A cable of length is stretched under tension between two fixed supports

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Oscillations, Waves & Acoustics · Wave motion, the wave equation and wave energy

Conditions of Validity, Ch Sturm-Liouville theory: the spectral theorem for a differential operator

Problem

A cable of length L=2.00 mL=2.00\,\mathrm m is stretched under tension T=80 NT=80\,\mathrm N between two fixed supports. Its mass per unit length varies linearly along its length, ρ(x)=(2.0+2.0x) g m−1,0≤x≤L,\rho(x) = (2.0+2.0x)\,\mathrm{g\,m^{-1}}, \qquad 0\le x\le L, with xx measured from one support. ((a)) Starting from the discretised bead model in which NN beads of mass mi=ρ(xi) Δxm_i=\rho(x_i)\,\Delta x connected by massless string segments obey Kv=ω2Mv\mathsf K\mathbf v = \omega^2\mathsf M\mathbf v with M=diag(mi)\mathsf M=\mathrm{diag}(m_i), take the limit Δx→0\Delta x\to0 and state the resulting Sturm-Liouville eigenvalue problem for the transverse modes Y(x)Y(x) of the cable, identifying pp, qq and ww for this cable. ((b)) State the inner product (weight and limits) in which two modes YmY_m, YnY_n belonging to distinct frequencies ωm≠ωn\omega_m\ne\omega_n are guaranteed orthogonal, without evaluating the integral. ((c)) A laboratory partner numerically integrates the two lowest modes, each normalised so that ∫0LYn2 dx=1\int_0^L Y_n^2\,\mathrm dx=1, and reports ∫0LY1(x)Y2(x) dx=0.17≠0\int_0^L Y_1(x)Y_2(x)\,\mathrm dx = 0.17 \ne 0, concluding that the Sturm-Liouville theorem must be wrong for this cable. Explain the error in this conclusion and state what integral should vanish instead.

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