Answer to Question #102618 in Classical Mechanics for BIVEK SAH

Question

Explain how a linear monoatomic chain behaves as a low pass filter

Answer 1

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

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

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

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

relation.