Sinus · living wave
f(x) = 1·sin(1x + 0) + 0
A harmonic wave is a periodic displacement y = A·sin(kx − ωt + φ): amplitude A sets the transported energy (∝ A²), angular frequency ω = 2πf the repetition rate, phase φ the shift in time. Mains voltage, a plucked string and a spoken vowel are all superpositions of exactly this function.
right-click · learn & record
f(x) = 1·sin(1x + 0) + 0
Periodic, odd function. Period = 2π/b, range [d−a, d+a]. It is the projection of uniform circular motion onto the vertical axis — the base of every soundwave in this engine.
Fourier synthesis · spectrum
f(t) = a₀/2 + Σ [aₖ·cos(2πkt) + bₖ·sin(2πkt)] — stack coefficients and read both domains at once.
b_k = 2/(πk) for every k. All harmonics present with 1/k decay — the brightest classic waveform; Gibbs overshoot ≈ 9 % stays at the jump no matter how many terms you add.
Fourier synthesis · living signal
f(t) = a₀/2 + Σ aₖcos(2πkt) + bₖsin(2πkt) · N = 8
Every periodic signal of period T decomposes into harmonics at integer multiples of f₀ = 1/T, with amplitudes aₖ and bₖ. Each rotating circle is one harmonic; their sum reproduces the signal exactly. Audio equalisers, MP3 coding and MRI reconstruction all operate in this spectral domain.
right-click · learn & record
Reconstructed signal · one periodMagnitude spectrum |cₖ| - Harmonics N
- 8
- Peak
- 1.066
- RMS
- 0.556
- Energy (Parseval)
- 0.310
- THD
- 72.6 %
Live coefficients · aₖ (cos) / bₖ (sin)
Every drag rewrites the series in place — the reconstruction, the |cₖ| spectrum and the Parseval metrics recompute on the same frame.
