The mechanism of Helmholtz resonance and a simple proof of the resonance frequency theory

Table of Contents

Introduction
Mechanism of Sound Production

Introduction

When you blow air across the mouth of an empty bottle, you hear a “booo” sound. This phenomenon is called Helmholtz resonance. In this article, we will explain the mechanism behind Helmholtz resonance.

The frequency of the “booo” sound produced by Helmholtz resonance (the resonance frequency) can be expressed as follows.

Resonance Frequency of Helmholtz Resonance

When the speed of sound is $v$, the volume of the bottle body is $V_0$, and the cross-sectional area and length of the neck are $S, L$ respectively, the resonance frequency of Helmholtz resonance $f_{\rm H}$ is given as follows.

\begin{eqnarray*}
f_{\rm H} = \frac{v}{2\mathrm{\pi}}\sqrt{\frac{S}{V_0 L}}\tag{☆}
\end{eqnarray*}

Let’s derive this expression.

Mechanism of Sound Production

The mechanism of Helmholtz resonance is relatively simple and works as follows.

① When you blow air across the mouth of the bottle, the air in the neck is pushed into the bottle.

② The pressure inside the bottle increases, pushing the air back out.

The repetition of ①　⇄　② causes the air to vibrate, producing sound.

To derive (☆), let’s first consider the air in the neck as a piston with mass $m$. If the cross-sectional area and length of the neck are $S, L$ respectively, and the density of air is $\rho$, then

\begin{eqnarray}
m \approx \rho S L
\end{eqnarray}

can be expressed approximately.

Now, when no force is applied to the piston, the pressure of the internal gas equals the external atmospheric pressure $P_0$, and its volume is $V_0$.

If we now push the piston in by $x$, changing the pressure to $P$ and the volume to $V$, then the piston experiences a restoring force of

\begin{eqnarray}
F = -S(P – P_0)
\end{eqnarray}

Also, let’s assume that the expansion and compression of the gas due to the piston’s motion is an adiabatic process. Then, from Poisson’s relation,

\begin{eqnarray}
PV^{\gamma} = P_0V_0^{\gamma}
\end{eqnarray}

holds. Here, $\gamma$ is the heat capacity ratio. Also,

\begin{eqnarray}
V = V_0 – Sx
\end{eqnarray}

holds, so substituting (4) into (3),

\begin{eqnarray*}
&P&(V_0 – Sx)^{\gamma} = P_0V_0^{\gamma} \\
\\
\therefore\ &P& = P_0V_0^{\gamma} \cdot (V_0 – Sx)^{-\gamma} \\
&&= P_0V_0^{\gamma} \cdot V_0^{-\gamma} \left(1 – \frac{Sx}{V_0}\right)^{-\gamma} \\
&&= P_0 \left(1 – \frac{Sx}{V_0}\right)^{-\gamma} \\
\end{eqnarray*}

is obtained.

Here, considering that $\frac{Sx}{V_0}$ is a small quantity, we can use the relation $(1 + \varepsilon)^{\alpha} \approx 1 + \alpha\varepsilon$ that holds when $|\varepsilon| \ll 1$ to write

\begin{eqnarray}
\begin{split}
P &=& P_0 \left(1 – \frac{Sx}{V_0}\right)^{-\gamma} \\
&\approx& P_0 \left(1 + \gamma \frac{Sx}{V_0}\right)
\end{split}
\end{eqnarray}

Therefore, substituting (5) into (2),

\begin{eqnarray*}
F &=& -S\left\{ P_0 \left(1 + \gamma \frac{Sx}{V_0}\right) – P_0\right\} \\
&=& – \gamma \frac{P_0S^2}{V_0} \cdot x
\end{eqnarray*}

is obtained.

Therefore, the equation of motion governing the air in the neck with mass $m$ is

\begin{eqnarray*}
m \frac{\mathrm{d^2}x}{\mathrm{d}t^2} =\, – \gamma \frac{P_0S^2}{V_0} \cdot x
\end{eqnarray*}

and this can be regarded as simple harmonic motion with spring constant $k = \gamma\frac{P_0S^2}{V_0}$.

Therefore, the frequency of air vibration, that is, the resonance frequency $f_{\rm H}$, is

\begin{eqnarray*}
f_{\rm H} &=& \frac{1}{2\pi} \sqrt{\frac{k}{m}} \\
&=& \frac{1}{2\pi} \sqrt{\frac{\gamma P_0S^2}{mV_0}}
\end{eqnarray*}

as calculated.

Substituting (1): $m \approx \rho S L$ and the relation for the speed of sound $v$: $v = \sqrt{\frac{\gamma P_0}{\rho}}$ into the above expression,

\begin{eqnarray*}
f_{\rm H} = \frac{v}{2\mathrm{\pi}}\sqrt{\frac{S}{V_0 L}}
\end{eqnarray*}

can be derived.