Appendix 5: Deriving the Reverberation Time Equation for Different Frequencies and Surfaces

In real rooms, we must also allow for the presence of a variety of different materials, as well as accounting for their variation of absorption as a function of frequency. This is complicated by the fact that there will be different areas of material with different absorption coefficients, and these will have to be combined in a way that accurately reflects their relative contribution. For example, a large area of a material with a low value of absorption coefficient may well have more influence than a small area of material with more absorption. In the Sabine equation, this is easily done by multiplying the absorption coefficient of the material by its total area and then adding up the contributions from all the surfaces in the room. This resulted in a figure Sabine called the “equivalent open window area,” as he assumed and experimentally verified that the absorption coefficient of an open window was equal to 1. It is therefore easy to incorporate the effects of different materials by simply substituting the total open window area for different materials calculated using the method described earlier for the open window area calculated using the average absorption coefficient in the Sabine equation. This gives a modified equation that allows for a variety of frequency-dependent materials in the room:

• where α_i(f) = the absorption coefficient for a given material
•     and S_i = its area

For the Norris-Eyring reverberation time equation, the situation is a little more complicated, because the equation does not use open window area directly. There are two possible approaches. The first is to calculate a weighted average absorption coefficient by calculating the effective open window area, as done in the Sabine equation, and then dividing the result by the total surface area. This gives the following equation for the average absorption coefficient:

which can be substituted for α in the Norris-Eyring reverberation time equation to give a modified equation that allows for different materials in the room:

Equation A5.2 can be used to calculate the effect of a variety of frequency-dependent materials in the room. However, there is an alternative way of looking at the problem that is more in the spirit of the reasoning behind the Norris-Eyring reverberation time equation. This second approach can be derived by considering the effect on the sound energy amplitude of successive reflections that hit surfaces of differing absorption coefficient. In this case, the proportion of the original sound energy that is reflected back will vary with each reflection as follows:

• Energy_{After one reflection} = Energy_{Before reflection} (1 − α₁)
• Energy_{After two reflections} = Energy_{Before reflections} (1 − α₁)(1 − α₂)
• Energy_{After three reflections} = Energy_{Before reflections} (1 − α₁)(1 − α₂)(1 − α₃)
• Energy_{After n reflections} = Energy_{Before reflections} (1 − α₁)(1 − α₂)(1 − α₃) × … × (1 − α_n)

This can be couched in terms of an average α by taking the geometric mean of the different reflection coefficients (1 − α). For example, after two reflections, the energy is at a level that would be the same as if there had been two reflections from a material whose reflected energy was given by:

After three reflections, the average reflection coefficient would be given by:

And after n reflections, the average reflection coefficient would be given by:

Because there are only a finite number of different materials in the room, but of differing areas, it is necessary only to consider an average based on just the number of different materials but weighted to allow for their differing surface areas. Because logarithms convert products to additions, this weighted geometric mean can be simply expressed as a sum of the individual absorption terms, and so the Norris-Eyring reverberation time equation can be rewritten in a modified form, which allows for the variation in material absorption due to both nature and frequency, as:

Equation A5.3 is also known as the Millington-Sette equation. Although Equation A5.3 can be used irrespective of the absorption level, it is still more complicated than the Sabine equation, and if the absorption coefficient is less than 0.3, it can be approximated very effectively by it, as discussed previously. Thus, in many contexts, the Sabine equation, Equation A5.1, is preferred.