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Reverberation Formula

Sabine and Reverberation Time

W C Sabine.
W C Sabine.
Wallace Clement Sabine is widely considered to be the founder of modern architectural acoustics. As a young faculty member of the Harvard University Physics Department, in 1895 Sabine was given the task of improving the acoustic performance of the lecture hall in the Fogg Art Museum. At that point, the acoustics of the lecture theatre were considered so awful that other faculty members believed it would be an impossible task.

W.C. Sabine carried out a considerable amount of research in this area and arrived at an empirical relationship between the volume of an auditorium, the amount of absorptive material within it and a quantity which he called the Reverberation Time (RT).

As defined by Sabine, the RT is the time taken for a continuous sound within a room to decay by 60 dB after being abruptly switched off and is given by:

where
V = the volume of the enclosure (m³), and
A = the total absorption within the enclosure (sabine).

The term A is calculated as the sum of the surface area (in m²) times the absorption coefficient (a) of each material used within the enclosure.

The absorption coefficient of any material, as originally defined by Sabine, is the ratio of the sound absorbed by that material to that absorbed by an equivalent area of open window. Thus a perfectly absorbent material would have an absorption coefficient of 1, where an absorption unit of 1 sabine represents a surface capable of absorbing sound at the same rate as 1m² of open window.

Using this simple formula, and providing one knows the surface areas and absorption coefficients of the materials to be used, the reverberation time of a room can be determined at the design stage using something as simple as a spreadsheet.

Optimum Reverberation Times

Such a formulation is particularly useful as it starts to recommend the most effective volume of rooms for particular reverberation times. Given that we know the range of values of RT for specific purposes, we can determine a relationship between room volume and internal surface area. This assumes the use of standard auditorium construction materials. Obviously, if you choose to clad the internal surfaces with marble, you will need to make additional allowances for the reduced absorption.

Figure 6 - An empirical relationship between room volume and the optimum reverberation time based on standard materials and field measurements.
Figure 6 - An empirical relationship between room volume and the optimum reverberation time based on standard materials and field measurements.

From this ratio, and the fact that each member of the audience increases the amount of absorption in the auditorium, volumes of rooms can be specified in m³ per person, which is a very useful figure at the initial stage of design.

Improving the Accuracy of Sabine's Equation

For fairly reverberant rooms with a uniform distribution of absorptive material, Sabine's formula gives a good indication of the expected behaviour. This is because Sabine assumes the sound decays continuously and smoothly, a situation requiring a homogeneous and diffuse sound field without great variation in room surfaces. As the absorption in a room is increased, however, the results obtained by this formula become less accurate. In the limiting case of a completely dead room (an anechoic chamber), where the absorption coefficients of the boundaries are 1.0 and the reverberation time should obviously be 0.0, Sabine's formula results in a finite RT.

Several different approaches have been used to derive equations which give values of reverberation time in better agreement with measured results from less reverberant rooms. One of these, the Norris-Eyring formula, assumes an intermittent decay with the arrival of fewer and fewer reflections. This gives the following formula:

where
S = the total surface area (m²), and
a = the average absorption coefficient.

This equation gives the correct value of 0.0 for a completely dead room but is more complex and only strictly valid for rooms with the same value of a for all surfaces.

When the materials of a room have a wide variety of absorption coefficients, the best predictions are obtained by the Millington-Sette equation. This is simply a matter of substituting an effective absorption coefficient ae = -ln(1-ai) into Sabine's equation to give:

where
si = the surface area of the ith material,
AI = the actual absorption coefficient, and
ln() = the natural logarithm to the base e.

NOTE: This formula indicates that highly absorbing materials are far more effective than would be anticipated in influencing the reverberation time. For example, when the actual absorption coefficient is greater than 0.63, the effective absorption coefficient is greater than one.

The Validity of Statistical Formula

It is clear that all of the equations described so far are purely statistical in nature and, as such, neglect all of the geometric information about the room (its shape, the position of absorbing materials, the use of reflectors, etc.). Thus, whilst they can closely indicate the reverberation time, they cannot be used to predict any acoustic anomalies within a room, such as discernible echoes, acoustic shadows, etc.

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