🎸 String Vibration
The vibrating string is governed by the 1D wave equation: ∂²y/∂t² = c² ∂²y/∂x², where c = √(T/μ) is wave speed, T is string tension and μ is linear mass density. With fixed boundary conditions, only discrete natural frequencies f_n = n·c/(2L) are sustained — these are the harmonics (n=1: fundamental, n=2,3,…: overtones). Click or drag the string to pluck it, or select a harmonic mode. Observe nodes (zero displacement) and antinodes (maximum amplitude) of each mode. 🇺🇦 Українська
Physical parameters
Modes
Wave Equation and Standing Waves
Numerically integrated with the finite-difference leapfrog scheme: y(x,t+dt) = 2y(x,t) − y(x,t−dt) + (c·dt/dx)²(y(x+dx,t) − 2y(x,t) + y(x−dx,t)). Stability requires the Courant number c·dt/dx ≤ 1 (set to 0.9 here). Mode n has n antinodes and n+1 nodes (including endpoints). Musical instruments (guitar, violin) rely on these standing wave patterns; timbre depends on the relative amplitudes of each harmonic.