different frequencies combine. The purple line shows the superposition of all enabled waves.
The superposition principle states that waves can be added together point by point. This is the heart of Fourier analysis: complex signals are simply sums of simple sine waves.
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