Blowing gently across the neck of a bottle sets the air in the neck oscillating like…

PHYSICS · Multi-part · Fermi University Physics · Oscillations, Waves & Acoustics · Simple, damped and driven oscillations · Vibrating bodies, musical and architectural acoustics

Mathematical Methods for Physics, Ch Second-order linear equations: damping, driving and resonance

Problem

Blowing gently across the neck of a bottle sets the air in the neck oscillating like a mass on a spring: the plug of air in the neck (length

L_n = 4.0\,\mathrm{cm}

, cross-sectional area

A = 1.2\,\mathrm{cm}^2

) has effective mass

m = \rho A L_n

, and the air trapped in the cavity of volume

V = 500\,\mathrm{cm}^3

acts as a spring of stiffness

k = \rho c^2 A^2/V

for small compressions made too quickly for heat to flow in or out (adiabatic compressions), where

\rho = 1.20\,\mathrm{kg\,m^{-3}}

is the density of air and

c = 343\,\mathrm{m\,s^{-1}}

is the speed of sound. Viscous drag on the moving air gives a damping coefficient

b = 2.5\times10^{-4}\,\mathrm{kg\,s^{-1}}

, so once a puff of air has passed the neck displacement

x

obeys

m\ddot{x} + b\dot{x} + kx = 0

. Treat the air in the neck as a rigid plug moving without end corrections. ((a)) Compute the effective mass

m

and the stiffness

k

. ((b)) Write the equation in standard form and state

\omega_0

\gamma

, and the damping regime. ((c)) A designer wants a mechanical mass-spring-damper with the same

\omega_0

and

\gamma

, built with a mass

M = 10\,\mathrm{g}

. Find the spring stiffness

k'

and damping coefficient

b'

this requires.

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