- Reverberation time
- Calculating and predicting reverberation time
- The effect of room size on reverberation time
- The problem of short reverberation times
- A simpler reverberation time equation
- Reverberation faults
- Reverberation time variation with frequency
- Reverberation time calculation with mixed surfaces
- Reverberation time design
- Ideal reverberation time characteristics
- Early decay time
- Lateral reflections
- Performer support
- The effect of air absorption
- Brain Bullets

Another aspect of the reverberant field is that sound energy which enters it at a particular time dies away. This is because each time the sound interacts with a surface in the room it loses some of its energy due to absorption. The time that it takes for sound at a given time to die away in a room is called the reverberation time. Reverberation time is an important aspect of sound behaviour in a room. If the sound dies away very quickly we perceive the room as being 'dead' and we find that listening to, or producing, music within such a space unrewarding. On the other hand when the sound dies away very slowly we perceive the room as being 'live'. A live room is preferred to a dead room when it comes to listening to, or producing, live music. On the other hand when listening to recorded music, which already has reverberation as part of the recording, a dead room is often preferred. However, as in many pleasurable aspects of life, reverberation must be taken in moderation. In fact the most appropriate length of reverberation time depends on the nature of the music being played. For example fast pieces of contrapuntal music, like that of Scarlatti or Mozart, require a shorter reverberation time compared with large romantic works, like that of Wagner or Berlioz, to be enjoyed at their best. The most extreme reverberation times are often found in cathedrals, ice rinks, and railway stations and these acoustics can convert many musical events to 'mush' yet to hear slow vocal polyphony, for example works by Palestrina, in a cathedral acoustic can be ravishing! This is because the composer has made use of the likely performance acoustic as part of the composition. Because of the importance of reverberation time in the perception of music in a room, and because of the differing requirements for speech and different types of music, much effort is focused on it. In fact a major step in room acoustics occurred when Wallace Clement Sabine enumerated a means of calculating, and so predicting, the reverberation time of a room in 1898. Much design work on auditoria in the first half of this century focused almost exclusively on this one parameter, with some successes and some spectacular failures. Nowadays other acoustical and psychoacoustical factors are also taken into consideration.

Clearly the length of time that it takes for sound to die is a function not only of the absorption of the surfaces in a room but is also a function of the length of time between interactions with the surfaces of the room. We can use these facts to derive an equation for the reverberation time in a room. The first thing to determine is the average length of time that a sound wave will travel between interactions with the surfaces of the room. This can be found from the mean free path of the room which is a measure of the average distances between surfaces, assuming all possible angles of incidence and position. For an approximately rectangular box the mean free path is given by the following equation:

MFP = 4V/S;

(6.15)

where

- MFP = the mean free path (in m)
- V = the volume (in m3) and
- S = the surface area (in m2)

The time between surface interactions may be simply calculated from Equation 6.15 by dividing it by the speed of sound to give:

t = 4V/ Sc

where

- t = the time between reflections (in s) and
- c = the speed of sound (in ms-1)

Equation 6.16 gives us the time between surface interactions and at each of these interactions α is the proportion of the energy absorbed, where α is the average absorption coefficient discussed earlier. If α of the energy absorbed at the surface then (1 - α) is the proportion of the energy reflected back to interact with further surfaces. At each surface a further proportion, α, of energy will be removed so the proportion of the original sound energy that is reflected will reduce exponentially. The combination of the time between reflections and the exponential decay of the sound energy, through progressive interactions with the surfaces of the room, can be used to derive an expression for the length of time that it would take for the initial energy to decay by a given ratio. See Appendix 3 for details.

There are an infinite number of possible ratios that could be used. However, the most commonly used ratio is that which corresponds to a decrease in sound energy of 60 dB, or 10^6. This gives an equation for the 60 dB reverberation time, known as T_{60} which is, from Appendix 3:

T_{60} = (-0.161V)/S ln(1 - α)

(6.17)

where

- T
_{60}= the 60 dB reverberation time (in s) - ln = log natural

Equation 6.17 is known as the Norris-Eyring reverberation formula, the negative sign in the top compensates for the negative sign that results from the narural logarithm resulting in a reverberation time which is positive. Note that it is possible to calculate the reverberation time for other ratios of decay and that the only difference between these and Equation 6.17 would be the value of the constant. The argument behind the derivation of reverberation time is a statistical one and so there are some important assumptions behind Equation 6.17. These assumptions are:

- that the sound visits all surfaces with equal probability, and at all possible angles of incidence. That is, the sound field is diffuse. This is required in order to invoke the concept of an average absorption coefficient for the room. Note that this is a desirable acoustic goal for subjective reasons as well; we prefer to listen to and perform music in rooms with a diffuse field.
- that the concept of a mean free path is valid. Again this is required in order to have an average absorption coefficient but in addition it means that the room's shape must not be too extreme. This means that this analysis is not valid for rooms which resemble long tunnels. However most real rooms are not too deviant and the mean free path equation is applicable.

The result in Equation 6.17 also allows some broad generalisations to be made about the effect of the size of the room on the reverberation time, irrespective of the quantity of absorption present. Equation 6.17 shows that the reverberation time is a function of the surface area, which detem1ines the total amount of absorption, and the volume, which determines the mean time between reflections in conjunction with the surface area. Consider the effect of altering the linear dimensions of the room on its volume and surface area. These clearly vary in the following way:

- V is proportional to (Linear dimension)
^{3}and - S is proportional to (Linear dimension)
^{2}

However, both the mean time between reflections, and hence the reverberation time, vary as:

V/S is proportional to ((Linear dimension)^{3}/ (Linear dimension)^{2}) is proportional to Linear Dimension

Hence as the room size increases the reverberation time increases proportionally, if the average absorption remains unaltered. In typical rooms the absorption is due to architectural features such as carpets, curtains, people, etc., and so tends to be a constant fraction of the surface area. The net result is that in general large rooms have a longer reverberation time than smaller ones and this is one of the cues we use to ascertain the size of a space, in addition to the initial time delay gap. Thus one often hears people referring to the sound of a 'big' or 'large', acoustic as opposed to a 'small' one when they are really referring to the reverberation time. Interestingly, now that it is possible to provide a long reverberation time in a small room, via electronic reverberation enhancement systems, with good quality, people have found that long reverberation times in a small room sound 'wrong' because the visual cues contradict the audio ones. That is, the listener, on the basis of the apparent size of the space and their experience, expects a shorter reverberation time than they are hearing. Apparently closing one's eyes restores the illusion by removing the distracting visual cue!

Let us use Equation 6.17 to calculate some reverberation times.

Example 6.7 What is the reverberation time of a room whose surface area is 75 m2, whose volume is 42 m3, and whose average absorption coefficient is 0.9, 0.2? What would be the effect of doubling all the dimensions of the room while keeping the average absorption coefficients the same?

Using Equation 6.17 and substituting in the above values gives, for α = 0.9:

T_{60} = (-0.161V)/(S ln(1-α) = (-0.161 x 42m-3)/(75 m^{2} x ln (1-0.9) ) = 0.042s (42 x10-3s)

which is very small!

For a = 0.2 we get:

T_{60} = (-0.161V)/(S ln(1-α) = (-0.161 x 42m-3)/(75 m^{2} x ln (1-0.2) ) = 0.43 s

which would correspond well with the typical T6o of a living room, which is in fact what it is.

If the room dimensions are doubled then the ratio of volume with respect to the surface area also doubles so the new reverberation times are given by:

Vdoubled/S doubled= Linear Dimension doubled= 2

so the old reverberation times are increased by a factor of 2:

T_{60} doubled = T_{60} X 2

which gives a reverberation time of:

T_{60} doubled = T_{60} X 2 = 0.042 x 2 = 0.084 s

when α = 0.9 and:

T_{60} doubled = T_{60} X 2 = 0.43 x 2 = 0.86 s

when α = 0.2.

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The very short reverberation times that occur when the absorption is high pose an interesting problem. Remember that one of the assumptions behind the derivation of the reverberation time calculation was that the sound energy visited all the surfaces in the room with equal probability. For our example room the mean time between reflections, using Equation 6.16 is given by:

= 4V/S*c* = (4 x42m^{3})/(75 m^{2} x *c*) =2.24m/344 ms^{-1} = 6.51 ms^{-1} (0.00651 s)

If the reverberation time calculated in Example 6.7, when a = 0.9, is divided by the mean time between reflections then the average number of reflections that have occurred during the reverberation time can be calculated to be:

N_{reflections} = T_{60}/τ = (42 x 10^{-3}s)/(6.51 x 10^{-3} s) = 6.45 reflections

This is barely enough reflections to have hit each surface once! In this situation the reverberant field does not really exist; instead the decay of sound in the room is really a series of early reflections to which the concept of reverberant field or reverberation does not really apply. In order to have a reverberant field there must be much more than 6 reflections. A suitable number of reflections, in order to have a reverberant field, might be nearer 20, although this is clearly a hard boundary to accurately define. Many studios and control rooms have been treated so that they are very 'dead' and so do not support a reverberant field.

Although the Norris-Eyring reverberation formula is often used to calculate reverberation times there is a simpler formula known as the Sabine formula, named after its developer Wallace Clement Sabine, which is also often used. Although it was originally developed from considerations of average energy loss from a volume, a derivation which involves solving a simple differential equation, it is possible to derive it from the Norris-Eyrin: reverberation formula. This also gives a useful insight into th contexts in which the Sabine formula can be reasonably applied Consider the Norris-Eyring reverberation formula below:

T_{60} = (-0.161 V)/
(S ln(1 - α))

The main difficulty in applying this formula is due to the need to take the natural logarithm of (1 - a). However, the natural logarithm can be expanded as an infinite series to give:

T_{60} = -0.161V/S
( α - (α^{2}/2) - (α^{3}/3) ... - (α^n)/∞ )

(6.18)

Because α < 1 the sequence always converges. However if (α< 0.3 then the error due to all the terms greater than -α is less than 5.7%. This means that Equation 6.18 can be approximated as:

T_{60(α < 0.3)} = ( -0.161V)/S (-α) = 0.161V/S α

(6.19)

Equation 6.19 is known as the Sabine reverberation formula and apart from being useful, was the first reverberation formula. It was developed on the basis of experimental measurements made by W.C.Sabine, thus initiating the whole science of architectural acoustics. Equation 6.19 is much easier to use and gives accurate enough results providing the absorption, α is less than about 0.3. In many real rooms this is a reasonable assumption However it becomes increasingly inaccurate as the average absorption increases and in the limit predicts a reverberation time when α = 1, that is reverberation without walls!

As stated previously, the basic assumption behind these equations is that the reverberant field is statistically random, that is a diffuse field. There are however acoustic situations in which this is not the case. Figure 6.13 shows the decay of energy, in dB, as a function of time for an ideal diffuse field reverberation. In this case the decay is a smooth straight line representing an exponential decay of an equal number of dBs per second. Figure 6.14 on the other hand shows two situations in which the reverberant field is no longer diffuse. In the first situation all the absorption is only on two surfaces, for example an office with acoustic tiles on the ceiling, carpets on the floor, and nothing on the walls.

**Fig 6.13 The ideal decay versus time curve for reverberation**

**Fig 6.14 Two situations which give poor reverberation decay curves**

**Fig 6.15 A double break reverb decay curve**

Here the sound between the absorbing surfaces decays quickly whereas the sound between the walls decays much more slowly, due to the lower absorption. In the second case there are two connected spaces, such as the transept and nave in a church, or under the balconies in a concert hall. In this case the sound energy does not couple entirely between the two spaces and so they will decay at different rates which depend on the level of absorption in them. In both of these cases the result is a sound energy curve as a function of time which has two or more slopes, as shown in Figure 6.15. This curve arises because the faster decaying waves die away before the longer decaying ones and so allow them to dominate in the end. The second major acoustical defect in reverberant decay occurs when there are two precisely parallel and smooth surfaces, as shown in Figure 6.16. This results in a series of rapidly spaced echoes, onomatopoeiacally called flutter echoes, which

**Fig 6.16 A situation which can cause flutter **

**Fig 6.17 The decay versus time curve for flutter**

which result as the energy shuttles forwards and backwards between the two surfaces. These are most easily detected by clapping one's hands between the parallel surfaces to provide the packet of sound energy to excite the flutter echo. The decay of energy versus time in this situation is shown in Figure 6.17 and the presence of the flutter echo manifests itself as a series of peaks in the decay curve. Note that this behaviour is also often associated with the double-slope decay characteristic shown in Figure 6.15 because the energy shuttling between the parallel surfaces suffers less absorption compared with a diffuse sound.

Equations 6.17 and 6.18 show that the reverberation time depends on the volume, surface area, and the average absorption coefficient in the room. However, the absorption coefficients of real materials are not constant with frequency. This means that, assuming that the room's volume and surface area are constant with frequency which is not an unreasonable assumption, the reverberation time in the room will also vary with frequency. This will subjectively alter the timbre of the sound in the room due to both the effect on the level of the reverberant field discussed earlier and the change in timbre as the sound in the room decays away. As an extreme example, if a particular frequency has a much slower rate of decay compared with other frequencies, then as the sound decays away this frequency will ultimately dominate and the room will 'ring' at that particular frequency. The sound power for steady state sounds will also have a strong peak at that frequency because of the effect on the reverberant field level.

Table 6.1 shows some typical absorption coefficients for some typical materials which are used in rooms as a function of frequency. Note that they are measured over octave bands. One could argue that third octave band measurements would be more appropriate psychacoustically, as the octave measurement will tend to blur variations within the octave which might be perceptually noticeable. In many cases, because the absorption coefficient varies smoothly with frequency, octave measurements are sufficient. However, especially when considering resonant structures, more resolution would be helpful. Note also that there are often no measurements of the absorption coeffi. cients below 125 Hz, this is due to both the difficulty in making such measurements and the fact that below 125 Hz other factors in the room become more important, as we shall see later.

In order to take account of the frequency variation of the absorption coefficients we must modify the equations which calculate the reverberation time as follows:

T_{60} = ( -0.161 V)/
(S ln(1-α(f))

where α(f) = frequency dependent absorption coefficient for the Norris-Eyring reverberation time equation and:

T_{60 (α<.3)} = ( 0.161 V)/S &alpha(f)
for the Sabine reverberation time equation.

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. These 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 one. The denominator in the Sabine reverberation equation, Equation 6.19, is also equivalent to the open window area of the room, but has been calculated using the average absorption coefficient in the room. It is therefore easy to incorporate the effects of different materials by simply calculating the total open window area for different materials, using the method described above, and substituting it for S α in Equation 6.19. This gives a modified equation which allows for a variety of frequency-dependent materials in the room as:

T_{60(α<.3)} = (0.161 V)
/( Σ_{all surfaces Si}
S_{i}α_{i}(f) )

(6.20)

where

- α
_{i}(f) = 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 the open window area directly. However the Norris-Eyring reverberation time equation can be rewritten in a modified fonn, as shown in Appendix 4, which allows for the variation in material absorption due to both nature and frequency, as:

T_{60(α<.3)} = (i0.161 V) / (Σ_{all surfaces Si} S_{i}α_{i}(f) )

(6.21)

Equation 6.21 is also known as the Millington-Sette equation. Although Equation 6.21 can be used irrespective of the absorption level it is still more complicated than the Sabine equation and, if the average 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 6.20, is preferred.

Equation 6.20 is readily used in conjunction with tables of absorption coefficients to calculate the reverberation time and can be easily programmed into a spreadsheet. As an example consider the reverberation time calculation for a living room outlined in Example 6.8.

Example 6.8 What is the 60 dB reverberation time (T_{60}) of a living room as a function of frequency whose surface area is 75 m2 and whose volume is 42 m3. The floor is carpet on concrete, the ceiling is plaster on lath, and both have an area of 16.8 m2. There are 6 m2 of windows and the rest of the surfaces are painted plaster on brick, ignore the effect of the door.

**Fig 6.18 The reverberation time for the untreated room as a function of frequency **

**Fig 6.19 The reverberation field level as a function of frequency**

Using the data in Table 6.1 set up a spreadsheet or table, as shown in Table 6.2 and calculate the equivalent open window area for each surface as a function of frequency. Having done that add up the individual surface contributions for each frequency band and apply the Equation 6.20 to the result in order to calculate the reverberation time.

From the results shown in Table 6.2, which are also plotted in Figure 6.18, one can see that the reverberation varies from 1.49 seconds at low frequencies to 0.55 seconds at high frequencies. This is a normal result for such a structure and would tend to sound a bit 'woolly' or 'boomy'. The relative level of reverberant field for this room, is also shown in Figure 6.19 and this shows approximately a 5 dB increase in the reverberant field at low frequencies.

The results of Example 6.8 beg the question: 'How can we improve the evenness of the reverberation time?' The answer is to either add, or remove, additional absorbing materials into the room in order to achieve the desired reverberation characteristic. Here the concept of an open window area budget is useful. The idea is that, given the volume of the room, and the desired reverberation time, the necessary open window area required is calculated. The open window area already present in the room is then examined and, depending on whether the room is over or under budget, appropriate materials are added or removed. Consider Example 6.9 which tries to improve the reverberation of the previous room.

Example 6.9 Which single material could be added to the room in Example 6.8 which would result in an improved reverberation time, and what amount would be required to effect the improvement?

A material which has a high absorption at low frequencies, such as wood panelling, needs to be added to the room. If the absorption budget is set as being equivalent to the open window area at 4 kHz then we must achieve an open window area of 12.5 m over the whole frequency range. The worst frequency in the previous example is 250 Hz, which only has 4.5 m of open window area at that frequency. This means that any additional absorber must add 12.5 - 4.5 = 8 m of open window area at that frequency. The absorption of wood panelling, from Table 6.1, at 250 Hz is 0.25. Therefore the amount of wood panelling required is:

Area (wood panelling) = Required open window area/Absorption coefficient = 8m/0.25 = 32m

Table 6.3, Figure 6.20 and Figure 6.21 show the effect of applying the treatment which dramatically improves the reverberation time characteristics. The reverberation time now only varies from 0.59 to 0.41 s which is a much smaller variation than before. The peak-to-peak variation in the level of the reverberant field has also been reduced to less than 2 dB.

**Fig 6.20 The reverberation time for the treated room as a function of frequency **

**Fig 6.21 The reverberation field level for the treated room as a function of frequency**

However the overall reverberation time has gone down, especially at the lowest frequencies, because of the effect of the wood panelling at frequencies other than the one being concentrated on. Thus in practice an iterative approach to deciding on the most suitable treatment for a room is often required. Another point to consider is that the treatment proposed only just fits in the room, and sometimes it proves impossible to achieve a desired reverberation characteristic due to physical limitations.

6.1.19 Ideal reverberation time characteristics

What is an ideal reverberation characteristic? We have seen that the decay should be a smooth exponential of a constant number of decibels of decay per unit time. We also know that different sorts of music require different reverberation times. In many cases the answer is, 'it depends on the situation'. However there are a few general rules which seem to be broadly accepted. Firstly, there is a range of reverberation times which are a function of the type of music being played; music with a high degree of articulation needs a drier acoustic than music which is slower and more harmonic. Secondly, as the performance space gets larger the reverberation time required for all types of music becomes longer. This result is summarised in Figure 6.22 which shows the 'ideal' reverberation time as a function of both music and room volume. Thirdly, in general, listeners prefer a rise in reverberation time in the bass (125 Hz) of about 40% relative to the midrange (1 kHz) value as shown in Figure 6.23. This rise in bass reverberation adds 'warmth' and it also helps increase the sound level of bass instruments, which often have weak fundamentals, by raising the level of the reverberant field at low frequencies. However, when recording instruments, or when listening to recorded music, this bass lift due to the reverberant field may be undesirable and therefore a flat reverberation characteristic preferred.

There are many other aspects of reverberation, too numerous to mention here, which must be considered when designing acoustic spaces. However, there are four aspects that are worthy

**Fig 6.22 Ideal reverberation times as a function of room volume and musical style **

**Fig 6.23 The ideal reverberation time versus frequency curves**

of mention as they have proved to be the downfall of more than one acoustic designer, or manufacturer of reverberation units.

The first aspect is that the measure of reverberation time as being the time it takes the sound to fall by 60 dB is not particularly relevant psychoacoustically; it is also very difficult to measure in situ. This is due to the presence of background noise, either unwanted or the music being played, which often results in less than 60 dB of energy decay before the decay sound becomes less than the residual noise in the environment. Even in the quieter environment of a Victorian town in the days before road traffic, Sabine had to do measurements, using his ears, at night to avoid the results being affected by the level of background noise. Because we rarely hear a full reverberant decay, our ears and brains have adapted, quite logically, to focus on what can be heard. Thus we are more sensitive to the effects of the first 20 to 30 dB of the reverberant decay curve. In principle, providing we have an even exponential decay curve, the 60 dB reverberation is directly proportional to the earlier curves and so this should not cause any problems. However if the curve is of the doubleslope form shown in Figure 6.15 then this simple relationship is broken. The net result is that, although the T_{60} reverberation time may be an appropriate value, because of the faster early decay to below 30 dB we perceive the reverberation as being shorter than it really is. The psychoacoustic effect of this is that the space sounds 'drier' than one would expect from a simple measurement of T_{60}' Modern acoustic designers therefore worry much more about the early decay time (EDT) than they used to when designing concert halls.

**Fig 6.24 Lateral reflections in a concert hall **

The second factor which has been found to be important for the listener is the presence of dense diffuse reflections from the side walls of a concert hall, called lateral reflections, as shown in Figure 6.24. The effect of these are to envelop or bathe the listener in sound and this has been found to be necessary for the listener to experience maximum enjoyment from the sound. It is important that these reflections be diffuse, as specular reflections will result in disturbing comb filter effects, as discussed in Chapter 1, and distracting images of the sound sources in unwanted and unusual directions. Providing diffuse reflections is thus important and this has been recognised for some time. Traditionally, the use of plaster mouldings, niches and other decorative surface irregularities have been used to provide diffusion in an ad hoc manner. More recently diffusion structures based on patterns of wells whose depths are formally defined by an appropriate mathematical sequence have been proposed and used. However it is not just the provision of diffusion on the side walls that must be considered.

**Fig 6.25 Lateral reflections in a shoe box concert hall **

The traditional concert hall is called a shoe-box hall, because of its shape, as shown in Figure 6.25, and this naturally provides a large number of lateral reflections to the audience. This shape, combined with the Victorian penchant for florid plaster decoration, resulted in some excellent sounding spaces. Unfortunately shoe-box halls are harder to make a profit with because they cannot seat as many people as some other structures. Another popular structure, which has a different behaviour as regards to lateral reflections

**Fig 6.26 Lateral reflections in a fan shaped concert hall **

**Fig 6.27 Lateral reflection from ceiling diffusion in a concert hall**

is the fan-shaped hall shown in Figure 6.26. This structure has the advantage of being able to seat more people with good sightlines but unfortunately it directs the lateral reflections away from the audience and those few that do arrive are very weak at the wider part of the fan. The situation can be improved via the use of explicit diffusion structures on the walls, ceilings, and mid-air as floating shapes, as shown in Figure 6.27. However it has been found that the pseudo-lateral diffuse reflections from the ceiling are not quite comparable in effect to reflections from the side walls, and so the provision of a good listening environment within the realities of economics is still a challenge.

**Fig 6.28 Early reflections to provide acoustic foldback to the performer **

**Fig 6.29 The effect of diffusion on the acoustic foldback fo the performer**

A third factor, which is often ignored, is the acoustics that the performers experience. Pop groups have known about this for years and take elaborate precautions to provide each performer on stage with their own individual balance of acoustic sounds via a technique known as foldback. In fact some performers now receive their foldback directly into their ears via a technique known as in-ear monitoring' and in many large gigs the equipment providing foldback to the performer can rival or even exceed, that which provides the sound for the audience. The classical musician, however, only has the acoustics of the hall to provide them with 'fold back'. Thus the musicians on the stage must rely on reflections from the nearby surfaces to provide them with the necessary sounds to enable them to hear themselves and each other. There are two requirements for the sound reaching the performer on stage. Firstly, it must be at a sufficient level and arrive soon enough to be useful. To begin with it is important that the surfaces surrounding the performers direct some sound back to them. Note that there is a conflict between this and providing a maximum amount of sound to the audience so some compromise must be reached. The usual compromise is to make use of the sound which radiates behind the performers and direct it out to the audience via the performers, as shown in Figure 6.28. This has the twofold advantage of providing the performers with acoustic foldback and redirecting sound energy that might have been lost back to the audience. Ideally the sound that is redirected back to the performers should be diffuse as this will blend the sounds of the different instruments together for all the performers, whereas specular reflectors can have hot and cold spots for a given instrument on the stage, as shown in Figure 6.29. An important aspect of acoustic fold back, however, is the time that it takes to arrive back at the performers. Ideally it should arrive immediately, and some does via the floor and direct sound from the instrument. However, the majority will have to travel to a reflecting or diffusing surface and back to the performers. There is evidence to show that, in order to maintain good ensemble comfortably, the musicians should receive the sound from other musicians within about 20 ms of the sound being produced. This means that ideally there should be a reflecting or diffusing surface within 10 ms (3.44 m or 11.5 ft) of the performer; the time is divided by 2 to allow for going to the reflecting surface and back. In practice some of the surfaces may have to be further away when large orchestral forces are being mustered, although the staging used can assist the provision of acoustic foldback. Sometimes, however, the orchestra enclosure is so large that the reflections arrive later than this. If they arrive later than about 50 ms the musicians will perceive them as echoes and ignore them. On the other hand if these reflections arrive at the boundary between perceiving it as part of the sound or an echo of a previous sound it can cause severe disruption of the performers' perception of it. The net effect of these 'late early reflections' is to damage the performers' ability to hear other instruments close to them and this further reduces their ability to maintain ensemble. In one prestigious hall, the reason musicians used to complain that they couldn't hear each other and so hated playing there was traced to the problem of late early reflections. As a postscript it is interesting to note that the orchestra enclosure in shoe-box halls often did the right things. However, in modern multipurpose facilities it is often a challenge to provide the necessary acoustic foldback while allowing space for scenery and machinery, etc.

The fourth aspect of reverberation, which caught early reverberation unit designers by surprise, is an observation. The observation is that, as well as suffering many reflections, the sound energy in a reverberant decay will have travelled through a lot of air. In fact the distance that the sound will have travelled will be directly proportional to the reverberation time, so a one second reverberation time implies that the sound will have travelled 344 m by the end of the decay. Although for low frequencies air absorbs a minimal amount of sound energy, at high frequencies this is not the case. In particular humidity, smoke particles and other impurities will absorb high-frequency energy and so reduce the level of high frequencies in the sound. This is one of the reasons that people sound duller when they are speaking at a distance. In terms of reverberation time, and also the level of the reverberant field, the effect of this extra absorption is to reduce the reverberation time, and the level of the reverberant field, at high frequencies. Fortunately this effect only becomes dominant at higher frequencies, above 2 kHz. Unfortunately it is dependent on the level of humidity and smoke in the venue and so the high-frequency reverberation time, and the reverberant field level, will change as the audience stays in the space. Note this is an additional dynamic effect over and above the static absorption simply due to the presence of a clothed person in a space and is due to the fact that people exhale water vapour and perspire. Clearly then the degree of change will be a function of both the physical exertions of the audience and the quality of the ventilation system! As the effect of air absorption is determined by the distance the sound has travelled, rather than its interaction with a surface, it is difficult to incorporate the effect into the reverberation time equations discussed earlier. An approximation that seems to work is to convert the effect of the air absorption into an equivalent absorption area by scaling an air absorption coefficient by the volume of the space. This is reasonable because as the volume of the room increases the more air that the wave travels through and the longer the distance that it travels. This coefficient is shown at the bottom of Table 6.1 and from it one can see that for small rooms the effect can be ignored because until the volume becomes greater than 40 m3 the equivalent absorbing area is less than 1 m2. The effect does become significant if one is designing artificial reverberation units because, if it is not allowed for, the result will be an overbright reverberation, which sounds unnatural.

In this section the concept of reverberation time and reverberant field has been discussed. The assumption behind the equations has been that the sound field is diffuse. However, if this is not the case then the equations are invalid. Although at mid and high audio frequencies a diffuse field can be possible, either by accident or design, at low frequencies this is not the case due to the effect of the room's boundaries causing standing waves.

You Need to Know

Another aspect of the reverberant field is that sound energy which enters it at a particular time dies away. This is because each time the sound interacts with a surface in the room it loses some of its energy due to absorption. The time that it takes for sound at a given time to die away in a room is called the reverberation time.

Reverberation time is an important aspect of sound behaviour in a room. If the sound dies away very quickly we perceive the room as being 'dead' and we find that listening to, or producing, music within such a space unrewarding. On the other hand when the sound dies away very slowly we perceive the room as being 'live'. A live room is preferred to a dead room when it comes to listening to, or producing, live music. On the other hand when listening to recorded music, which already has reverberation as part of the recording, a dead room is often preferred.

The most extreme reverberation times are often found in cathedrals, ice rinks, and railway stations and these acoustics can convert many musical events to 'mush' yet to hear slow vocal polyphony, for example works by Palestrina, in a cathedral acoustic can be ravishing!

Calculating and predicting reverberation time

the length of time that it takes for sound to die is a function not only of the absorption of the surfaces in a room but is also a function of the length of time between interactions with the surfaces of the room. We can use these facts to derive an equation for the reverberation time in a room. The first thing to determine is the average length of time that a sound wave will travel between interactions with the surfaces of the room. This can be found from the mean free path of the room which is a measure of the average distances between surfaces, assuming all possible angles of incidence and position. For an approximately rectangular box the mean free path is given by the following equation:

MFP = 4V/S;

(6.15)

where

- MFP = the mean free path (in m)
- V = the volume (in m3) and
- S = the surface area (in m2)

The time between surface interactions may be simply calculated from Equation 6.15 by dividing it by the speed of sound to give:

t = 4V/ Sc

where

- t = the time between reflections (in s) and
- c = the speed of sound (in ms-1)

at each of these interactions α is the proportion of the energy absorbed, where α is the average absorption coefficient discussed earlier. If α of the energy absorbed at the surface then (1 - α) is the proportion of the energy reflected back to interact with further surfaces. At each surface a further proportion, α, of energy will be removed so the proportion of the original sound energy that is reflected will reduce exponentially. The combination of the time between reflections and the exponential decay of the sound energy, through progressive interactions with the surfaces of the room, can be used to derive an expression for the length of time that it would take for the initial energy to decay by a given ratio.

There are an infinite number of possible ratios that could be used. However, the most commonly used ratio is that which corresponds to a decrease in sound energy of 60 dB, or 10^6. This gives an equation for the 60 dB reverberation time, known as T_{60} which is, from Appendix 3:

T_{60} = (-0.161V)/S ln(1 - α)

(6.17)

where

- T
_{60}= the 60 dB reverberation time (in s) - ln = log natural

Equation 6.17 is known as the Norris-Eyring reverberation formula, the negative sign in the top compensates for the negative sign that results from the narural logarithm resulting in a reverberation time which is positive.

**The effect of room size on reverberation time**

as the room size increases the reverberation time increases proportionally, if the average absorption remains unaltered. In typical rooms the absorption is due to architectural features such as carpets, curtains, people, etc., and so tends to be a constant fraction of the surface area. The net result is that in general large rooms have a longer reverberation time than smaller ones and this is one of the cues we use to ascertain the size of a space, in addition to the initial time delay gap. Thus one often hears people referring to the sound of a 'big' or 'large', acoustic as opposed to a 'small' one when they are really referring to the reverberation time. Interestingly, now that it is possible to provide a long reverberation time in a small room, via electronic reverberation enhancement systems, with good quality, people have found that long reverberation times in a small room sound 'wrong' because the visual cues contradict the audio ones. That is, the listener, on the basis of the apparent size of the space and their experience, expects a shorter reverberation time than they are hearing. Apparently closing one's eyes restores the illusion by removing the distracting visual cue!

The problem of short reverberation times

In order to have a reverberant field there must be much more than 6 reflections. A suitable number of reflections, in order to have a reverberant field, might be nearer 20, although this is clearly a hard boundary to accurately define. Many studios and control rooms have been treated so that they are very 'dead' and so do not support a reverberant field.

Although the Norris-Eyring reverberation formula is often used to calculate reverberation times there is a simpler formula known as the Sabine formula, named after its developer Wallace Clement Sabine, which is also often used. Although it was originally developed from considerations of average energy loss from a volume, a derivation which involves solving a simple differential equation, it is possible to derive it from the Norris-Eyrin: reverberation formula. This also gives a useful insight into th contexts in which the Sabine formula can be reasonably applied Consider the Norris-Eyring reverberation formula below:

T_{60} = (-0.161 V)/ (S ln(1 - α))

The main difficulty in applying this formula is due to the need to take the natural logarithm of (1 - a). However, the natural logarithm can be expanded as an infinite series to give:

T_{60} = -0.161V/S ( α - (α^{2}/2) - (α^{3}/3) ... - (α^n)/∞ )

(6.18)

Because α < 1 the sequence always converges. However if (α< 0.3 then the error due to all the terms greater than -α is less than 5.7%. This means that Equation 6.18 can be approximated as:

T_{60(α < 0.3)} = ( -0.161V)/S (-α) = 0.161V/S α

(6.19)

Equation 6.19 is known as the Sabine reverberation formula and apart from being useful, was the first reverberation formula. It was developed on the basis of experimental measurements made by W.C.Sabine, thus initiating the whole science of architectural acoustics. Equation 6.19 is much easier to use and gives accurate enough results providing the absorption, α is less than about 0.3. In many real rooms this is a reasonable assumption However it becomes increasingly inaccurate as the average absorption increases and in the limit predicts a reverberation time when α = 1, that is reverberation without walls!

As stated previously, the basic assumption behind these equations is that the reverberant field is statistically random, that is a diffuse field. There are however acoustic situations in which this is not the case. Figure 6.13 shows the decay of energy, in dB, as a function of time for an ideal diffuse field reverberation. In this case the decay is a smooth straight line representing an exponential decay of an equal number of dBs per second. Figure 6.14 on the other hand shows two situations in which the reverberant field is no longer diffuse. In the first situation all the absorption is only on two surfaces, for example an office with acoustic tiles on the ceiling, carpets on the floor, and nothing on the walls.

Reverberation time variation with frequency

the absorption coefficients of real materials are not constant with frequency. This means that, assuming that the room's volume and surface area are constant with frequency which is not an unreasonable assumption, the reverberation time in the room will also vary with frequency. This will subjectively alter the timbre of the sound in the room due to both the effect on the level of the reverberant field discussed earlier and the change in timbre as the sound in the room decays away. As an extreme example, if a particular frequency has a much slower rate of decay compared with other frequencies, then as the sound decays away this frequency will ultimately dominate and the room will 'ring' at that particular frequency. The sound power for steady state sounds will also have a strong peak at that frequency because of the effect on the reverberant field level.

Table 6.1 shows some typical absorption coefficients for some typical materials which are used in rooms as a function of frequency. Note that they are measured over octave bands. One could argue that third octave band measurements would be more appropriate psychacoustically, as the octave measurement will tend to blur variations within the octave which might be perceptually noticeable. In many cases, because the absorption coefficient varies smoothly with frequency, octave measurements are sufficient. However, especially when considering resonant structures, more resolution would be helpful. Note also that there are often no measurements of the absorption coeffi. cients below 125 Hz, this is due to both the difficulty in making such measurements and the fact that below 125 Hz other factors in the room become more important, as we shall see later.

In order to take account of the frequency variation of the absorption coefficients we must modify the equations which calculate the reverberation time as follows:

T_{60} = ( -0.161 V)/ (S ln(1-α(f))

where α(f) = frequency dependent absorption coefficient for the Norris-Eyring reverberation time equation and:

T_{60 (α<.3)} = ( 0.161 V)/S &alpha(f) for the Sabine reverberation time equation.

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. These 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 one. The denominator in the Sabine reverberation equation, Equation 6.19, is also equivalent to the open window area of the room, but has been calculated using the average absorption coefficient in the room. It is therefore easy to incorporate the effects of different materials by simply calculating the total open window area for different materials, using the method described above, and substituting it for S α in Equation 6.19. This gives a modified equation which allows for a variety of frequency-dependent materials in the room as:

T_{60(α<.3)} = (0.161 V) /( Σ_{all surfaces Si} S_{i}α_{i}(f) )

(6.20)

where

- α
_{i}(f) = 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 the open window area directly. However the Norris-Eyring reverberation time equation can be rewritten in a modified fonn, as shown in Appendix 4, which allows for the variation in material absorption due to both nature and frequency, as:

T_{60(α<.3)} = (i0.161 V) / (Σ_{all surfaces Si} S_{i}α_{i}(f) )

(6.21)

Equation 6.21 is also known as the Millington-Sette equation. Although Equation 6.21 can be used irrespective of the absorption level it is still more complicated than the Sabine equation and, if the average 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 6.20, is preferred.

Equation 6.20 is readily used in conjunction with tables of absorption coefficients to calculate the reverberation time and can be easily programmed into a spreadsheet. As an example consider the reverberation time calculation for a living room outlined in

The results of Example 6.8 beg the question: 'How can we improve the evenness of the reverberation time?' The answer is to either add, or remove, additional absorbing materials into the room in order to achieve the desired reverberation characteristic. Here the concept of an open window area budget is useful. The idea is that, given the volume of the room, and the desired reverberation time, the necessary open window area required is calculated. The open window area already present in the room is then examined and, depending on whether the room is over or under budget, appropriate materials are added or removed.

What is an ideal reverberation characteristic? We have seen that the decay should be a smooth exponential of a constant number of decibels of decay per unit time. We also know that different sorts of music require different reverberation times. In many cases the answer is, 'it depends on the situation'. However there are a few general rules which seem to be broadly accepted. Firstly, there is a range of reverberation times which are a function of the type of music being played; music with a high degree of articulation needs a drier acoustic than music which is slower and more harmonic. Secondly, as the performance space gets larger the reverberation time required for all types of music becomes longer. This result is summarised in Figure 6.22 which shows the 'ideal' reverberation time as a function of both music and room volume. Thirdly, in general, listeners prefer a rise in reverberation time in the bass (125 Hz) of about 40% relative to the midrange (1 kHz) value as shown in Figure 6.23. This rise in bass reverberation adds 'warmth' and it also helps increase the sound level of bass instruments, which often have weak fundamentals, by raising the level of the reverberant field at low frequencies. However, when recording instruments, or when listening to recorded music, this bass lift due to the reverberant field may be undesirable and therefore a flat reverberation characteristic preferred.

The first aspect is that the measure of reverberation time as being the time it takes the sound to fall by 60 dB is not particularly relevant psychoacoustically; it is also very difficult to measure in situ. This is due to the presence of background noise, either unwanted or the music being played, which often results in less than 60 dB of energy decay before the decay sound becomes less than the residual noise in the environment. Even in the quieter environment of a Victorian town in the days before road traffic, Sabine had to do measurements, using his ears, at night to avoid the results being affected by the level of background noise. Because we rarely hear a full reverberant decay, our ears and brains have adapted, quite logically, to focus on what can be heard. Thus we are more sensitive to the effects of the first 20 to 30 dB of the reverberant decay curve. In principle, providing we have an even exponential decay curve, the 60 dB reverberation is directly proportional to the earlier curves and so this should not cause any problems. However if the curve is of the doubleslope form shown in Figure 6.15 then this simple relationship is broken. The net result is that, although the T_{60} reverberation time may be an appropriate value, because of the faster early decay to below 30 dB we perceive the reverberation as being shorter than it really is. The psychoacoustic effect of this is that the space sounds 'drier' than one would expect from a simple measurement of T_{60}' Modern acoustic designers therefore worry much more about the early decay time (EDT) than they used to when designing concert halls.

**Lateral reflections **

The second factor which has been found to be important for the listener is the presence of dense diffuse reflections from the side walls of a concert hall, called lateral reflections, as shown in Figure 6.24. The effect of these are to envelop or bathe the listener in sound and this has been found to be necessary for the listener to experience maximum enjoyment from the sound. It is important that these reflections be diffuse, as specular reflections will result in disturbing comb filter effects, as discussed in Chapter 1, and distracting images of the sound sources in unwanted and unusual directions. Providing diffuse reflections is thus important and this has been recognised for some time. Traditionally, the use of plaster mouldings, niches and other decorative surface irregularities have been used to provide diffusion in an ad hoc manner. More recently diffusion structures based on patterns of wells whose depths are formally defined by an appropriate mathematical sequence have been proposed and used. However it is not just the provision of diffusion on the side walls that must be considered.

The traditional concert hall is called a shoe-box hall, because of its shape, as shown in Figure 6.25, and this naturally provides a large number of lateral reflections to the audience. This shape, combined with the Victorian penchant for florid plaster decoration, resulted in some excellent sounding spaces. Unfortunately shoe-box halls are harder to make a profit with because they cannot seat as many people as some other structures

Early reflections and performer support

A third factor, which is often ignored, is the acoustics that the performers experience. Pop groups have known about this for years and take elaborate precautions to provide each performer on stage with their own individual balance of acoustic sounds via a technique known as foldback. In fact some performers now receive their foldback directly into their ears via a technique known as in-ear monitoring' and in many large gigs the equipment providing foldback to the performer can rival or even exceed, that which provides the sound for the audience. The classical musician, however, only has the acoustics of the hall to provide them with 'fold back'. Thus the musicians on the stage must rely on reflections from the nearby surfaces to provide them with the necessary sounds to enable them to hear themselves and each other.

There are two requirements for the sound reaching the performer on stage. Firstly, it must be at a sufficient level and arrive soon enough to be useful. To begin with it is important that the surfaces surrounding the performers direct some sound back to them. Note that there is a conflict between this and providing a maximum amount of sound to the audience so some compromise must be reached. The usual compromise is to make use of the sound which radiates behind the performers and direct it out to the audience via the performers, as shown in Figure 6.28. This has the twofold advantage of providing the performers with acoustic foldback and redirecting sound energy that might have been lost back to the audience. Ideally the sound that is redirected back to the performers should be diffuse as this will blend the sounds of the different instruments together for all the performers, whereas specular reflectors can have hot and cold spots for a given instrument on the stage, as shown in Figure 6.29. An important aspect of acoustic fold back, however, is the time that it takes to arrive back at the performers. Ideally it should arrive immediately, and some does via the floor and direct sound from the instrument. However, the majority will have to travel to a reflecting or diffusing surface and back to the performers. There is evidence to show that, in order to maintain good ensemble comfortably, the musicians should receive the sound from other musicians within about 20 ms of the sound being produced.

This means that ideally there should be a reflecting or diffusing surface within 10 ms (3.44 m or 11.5 ft) of the performer; the time is divided by 2 to allow for going to the reflecting surface and back. In practice some of the surfaces may have to be further away when large orchestral forces are being mustered, although the staging used can assist the provision of acoustic foldback. Sometimes, however, the orchestra enclosure is so large that the reflections arrive later than this. If they arrive later than about 50 ms the musicians will perceive them as echoes and ignore them. On the other hand if these reflections arrive at the boundary between perceiving it as part of the sound or an echo of a previous sound it can cause severe disruption of the performers' perception of it. The net effect of these 'late early reflections' is to damage the performers' ability to hear other instruments close to them and this further reduces their ability to maintain ensemble. In one prestigious hall, the reason musicians used to complain that they couldn't hear each other and so hated playing there was traced to the problem of late early reflections.

The effect of air absorption

Although for low frequencies air absorbs a minimal amount of sound energy, at high frequencies this is not the case. In particular humidity, smoke particles and other impurities will absorb high-frequency energy and so reduce the level of high frequencies in the sound. This is one of the reasons that people sound duller when they are speaking at a distance. In terms of reverberation time, and also the level of the reverberant field, the effect of this extra absorption is to reduce the reverberation time, and the level of the reverberant field, at high frequencies. Fortunately this effect only becomes dominant at higher frequencies, above 2 kHz. Unfortunately it is dependent on the level of humidity and smoke in the venue and so the high-frequency reverberation time, and the reverberant field level, will change as the audience stays in the space. Note this is an additional dynamic effect over and above the static absorption