The Resonant Ice Crystal · AI Generator #GEN-F4B3405E

Physics

Problem

A cylindrical ice crystal of length

L = 12.0\;\text{cm}

and radius

r = 2.0\;\text{mm}

is suspended horizontally in air. When struck, it vibrates as an acoustic resonator with both ends effectively open to the surrounding air. The speed of sound in ice at the crystal's temperature is

v_{\text{ice}} = 3200\;\text{m/s}

, while the speed of sound in air is

v_{\text{air}} = 343\;\text{m/s}

. Due to the large acoustic impedance mismatch between ice and air, only a small fraction of the sound energy escapes from each end of the crystal. Specifically, the amplitude transmission coefficient for sound waves traveling from ice to air is

t = 0.12

. When the crystal vibrates in its fundamental longitudinal mode, it creates standing waves inside the ice with displacement nodes at both ends. The crystal also radiates sound into the surrounding air. (a) Find the fundamental resonant frequency

f_0

of the ice crystal. (b) Calculate the quality factor

Q

of this resonator, given that

Q = \omega_0 \tau

, where

\tau

is the energy decay time constant. The energy decay is dominated by acoustic radiation losses through the ends. (c) If the crystal is struck and begins oscillating with an initial stored energy

E_0 = 2.5 \times 10^{-6}\;\text{J}

, how long will it take for the oscillation amplitude to decrease to

1/e

of its initial value? Note: The acoustic power radiated from each end is

P = \frac{1}{2}\rho_{\text{air}} v_{\text{air}} A t^2 v_{\text{max}}^2

, where

A

is the cross-sectional area and

v_{\text{max}}

is the maximum particle velocity amplitude at the end.

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