Sources of Musical Sound — Question 7

Question 7

A bottle resonator uses a gas spring and a neck mass

A Helmholtz model treats a neck air slug of effective length $\ell$ and area $A$ as mass $m=\rho A\ell$. Its displacement $x$ compresses a cavity of volume $V$, giving $\Delta p=B Ax/V$. Let $c^2=B/\rho$, $A=3.0\times10^{-4}\,\mathrm{m^2}$, $\ell=0.030\,\mathrm m$, $V=1.0\times10^{-3}\,\mathrm{m^3}$ and $c=340\,\mathrm{m/s}$.

1. Derive the resonance frequency and evaluate it.

2. How does frequency change if cavity volume quadruples? State the main wavelength condition behind this lumped model.

Question 7 - Solution

A bottle resonator uses a gas spring and a neck mass

(a) Method and calculation

The restoring force is $-A\Delta p$, so$$k_{\mathrm{eff}}=\frac{BA^2}{V},\quad\omega^2=\frac{k_{\mathrm{eff}}}m=\frac{c^2A}{V\ell}.$$$$f=\frac c{2\pi}\sqrt{\frac{A}{V\ell}}=\frac{340}{2\pi}\sqrt{10}=171.12\,\mathrm{Hz}.$$

(b) Interpretation and further calculation

With the neck unchanged, $f\propto V^{-1/2}$, so quadrupling volume halves the frequency to $85.56\,\mathrm{Hz}$. The cavity dimensions and neck length should be small relative to wavelength so cavity pressure is nearly uniform and neck air moves approximately as one slug. Here $\lambda=c/f\simeq1.99\,\mathrm m$.

Physical check

The expression has units of frequency because $A/(V\ell)$ has units $\mathrm{m^{-2}}$. Radiation loss and viscous damping are neglected.