Fourier Harmonics

Fourier analysis, introduced by Joseph Fourier in 1822, expresses a periodic signal as a sum of sines and cosines at integer multiples of a fundamental frequency. Each component is called a harmonic. Decomposing a signal into harmonics underlies audio compression, image filtering, vibration analysis, and the physics of musical timbre — the reason a violin and a flute playing the same pitch sound different is the relative amplitudes of their harmonics.

The plot shows a simple composite wave, \sin(x) + 0.4\cos(2x), together with its two component curves \sin(x) and 0.4\cos(2x), all drawn over -2\pi to 2\pi. Plotted side by side, the second harmonic (twice the fundamental frequency, smaller amplitude) is visibly distinct from the base sine, and the composite shows how a higher-frequency term subtly reshapes each cycle without removing the underlying periodicity. This is the elementary building block of how arbitrarily complex waveforms can be assembled from simple sinusoids.