Hao Tang 唐顥

Helmholtz resonators

This note derives the resonance frequency of Helmholtz resonators. The result is useful for building intuitions in acoustic phonetics, in particular, how the vocal tract shape affects the formant frequencies of vowels. In certain cases, Helmholtz resonators can actually give a good approximation of the frequency values.

A Helmholtz resonator is a bottle with a large cavity and a short neck. The cavity of the bottle is of length $\ell_1$ and the cross-sectional area is $A_1$, while the neck is of length $\ell_2$ and the cross-sectional area of the neck is $A_2$. A figure of the resonator is depicted below.

When there is a pressure difference between the inside and the ouside of the neck (for example, when one blows across the tip of the neck), the small mass of air within the neck is pushed with a force proportional to the cross-sectional area of the neck and the pressure difference.

Suppose the small mass of air within the neck is pushed inside a little. The air in the cavity is slightly compressed. Since the compression is relatively fast, there is no time for the air in the cavity to exchange heat with others. Instead, the pressure inside the cavity increases, pushing the small mass of air within the neck out and transferring the energy as work back to the small mass of air in the neck. This process is known as the adiabatic process. When the small mass of air is pushed out of the neck, the air in the cavity is expanded, and the pressure inside decreases, pulling the small mass of air inside. The small mass of air within the neck oscillates (as shown in the figure below) following the simple harmonic motion, and generates sound waves.

Since the small mass of air within the neck follows the simple harmonic motion, we can model it as a mass attached to a string with spring constant $k$ (such as the one below).

Before deriving the resonance frequency, let us review the assumptions used so far. We assume the air inside the cavity follows the adiabatic process. We assume the small mass of air follows the simple harmonic motion. In other words, the force applies on the small mass follows Hooke's Law. Finally, we will use Newton's second law to relate the force with the mass. In the derivation below, anything related to the cavity is subscripted with 1's, and anything related to the neck is subscripted with 2's.

Based on these assumptions, we can derive the following. \begin{align} \omega^2 = \frac{k}{m_2} = - \frac{1}{m_2} \frac{dF}{dx} = - \frac{A_2}{m_2} \frac{dP}{dx} = - \frac{A_2^2}{m_2} \frac{dP}{dV} \end{align} The first equality follows from the simple harmonic motion, and the second follows from Hooke's Law. The third equality converts the force to pressure, the fourth converts the shift in distance to the amount of volume compressed (or expanded).

Since we assume the air in the cavity follows the adiabatic process, we have the following equation. (For the air, the constant $\gamma$ is around 1.4.) \begin{align} P V^\gamma = \text{const} \end{align} We can take the derivative of both sides with respect to $V$ and have \begin{align} \frac{dP}{dV} V^\gamma + P \gamma V^{\gamma-1} = 0, \end{align} or simply, \begin{align} \frac{dP}{dV} = - \frac{\gamma P}{V}. \end{align}

If we plug the pressure and volume relationship back to our derivation of the resonance frequency, we have \begin{align} \omega^2 = - \frac{A_2^2}{m_2} \frac{dP}{dV} = \frac{A_2^2}{m_2} \frac{\gamma P_1}{V_1} = \frac{A_2^2}{A_2 \ell_2 \rho} \frac{\gamma P_1}{V_1} = \frac{A_2}{\ell_2 \rho} \frac{\gamma P_1}{A_1 \ell_1}, \end{align} where we assume the density of the air is a constant $\rho$. As a result, we have that the resonance frequency is inversely proportional to the square root of the volume in the cavity, inversely proportional to the square root of the cross-sectional area of the neck, and proportional to the square root of the length of the neck. \begin{align} \omega \propto \sqrt{\frac{A_2}{\ell_2}\frac{1}{A_1 \ell_1}} \end{align}

This derivation appears in the book Simple Nature by Benjamin Crowell, and is quite accessible, requiring only high-school physics. If you are not willing to make the above assumptions, you can derive everything from first principles, i.e., wave equations and transport phenomena. Many derivations exist, such as those in the following books. (In fact, the three derivations are all different.)