Room Acoustics

Most music is listened to in enclosed spaces, and is influenced by the presence of boundaries.

• Delay a function of the distance, as sound travels at 344ms^-1.
• 'Uncontaminated' information; high level required for clear sound.
• Behaves as sound in free space as it has not met any boundaries.

I_{direct sound} = Qw_source ÷ 4πr²

This is with reference to the inverse square law.

• Reflections off one or more surfaces.
• Separated from direct sound in both time and direction.
• Vary as source or listener move; changes used to judge size of space and position of source.
• Reflections over 30ms perceived as echoes.
• Interference effects caused; affect clarity and timbre of music.
• Intensity levels affected by distance and reflective surface.
• Most surfaces absorb some sound energy; some surfaces can 'focus' sound (e.g. parabolic reflector as shown below…).

• Absorption measured by a material's absorption coefficients.
• Defines the energy or power lost as sound strikes a given material.
• In reality different frequencies have different coefficients (shown in this table).

Intensity_reflected = Intensity_incident × (1 - α)

• Dense reflections from many possible paths.
• Adds richness and supports musical sounds; integrates the different sound elements from an instrument. Rooms with very little reverberation sound 'bad'; bathrooms are great to sing in!
• Reverb time is a function of the size of the room; eventually sound is fully absorbed.
• Initial delay between direct sound and reverberation is an important cue to the space.
• The time the reverberation takes to die away is known as the reverberation time and is dependant on both the size of the space and the amount of sound absorbed at each reflection. The time it takes for a sound's reverberation field to attenuate by 60dB is known as the RT60.

The room space affects 3 aspects of the reverberant field;

1. Increase of the reverberant field level; room size affects the time between reflections. Absorption also affects the rate.
2. Steady state level of the reverberant field is inversely proportional to the amount of absorption in the room; sound power input = power lost by absorption (needs constant held note).
3. Decay of the reverberant field level; the decay rate is determined by the amount of sound energy absorbed at each reflection.

Bigger spaces tend to have longer reverberation times and well furnished spaces tend to have shorter reverberation times.

Furnished living room	0.2s
Cathedral (stone and glass)	10s

• Direct sound & early reflections follow the inverse square law (plus absorption effects for early reflections), and so their amplitude varies with position.
• Later reflections remain constant; waves arrive from all directions. The steady state level, at a given point, is an integrated sum of all sound intensities.

• The reverberant field is independent of the position of the listener with respect to the source; the direct sound is dependant on this distance.
• At a given distance the reverberant sound will dominate; this is the critical distance.

• The reverb is the sum of all reverb tail energy, therefore the overall sound level is increased by reverberation.
• Level of reverb is affected by the sound absorption in the room; low level of absorption will result in a higher level of reverberant field.

W_{reverberation} = W_source (1 - α ÷ Sα)

• How far from a sound source is the critical distance? In the average living room, the critical distance for a loudspeaker is about 86cm! That means we listen to most music well into the reverberant field. The quality of the reverberant field is therefore very important.
• The level and quality of the reverberant field is a function of the average absorption coefficient in the room.
• Absorption coefficients change with frequency; the reverberant level therefore also changes with frequency. When the reverberant field is dominant it will determine the perceived timbre of the sound.
• In addition, speaker drivers (both bass and treble) become more directional at the high ends of their frequency range. This means that they contribute less to the reverberant field, and therefore these frequencies seem 'duller'.

• A 'dead' sounding room is a room in which the sound dies away (by being partially absorbed by each surface it interacts with) very quickly. A 'live' sounding room is one in which the sound dies away slowly.
• Listening to and producing live music in a dead room doesn't normally sound good to us. Live rooms tend to sound much better for performance and listening to a performance.
• Recordings include reverberation; therefore dead rooms are more suited to the playback of recordings.
• Many types of music have been written to take advantage of a particular type of reverberation. For example, early polyphonic vocal pieces sound incredible in a cathedral.
• The reverb time of a room is a product of both the absorptive properties of the room and the length of time between surface interactions.
• The mean free path of a room is the average distance between surfaces.

MFP = 4V ÷ S

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

• The time between surface interactions can be found by adding in the speed of sound…

τ = 4V ÷ Sc

τ = the time between reflections (s)
c = the speed of sound (ms^-1)

• At each interaction with a surface there is some energy which is absorbed by the surface. What proportion of the energy is absorbed is shown by the absorption coefficient. The average absorption coefficient is shown by α.
• 1 - α therefore shows us what proportion of the wave's energy is reflected back into the room. These reflections will hit other surfaces, making the decay exponential.
• We should therefore be able to work out how quickly the reverberant field's energy will reduce by 60dB; this is the RT60 (sometimes called the T₆₀).

The Norris-Eyring Reverberation Formula

RT60 = -0.161V ÷ S In(1 - α)

This formula assumes that the reverb field is diffuse (that sound visits all surfaces with equal probability and at all possible angles of incidence) and that there is a valid mean free path (i.e. the room isn't an extreme shape).

• As the room size increases the reverberation time increases proportionally, as long as the average absorption coefficient remains the same.
• Most rooms provide absorption in the form of carpets, curtains, people and so on; this tends to be a constant fraction of the surface area. Therefore, larger rooms tend to have longer reverberation times. We talk about "big" or "small" sounding spaces, when in fact we are referring to the reverberation time.

Small Rooms

• Small rooms can absorb sound energy very quickly. This can easily result in a very low number of reflections.
• If there are fewer than 20(ish) reflections we can say that there is no reverberant field. In these cases there are only early reflections after the initial sound. The reflections are not diffuse.
• Artificial reverbs tends to call patches that only have early reflections 'ambience'.

The Sabine Reverberation Formula

RT60_{(α < 0.3)} = 0.161V ÷ Sα

The Sabine formula is simpler to use than the Norris-Eyring formula above. It gives accurate results as long as the absorption (α) is less than around 0.3. This isn't a problem in most real rooms.

Reverberation Faults

• Some rooms are not capable of a diffuse reverberant field. For example, a room with absorptive materials on the floor and ceiling (e.g. carpet and foam tiles) and reflective surfaces (e.g. painted plaster) on the walls will see sound energy between the absorptive surfaces decay much faster than the sound between the reflective surfaces. Another example is when two rooms are connected by an opening; the two rooms have separate decay times.
  Both examples produce reverb tails with two (a 'double break' decay curve) or more slopes.
• Another fault occurs when there are two precisely parallel smooth surfaces in a room. This causes sound to rapidly reflect back and forth, and is known as a flutter echo.

• The absorption coefficients of real materials varies with frequency. This alters the timbre of sounds within the room, and also affects the timbre of the reverb tail.
• Absorption coefficients are measured in octaves.

Equivalent Open Window Area

• In a room there are differing areas of various materials, all having different absorptive properties.
• Sabine came up with the equivalent open window area to account for this. Multiply the absorption coefficient of the material by its total area and then add up the contributions from all surfaces in the room. Sabine figured (correctly) that an open window has an absorption coefficient of one.
• By altering both RT60 equations we can come up with formulæ which allow for a variety of frequency-dependant materials in the room…

Altered Sabine Formula

RT60_{(α < 0.3)} = 0.161V ÷ Σ_{all surfaces} S_iα_i(ƒ)

α_i(ƒ) = absorption coefficient for a given material
S_i = its area

The Millington-Sette Equation

RT60 = -0.161V ÷ Σ_{all surfaces} S_i In(1 - α_i(ƒ))

It's a pretty simple process to make a spreadsheet using these formulæ (probably just the easier Sabine version). I've made this Excel reverb calculator if you want to see an example.

Once the current reverberation time of a room has been calculated it can be related to the desired reverberation characteristics. The necessary open window area is then calculated.
If one frequency has the required open window area (and therefore reverberation time), the open window areas at other frequencies must be altered to match it. The difference between the required and actual open window areas can be used to work out the amount of extra absorptive material required.

Area_material = Required Open Window Area ÷ Absorption Coefficient

Remember that we normally have to replace one material with another, rather than just adding extra materials into a room. It's easy to forget this!

• Rhythmic music, or music with a high degree of articulation, needs a drier acoustic than slower, more harmonic music.
• As the performance space gets larger the required reverberation time gets longer.
• Listeners prefer the bass (125 Hz) reverberation time to rise by about 40% in relation to the midrange (1 kHz) reverberation time. This adds 'warmth' and increases the sound level of bass instruments. This bass lift is undesirable in recording environments.

Early Decay Time

It is hard for us to hear a reverberation tail when other sounds obscure it. We often cannot hear the reverb after the first 20 or 30 dB. We are therefore more sensitive to this early decay. If the room has a double slope then we pay attention to the first curve and not to any subsequent ones. As a result, the room sounds 'drier' than it really is. Acousticians use the Early Decay Time (EDT) as a key concept when designing spaces.

Lateral Reflections

Dense diffuse lateral reflections from the wall envelop the listener with sound, increasing their experience. Specular (non diffuse) reflections introduce comb filtering effects and distracting psychoacoustic images. Modern diffusion structures (diffusors) are based on patterns of wells which are mathematically defined.

Early Reflection Foldback

Acoustic performers need to hear themselves and each other, and use the room's early reflections to do this. They need to receive the reflected sound within 20ms, and so a reflecting or diffusing surface must be within 10ms of the performer(s).

Air Absorption

Reverberant sound has, by definition, travelled through a lot of air. A one second reverberation time means that the sound at the end of the tail has travelled 344 metres. High frequencies (above 2 kHz) are absorbed by air; this is increased by humidity, smoke and other impurities. This is why sounds are duller at a distance.
The sound travels further as the volume of the room increases; we therefore use an absorption coefficient scaled to the volume in m³.

• Not all energy is reflected in a random fashion (diffuse field).
• Some energy is reflected in cyclic paths.
• If the length of the path is a precise number of half wavelengths then they will form standing waves in the room.

Standing waves have pressure and velocity distributions that are spatially static, therefore…

1. They do not visit each surface with equal probability.
2. They do not strike these surfaces with random incidence; a particular angle of incidence is involved.
3. They follow a cyclic path; these are strongly frequency dependant and are determined by the room geometry.

• Otherwise known as room modes and modal frequencies.
• Modes are spatially static; SPL levels will vary dramatically around the room.

Axial Modes

ƒ_x(axial) = c ÷ 2 (x ÷ L)

ƒ_x(axial) = the axial modal frequencies (Hz)
x = the number of half wavelengths which fit between the surfaces
L = the distance between the reflecting surfaces (m)
c = speed of sound (ms^-1)

Occur between two opposing surfaces, and so are a function of the linear dimensions of the room.

Tangential Modes

ƒ_{xy(tangential)} = √ c ÷ 2 (x ÷ L)² + (y ÷ W)²

ƒ_{xy(tangential)} = tangential modal frequencies (Hz)
x = the number of half wavelengths between one set of two surfaces
y = the number of half wavelengths between the other set of surfaces
L, W = the distance between the reflecting surfaces (m)

Occur between four surfaces, and so are a function of two of the dimensions in the room.

Oblique Modes

ƒ_xyz(oblique) = c ÷ 2 √ (x ÷ L)² + (y ÷ W)² + (z ÷ H)²

ƒ_xyz(oblique) = oblique modal frequencies (Hz)
x, y, z = the number of half wavelengths between the surfaces
L, W, H = the distance between the reflecting surfaces (m)

Occur between all six surfaces, and so are a function of all three dimensions of the room.

Universal Mode Frequency Equation

Gives the frequencies of all possible modes in the room.

ƒ_xyz = c ÷ 2 √ (x ÷ L)² + (y ÷ W)² + (z ÷ H)²

x, y, z = the number of half wavelengths between the surfaces

If any of the dimensions are integer multiples of each other then some of the modal frequencies will be the same; this can cause problems.

Favourable Room Dimensions

	Height	Width	Length
A	1.00	1.14	1.39
B	1.00	1.28	1.54
C	1.00	1.60	2.33

• As modes are not diffuse, they visit fewer surfaces and are therefore not absorbed as quickly as diffuse reflections.
• This behaviour is frequency dependant.
• The room's decay time is therefore made up of several decay times. The diffuse field is normally the first to decay, with modal reflections taking longer to decay.

• The number of modes within a given frequency bandwidth increases with frequency; modal density increases with frequency.
• All rooms have modes in their low frequency ranges, even anechoic spaces. This results in a frequency below which modes dominate and the field is not diffuse. Standard reverb time calculations therefore cannot be used.

There are three frequency regions…

The Cut-Off Region

The region below the lowest mode. The room is smaller than a half wavelength in all dimensions. This results in reduced sound levels at these frequencies.

ƒ_cut-off = c ÷ 2 × Longest Dimension

The Modal Region

The modal behaviour of the room dominates. No analysis based on a diffuse field can be undertaken.

The Diffuse Field Region

The region in which a diffuse field can exist. Reverberation times can be calculated. Modal effects are minimal, making the room sound pleasing.

The boundary between the modal and diffuse field regions is the critical frequency.

In an acoustically large room the critical frequency is below the lowest frequency of the sound that will be generated in the room. In acoustically small rooms the critical frequency will occur within the frequency range of the sounds being produced in it.

The critical frequency can be calculated either by using the Mean Free Path or the room's RT60.

ƒ_critical = (3 ÷ 2) c ÷ MFP

ƒ_critical = 2102 √ (RT60 ÷ V)

• A gap between the direct sound and the first reflection (initial time delay or ITD); this should be under 30ms so that it isn't perceived as an echo. A short delay makes the space sound 'intimate'.
• High level diffuse lateral early reflections with a flat frequency response.
  
• A smoothly decaying diffuse reverberant field with no modal behaviour or other defects. The decay time should match the musical material.

• The control room should allow the listener to hear the acoustic of the recorded space.
• The recorded space is usually larger than the playback space. Therefore, the first reflection the listener hears is from the playback space, not from the recorded space. The precedence effect means that the sound is perceived as being in the smaller space.
  
• To overcome this, the early reflections from the playback space need to be suppressed. This can be achieved by using absorption. It can also be achieved by using angled or shaped walls. This is known as the reflection-free zone technique.
• The effect only works in a small area of the room.
• The listening room still needs to be diffuse. The rear wall normally has a diffusion structure to ensure this. This wall provides the first reflection in the playback room, and therefore defines the ITD. This delay would ideally be about 20ms; if it were any longer it could be perceived as an echo. In practice, an ITD between 8 and 20ms works well.
• In order to suppress the early reflections, material with an absorption coefficient of about 0.9 should be used on the contributing surfaces.
• Remember that this sort of absorptive treatment only works on high and mid frequencies, as lower frequencies have longer wavelengths that are not affected as much by the absorptive materials.

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