PhysSandbox
Classical MechanicsWaves & SoundElectricity & MagnetismOptics & LightGravity & OrbitsLabs
🌙Astronomy & The Sky🌡️Thermodynamics🌍Biophysics, Fluids & Geoscience📐Math Visualization🔧Engineering🧪Chemistry

More from Math Visualization

Other simulators in this category — or see all 78.

View category →
NewSchool

FFT Magnitude Spectrum

Paint or preset a 256-point signal; radix-2 FFT shows |X[k]| vs bin (DC to Nyquist).

Launch Simulator
NewUniversity / research

2D Phase Portrait (ODE)

Direction field and click-to-trace trajectories for planar systems: harmonic, damped, saddle, nodes, foci, pendulum (RK4).

Launch Simulator
School

Lissajous Curves

Beautiful patterns from two frequencies with adjustable ratio.

Launch Simulator
NewKids

Harmonograph

Two damped harmonic sums in x and y: decaying rosette trace vs Lissajous loops.

Launch Simulator
NewKids

Spirograph (Trochoids)

Hypo- or epitrochoid: fixed R, rolling r, pen d; hue trail and period hints.

Launch Simulator
NewKids

Sorting Algorithms (parallel)

Bubble, insertion, merge, quicksort, and heapsort on the same shuffled permutation — five bar rows advance in lockstep so you can compare how each method moves values.

Launch Simulator
PhysSandbox

Interactive physics, chemistry, and engineering simulators for students, teachers, and curious minds.

Physics

  • Classical Mechanics
  • Waves & Sound
  • Electricity & Magnetism

Science

  • Optics & Light
  • Gravity & Orbits
  • Astronomy & The Sky

More

  • Thermodynamics
  • Biophysics, Fluids & Geoscience
  • Math Visualization
  • Engineering
  • Chemistry

© 2026 PhysSandbox. Free interactive science simulators.

PrivacyTermsContact
Home/Math Visualization/Fourier Series

Fourier Series

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

Waveform

12

Shortcuts

  • •Space / Enter — reset phase

Measured values

SeriesSquare
Terms12

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.