A beaded string with identical beads spaced apart hangs between two fixed supports…

PHYSICS · Multi-part · Fermi University Physics · Oscillations, Waves & Acoustics · Coupled oscillators, normal modes and Fourier analysis · Coupled oscillators, normal modes and Fourier analysis

Modes, Packets and Media, Ch The continuum limit, Fourier series and the spectrum of a measured signal

Problem

A beaded string with N=8N=8 identical beads spaced aa apart hangs between two fixed supports and obeys the exact chain dispersion relation ω=2ω0∣sin⁡(ka/2)∣\omega = 2\omega_0|\sin(ka/2)|, with ω0\omega_0 the single-bead frequency and v=ω0av=\omega_0a the continuum wave speed. Between fixed ends the nnth mode has ka=nπ/(N+1)ka = n\pi/(N+1). ((a)) For the n=6n=6 mode, compute the exact fractional difference 1−ω/(vk)1-\omega/(vk) between the continuum-limit frequency vkvk and the true chain frequency ω\omega. ((b)) Compute the same fractional difference using the second-order approximation ω≈vk[1−(ka)224]\omega \approx vk\left[1-\dfrac{(ka)^2}{24}\right], and comment on whether this approximation is still trustworthy at n=6n=6.

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