When a room is excited by an impulse, the sound energy is reflected from its surfaces. At each reflection some of the sound is absorbed and therefore the sound energy decays exponentially. Ideally the sound should be reflected from each surface with equal probability, forming a diffuse field. This results in a single exponential decay with a time constant proportional to the average absorption in the room.
Fig 6.30 Cyclic reflection paths in a room
However in practice not all the energy is reflected in a random fashion. Instead some energy is reflected in cyclic paths, as shown in Figure 6.30. If the length of the path is a precise number of half wavelengths then they will form standing waves in the room. These standing waves have pressure and velocity distributions which are spatially static and so behave differently to the rest of the sound in the room in the following ways:
Another name for these standing waves in a room are room modes and the frequencies at which they occur are known as modal frequencies. Because the modes are spatially static there will be a strong variation of sound pressure level as one moves around the room, which is undesirable. There are three basic types of room mode which are outlined in Sections 6.2.1 to 6.2.3.
Fig 31 Axial Mode Paths
These modes occur between two opposing surfaces, as shown in Figure 6.31, and so are a function of the linear dimensions of the room. The frequencies of an axial mode are given by the following equation:
fx(axial) = C/2*(x/L)
where
This equation shows that there are an infinite number of possible modal frequencies at which an integer number of wavelengths fit into the room with lowest modal frequency occurring when just one half wavelength fits into the space between the reflecting surfaces.
Fig 6.32 Tangential mode paths
These modes occur between four surfaces, as shown in Figure 6.32, and so are a function of two of the dimensions of the room. The frequencies of the tangential modes are given by the following equation:
fxy(tangential) = sqrt
[c/2(x/L)2+(y/W)2]
where
There is also an infinite number of tangential modes, but they must fit an integral number of half wavelengths in two dimensions. This has the interesting consequence that the lowest modal frequencies are higher than the axial modes, despite the fact that the apparent path length is greater. The reason is that the standing waves must fit between the opposing surfaces, that is on the sides rather than the hypotenuse of the triangular path, and as the propagating wave travels down the hypotenuse, the effective wavelength, or phase velocity, on the sides of the room is larger, as shown in Figure 6.33. The lowest modal frequency for a tangential mode occurs when precisely one half wavelength, at the phase velocity, fits into each dimension.
Fig 6.33 The phase velocity of tangential modes
Fig 6.34 An oblique modal path
These modes occur between all six surfaces, as shown in Figure 6.34, and so are a function of all three dimensions of the room. The frequencies of the oblique modes are given by the following equation:
fxyz(oblique) = C/2 [(x/L)2 + (y/W)2 + (z/H)2]
where
The lowest frequencies of these modes are also higher than the lowest axial modes, for the reasons discussed earlier. where !,yztublique) X, y, Z and L, W, H
The combination of these three types of modes form a dense set of possible standing wave frequencies in the room and they can be combined into one equation by simply allowing x, y, and z. in the oblique mode equation to range from 0, 1, 2 to infinity, giving the following equation which will give the frequencies of all possible modes in the room:
fxyzp = C/2 [ (x/L)2 + (y/W)2 + (z/H)2]
(6.22)
where x, y, z = the number of half wavelengths between the surfaces (0, 1, 2, ..., infinity)
The above equation also shows that if any of the dimensions are integer multiples of each other then some of the modal frequencies will be the same and this can cause problems. It is therefore better to choose non-commensurate ratios for the wall dimensions to ensure that the modes are spread out as much as possible. Much work has been done on ideal room ratios and one set of favourable room dimensions is shown in Table 6.4. However, these dimensions are not necessarily the only optimum ones for all room sizes. It is also important to realise that room modes are inherent in any structure which encloses the sound sources. This means that changing the shape of the room, for example by angling the walls, does not remove the resonances, it merely changes their frequencies from values which are easily calculated to ones that are not.
In general the number of resonances within a given frequency bandwidth increases with frequency. In fact it can be shown that they increase proportional to the square of the frequency, and in large well-behaved acoustical spaces, that sound good, this increase in mode density with frequency is smooth. This is the rationale behind a method for assessing the modal behaviour in a room known as the Bonello criteria. These criteria try to ascertain how significant the modal behaviour of a room is in perceptual terms. It does this by dividing the audio frequency spectrum into third octave bands, as an approximation of critical bands, and then counting the number of modes per band. If the number of modes per third octave band increase monotonically then there is a good chance that we will perceive the room as having a 'smooth' frequency response despite the resonances. If the number of resonances per third octave drops as the frequency rises then there will be a perceptually noticeable peak in the frequency response. Coincident modes are also another way of creating a perceptually noticeable frequency response peak and the Bonello criteria does further stipulate that there should be no modal coincidence within a third octave band unless there are at least three additional non-coincident resonances to balance the two that are coincident. As an example of the calculation of mode frequencies let us calculate some for a typical living room.
Example 6.10 Calculate the lowest frequency mode in a room which measures 3.5 m x 5 m x 2.5 m. At what frequency would a tangential mode with one half wavelength along the 3.5 m dimension and three half wavelengths along the 5 m dimension occur, at what frequency would the (2 2 2) oblique mode occur, and at what frequency is the first coincident mode?
Using Equation 6.21 calculate the modes as follows. The lowest frequency mode is the first axial mode along the longest dimension of the room which is the (0 1 0) or axial mode in this example so the lowest modal frequency in the room is:
f010 =C/2 [(
0/3.5m
)2 +
(1/5m)2 +
(
0/2.5m)2] = (344 ms-1)/2 [(
1/5m)2
= 34.4 Hz
The mode with one half wavelength along the 3.5 m dimension and three half wavelengths along the 5 m dimension is the (1 3 0) or tangential mode so its frequency is:
f130 =C/2 [( 1/3.5m )2 + (3/5m)2 + ( 0/2.5m)2] = 172 ms-1 [ 0.082 + 0.36]= 114.4 Hz
The frequency of the (2 2 2) or oblique mode is:
f222 =C/2 [( 2/3.5m )2 + (2/5m)2 + ( 2/2.5m)2] = 172 ms-1 [ 0.327 +0.16 + 0.64]= 182.6 Hz
The dimensions of 2.5 m and 5 m are related by a factor of 2 so the second axial mode along the 5 m dimension will be at the same frequency as the first axial mode along the 2.5 m dimension. That is:
f020 = f001
The (020) mode has a frequency of:
f020 =C/2 [( 0/3.5m )2 + (2/5m)2 + ( 0/2.5m)2] = 172 ms-1 [ (2/5m)2] = 68.8 Hz
and the (0 0 1) mode has a frequency of:
f001 =C/2 [( 0/3.5m )2 + (0/5m)2 + ( 1/2.5m)2] = 172 ms-1 [ (1/2.5m)2] = 68.8 Hz
2.5m
which are both at the same frequency and are therefore coincident.
As has been already discussed, modes behave differently to diffuse sound and this has the following consequences:
How does the energy in a mode decay as a function of time, how can it be related to the reverberation, and what is the effect of absorption in a mode on the frequency response?
The decay of sound energy in modes, is in many respects, identical to the decay of sound energy which is analysed in Appendix 3. The main difference is that the absorption coefficient is sometimes smaller, because the modal sound wave does not have random incidence, it will also be specific to the surfaces involved instead of being an average value for the whole room. In addition the time between reflections will be dependent on the length of the modal path rather than the mean free path. This means that the decay time for a mode is likely to be different to the diffuse sound.
For example the length of an axial mode path is determined by the distance between the two reflecting surfaces that support the mode, which will be one of the room's dimensions. Thus for an axial mode the energy left after a given time period is given by modifying Equation A3.5 in Appendix 3 using the distance between the surfaces instead of the mean free path to give:
Modal energy after t seconds = Modal energyinitial (1 - &α;mode)t5(c/Lmode)
where
The above equation can be manipulated to give a 60 dB decay time, analogous to Equation 6.17, which is:
T60(modal) = (Lmode /c ) (ln(10-6)/ln(1-αmode)= (Lmode /ln(1-αmode) (-13.82/344ms-2) =-0.04Lmode/ln(1-αmode)
where T60(modal) = the 60 dB decay time of the mode (in s)
This expresses a similar result to Equation 6.17 except for the difference caused by the differing length of the modal structure with respect to the mean free path. If the length of the modal structure is longer than the mean free path then, assuming similar levels of absorption, the decay time for the mode will be longer than the diffuse field whereas if the length is smaller then the modal decay will be shorter than the diffuse field. The length between reflections is both a function of the surfaces that support the mode and the type of mode - axial, tangential, or oblique - that occurs. For axial modes the mode length, L mode, is simply the relevant room dimension.
For the other types of mode the situation is more complicated, as shown in Figure 6.35 for a tangential mode.
Fig 6.35 The path length for a tangential mode
However, one could argue that the path length for this type of mode is given by the length of the hypotenuse of the triangle formed by half the length and half the width of the four walls that support the mode. That is, the modal length is given by:
Lmode(tangential) = [(Length/2)2 + (Width/2)2
= 1/2 (Length2 + Width2)
This equation shows that the distance between reflections for a tangential mode is essentially the diagonal dimension between the four surfaces that support the mode divided by two to allow for the fact that the wave suffers two reflections along this path. The length so derived can then be used as the modal length in the equation for the modal decay. A similar argument can be applied to the oblique modes, which visit all six surfaces. Because of this the modal supporting structure is a cuboid and the diagonal between opposing corners must be used. In addition the wave will suffer three reflections along this path. This gives a path length for the oblique mode as:
L mode(oblique) =
[(
Length/3
)2 +
(
Width/3
)2
(
Height/3
)2 = 1/3 [length2 + width2 + height2]
Fig 6.36 The bandwidth of modes for a given value of absorption
The absorption does more than cause the mode to decay; it reduces the total energy stored in the mode, in a similar manner to the effect of absorption on the reverberant field, and also causes the mode to have a finite bandwidth which is proportional to the amount of absorption, as shown in Figure 6.36. The absorption also reduces the peak to minimum variation in the standing wave pattern, and so reduces the spatial variation of the sound pressure, as shown in Figure 6.37. The bandwidth of a mode can be calculated from the 60 dB decay time using the following equation:
Fig 6.37 The spatial variation in the amplitude of modes for a given value of absorption
Fig 6.38 Typical variation in the amplitudes and bandwidths for the different mode types assuming an even distribution of absorption.
Bωmode = 2.2/T60 (modal)
where
Bωmode= the -3 dB bandwidth of the mode (in Hz)
Because it is not always possible to calculate the true modal decay time this equation can be dangerously approximated using the reverberation time of the room as:
Bωmode mode approximately equals 2.2/T60
This assumption is dangerous because the mode will not be diffuse whereas the reverberation time calculation assumes a diffuse sound field. In general the bandwidths and intensity levels of a mode are proportional to the number of reflections required to support them. This means that axial modes tend to be the strongest followed by tangential and then oblique modes in order of strength, as shown in Figure 6.38. However, this is not always the case as a tangential mode in a room with four reflecting surfaces could be stronger than the axial mode between the other two absorbing surfaces.
Example 6.11 Calculate the approximate modal bandwidth in a room which has a reverberation time (T60) of 0.44 seconds. What would be the modal bandwidth of axial modes along the 5 m dimension of the room if the absorption coefficients on the opposing walls were equal to the average room absorption coefficient of 0.2?
The approximate modal bandwidth can be calculated as:
Bw mode approx = 2.2/T60 = 2.2/0.44s = 5HZ
To answer the second question one must calculate the modal decay time, T60 (modal), which is given by:
T60 modal = -O.04L /ln(1-alphamode)= (-0.04 x 5 m)/ln(1-0.2) = 0.9 s
This value of decay can be used to calculate the actual modal bandwidth as:
Bw mode = 2.2/T60 modsl = 2.2/0.9s = 24 hz
Clearly care must be taken when calculating modal bandwidths due to the fact that the diffuse field assumptions no longer apply. In the case above, even though an even distribution of absorption was assumed, the decay time and bandwidth were radically different to that predicted by the diffuse field assumptions simply because the path travelled by the sound wave was longer than the mean free path. In practice the absorption coefficient is likely to be different as well, making prediction even more difficult. Note that, if the absorption remains constant with frequency, the bandwidths of a mode are independent of their frequency - they are simply a function of the modal decay time.
Because all rooms have modes in their lower frequency ranges there will always be a frequency below which the modal effects dominate and the room can no longer be treated as diffuse. Even anechoic rooms have lower frequency limits to their operation. One of the effects of room modes is to cause variations in the frequency response of the room, via its effect on the reverberant field. The frequency response due to modal behaviour will also be room position dependent, due to the spatial variation of standing waves. An important consequence of this is that the room no longer supports a diffuse field in the modal region and so the reverberation time concept is invalid in this frequency region. Instead an approach based on modal decay should be used. But at what frequency does the transition occur, can it be even calculated? Consider the typical frequency response of a room, shown in Figure 6.39. In it, three different frequency regions can be identified.
Fig 6.39 The frequency response of a typical room
The transition boundary between the region of modal behaviour and the region of diffuse behaviour is known as the critical frequency. As is usual in these situations, although the critical frequency is a single frequency it is not a sharp boundary, it represents some defined point in a transition region between the two regions.
The concept of critical frequency allows us to define the difference between rooms which are 'large' and 'small' in acoustical terms. In an acoustically large room the critical frequency is below the lowest frequency of the sound that will be generated in the room whereas an in an acoustically small room the critical frequency will occur within the frequency range of the sounds being produced in it. Examples of acoustically large rooms would be concert halls, cathedrals and large recording studios. Most of us listen to and produce music in acoustically small rooms such as bedrooms, bathrooms, living rooms, etc., and there is an increasing trend-due to the effect of computer recording and editing technology and because it's cheaper-to perform more and more music and sound production tasks in small rooms.
How can the critical frequency be calculated? There are two main approaches. The first is to recognise that when the wavelength of the sound approaches the mean free path of the room then the likelihood of modal behaviour increases, because a sound wave is 'in touch' with all the walls in the room. This approach can be used to set an approximate lower frequency bound on the critical frequency below which it is likely to be difficult to prevent modal effects from dominating the acoustics without extreme measures being taken. This approach gives the following expression for calculating the critical frequency, which assumes that modal behaviour dominates once the mean free path is equal to one and a half wavelengths:
fcritical = (3/2)(c/MFP) = (3/2)(344ms-1/MFP)
This expression is useful for making a rapid assessment of the likelihood of achieving a particular critical frequency in a given room. However, the real critical frequency may well be higher because a room can have significant modal behaviour at high frequencies, if the absorption is low. Because of this the accepted definition of critical frequency is based on the mode bandwidth,
Fig 6.40 The composite effect of adjacent modes on the spatial variation in a room
although this can result in a chicken and egg situation at the initial design stages, hence the earlier equation. The rationale for this is as follows. The main consequence of modal behaviour is the frequency and spatial variation caused by it. This means that if a given frequency excites only one mode, then this variation will be very strong. However, if a given frequency excites more than one mode, both the spatial and frequency variation will be reduced. Figure 6.40 shows the effect of adding three adjacent modes together and it shows that once more than three adjacent modes are added together the variation is considerably reduced. The way to excite adjacent modes with a single frequency is to increase their bandwidth until the three bandwidths associated with the three modes overlap a given frequency point, as shown in Figure 6.41. The critical frequency is defined as when the modal overlap equals three, so at least three modes are excited by a given frequency, and is given by:
fcritical = 2102 (T60/V)
This equation shows that the critical frequency is inversely proportional to the square root of the room volume and is proportional to the square root of the reverberation time, which is also proportional to the cube root of the volume, if the absorption remains constant, as discussed earlier. The net result of this is that, as expected, the critical frequencies for larger rooms are
Fig 6.41 The way to guarantee that at least three modes are excited by a given frequency
generally lower than those of smaller ones. Thus big rooms are acoustically 'large' as well.
As an example let us calculate the critical frequency of our typical living room:
Example 6.12 What is the critical frequency of a room whose surface area is 75 m2, whose volume is 42 m3? What is the critical frequency of the same room if the average absorption coefficient is 0.2?
Using the first equation calculate the lowest bound on the critical frequency using the mean free path as:
fcritical = (3/2)c/MFP = (3/2)((c/4V)/S) = 1.5 ((344ms-1/(4 x 42m-3))/75m2) = 1.5 (344ms-1/2.24m) = 230Hz
Using the second equation calculate the critical frequency using the reverberation time. First calculate the reverberation time as:
T60 = ( -0.161 V)/S ln(1-α)= (-0.161 x 42 m-3) /75m2 ln(1-0.2) = 0.43 s
Then, using the second equation, calculate the critical frequency as:
f critical = 2102 (T60 /V) = 2102 (0.43s/42m-3)=213Hz
The second equation predicts a slightly lower critical frequency compared to the first one. However, the agreement is surprisingly good. Although the modal overlap has been calculated using a reverberation time, and hence a diffuse field assumption, this is probably just valid at this frequency which represents the boundary between the two regions. The critical frequency results show that, for this room, frequencies below 213 Hz must be analysed using modal decay time rather than reverberation time.
You Need to Know
When a room is excited by an impulse, the sound energy is reflected from its surfaces. At each reflection some of the sound is absorbed and therefore the sound energy decays exponentially. Ideally the sound should be reflected from each surface with equal probability, forming a diffuse field. This results in a single exponential decay with a time constant proportional to the average absorption in the room.
However in practice not all the energy is reflected in a random fashion. Instead 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. These standing waves have pressure and velocity distributions which are spatially static and so behave differently to the rest of the sound in the room in the following ways:
Another name for these standing waves in a room are room modes and the frequencies at which they occur are known as modal frequencies. Because the modes are spatially static there will be a strong variation of sound pressure level as one moves around the room, which is undesirable.
There are three basic types of room mode which are outlined in Sections 6.2.1
These modes occur between two opposing surfaces, as shown in Figure 6.31, and so are a function of the linear dimensions of the room.
These modes occur between four surfaces, as shown in Figure 6.32, and so are a function of two of the dimensions of the room.
There is an infinite number of tangential modes, but they must fit an integral number of half wavelengths in two dimensions. This has the interesting consequence that the lowest modal frequencies are higher than the axial modes, despite the fact that the apparent path length is greater. The reason is that the standing waves must fit between the opposing surfaces, that is on the sides rather than the hypotenuse of the triangular path, and as the propagating wave travels down the hypotenuse, the effective wavelength, or phase velocity, on the sides of the room is larger, as shown in Figure 6.33. The lowest modal frequency for a tangential mode occurs when precisely one half wavelength, at the phase velocity, fits into each dimension.
These modes occur between all six surfaces, as shown in Figure 6.34, and so are a function of all three dimensions of the room.
In general the number of resonances within a given frequency bandwidth increases with frequency. In fact it can be shown that they increase proportional to the square of the frequency, and in large well-behaved acoustical spaces, that sound good, this increase in mode density with frequency is smooth. This is the rationale behind a method for assessing the modal behaviour in a room known as the Bonello criteria These criteria try to ascertain how significant the modal behaviour of a room is in perceptual terms. It does this by dividing the audio frequency spectrum into third octave bands, as an approximation of critical bands, and then counting the number of modes per band. If the number of modes per third octave band increase monotonically then there is a good chance that we will perceive the room as having a 'smooth' frequency response despite the resonances.
Definition: A function from a partially ordered domain to a partially ordered range such that x > y implies f(x) ≥ f(y).
If the number of resonances per third octave drops as the frequency rises then there will be a perceptually noticeable peak in the frequency response. Coincident modes are also another way of creating a perceptually noticeable frequency response peak and the Bonello criteria does further stipulate that there should be no modal coincidence within a third octave band unless there are at least three additional non-coincident resonances to balance the two that are coincident. As an example of the calculation of mode frequencies let us calculate some for a typical living room.
As has been already discussed, modes behave differently to diffuse sound and this has the following consequences:
The decay of sound energy in modes, is in many respects, identical to the decay of sound energy which is analysed in Appendix 3. The main difference is that the absorption coefficient is sometimes smaller, because the modal sound wave does not have random incidence, it will also be specific to the surfaces involved instead of being an average value for the whole room. In addition the time between reflections will be dependent on the length of the modal path rather than the mean free path. This means that the decay time for a mode is likely to be different to the diffuse sound.
If the length of the modal structure is longer than the mean free path then, assuming similar levels of absorption, the decay time for the mode will be longer than the diffuse field whereas if the length is smaller then the modal decay will be shorter than the diffuse field. The length between reflections is both a function of the surfaces that support the mode and the type of mode - axial, tangential, or oblique - that occurs. For axial modes the mode length, L mode, is simply the relevant room dimension.
Because all rooms have modes in their lower frequency ranges there will always be a frequency below which the modal effects dominate and the room can no longer be treated as diffuse. Even anechoic rooms have lower frequency limits to their operation. One of the effects of room modes is to cause variations in the frequency response of the room, via its effect on the reverberant field. The frequency response due to modal behaviour will also be room position dependent, due to the spatial variation of standing waves. An important consequence of this is that the room no longer supports a diffuse field in the modal region and so the reverberation time concept is invalid in this frequency region.
The transition boundary between the region of modal behaviour and the region of diffuse behaviour is known as the critical frequency. As is usual in these situations, although the critical frequency is a single frequency it is not a sharp boundary, it represents some defined point in a transition region between the two regions.
The concept of critical frequency allows us to define the difference between rooms which are 'large' and 'small' in acoustical terms. In an acoustically large room the critical frequency is below the lowest frequency of the sound that will be generated in the room whereas an in an acoustically small room the critical frequency will occur within the frequency range of the sounds being produced in it. Examples of acoustically large rooms would be concert halls, cathedrals and large recording studios. Most of us listen to and produce music in acoustically small rooms such as bedrooms, bathrooms, living rooms, etc., and there is an increasing trend-due to the effect of computer recording and editing technology and because it's cheaper-to perform more and more music and sound production tasks in small rooms.