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Helmholtz resonator calculator: what note does a bottle make?

Enter the neck diameter and length, the cavity volume and the air temperature to get the resonant frequency of a Helmholtz resonator, the period and the effective neck length.

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Results

Resonant frequency

145.7 Hz

Period
6.86 ms
Neck area
5.07 cm²
Effective neck length (with end correction)
7.24 cm
Speed of sound at that temperature
343.2 m/s

Blow across the top of an empty bottle and it hums at a pitch set by its shape. That is a Helmholtz resonator. This Helmholtz resonator calculator takes the neck diameter, the neck length, the cavity volume and the air temperature to give the resonant frequency, the period, the neck area, the effective neck length and the speed of sound.

How the calculation works

  • Neck area A = π × (diameter ÷ 2)², and effective length L = neck length + 1.7 × radius, the end correction for the air moving beyond the ends of the neck.
  • Speed of sound c = 331.3 × √(1 + T ÷ 273.15) m/s, with T the air temperature in °C.
  • Resonant frequency f = c ÷ (2π) × √[A ÷ (V × L)], with the area, length and volume in consistent units.
  • Period = 1,000 ÷ f, in milliseconds.

Example

A bottle-like cavity of 60 in³ (983 cm³) has a neck 1 in (2.54 cm) wide and 2 in (5.08 cm) long, at 68 °F. The neck area is 5.07 cm² and the effective length is 5.08 + 1.7 × 1.27 = 7.24 cm. The speed of sound is 343.2 m/s, so the resonant frequency is 145.7 Hz, a period of 6.86 ms.

Why it matters

Helmholtz resonators are used wherever a particular frequency must be boosted or absorbed. A speaker designer tunes a port to extend the bass, an acoustician adds bottle-shaped absorbers to cut a room mode and an exhaust designer tunes a side branch to remove a drone. Knowing the frequency before building saves trial and error.

Good to know

The formula is for a small neck and cavity compared with the wavelength, as in a bottle or a small enclosure. A larger neck, a cavity that is not compact and a neck with a flared end change the result, and the end correction depends on whether the ends are flanged. The temperature matters since the speed of sound rises about 0.6 m/s per degree Celsius. Losses from friction in the neck lower the sharpness of the resonance. Treat the result as a first estimate and tune by ear or with a measurement.

The pipe resonance calculator treats open and closed tubes, the speed of sound calculator gives the sound speed and the reverberation time calculator covers room acoustics.

Frequently asked questions

What is a Helmholtz resonator?

It is a cavity of air with a short neck, such as a bottle, an ocarina or a ported speaker box. The air in the neck acts as a mass and the air in the cavity as a spring, so they oscillate at a natural frequency.

What is the end correction?

The moving air extends a little beyond each end of the neck, which makes the neck act longer than it is. A common value is 0.85 × the diameter at each flanged end, or about 1.7 × the radius in total.

How do I lower the frequency?

Increase the cavity volume, lengthen the neck or reduce the neck area. The frequency goes with the square root of the area over the volume and length.

Where are Helmholtz resonators used?

In musical instruments (guitar bodies, ocarinas), in bass reflex speaker enclosures, in car mufflers to cancel a drone, in room acoustics as sound absorbers and in engine intake systems.

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Results are estimates provided for information purposes. They do not replace professional advice.

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