# Helmholtz Resonator Calculator

Created by Davide Borchia
Reviewed by Anna Szczepanek, PhD and Rijk de Wet
Last updated: Sep 23, 2022

Take a box, poke a hole in it: that's how you can build a rudimental resonating chamber, then use our Helmholtz resonator calculator to find the frequency it operates!

Since antiquity, humanity has built tools that help amplify or absorb sounds: the Helmholtz resonator is one of these devices. If you want to discover more about them, keep reading! You will learn:

• What a Helmholtz resonator is;
• How a Helmholtz resonator works;
• How to calculate the frequency of a Helmholtz resonator;
• The applications of Helmholtz resonance: from exhausts to musical instruments; and
• A neat experiment you can try at home.

Tune in to our Helmholtz frequency calculator!

## What is a Helmholtz resonator?

Resonators are devices that use resonance, the property of objects to prefer a specific frequency of oscillations at which the energy transfer is particularly effective to enhance or dampen a wave.

🙋 Resonators can exist wherever there is a wave-like behavior: water, electromagnetism, acoustics, the list goes on. Our calculator focuses on acoustic resonators!

A Helmholtz resonator is a closed (or partially closed) cavity where air oscillates at a particular standing frequency controlled by a few parameters. The result is a resonance effect widely used in acoustics: from sound absorbers to musical instruments. Let's take a look at the design of a Helmholtz resonator!

The shape of the resonator itself doesn't really matter; theoretically, any shape of the cavity works. Helmholtz (a German physicist) invented the original resonator, which used an almost spherical cavity to pick up a specific frequency from a complex sound, isolating it from the rest. A small opening in the cavity allowed the experimenter to listen to that specific frequency.

## How does a Helmholtz resonator work?

When the Helmholtz resonator is used to pick up a sound, the cavity should have a larger opening from which the sound "enters". Inside the cavity, the pressure increases because of the soundwave entering it, and decreases thanks to the inertia of air when it bounces in the cavity, doing so at a specific frequency. The user can use a smaller opening to listen to the sound inside the cavity.

Helmholtz resonators can also work as sound absorbers: in this case, there is no "listening" opening. The resonator design makes it "focus" at a specific frequency, trapping it in the chamber where it eventually gets dissipated after multiple reflections inside the resonator.

If appropriately designed, a Helmholtz resonator can almost entirely erase a frequency from a room where a complex sound is reproduced.

## How to calculate the Helmholtz resonator frequency

To calculate the Helmholtz resonator frequency, you need to know a few parameters of the resonator's design. Let's take at the formula first:

$f_{\text{res}} = \frac{c}{2\pi}\sqrt{\frac{A_0}{V \cdot L_0}}$

Where:

• $c$ is the speed of the sound in the medium (i.e., air). Its value is taken as $344\ \text{m}\!\cdot\!\text{s}^{-1}$;
• $A_0$ is the cross-section of the opening that picks the sound;
• $V$ is the volume of the resonator; and
• $L_0$ is the depth of the opening.

🔎 The frequency is associated with the geometrical measurements of the resonator: remember that you can describe oscillating phenomena in terms of wavelengths!

The depth of the opening may take into account the end correction of a hole in acoustics: that's a factor added to the length to justify the different starting point of the soundwave in the opening itself. The following formula gives the end correction:

$L_0=L+ΔL$

Where $L$ is the true length of the opening and $ΔL$ is the end correction.

🔎 The concept of resonance frequency is not limited to acoustics: try our resonant frequency calculator and our RLC circuit calculator to learn how engineers (and not only) design LC circuits!

## Examples of Helmholtz resonators — and an experiment!

The name Helmholtz resonator may sound highly technical, but we are almost sure you met them already without knowing it!

The first example of a Helmholtz resonator is... a bottle! When you take a bottle, and blow over the open neck, you hear a sound, a single note, usually pretty low: that's the resonating frequency of the bottle. Blowing right above the neck creates the difference in pressure necessary to make the air inside the bottle vibrate: that's an interesting variation on the functioning of a Helmholtz resonator.

Fill the bottle (or empty it) with a liquid, and you will hear the frequency shift: the emptier the bottle, the lower the frequency. Check the formula for the resonating frequency: the volume of the resonator is at the denominator of the fraction!

Many musical instruments make use of Helmholtz resonators. The examples range from acoustic guitars, where the wooden body helps amplify the sounds, to ocarinas, which are almost ideal Helmholtz resonators.

In ocarinas, the different notes are obtained by closing or opening holes with the fingers, thus changing the value of $A_0$.

Helmholtz resonators excel in muffling sounds when designing concert rooms or studios. In those cases, once sound engineers identify a problematic frequency, they design the proper resonator that absorbs that particular tone. Another application of Helmholtz resonance is in the exhausts of vehicles: creating a suitably sized exhaust allows to filter the unwanted noises from the engine. Or to amplify them, even if no one likes that.

One last example! Have you ever tried listening to a seashell? The sound you hear, commonly called the "sound of the sea", is the effect of Helmholtz's resonance: the internal structure of the shell makes it an astonishingly good resonator, and you can hear only some specific frequencies. 🐚

## How to use our Helmholtz resonator calculator?

Our Helmholtz resonator calculator allows you to calculate the value of the Helmholtz resonance frequency for various combinations of shapes and openings. Let's take a look at it.

1. Choose the type of chamber you are considering. We offer you four different possibilities:

• Arbitrary shape;
• Parallelepipedal shape;
• Spherical shape; and
• Cylindrical shape.

In the first one, you have to manually insert the volume (you can calculate it for many shapes with our volume calculator, while for the others, we are the ones doing the math — you only have to insert the measurement of the chamber.

1. Then you have to choose the type of opening you are considering. Here you have three choices:

• Arbitrarily shaped opening;
• Circular opening; and
• Rectangular opening.

Choose the one that fits your problem better and insert the measurements if necessary.

🙋 We added the end correction for a circular opening. We set it by default at $0.00$, but if you want to modify, simply click on advanced mode and tune its value!

1. Finally, insert the value of the length of the opening. It is crucial for a satisfying outcome of the calculations.

## An example of a Helmholtz resonator

We got really excited to see that there is an exceptionally simple experiment you can make at home to test our Helmholtz resonator calculator. You will only need an empty bottle and a tape meter.

The bottle should have a shape made of a composition of elementary shapes, like sphere and cylinder. Take the measurements you will need:

• Circumference of the bottle;
• Height of the cylindrical part;
• Diameter of the opening (where the cork sits, to be clear);
• Length of the neck.

Now calculate the volume of the bottle. You can use our calculators to do it faster: go to our sphere volume calculator and cylinder volume calculator!

We found that our bottle, with a circumference of $23.5\ \text{cm}$, and a height of the cylindrical section equal to $16.5\ \text{cm}$ has a volume of:

\begin{align*} \footnotesize V_\text{bottle}& = \footnotesize V_\text{sph}+V_\text{cyl}\\ & \footnotesize = 108.58\ \text{cm}^{3}+725.12\ \text{cm}^{3} \\ & \footnotesize = 833.7\ \text{cm}^{3} \end{align*}

The neck has the radius of $0.95\ \text{cm}$ and the length of $7.5\ \text{cm}$.

Insert these measurements in our calculator after selecting arbitrary shape in the resonator shape section, and circular opening in the opening section. The result you will find is:

\begin{align*} \footnotesize f_\text{res} & \footnotesize =\frac{344}{2\pi} \sqrt{\frac{A_0}{V\cdot L_0}}\\ & \footnotesize = \frac{344}{2\pi}\sqrt{\frac{2.835\cdot 10^{-4}}{8.337\cdot 10^{-4}\cdot 0.075}} \\ & \footnotesize =116.58\ \text{Hz} \end{align*}

Install an application that allows you to detect the frequency of a sound. We used , but you can easily use an online tool like .

Blow on the bottle and read the measurement: we got $117\ \text{Hz}$! Science works!

🔎 Fill the bottle with a bit of tap water and change the height of the cylindrical part according to the new measurement. Blowing on the bottle will produce a sound with a higher pitch: this is an adjustable Helmholtz resonator. For a new height of $12.2\ \text{cm}$, we calculated a resonant frequency of $132.57\ \text{Hz}$, and measured $134\ \text{Hz}$. Mind-blowing bottle-blowing!

## FAQ

### How do I calculate the Helmholtz resonance frequency?

The Helmholtz resonator frequency equation is fᵣₑ= c/2π × √(A₀/(V × L₀)), where c is the speed of sound in air (344 m/s), and A₀, L₀, V are the geometrical parameters of the resonator: the area and length of the opening, and the volume of the chamber.

### How does a Helmholtz resonator work?

A Helmholtz resonator is a device able to pick up a single frequency (the Helmholtz resonance frequency), and amplify or suppress it, depending on the configuration of the resonance chamber.

The chamber's volume and one or more openings determine the frequency at which the air inside the resonator vibrates according to the standing waves contained there.

Find out more at omnicalculator.com!

### What is a Helmholtz resonator exhaust?

The noise of an engine can be reduced noticeably using a carefully designed Helmholtz resonator exhaust able to absorb and dissipate the problematic frequencies. Helmholtz resonators in exhausts are also used to enhance a particular note from the engine: this is a common modification in car tuning.

### Where do you find the Helmholtz resonance?

Helmholtz resonance is a widespread acoustic phenomenon. You can experience it in many situations: from blowing on a bottle to keeping your car window open while driving, the appearance of a fixed frequency sound is associated with the existence of a resonance frequency for every cavity, which can produce a sound in the right conditions.

Davide Borchia
Resonator shape
Type of shape
Arbitrary shape
Volume
cu ft
Opening
Type of opening
Arbitrary shape
Area
in²
Length of the opening
in
Result
Resonance frequency
Hz
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