Theory

This chapter explains some aspects of room acoustics theory, which will help to clarify the implementation decisions governing the simulation techniques discussed in later chapters. The account given here deals only with topics which are directly applied in Wayverb. For more detailed derivations of equations, along with information about more advanced acoustic phenomena such as diffraction, refraction, and the Doppler effect, the books by Vorländer [1] and Kuttruff [2] are recommended.

Waves and Media

Sound waves can be described completely by specifying the instantaneous velocity, \(\vec{v}\), of each particle in the propagation medium. Not all particles will have the same velocity, which causes fluctuations in density, pressure, and temperature, which are dependent upon time, and position. Sound pressure, \(p\), is the difference between the “at rest” pressure \(p_0\), and the pressure measured in the medium at a particular position and time \(p_\text{tot}\) [1, p. 9]:

\[p = p_\text{tot} - p_0\](1)

Sound pressure is measured in pascals (1 Pa = 1 N/m²) [1, p. 16]. The goal of room acoustic simulation is to predict the change in sound pressure over time at particular points in a sound field. A similar equation to eq. 1 can be written for change in density due to sound \(\rho\), where \(\rho_0\) is the static density of the medium, and \(\rho_\text{tot}\) is the instantaneous density:

\[\rho = \rho_\text{tot} - \rho_0\](2)

For the purposes of room acoustics, it may be assumed that the propagation medium is air. Sound propagation through liquids and solid structures will be ignored. The speed of sound in air, \(c\), is approximately

\[c=(331.4 + 0.6\theta) \text{m/s}\](3)

where \(\theta\) is the temperature in degrees Celsius [2, p. 7]. In most real-world acoustics problems, variations in temperature will be very small, and can be ignored. That is, the propagation medium is assumed to be homogeneous. A propagation medium can be specified by its characteristic impedance or wave impedance \(Z_0\) [1, p. 14]:

\[Z_0 = \rho_0 c\](4)

This quantity denotes the medium’s resistance to pressure excitation, or alternatively the pressure required to induce movement in the medium’s particles. For air, the characteristic impedance is generally around 400 kg/m²s (although this depends on the speed of sound, and therefore the air temperature) [1, p. 15].

The difference in sound pressure between the quietest audible sound and the threshold of pain is around 13 orders of magnitude. To make working with the values involved more manageable, sound pressure is usually given in terms of the sound pressure level (SPL), which is measured on a logarithmic scale in decibels (dB) [2, p. 23]:

\[\text{SPL} = 20\log_{10}\frac{p_\text{rms}}{p_0} \text{dB}\](5)

Here, \(p_\text{rms}\) is the root mean square sound pressure, and \(p_0\) is a reference pressure of \(2 \times 10^{-5}\) Pa.

Wave propagation in a medium causes energy to be transported through that medium. This energy flow can be measured in terms of the energy transported per second through an area of 1m² (W/m²), and is called sound intensity, \(I\) [1, p. 19]:

\[\vec{I} = \overline{p \vec{v}}\](6)

Note that the overline notation signifies time-averaging. The sound intensity can also be given in terms of the intensity level, \(L_I\) [1, p. 20]:

\[L_I = 10\log_{10}\frac{|\vec{I}|}{I_0}\](7)

where \(I_0 = 10^{-12} W/m^2\) is a reference intensity chosen to match the levels of sound pressure and intensity in a plane wave.

For a harmonic wave, the temporal and spatial periods of the wave are related by the speed of sound, \(c\):

\[c = \lambda f = \frac{\lambda}{T}\](8)

where \(f\) is the frequency of the wave in Hz, \(T\) is the temporal period, and \(\lambda\) is the spatial period or wavelength.

In the simulation presented in this project, sound waves are assumed to propagate equally in all directions from a point-like source, or monopole. The pressure wave, propagating with increasing radius, is known as a spherical wave.

The pressure \(p\) observed at distance \(r\) and time \(t\) from a monopole source with signal strength \(Q(t)\) is given by [1, p. 24]:

\[p(r, t) = \frac{\rho_0}{4\pi r}\dot{Q}\left(t-\frac{r}{c}\right)\](9)

For harmonic waves, the total radiated sound power \(P\) of a monopole source is related to the sound intensity by [2, p. 15]:

\[P = 4 \pi r^2 I\](10)

Boundary Characteristics

Room acoustics is not just concerned with the behaviour of pressure waves in air. Additionally, room acoustics problems normally bound the air volume with a set of surfaces (walls, floors, baffles etc.), from which an incident pressure wave may be reflected and/or absorbed. The reflected pressure waves generally lead to complex sound fields, which in turn contribute to the particular sonic “character” or acoustic of an enclosure.

Several assumptions are made to simplify the equations governing wave behaviour at a boundary. First, incident waves are assumed to be plane waves. This is never true in the Wayverb app, as simulations use point sources which produce spherical waves. However, the curvature of the wave front may be ignored so long as the source is not close to the reflecting surface [2, pp. 35–36] (“close” here will depend on the error constraints of the particular simulation). Secondly, boundary surfaces are assumed to be flat, and infinitely large [2, p. 35]. This is a valid approximation only if the size of each surface is large relative to the longest wavelength in the simulation.

Magnitude and Phase

The reflection factor \(R\) of a boundary is a complex value given by

\[R=|R|\exp(i\chi)\](11)

which describes a modification to the amplitude and phase of a wave reflected in a boundary (\(|R|\) is the magnitude term, \(\chi\) is phase).

This factor depends both on the frequency and direction of the incident wave. When \(\chi = \pi\), \(R\) is negative, corresponding to a phase reversal. This is known as a “soft” wall, but is rarely seen in room acoustics. It is reasonable to assume that reflections are in-phase in the majority of architectural acoustics problems [2, p. 36].

The wall impedance \(Z\) is defined as the ratio of sound pressure to the normal component of particle velocity at the wall surface. It is related to the reflection factor by

\[R=\frac{Z\cos\theta-Z_0}{Z\cos\theta+Z_0}\](12)

where \(\theta\) is the angle of incidence, and \(Z_0\) is the characteristic impedance of the propagation medium, normally air. In the case that the wall impedance is independent of the wave angle-of-incidence, the surface is known as locally reacting. A locally reacting surface does not transmit waves tangentially along the wall surface. In Wayverb, all surfaces are modelled as locally reacting.

The absorption coefficient \(\alpha\) of a wall describes the proportion of incident energy which is lost during reflection. It is defined as

\[\alpha =1-|R|^2\](13)

The specific acoustic impedance \(\xi\) for a given surface is defined as the impedance of that surface \(Z\) divided by the acoustic impedance of the propagation medium (air) \(Z_0\).

\[\xi=\frac{Z}{Z_0}\](14)

Inserting this equation into eq. 12 gives:

\[R=\frac{\xi\cos\theta-1}{\xi\cos\theta+1}\](15)

where \(\theta\) is the angle of incidence [3].

For a general surface, \(\xi\) will be a function of the incident angle. However, in the case of a locally reacting surface, the impedance is independent of the angle of incidence. The \(\xi\) term in eq. 15 can then be replaced by \(\xi_0\) which represents the normal-incidence specific impedance (which will be constant for all angles). Thus, the reflection factor of a locally reacting surface is

\[R=\frac{\xi_0\cos\theta-1}{\xi_0\cos\theta+1}\](16)

Surfaces in Wayverb are defined in terms of absorption coefficients. To express the reflectance in terms of absorption, an equation for \(\xi_0\) in terms of absorption must be found, and substituted into eq. 16.

Assuming that the absorption coefficients denote normal-incidence absorption, then by rearranging eq. 13, the normal-incidence reflection coefficient \(R_0\) is given by

\[R_0=\sqrt{1-\alpha}\](17)

\(R_0\) can also be expressed by setting \(\theta\) to 0 in eq. 16:

\[R_0=\frac{\xi_0 -1}{\xi_0 +1}\](18)

To express \(\xi_0\) in terms of \(\alpha\), eq. 18 is rearranged in terms of the normal-incidence reflection coefficient:

\[\xi_0=\frac{1+R_0}{1-R_0}\](19)

Then, eq. 17 may be substituted into eq. 19 to give \(\xi_0\) in terms of \(\alpha\). This new definition of \(\xi_0\) can then be used in conjunction with eq. 16 to define the angle-dependent reflection factor of a locally reacting surface.

Scattering

The reflection factor, absorption coefficient, and wall impedance describe the behaviour of perfectly-reflected (specular) waves. If the reflecting surface has imperfections or details of the same order as the wavelength, as many surfaces in the real world do, then some components of the reflected wave will be scattered instead of specularly reflected.

Describing the nature of the scattered sound is more complicated than specular reflections. A common method is to use a scattering coefficient, which describes the proportion of outgoing energy which is scattered, and which may be dependent on frequency (see fig. 1). The total outgoing energy \(E_\text{total}\) is related to the incoming energy \(E_\text{incident}\) by

\[E_{\text{total}}=E_{\text{incident}}(1-\alpha)\](20)

Then the scattering coefficient \(s\) defines the proportion of this outgoing energy which is reflected specularly \(E_\text{specular}\) or scattered \(E_\text{scattered}\):

\[ \begin{aligned} E_{\text{specular}} & =E_{\text{incident}}(1-\alpha)(1-s) \\ E_{\text{scattered}} & =E_{\text{incident}}(1-\alpha)s \end{aligned} \](21)

Alone, the scattering coefficient fails to describe the directional distribution of scattered energy. In the case of an ideally-diffusing surface, the scattered energy is distributed according to Lambert’s cosine law. That is, the intensity depends only on the cosine of the outgoing scattering angle, and is independent of the angle of incidence (see fig. 2). More complex scattering distributions, which also depend on the outgoing direction, are possible [4], [5], but there is no single definitive model to describe physically-accurate scattering.

Impulse Response Metrics

Sabine’s Equation

If an air-filled enclosure is excited impulsively, the sound field in the enclosure will not die away immediately, unless the boundaries are perfectly anechoic. Usually, the boundaries will reflect sound pressure waves back and forth inside the enclosure, and at each reflection some energy will be lost to the boundary. In this way the sound energy in the simulation will decrease gradually over time.

Assuming that the sound field is diffuse (that is, isotropic; having the same intensity in all directions [2, p. 52]), there will be a constant number of reflections per unit time. Under the additional assumption that the boundary absorption, \(\alpha\), has a homogeneous distribution, each reflection will cause that wave front to lose a proportion of its incoming energy equal to the absorption coefficient \(\alpha\).

The energy density, \(w\) of a sound field is defined as the amount of energy contained in one unit volume [2, p. 14]. In a diffuse sound field, the energy density at time \(t\) can be given by

\[w(t) \approx w_0 (1 - \alpha)^{\overline{n} t}\](22)

where \(w_0\) is the initial energy density when \(t=0\) (the time of the impulsive excitation), and \(\overline{n}\) is the average number of reflections per unit time. Using the definition of sound pressure level, the energy density decay can be rewritten as a linear level decay [1, p. 61]:

\[L(t) = L_0 + 10\log_{10} \frac{w(t)}{w_0} = L_0 + 4.34 \overline{n} t \ln (1 - \alpha)\](23)

A common measure of reverberation time is the RT60, the time taken for the level to decrease by 60dB. By rearranging eq. 23, the RT60 can be expressed [1, p. 61]:

\[\text{RT60} = -\frac{60}{4.34\overline{n}\ln(1-\alpha)}\](24)

The average number of reflections per second, \(\overline{n}\) can be approximated by

\[\overline{n} = \frac{cS}{4V}\](25)

where \(S\) is the total surface area of the enclosure boundaries, and \(V\) is the enclosure volume [2, p. 112].

By substituting eq. 25 into eq. 24 and simplifying (assuming \(\alpha\) is very small), and assuming \(c = 343 m/s\), Sabine’s equation is found [1, p. 61]:

\[\text{RT60} = 0.161\frac{V}{S\alpha} = 0.161\frac{V}{A}\](26)

The term \(A = S\alpha\) is known as the equivalent absorption area of the room [2, p. 130]. Sabine’s equation can be used to estimate the reverberation time of an enclosure, given the volume, surface area, and homogeneous absorption of that enclosure, and assuming that the sound field is diffuse.

The Sabine equation has some important limitations. Firstly, it fails at high absorptions. With the absorption coefficient set to 1, it estimates a finite reverb time, even though a completely absorptive enclosure cannot reverberate. Secondly, the assumption that the sound field in the enclosure is perfectly diffuse is untrue in practice. At low frequencies rooms behave modally, concentrating sound energy at specific points in the room. Under such circumstances, the sound field is clearly not diffuse, and so the Sabine equation is a poor predictor of reverb time at low frequencies.

Computing Reverb Times from Measurements

It is possible to compute the RT60 of a measured impulse response (IR). This is useful for comparing the reverb times of different IRs, or for evaluating an IR against a predicted reverb time.

Methods for evaluating the RT60 from measurements are given in the ISO 3382 standard [6]. Normally, the decay time is measured over a shorter range than 60dB, and then extrapolated. For example, the T20 measurement is an estimate of the RT60 based on a 20dB level drop, and the T30 is based on a 30dB drop.

The analysis itself proceeds as follows: the IR is reversed, each sample value is squared, then the squared sample values are integrated/accumulated. The backward-integrated IR can then be reversed once more to produce a decay curve. For a T20 measurement, the -5dB and -25dB points on the curve (relative to the maximum level) are found, and a regression line is fitted to the region between these points using the least-squares method. The gradient of this line, \(d\), is a value measured in dB/s. The T20, in seconds, is given by \(60/d\). The T30 is found in the same way, but the regression line is fitted to the region between -5dB and -35dB instead. To find the reverb time in a particular frequency band, the IR may be band-pass filtered before computing the decay curve.

A final measurement, known as the early decay time (EDT) is also based on fitting a regression line to the region between 0dB and -10dB, and then extrapolating to 60dB as usual. This measurement is useful as a descriptor of the ratio of direct to reverberant sound: an IR with a louder direct contribution and quieter reverberation will have a steep early gradient, and therefore a relatively short EDT. EDT is also useful as a measure of “perceived reverberance”: the transients in music and speech normally have a much shorter period than the time taken for the level to drop 60dB, so the early part of the IR has greater perceptual significance than the late [1, p. 94].

ISO 3382 also defines the term just noticeable difference (JND), which is the smallest perceivable difference between two acoustic measurements [1, pp. 99–100]. That is, in order for simulation results to be indistinguishable from predictions or real-world measurements, the difference in each characteristic should be within the JND for that characteristic. For example, the JND for reverb time measurements (EDT, T20, T30) is 5%, and the JND for level is 1dB.

Summary

An overview of the properties governing the behaviour of a sound field in an enclosed space has been given. The relationships between different physical properties have been shown. Sound behaviour at boundaries has been explained, along with methods for deriving boundary characteristics from absorption and scattering coefficients. Finally, Sabine’s equation for estimating reverb time has been derived, and methods for computing statistics from recorded impulse responses have been set out.

References