A string of length is fixed at both ends

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

A string of length L=2.00 mL=2.00\,\mathrm{m} is fixed at both ends, so its transverse displacements obey the regular Sturm-Liouville problem −y′′=λy-y''=\lambda y on [0,L][0,L] with y(0)=y(L)=0y(0)=y(L)=0, p=1p=1, q=0q=0, w(x)=1w(x)=1. Its eigenfunctions are yn(x)=sin⁡(nπx/L)y_n(x)=\sin(n\pi x/L). A sawtooth-shaped initial displacement f(x)=xf(x)=x (in metres) for 0≤x≤L0\le x\le L is to be written as f(x)=∑n=1∞cnyn(x)f(x)=\sum_{n=1}^\infty c_n y_n(x). ((a)) Write down ∥yn∥w2=∫0Lyn2 dx\|y_n\|_w^2=\int_0^L y_n^2\,\mathrm{d}x. ((b)) Compute c1c_1, c2c_2 and c3c_3 from cn=⟨yn,f⟩w/∥yn∥w2c_n=\langle y_n,f\rangle_w/\|y_n\|_w^2.

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