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Home/Math Visualization/Fourier Series

Fourier Series

Build waveforms from sine waves. Add harmonics one by one.

Waveform

12

Shortcuts

  • •Space / Enter — reset phase

Measured values

SeriesSquare
Terms12

About this model

Sum harmonics to approximate periodic signals. Adjust amplitudes of sine/cosine terms and see convergence toward square, sawtooth, or custom shapes.

Who it's for: Signals, acoustics, and advanced math; synthesis vs analysis view.

Key terms

  • Fourier series
  • harmonics
  • superposition
  • periodic function
  • synthesis

How it works

Finite Fourier sums approximating periodic targets: square (odd sines), sawtooth, and triangle (odd sines with 1/k²). Raise the term count to see convergence; near jump discontinuities you still see Gibbs overshoot (square/saw). The dashed curve is the ideal limit; cyan is the partial sum. Toggle animate to slide phase in time.

Key equations

Square: (4/π) Σ sin((2k−1)t)/(2k−1)

Frequently asked questions

Why do sharp corners need many harmonics?
Discontinuities or sharp bends require high-frequency content to approximate; smooth waves need fewer terms for a good fit.