A uniform string of length

PHYSICS · Subjective · Fermi University Physics · Oscillations, Waves & Acoustics · Coupled oscillators, normal modes and Fourier analysis · Superposition, boundaries and interference

Modes, Packets and Media, Ch The continuum limit, Fourier series and the spectrum of a measured signal

Problem

A uniform string of length LL, fixed at both ends, is plucked at x=L/4x=L/4 to a height hh and released from rest, giving the usual triangular initial shape. Recall from the first course that the ordinary Fourier sine coefficients of a pluck at x=pLx=pL to height hh are bn=2hn2π2p(1−p)sin⁡(nπp)b_n = \dfrac{2h}{n^2\pi^2p(1-p)}\sin(n\pi p), and that the energy in mode nn is proportional to n2bn2n^2b_n^2. Use also ∑n=1∞sin⁡2(nπp)/n2=π2p(1−p)/2\sum_{n=1}^{\infty}\sin^2(n\pi p)/n^2 = \pi^2p(1-p)/2. Find the fraction of the total energy carried by each of the first four modes (n=1,2,3,4n=1,2,3,4) for a pluck at p=1/4p=1/4.

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