A string of length

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Partial differential equations and boundary-value problems · Fourier analysis and integral transforms

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

Problem

A string of length

L = 0.80\ \mathrm{m}

, held fixed at both ends, has tension

T = 90\ \mathrm{N}

and mass per unit length

\rho = 4.0\ \mathrm{g\,m^{-1}}

. It is driven by a spatially uniform transverse force per unit length

F(x,t) = F_0\sin(\Omega t)

, with

F_0 = 6.0\ \mathrm{N\,m^{-1}}

and driving angular frequency

\Omega = 250\ \mathrm{rad\,s^{-1}}

, which is not equal to any natural angular frequency of the string. ((a)) Write the steady-state spatial equation for

Y(x)

, where

y(x,t)=Y(x)\sin(\Omega t)

, in the inhomogeneous Sturm-Liouville form

(\mathcal{L}-\lambda w)Y = wf

, and identify

\lambda

and

f

. ((b)) Expand

f

in the string's (unnormalised) sine eigenfunctions

y_n(x)=\sin(n\pi x/L)

and give the coefficients for

n=1,3,5

. ((c)) Hence estimate the amplitude of the steady-state displacement at the centre of the string, keeping terms up to

n=5

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