A Helmholtz resonator is the acoustic equivalent of a spring-mass oscillator. The air mass in a narrow neck acts as the mass; the trapped air in a sealed cavity acts as the spring (compressing/expanding as the neck plug moves). Like all single-degree-of-freedom oscillators, it has one resonant frequency where it absorbs / radiates strongly — and almost nothing elsewhere. Blow across the top of an empty bottle and you hear exactly that resonance.
The classical formula
f₀ = (c / 2π) · √(A / (V · L_eff)), where c is sound speed, A is the neck cross-sectional area, V is the cavity volume, and L_eff is the effective neck length including end corrections. Notice the trade-offs: wider neck raises f₀, longer neck lowers it; bigger cavity lowers it. Bass absorbers thus tend to need large cavities (50+ L) for low frequencies.
End correction
Air doesn't stop moving exactly at the neck opening — it bulges out into the surrounding air, adding a small effective length. For a flanged neck (one mounted in a flat baffle, like a port in a speaker box wall): δ ≈ 0.85·r at the open end. For an unflanged opening (like a bottle's pure tube): δ ≈ 0.61·r. A typical Helmholtz absorber mounted in a wall is flanged on the outside and partly flanged on the inside, giving δ ≈ 1.7·r_eq (close to the "0.85 + 0.85" sum used in this tool). Without end correction, predicted frequencies are several percent too high.
Bandwidth and Q
The quality factor Q measures how sharply tuned the resonance is. High Q (50+): very narrow bandwidth, deep absorption at f₀ only, useful for treating a single problem room mode. Low Q (3-10): wider bandwidth, less peak absorption, treats a broader frequency range. Q is controlled by damping in the resonator: stuffing the cavity with fiberglass or rockwool lowers Q dramatically. Bandwidth at the −3 dB points is Δf = f₀/Q.
Applications
Helmholtz designs serve two engineering communities. Room acoustics: tune absorbers to problem bass modes (35-150 Hz typical), where porous absorbers would need impractical thickness. Loudspeaker ports: a bass-reflex speaker is a Helmholtz resonator where the port + cabinet tunes a low-frequency peak that augments the driver's natural roll-off — at the cost of phase complications and group delay. Same equation, different intent.
Why does a wine bottle sound a low note when you blow across it?
You're exciting the bottle's Helmholtz resonance. The "neck" is the bottle's narrow top, the "cavity" is the air-filled body. Blowing across the lip creates a turbulent jet that periodically pushes the air plug in and out — when your jet's frequency content matches the bottle's f₀, the resonance locks onto it. A standard 750 mL wine bottle resonates around 150-200 Hz when empty. Pour wine in to reduce the cavity volume and the pitch rises — by the equation, f ∝ 1/√V.
How do I design a 100 Hz bass trap for a room mode?
For 100 Hz, typical dimensions are around: neck 100 mm × 20 mm slot, neck length 50 mm, cavity 20-30 liters. Many designs use a slot in plywood (the "panel") covering a sealed box with fiberglass damping inside. Q around 5-10 gives a useful 10-20 Hz bandwidth. Place it at a pressure maximum for the target room mode — usually a corner. You'll likely need 2-4 such resonators to make a measurable difference; one Helmholtz absorber is rarely enough.
What's the relationship between port tuning and the Helmholtz formula?
A bass-reflex speaker port IS a Helmholtz resonator. The port is the neck, the cabinet interior is the cavity, the driver provides the energy source. Port tuning frequency from the manufacturer = exactly the Helmholtz f₀ predicted by this formula. A 50 mm diameter × 150 mm long port in a 30 L cabinet tunes to about 56 Hz — close to where a typical bookshelf 2-way is voiced. Designers tweak port length to shift tuning without changing the cabinet.
Why does the cavity have to be sealed?
If the cavity leaks, the "air spring" can't compress — the resonance disappears. Real designs need an airtight seal except through the neck. For bass-reflex speakers this is usually achieved with caulking and gaskets. For bass traps, plywood panels are screwed down tight with rubber gaskets at the edges. Even small leaks (a screw missing, a slit on one edge) drop the Q dramatically and shift the tuning slightly.
How accurate is the formula?
For a resonator with neck diameter much smaller than its length, and cavity much larger than the neck volume, the formula is accurate within a few percent. Errors grow when (a) the cavity has dimensions comparable to a wavelength at f₀ (then standing waves develop inside), (b) the neck is very short relative to its diameter (end corrections dominate), or (c) the boundary geometry departs from the assumed flange/unflange. For typical bass absorbers and speaker ports, the formula is the right starting point — but final tuning is done by measurement.
What's a "slot resonator"?
A Helmholtz resonator with a long, thin slot instead of a circular hole as the neck. The equivalent area is just w × h. Slot resonators are popular in wall treatment because they hide visually behind decorative slats. Acoustic principle is identical — same Helmholtz equation. Be aware that very thin slots (height < 5 mm) develop viscous damping inside that lowers Q significantly, which can be either a feature (broader bandwidth) or a bug depending on your goals.