A string of length

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

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

Problem

A string of length L=1.2 mL = 1.2\ \mathrm{m}, fixed at both ends, has tension T=64 NT = 64\ \mathrm{N} and mass per unit length ρ=1.0 g m−1\rho = 1.0\ \mathrm{g\,m^{-1}}. Its first natural angular frequency is ω1\omega_1. ((a)) The string is driven by F(x,t)=F0sin⁡(πx/L)sin⁡(ω1t)F(x,t) = F_0\sin(\pi x/L)\sin(\omega_1 t), with F0=0.30 N m−1F_0=0.30\ \mathrm{N\,m^{-1}}. Decide whether a steady periodic solution exists, giving the reason. ((b)) Instead the string is driven at the same frequency ω1\omega_1 but with the spatial profile of the second mode, F(x,t)=F0sin⁡(2πx/L)sin⁡(ω1t)F(x,t)=F_0\sin(2\pi x/L)\sin(\omega_1 t). Decide whether a steady periodic solution now exists and, if so, find the amplitude of the second-mode component of the response.

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