Sine waves, superposition, and the building blocks of Fourier analysis
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.
This page explores what makes sine waves special and how they combine to form more complex signals.
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.
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.
Every sine wave is fully described by three parameters. Adjust them below to see how each affects the wave:
How many oscillations per unit
Height of the wave peaks
Horizontal shift
How many complete cycles occur per unit time. Higher frequency = faster oscillation.
The maximum height of the wave from the center line. Controls the "loudness" or intensity.
The horizontal shift of the wave. Determines where the wave "starts" in its cycle.
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.
Add waves together to see how different frequencies combine. The purple line shows the superposition of all enabled waves.
Where:
A = amplitudef = frequency (cycles per second, or Hz)t = timeφ = phase (initial angle in radians)