Auditory Theory: Acoustics

Lecture 022 Environment III

Reading Assignment for Lecture 023

Before next lecture please read Sections

• 6.3 Absorption materials 305
  6.4 Diffusion materials 311
  6.5 Sound isolation 316
  6.6 Energy-time considerations 321

pages 305 to 321 of Acoustics and Psychoacoustics. We may have a brief quiz on these sections at the beginning of the next class.

Brain Bullets

Room modes and standing waves

• 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:
• They do not visit each surface with equal probability. Instead a subset of the surfaces are involved.
• They do not strike these surfaces with random incidence. Instead a particular angle of incidence is involved in the reflection of the standing wave.
• They require a coherent return of energy back to an original surface, a cyclic path. This is of necessity strongly frequency dependent and so these paths only exist for discrete frequencies which are determined by the room geometry.
• 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

Axial modes

• 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.

Tangential modes

• 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.

Oblique modes

• 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 Bonello criteria

• 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.

The behaviour of modes

• As has been already discussed, modes behave differently to diffuse sound and this has the following consequences:
• The standing wave is not absorbed as strongly as sound which visits all surfaces. This is due to both the reduction in the nunber of surfaces visited and the change in absorption due to non-random incidence.
• This reduction in absorption is strongly frequency-dependent and results in less absorption and therefore a longer decay time at the frequencies at which standing waves occur.
• The decay of sound energy in the room is no longer a single exponential decay with a time constant proportional to the average absorption in the room. Instead there are several decay times. The shortest one tends to be due to the diffuse sound field whereas the longer ones tend to be due to the modes in the room. This results in excess energy at those frequel1des with the attendant degradation of the sound in the room.

The decay time of axial modes

• 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.

Critical frequency

• 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 cut-off region: the region below the lowest resonance, sometimes called the room cut-off region. In this region the room is smaller than a half wavelength in all dimensions. This does not mean that the room does not support sound propagation, in fact it behaves more like the air in a bicycle pump when the end is blocked. This means that the environment 'loads' any sources of sound in the room differently (such as loudspeakers or musical instruments), and often the effect of this loading is to reduce the ability of the source to radiate sound into the room and so results in reduced sound levels at these frequencies. The low frequency cut-off can be calculated simply from:
  • f_cutoff = c/(2 x Longest dimension)
  • =344 ms^-1/2 x Longest dimension (m)
• The modal region: the next region is the modal region in which the modal behaviour of the room dominates its acoustic performance. In this region the analysis based on the assumption of a diffuse field is doomed to fail.
• The diffuse field region: the final region is the region in which a diffuse field can exist and therefore the concept of reverberation time is valid. In general this region of the frequency range is the one that will sound the best, providing the reverberation characteristics are good, because the effects of room modes are minimal and so the listener experiences an even reverberant sound level throughout the 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.

Acoustically 'large' and 'small' rooms

• 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.