A string of length fixed at both ends is treated with the Dirichlet eigenfunctions

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 LL fixed at both ends is treated with the Dirichlet eigenfunctions yn(x)=sin⁡(nπx/L)y_n(x)=\sin(n\pi x/L), weight w=1w=1, eigenvalues λn=(nπ/L)2\lambda_n=(n\pi/L)^2. Consider the smooth (infinitely differentiable) function f(x)=sin⁡ ⁣(3πx2L)f(x)=\sin\!\left(\dfrac{3\pi x}{2L}\right) on [0,L][0,L]. ((a)) Without evaluating any coefficient integral, decide whether ff satisfies the boundary conditions y(0)=y(L)=0y(0)=y(L)=0 of the eigenvalue problem, and use the boundary-matching argument for coefficient decay to predict the power of nn with which the expansion coefficients cnc_n fall off for large nn. ((b)) State whether the fact that ff is infinitely differentiable is, by itself, enough to guarantee fast decay of cnc_n, and give the reason.

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