Auditory Theory: Acoustics

Lecture 020 Environment I

Reading Assignment for Lecture 021

Before next lecture please read Sections

  • 6.1.10 Reverberation time 267

pages 267 to 288 of Acoustics and Psychoacoustics. We may have a brief quiz on these sections at the beginning of the next class.

Brain Bullets

  • The direct sound

  • After a short delay the listener in the space will hear the sound of the starting pistol, which will have travelled the shortest distance between it and the listener. The delay will be a function of the distance, as sound travels 344 metres (1129 feet) per second or approximately 1 foot per millisecond. The shortest path between the starting pistol and the listener is the direct path and therefore this is the first thing the listener hears.
  • The direct component is important because it carries the information in the signal in an uncontaminated form.Therefore a high level of direct sound is required for a clear sound and good intelligibility of speech.
  • The direct sound also behaves in the same way as sound in free space, because it has not yet interacted with any boundaries. This means that we can use the equation for the intensity of a free space wave some distance from the source to calculate the intensity of the direct sound. The intensity of the direct sound is therefore given, from Chapter 1, by:
  • Idirect sound = (QWsource)/(4πr2)

Early reflections

  • A little time later the listener will then hear sounds which have been reflected off one or more surfaces (walls, floor, etc.)
  • These sounds are called early reflections and they are separated in both time and direction from the direct sound. These sounds will vary as the source or the listener move within the space. We use these changes to give us information about both the size of the space and the position of the source in the space. If any of these reflections are very delayed, total path length difference longer than about 30 milliseconds (33 feet), then they will be perceived as echoes. Early reflections can cause interference effects, as discussed in Chapter 1, and these can both reduce the intelligibility of speech, and cause unwanted timbre changes in music, in the space
  • The intensity levels of the early reflections are affected by both the distance and the surface from which they are reflected. In general most surfaces absorb some of the sound energy and so the reflection is weakened by the reflection process. However it is possible to have surfaces which 'focus' the sound, as shown in Figure 6.4, and in these circumstances the intensity level at the listener will be enhanced. It is important to note, however, that the total power in the sound will have been reduced by the interaction with the surface. This means that there will be less sound intensity at other positions in the room
  • The absorption coefficient of a material defines the amount of energy, or power that is removed from the sound when it strikes it. In general the absorption coefficient of real materials will vary with frequency but for the moment we shall assume they do not. The amount of energy, or power removed by a given area of absorbing material will depend on the energy, or power, per unit area striking it. As the sound intensity is a measure of the power per unit area this means that the intensity of the sound reflected is reduced in proportion to the absorption coefficient That is
  • Intensity reflected = Intensity incident x (1 - alpha)
    • where
    • Intensity reflected = the sound intensity reflected after absorption (in W m-2)
    • Intensity incident = the sound intensity before absorption (in W m-2)
    • alpha = the absorption coefficient
  • At an even later time the sound has been reflected many times and is arriving at the listener from all directions, as shown in Figure 6.7. Because there are so many possible reflection paths, each individual reflection is very close in time to its neighbours and thus there is a dense set of reflections arriving at the listener. This part of the sound is called reverberation and is desirable as it adds richness to, and supports, musical sounds. Reverberation also helps integrate all the sounds from an instrument so that a listener hears a sound which incorporates all the instrument's sounds, including the directional parts. In fact we find rooms which have very little reverberation uncomfortable and generally do not like performing music in them; it is much more fun to sing in the bathroom compared to the living room. The time taken for reverberation to occur is a function of the size of the room and will be shorter for smaller rooms, due to the shorter time between reflections. In fact the time gap between the direct sound and reverberation is an important cue to the size of the space that the music is being performed in.
  • The time that it takes for the sound to die away is called the reverberation time and is dependent on both the size of the space and the amount of sound absorbed at each reflection. In fact there are three aspects of the reverberant field that the space affects, see Figure 6.8.
    • The increase of the reverberant field level: This is the initial portion of the reverberant field and is affected by the room size, which affects the time between reflections and therefore the time it takes the reverberant field to build up. The amount of absorption in the room also affects the time that it takes the sound to get to its steady state level. This is because, as shall be shown later, the steady state level is inversely proportional to the amount of absorption in the room. The sound level will take longer to reach a louder level than a smaller one, because the rate at which sound builds up depends on the time between reflections and the absorption.
    • The steady state level of the reverberant field: If a steady tone, such as an organ note, is played in the space then after a period of time the reverberant sound will reach a constant level because at that point the sound power input balances the power lost by absorption in the space. This means that the steady state level will be louder in rooms which have a small amount of absorption. Note that a transient sound in the space will not reach a steady state level.
    • The decay of the reverberant field level: When a tone in the space stops, or after a transient, the reverberant sound level will not reduce immediately but will instead decay at a rate determined by the amount of sound energy that is absorbed at each reflection. Thus in spaces with a small amount of absorption the reverberant field will take longer to decay.
  • Bigger spaces tend to have longer reverberation times and well furnished spaces tend to have shorter reverberation times. Reverberation time can vary from about 0.2 of a second for a small well furnished living room to about 10 seconds for a large glass and stone cathedral.

The behaviour of the reverberant sound field

  • The direct sound and early reflections follow the inverse square law, with the addition of absorption effects in the case of early reflections, and so their amplitude varies with position. However the reverberant part of the sound remains constant with the position of the listener in the room.This is not due to the sound waves behaving differently from normal waves; instead it is due to the fact that the reverberant sound waves arrive at the listener from all directions. The result is that at any point in the room there are a large number of sound waves whose intensities are being added together. These sound waves have many different arrival times, directions and amplitudes because the sound waves are reflected back into the room, and so shuttle forwards, backwards and sideways around the room as they decay. The steady state sound level, at a given point in the room, therefore is an integrated sum of all the sound intensities in the reverberant part of the sound, as shown in Figure 6.9. Because of this behaviour the reverberant part of the sound in a room is often referred to as the reverberant field.

The balance of reverberant to direct sound

  • This behaviour of the reverberant field has two consequences. Firstly the balance between the direct and reverberant sounds will alter depending on the position of the listener relative to the source. This is due to the fact that the level of the reverberant field is independent of the position of the listener with respect to the source, whereas the direct sound level is dependent on the distance between the listener and the sound source. These effects are summarised in Figure 6.10 which shows the relative levels of direct to reverberant field as a function of distance from the source. This figure shows that there is a distance from the source at which the reverberant field will begin to dominate the direct field from the source. The transition occurs when the two are equal and this point is known as the critical distance.

The level of the reverberant sound in the steady state

  • at equilibrium, the rate of energy removal from the room will equal the energy put into its reverberant sound field. As the sound is absorbed when it hits the surface, it is absorbed at a rate which is proportional to the surface area times the average absorption, or So.. This is similar to a leaky bucket being filled with water where the ultimate water level will be that at which the water runs out at the same rate as it flows in, see Figure 6.11. The amount of sound energy available for contribution to the reverberant field is also a function of the absorption because if there is a large amount of absorption then there will be less direct sound reflected off a surface to contribute to the reverberant field - remember that before the first reflection the sound is direct sound.
  • An interesting result from Equation 6.6 is that it appears that the level of the reverberant field depends only on the total absorbing surface area. In other words it is independent of the volume of the room. However in practice the surface area and volume are related because one encloses the other. In fact, because the surface area in a room becomes less as its volume decreases, the reverberant sound level becomes higher for a given average absorption coefficient in smaller rooms. Another way of visualising this is to realise that in a smaller room there is less volume for a given amount of sound energy to spread out in, like a pat of butter on a smaller piece of toast. Therefore the energy density, and thus the sound level, must be higher in smaller rooms.

Critical Distance

  • As most people would be about 2 m away from their loudspeakers when they are listening to them this means that in a normal domestic setting the reverberant field is the most dominant source of sound energy from the hi-fi, and not the direct sound. Therefore the quality of the reverberant field is an important aspect of the performance of any system which reproduces recorded music in the home. There is also an effect on speech intelligibility in the space as the direct sound is the major component of the sound which provides this. The level of the reverberant field is a function of the average absorption coefficient in the room. Most real materials, such as carpets, curtains, sofas and wood panelling have an absorption coefficient which changes with frequency. This means that the reverberant field level will also vary with frequency, in some cases quite strongly. Therefore in order to hear music, recorded or otherwise, with good fidelity, it is important to have a reverberant field which has an appropriate frequency response.

The effect of source directivity on the reverberant sound

  • There is an additional effect on the reverberation field, and that is the directivity of the source of sound in the room. Most hi-fi loudspeakers, and musical instruments, are omnidirectional at low frequencies but are not necessarily so at higher ones. As the level of the reverberant field is a function of both the average absorption and the directivity of the source, the variation in directivity of real musical sources will also have an effect on the reverberant sound field and hence the perception of the timbre of the sound.
  • The effect therefore of a directive source with constant on axis response is to reduce the reverberant field as the 'Q' gets higher. The subjective effect of this would be similar to reducing the high 'Q' regions via the use of a tone control which would not normally be acceptable as a sound quality. A typical reverberant response of a typical domestic hi-fi speaker is shown in Figure 6.12. Note that the reverberant response tends to drop in both the midrange and high frequencies. This is due to the bass and treble speakers becoming more directive at the high ends of their frequency range. The dip in reverberant energy will make the speaker less 'present' and may make sounds in this region harder to hear in the mix. The drop in reverberant field at the top end will make the speaker sound 'duller'. Some manufacturers try to compensate for these effects by allowing the on-axis response to rise in these regions, however this brings other problems. The reduction in reverberant field with increasing 'Q' is used to advantage in speech systems to raise the level of direct sound above the reverberant field and so improve the intelligibility.