Vibrating String

Fourier Analysis · Axiom Academy

EXAMPLE Vibrating String: Modal Solution Complete worked example solving the wave equation for a plucked string A string of length L is fixed at both ends and plucked at its center, creating a triangular initial displacement with maximum height h . Given: String length = L , initial height at center = h , wave speed = c Find: The displacement u(x,t) for all positions and times. The wave equation with fixed boundary conditions has normal mode solutions (n x/L) (n ct/L) Fourier coefficients are computed by projecting the initial condition onto each normal mode Symmetry of the initial condition determines which harmonics are present: plucking at the center eliminates even harmonics The solution is a superposition of standing waves, each oscillating at its natural frequency _n = n c/L This modal decomposition explains the characteristic sound of plucked string instruments Coefficients decay as 1/n^2 , making the series rapidly convergent

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