🎸 String Vibration — Harmonics & Standing Waves

Pluck a vibrating string and explore its fundamental frequency and overtones. See how string tension, length, and linear density determine pitch. Visualise the Fourier decomposition into harmonics, and the spectrum of partial amplitudes. Click on the string to pluck it at any position.

Click on the top string to pluck at that position.

Length L: 0.65 m

Tension T: 73 N

Lin. density μ: 0.000385 kg/m

Pluck pos x₀: 0.33

Damping: 0.995

f₁: — Hz

Note: —

f₂: — Hz

f₃: — Hz

Wave speed: — m/s

Physics

The wave equation: ∂²y/∂t² = (T/μ)·∂²y/∂x². The fundamental frequency is f₁ = (1/2L)·√(T/μ), and harmonics f_n = n·f₁. When you pluck the string at position x₀, the Fourier coefficients are b_n = (2/nπ)·(1/x₀(1−x₀))·sin(nπx₀), weighting each harmonic differently. The simulated string is computed as a sum of standing waves y(x,t) = Σ b_n·sin(nπx/L)·cos(2πf_nt).