Answer to Question #102618 in Classical Mechanics for BIVEK SAH

Question #102618
Explain how a linear monoatomic chain behaves as a low pass filter
1
Expert's answer
2020-02-11T09:53:52-0500

The propagation of harmonic waves in a linear chain consisting of equally spaced

masses m connected by linear springs of stiffness k has been studied extensively. The

chain behaves as a low pass filter so that waves can propagate without attenuation

below the frequency ω0=2km\omega_0=2\sqrt{\frac{k}{m}} . Above this frequency, the amplitude decays

exponentially and those waves are called evanescent waves. The [0,ω0][0,\omega_0] frequency range is called a pass-band and in that range the dispersion relation is given by ω=ω0sin0.5ka\omega=\omega_0 \sin{0.5ka}, where k is the wave number, and a is the distance between two consecutive masses. In that range, waves with different phase velocities c=ωkc=\frac{\omega}{k} and group velocities cg=dωdkc_g=\frac{d\omega}{dk}.

For long waves (k0)(k\to 0) , the chain behaves as a rod governed by the classical wave

equation. Several higher order continuum models are derived from the dispersion

relation.



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