A guitar string of length

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Fourier analysis and integral transforms · Function spaces and operator theory

Conditions of Validity, Ch Eigenfunction expansions, convergence and the Rayleigh quotient

Problem

L = 0.65\ \mathrm{m}

, clamped at both ends, is released from a shape well approximated by two terms of its sine (normal-mode) expansion,

y(x,0) = A\sin\!\left(\frac{\pi x}{L}\right) + \frac{A}{9}\sin\!\left(\frac{3\pi x}{L}\right),

with

A = 2.0\ \mathrm{mm}

. The weight for this problem is

w=1

. ((a)) Using Parseval's identity for the sine eigenfunctions, estimate the percentage of the total mean-square displacement

\int_0^L y(x,0)^2\,\mathrm{d}x

carried by the fundamental mode alone. ((b)) The stored elastic energy of the string is proportional to

\int_0^L\big(\partial y/\partial x\big)^2\,\mathrm{d}x

. What percentage of it does the fundamental carry, and why does it differ from the answer to (a)?

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