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Helmholtz Resonant Absorber

by Adam P. Salisbury August 30, 2004

1. Introduction.

A listening room, defined by its dimensions, can be mapped in terms of a series of pressure peaks and nulls, in all three dimensions. This refers to the creation of standing waves (modes), and the resultant sonic characteristic of the room at modal (essentially bass) frequencies. There are other considerations, such as the boundary effect, but this has less to do with specific modal treatment, and more to do with generalised treatment, and certainly with loudspeaker and listening position placement.

The problem with modal treatment comes in dealing with the very large wavelengths associated with modal frequencies. The provision of absorption at these frequencies, by simple absorptive panels of Rockwool or fibreglass, is not practical given the huge material depth required to provide pretty average levels of absorption. Even at 100Hz the depth of material required is:

1120 being the speed of sound in feet/second (ft/s) and 4f being 4 x the target frequency.

This means that a more dimensionally modest, and hence complex means of low frequency absorption is required. The three common means of providing such are the tube trap, the resonant panel (or membrane), and the Helmholtz resonator. The first two provide a comparatively wide operational bandwidth; at varying levels of absorption, and are particularly useful for generalised treatment. The Helmholtz resonator is very much function (frequency) specific. The narrow bandwidth of operation makes it ideal for treating single frequency anomalies.

The designation 'Helmholtz' resonator comes from the German physicist Hermann Von Helmholtz (1821 - 1894), who mathematically derived their design frequency in order to study the harmonic content of complex sounds. As a means of providing ambient resonance, simple resonators considerably pre-date Helmholtz, and are know to have been used in ancient Greek theatres. As a means of providing absorption by resonance, resonators were used in many early churches - from simple ash or fleece-filled jugs, and more latterly to specially designed walls or other structures, which remain in use today.

2. Calculations.

The often used, yet still appropriate analogy of an Helmholtz resonator is that of blowing across the top of a bottle - doing so yields a monotonic sound at the resonant frequency of the bottle. Taking the analogy a little further, the neck of the bottle represents the port, and the body of the bottle the cavity or enclosure.

Those familiar with ported loudspeaker/subwoofer enclosure design will understand the relationship of port area to enclosure area in terms of tuning (F B ). This is often referred to as the tuning frequency, but could also be called the Helmholtz frequency. The formula for calculating the resonant frequency of a Helmholtz resonator is:

[formula2] [formula3]

Where f is the resonant frequency (Hz), c is the speed of sound (1120ft/s), p =3.14, S is the area of the port (ft 2 ), L is the length of the port (ft), and V is the cavity volume (ft 3 ).

Note the requirement that the units of measurement are constant. The example uses feet, but could be substituted for meters. Despite the simplicity of the equation there are many measurement variations that the designer will face. Therefore, it is highly recommended that the equation be programmed into a spreadsheet, allowing the designer to see instantly the effect each measurement change has on the design. The following Excel spreadsheet includes such calculations (including port end-correction), as well as a basic axial room mode calculator, which is useful for initial room mode mapping.

http://www.apsalisbury.dsl.pipex.com/modecalc.xls

Excited by a frequency relative to it's tuning, the air mass in the port tries to leave. It's motion, in turn, lowers the pressure of the air mass in the enclosure - which consequently attempts to draw the port's air mass back into the cavity. These motions are, naturally, lossy - the activating frequency (which, in reality, is a room mode), in perpetuating these motions, itself becomes lossy - hence the absorbent action.

2.1 Port end-correction.

The Helmholtz resonator is a narrow-bandwidth device, designed and used to target specific single frequency anomalies. The margin for design or constructional error, therefore, is minimal. One area often overlooked is that of port end-correction, without which a design may be rendered ineffective. Basically speaking, the air immediately outside either end of the port acts in sympathy with the design air mass in the port. This has the effect of increasing the apparent length of the port (fig1), and hence affects the tuning frequency of the resonator.

fig1_th.jpg
Fig1.

The apparent increase in port length is calculated as 0.732 x the port diameter (for a flush-fitted port, with little or no flare). Naturally, then, any shift in design frequency is connected to the diameter of port used, as well as to the cavity volume, in consequent calculations. At the volumes likely, and with a 3" - 4" port, the uncorrected resonant frequency will be off (high) by around 10Hz - 15Hz. Which in real terms is sufficient for the device to lack proper excitation by the target room mode, and hence prove ineffective.

Where f is the resonant frequency (Hz), c is the speed of sound (1120ft/s), p =3.14, S is the area of the port (ft 2 ), L is the length of the port (ft), D is the diameter of the port (ft), and V is the cavity volume (ft 3 ).

Modified to include port end-correction the formula elicits a marked change in resonant frequency. For so function (frequency) specific a device, it is vital that port end-correction be taken into account.

3. Design.

The fundamental design is an enclosure, or cavity, into which there is a vent, or port. The dimensions of these two areas are critical in that combined they determine the resonant frequency of the resonator. The shape of the cavity itself has a function, in that a smooth round cavity has very little frictional loss. This leads to a resonator that is highly absorbent over an ultra-narrow bandwidth, perhaps only really effective at the centre frequency - and equating to a very high Q (centre frequency divided by bandwidth). A square or rectangular cavity with angled intersections has frictional losses that reduce effectiveness in favour of an increased bandwidth, and equating to a low Q (relatively speaking). A good compromise is the tube, which retains performance over a useable bandwidth.

It should be borne in mind that by useable (which of course depends on the application and the end-user), non-damped resonators still only have worthwhile losses over perhaps 5Hz or 10Hz either side of the centre frequency. This can be increased, at the expense of reduced absorbent effect, by adding designed frictional losses in the form of cavity or port wadding. These design variables allow for fine-tuning, which is an essential part of passive acoustic treatment.

In terms of practical design, the basis of the project was a calculated room mode at 62Hz (the above calculations being based on the dimensions of the finalised design, after test measurements and tuning). Not a stacked mode, but one disparate from the neighbouring (upper) mode by some degree. Initial fine tuning in terms of frequency selection were performed using a combination of test tone sweeps and single frequency test tones, measured using a Radioshack SPL meter. Important in any area of hifi, including acoustical treatment, measurements were combined with music listening tests (although a problem area had been identified previously, and was the causation of the project).

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