Slat Helmholtz Resonator

The formula for calculating the Helmholtz resonant frequency for a slat resonator is:

WRONG often published and in calculators used formula
fo = 2160*sqrt(r/((d*1.2*D)+(r+w)))

CORRECT formula
fo = 2160*sqrt(r/((d*1.2*D)*(r+w)))

  • fo = resonant frequency
  • r = slot width
  • d = slat thickness
  • 1.2 = mouth correction
  • D = cavity depth
  • w = slat width
  • 2160 = c/(2*PI) but rounded
  • c = speed of sound in inch/sec

What is this mouth correction?

A Helmholtz resonator is a mass-spring system, which is comparable with a panel or membrane resonator.
The system is based on a mass which vibrates in resonance on a spring.
The ratio of the mass versus the dynamic stiffness of this spring defines the resonance frequency.
The air layer in the cavity acts as a spring with a certain dynamic stiffness mainly defined by its volume.
The larger the Volume, the weaker the spring becomes (lowering resonance frequency) and vice versa.

For a panel resonator it's easy to imagine what the mass is: the panel.
The heavier this mass becomes the lower the resonance frequency and vice versa.
As such a panel resonator is mainly defined by the combination of both properties.
This isn't complete, since angle of incidence, weakness of spring, damping etc. will influence the resonance frequency and the Q-factor.

For a Helmholtz resonator this mass is represented by the mass of the air enclosed by the neck or slot of the resonator.
However this apparent mass extends outside the exact geometrical boundaries of this neck or slot.
This is covered by the mouth correction, which is in fact a correction factor increasing those geometrical boundaries.
In reality this phenomenon is much more complicated than the simple factor, used by the traditional formulas.
As such the distance between those necks or slots (interaction) and others will influence this correction.
For practical use however the standard formulas are a good approach.

Slot width:  mm
Slat width:  mm
Depth from wall:  mm
Slat depth:  mm
effective depth of Slat:  mm
Absorption Frequency:  Hz

If the gaps vary say 5mm, 10mm, 15mm, 20mm and the wall is angled as shown below, a broad band low mid resonator is created that still keeps the high frequencies alive.
Remember the cavity behind must be airtight!

By working out the different slat widths and slat gaps you can create a broadband low mid resonator at specific frequencies.

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