# Helmholtz Resonator

## Introduction

When one blows carefully across the opening of a bottle, a clear sound is generated. The sound is generated by oscillations in the air stream passing over the far edge of the opening.

The bottle is composed of an opening called the mouth, a narrow volume called the neck, and a large volume called the cavity. Conceptually speaking, by blowing over the mouth of the bottle, the inert mass of the air in the neck is pushed inward into the cavity. This compresses the air in the cavity and increases the pressure in the cavity. At a certain point the pressure in the cavity is high enough to push the inert mass of air in the neck back up and beyond its initial position outside. This then directs the flow of air across the mouth to flow over the edge. The airstream over the bottle then pushes the air in the neck back down and the cycle repeats. The airstream alternates quickly between going inside the bottle and going over the bottle and produces the sound.

The oscillation of the air stream happens at a specific frequency defined by the geometry of the bottle. This frequency is called the Helmholtz resonance frequency. To model this phenomena the theory describing a Helmholtz resonator can be used. While the geometry of a Helmholtz resonator is different from a bottle the same theoretical concepts can be applied.

Helmholtz resonators are commonly used to suppress noise in ducts of ventilation systems, and exhaust pipes of combustion engines [Li, et al., 2017].

A different way to understand the functioning of the Helmholtz resonator is by an analogy to a mechanical mass-spring system or electrical inductance-capacitance system. The mass (inductance) and spring (capacitance) are in series. In this conceptual model, the air in the neck acts as a mass (inductance), and air in the cavity is like a spring (capacitance) [Rahim & Johari, 2016].

In this notebook we simulate an axisymmetric Helmholtz resonator without resistance, due to the fact that the value of resistance does not affect the resonance frequency, and we reproduce the resonator frequency response.

## Domain

The bottle geometry consists of a neck of length and radius , and a spherical cavity of radius . Since this oscillator is rotationally symmetric about the -axis the geometry can be approximated by its 2D cross-section in the plane. The sketches below illustrates the simulation domain.

Define the geometric parameters to be used:
Define a RegionUnion between the neck and the cavity cross-section:

## Model Setup

The system will be modeled in the frequency domain to analyse its frequency response. The main PDE model of acoustics in the frequency domain is the Helmholtz equation. Since the domain and the boundary conditions are rotationally symmetric about the -axis an axisymmetric PDE model can be used. The axisymmetric Helmholtz equation is used to solve for the pressure field in , and to describe the propagation of harmonic sound waves at different frequencies.

The axisymmetric Helmholtz equation is given as:

where is the density in , is the speed of sound in the medium , and is the angular frequency .

The axisymmetric Helmholtz equation uses a truncated cylindrical coordinate system in 2D with independent variables instead of the cylindrical coordinates . The cylindrical coordinate variable disappears because the system is rotationally symmetric about the -axis.

The AcousticPDEComponent function can produce the axisymmetric form of the Helmholtz equation. To do so the parameter "RegionSymmetry" is set to "Axisymmetric".

The sound medium used, is air.

Find the density and sounds speed for the sound medium air:
Set up a 2D axisymmetric Helmholtz equation:

The goal is to obtain the frequency response of the oscillator. For this, the system is analyzed over a range of frequencies and the pressure level at those frequencies are recorded. The frequency sweep will be with frequencies ranging from to , which are evenly spaced, using a log scale. This range of frequencies allows one to see the different harmonics of the system.

Set the frequency range:

In the next section we show how to compute and visualize the frequency response of the system.

### Resonance frequency

As mentioned before the Helmholtz resonator can be viewed as equivalent to an oscillatory system. In electrical terminology the system behaves like a circuit in series. So, taking the circuit analogy one can find that the resonance frequency is given by [Blackstock, 2001]:

where the parameters and are given by:

and

Here, is the density of the medium, is the speed of sound in the medium, is the cross-sectional area of the neck, is the volume of the acoustic cavity and is the total length of the neck.

As mentioned before there exists a restoring force that causes the mass of air to leave the neck, this small distance covered by the mass of air is considered in the theory to still be part of the tube, as if the air had always remained inside the neck. This distance is added to the original neck length as a correction factor . Therefore, if the original length of the neck is , then the total length equals to .

The correction length of the neck depends on the geometry of the connection between the neck and the cavity. Here, the correction length will be assumed to be [Pierce, 2019].

With the above definitions one finds that the Helmholtz resonance frequency is given by:

where in the last equation the assumption is made that the volume is from a sphere with and the neck cross-sectional area is .

Define the total length :
Compute the resonance frequency :

The resonance frequency fr is part of the frequency range, fRange, for the frequency sweep.

Find the nearest frequency value to the resonance frequency:

## Boundary Conditions

At the mouth of the oscillator a harmonically oscillating pressure is applied with . At the rest of the boundaries a sound hard boundary condition is applied, and as this is the natural default boundary condition, this can be left out.

Specify the pressure at :

## Acoustic model

Specify the PDE with the model parameters and boundary conditions:

In acoustics simulations the wavelength of a sound wave needs to be resolved by a sufficiently fine mesh in order to get an accurate numerical solution. Here we use a helper function to set the MaxCellMeasure. More details are given in the Acoustics in the frequency domain monograph.

Specify a helper function for computing the max grid size:
Set up and solve the PDE at the different frequencies:

Next, we would like to visualize the pressure distribution at a specific frequency. For this we create a function that returns the nearest frequency to our query frequency.

Create a function to find the nearest frequency index for a given input:

Assuming we want to visualize the pressure distribution at , we find the closest corresponding frequency from the frequency list.

Get the position of the solution corresponding to :
Check value of the closest frequency:

Visualize the real value of the the pressure distribution at .

Plot the pressure distribution at using DensityPlot:

As a matter of convenience one can also visualize the sound pressure level. The sound pressure level is a logarithmic measure of the effective pressure of a sound relative to a reference value. In air the pressure reference value is . This value is near the absolute threshold of human hearing, which is the minimum sound level that a human can hear. The sound pressure level, denoted and measured in is given by:

where is the reference value and is the root mean square of pressure which is given by:

Define the reference value:

Visualize the sound pressure level at .

Get the position which corresponds to :
Plot it:

In order to visualize the frequency response of the oscillator one needs to get the average pressure over the cavity volume given by:

where is the absolute value of the pressure.

First, one needs to define the region of integration which in this case is the cavity. To make the limits of integration coincide with the limits of the solution one must convert the region of interest into a mesh.

Define the region of integration as a mesh:
Calculate the volume of the cavity :

Finally, define the integral and get the average pressure over all the frequencies using NIntegrate.

Compute the integral:
Visualize the frequency response of the model using a log-log scale and compare it with the theoretical resonance frequency :

The peaks shown in the plot represent the different harmonics of the system including the fundamental frequency, that has the highest peak.

The difference between the numerical and theoretical resonance frequency can be obtained for a more accurate comparison.

Get the index of the resonance frequency of the numerical solution:
Find the frequency:
Compute the error in percent:

As an additional step, one can verify that the system is actually oscillating by transforming the solution into the time domain with the harmonic wave relation:

Generate a list of values of the pressure evaluated at a certain point of the domain and at the resonance frequency , over a range of time:
Plot the list of values using ListLinePlot:

One sees that the temporal pressure variation at the point behave as a sine wave with frequency , in other words, shows an oscillatory behaviour.

Using the numerical resonance frequency one can play the sound that the bottle generates.

Play a sine wave for 3 seconds using the numerical resonance frequency:

## Wine bottle as a Helmholtz resonator

In this section we take the Helmholtz resonator one step further. This section shows how to analyze the acoustic resonance of a wine bottle. The purpose of this section is to also validate the FEM model by comparing numerical results with experimental ones. The experimental data were obtained by performing signal processing on the audio recorded from the bottle when blowing over the opening. The same steps as in the previous section are then used to build the model.

We start by set up the geometric model of the wine bottle.

Define the geometric parameters of the bottle in meters :

As the bottle wine is rotationally symmetric about the -axis the geometry can be approximated by its 2D cross-section in the plane.

Here we show the image of the wine bottle we used as a model to create the mesh. In principal it is possible to build the geometry from physical object by using functions like ImageMesh. To do so a picture of the bottle must be taken. In order to avoid the curved opening and bottle bottom distort the geometry a special lense, a telecentric lens, needs to be used.

Here the geometry is created manually. The cross-sectional area of the bottle consists of two parts: a neck and a cavity. The neck is made of a rectangle primitive region, while the cavity is of straight lines and curves.

Build the cavity boundary using Line and BSplineCurve:
Turn the cavity boundary into a graphic and discretize it:
Build the neck using Rectangle:

To join both regions, we must first convert each one into a boundary element mesh and then use BoundaryElementMeshUnion on them. This function is provided by the FEMAddOns paclet that can conveniently be installed with the resource function FEMAddOnsInstall.

Generate the boundary mesh of the cavity and neck:
Join both boundary element meshes using the function BoundaryElementMeshUnion:
Visualize the boundary mesh:
Turn the boundary element mesh into an element mesh:

Now, using the same setup as in the previous section, we solve the model.

Set up and solve the PDE at the different frequencies with a range from to :

To get the resonance frequency we need to compute the average pressure over the cavity volume.

First, define the region of integration as a mesh:
Compute the volume of the wine cavity:
Compute the average pressure:

In addition to the experimental result, we can also compare it with the theoretical value.

Define the total length and the height of the cavity:
Compute the resonant frequency with equation and the corresponding volume of a cylinder given by :
Visualize the frequency response of the model using a log-log scale and compare it with the theoretical resonance frequency :
Get the index of the resonance frequency of the numerical solution:
Find the resonance frequency:
Compute the error between the numerical and theoretical resonance frequencies:

Finally, we analyze the audio signal to get the experimental resonance frequency. The audio used was recorded using AudioCapture, which uses an audio input devices such as a built-in microphone.

Define the recorded sound:

The Audio object can be use directly in any computation.

Visualize the audio:

We can use Periodogram to get a visualization of the squared magnitude of the discrete Fourier transform, the power spectrum, of the audio given in .

Visualize the power spectrum and compare the highest peak with the theoretical resonance (red) and the numerical frequency fnumerical (brown):

The peak shown in the spectrum represents the bottle's experimental resonance frequency, which is around 121 .

To get an exact value of the resonance frequency we convert the signal into Fourier space. For this we can also use PeriodogramArray.

Compute the power spectrum in decibels :

To correctly visualize the power spectrum as a function of frequency, a list of frequencies needs to be created. This list of frequencies is based on the signal's sampling rate in the frequency space and the length of the audio.

Extract the sampling rate:
Extract the length of the audio:
Create a table that corresponds to the signal's sampling in frequency space:
Put the values in the form of :

This transform produces a mirror copy of the frequencies. These extra frequencies are the signal's negative frequency components, and we omit them.

Take the first elements from the spectrum:

Plotting the power spectrum as a function of frequency visualizes the signal's frequency components. Peaks correspond to a large power.

Plot the power spectrum:

The power spectrum shows that the audio has a frequency component of around .

Find peaks of the power spectrum:
Convert the peak location into the corresponding frequency:

We obtained three different values for the resonant frequency: the simulation value, the measured value, and the theoretical value.

Compare the three values:
Compute the error between the numerical and measured resonance frequencies:

## Nomenclature

 Symbol Description Unit ρ density of a medium [kg/m3] c speed of sound in a medium [m/s] p sound pressure [Pa] ω sound wave angular frequency [rad/s] f sound wave frequency [Hz] rneck radius of the neck [m] rcavity radius of the cavity [m] l length of the neck [m] lt total length of the neck [m] Ω computational domain N/A V cavity volume [m3] S neck cross-sectional area [m2] Lp sound pressure level [dB]

## References

1.  Blackstock, D. T. (2001). Fundamentals of physical acoustics.

2.  Li, L., Liu, Y., Zhang, F., & Sun, Z. (2017). Several explanations on the theoretical formula of Helmholtz resonator. Advances in Engineering Software, 114, 361-371.

3.  Pierce, A. D. (2019). Acoustics: an introduction to its physical principles and applications. Springer.

4.  Rahim, R. A., & Johari, M. J. B. (2016). Design and Simulation of MEMS Helmholtz Resonator for Acoustic Energy Harvester. 2016 International Conference on Computer and Communication Engineering (ICCCE).