Harmonic Analysis

Trigonometry · Axiom Academy

Discover how any periodic function can be built from simple sine and cosine waves— the foundation of Fourier analysis. The central principle of harmonic analysis, discovered by Joseph Fourier, states that any periodic function can be written as an infinite sum of sines and cosines: This is called a Fourier series . Each term is a "harmonic"—a sine or cosine wave with a specific frequency. The first term (n=1) is the fundamental frequency, and higher terms (n=2, 3, 4...) are harmonics that vibrate at integer multiples of that frequency. Let's see what happens when we add together multiple sine waves at different frequencies. Each additional harmonic contributes to shaping the final wave: 3. Approximating a Square Wave A square wave suddenly jumps between two values—it seems impossible to make from smooth sine waves! But watch what happens when we add the right harmonics: The secret: we only use odd harmonics (1st, 3rd, 5th, 7th...) with amplitudes that decrease as 1/n. Each term adds detail to approximate the sharp corners. 4. Approximating a Sawtooth Wave A sawtooth wave rises linearly then drops sharply. Unlike the square wave, the sawtooth uses all harmonics (both odd and even): Each harmonic contributes to smoothing out the linear rise and sharpening the sudden drop. Fourier series aren't just mathematical curiosities—they're fundamental to how we understand waves, signals, and vibrations in the real world:

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