Of the many available reverberation time prediction formulas, Sabine's and Eyring's equations are still widely used. The assumptions of homogeneity and isotropy of sound energy during the decay associated with those models are usually recognized as a reason for lack of agreement between predictions and measurements. At the same time, the inaccuracy in the estimation of the sound-absorption coefficient adds to the uncertainty of calculations. This paper shows that the error of incorrectly assumed sound absorption is more detrimental to the prediction precision than the inherent error in the formulas themselves. The proposed absorption calibration procedure reduces the differences between the measured and predicted reverberation time values, showing that an accuracy within ±10% from the target reverberation time values can be achieved regardless of the absorption distribution in a room. The paper also discusses the oft neglected air absorption of sound, which may introduce considerable bias to the measurement results. The need for an air-absorption compensation procedure is highlighted, and a method for the estimation of its parameters in octave bands is proposed and compared with other approaches. The results of this study provide justification for the use of the Sabine and Eyring formulas for reverberation time predictions.

## I. INTRODUCTION

One of the most important properties of sound in physical spaces is reverberation, which is affected by the size, the geometry, and even the materials of the surfaces of the enclosure. Reverberation is also a significant factor affecting the perception of sound, having an influence on, e.g., clarity of music and intelligibility of speech. A parameter commonly used to describe the sound decay in a room is called reverberation time (RT), already established in the literature over 100 years ago.^{1} Since then, several methods to estimate a room's RT have been developed, involving both simple formulas and more sophisticated models that take into account an enclosure's geometry and sound-absorption distribution.^{2–11} However, the basic models introduced by Sabine^{1} and Eyring^{2} are still the best known and the most frequently used.

The accuracy of RT prediction is crucial when designing spaces for speech and music, such as concert halls and auditoriums. Over the years, many studies have been published attempting to evaluate the correctness of RT estimation formulas, both in spaces with big volumes, such as concert halls,^{10} and in small rectangular rooms,^{12–14} mostly classrooms.^{15–19} The results generally reveal that the considered formulas are burdened with a certain error, often proving them to be too unreliable to use in acoustic design. This study identifies the sources of the uncertainty associated with two classical reverberation formulas and introduces ways to reduce the errors to achieve sufficient accuracy for practical designs.

One of the sources of error in reverberation calculations is the sound-absorption coefficient. Its insufficient measurement accuracy and reproducibility has been known since the 1930s^{20,21} and has earned a name for itself: “the absorption coefficient problem.”^{22,23} Even though the measurement procedure has been standardized and is occasionally updated,^{24,25} the issue still remains and is a subject of scientific debate.^{26–31}

Another source of uncertainty is the neglect of the influence that the absorption of sound in the air has on a room's RT, causing discrepancies between predicted and measured RT values. The early works by Sabine and Eyring considered the attenuation of sound in a room to be a result of surface absorption only.^{1,2} The neglected attenuating effect of the medium was introduced in RT formulas by Knudsen.^{32} However, some studies omit the air absorption in RT calculations^{10,33} or use it with selected formulas only.^{15}

The present work investigates the ability of Sabine's and Eyring's formulas to accurately predict the RT value for different sound-absorption conditions in a small rectangular room with variable acoustics. The RT values estimated from captured room impulse responses (RIRs) and from the aforementioned models are compared for different amounts and distributions of the absorbing and reflecting elements. The study presents the absorption calibration procedure that is adapted to the measured RT values.

Additionally, the study shows the effect of air absorption on the RT measurements and highlights the need for its compensation. A method to estimate the air-absorption coefficient of sound for full-octave bands, based on the standardized calculations for pure-tone absorption, is proposed.

The remainder of this paper is organized as follows: Sec. II presents the Sabine and Eyring RT prediction formulas and discusses the sound field conditions associated with each of them. The issues related to sound-absorption-coefficient calibration are elaborated upon in Sec. III. Section IV recapitulates the methods to calculate the air absorption for pure tones, proposes a new procedure that translates it to a coefficient for full-octave bands, and compares the new method with the other approaches. Section V describes comprehensive RT measurements conducted in a variable acoustic laboratory. Section VI presents the results of the measurements, compares them with the predictions obtained with Sabine's and Eyring's formulas, and shows the calibration of sound-absorption coefficients. Section VII discusses the results of the study, and Sec. VIII concludes the paper.

## II. RT ESTIMATION

This section reviews popular RT estimation formulas and models. It also discusses their development and assumptions regarding the properties of the sound field.

### A. Sound-decay model

Considering the loss of sound energy resulting from the absorption at the boundary of the enclosure, the rate of sound decay is described in terms of sound intensity,^{2,32,34}

where *I*(*t*) and *I*(0) are the sound intensities at times *t* and 0, respectively, and frequency *f*; *c* is the speed of sound propagation in air; and $l\xaf$ is the mean free path. Here, $\alpha \u0302$ represents the general absorption coefficient of the room's surfaces regardless of the approach adopted to calculate it. The RT value at time $t=T60(f)$ is obtained when $I(t,f)/I(0,f)=10\u22126$ (i.e., a 60-dB decay), resulting in the following formula:

Here, the propagation in an ideal gas is assumed with no air absorption.

### B. Reverberation time formulas

The earliest work on the RT estimation comes from Wallace Sabine, who also defined this parameter as the time needed for sound to become inaudible. As a result of his experimental work, Sabine introduced the following formula to predict the RT of concert halls:^{1}

where *V* is the volume of a space in $m3$, *S* is the room surface in $m2$, 0.164 is an experimentally determined coefficient (although different values between 0.16 and 0.164 are commonly used^{6,10,33,35,36}), and $\alpha \xaf$ is the average absorptivity in the room, defined as $\alpha \xaf=\u2211iSi\alpha i/S$, where *S _{i}* are the surface areas in $m2$ and

*α*are the corresponding absorption coefficients of each surface. Equation (3) is equivalent to Eq. (2) when the standard atmospheric conditions and a shoebox room are considered. To generalize Sabine's formula and make it applicable in other scenarios as well, the present study uses Eq. (3) in the form that is closer to Eq. (2), when

_{i}Although Sabine's formula is commonly used to predict the RT of different types of rooms, only very specific rooms meet the requirements that make these estimations accurate.^{10,37} Since the formula assumes continuous decay of sound,^{38} the key condition is that, at any given moment, sound energy is diffused equally throughout the space, i.e., it is homogeneous and isotropic. In practical terms, achieving the diffuse sound field is commonly translated to a number of additional guidelines for the enclosure: e.g., walls are not parallel, basic dimensions (height, width, and length) have no big differences between them, and the small absorption^{38} [$\alpha \xaf<0.2$ (Refs. 17 and 39)] is uniformly distributed on all surfaces.^{2,10,12,17}

To improve the RT predictions for rooms with considerable absorption, Eyring^{2} proposed another reverberation theory, which was based on the mean free path of sound particles, which for rectangular rooms is $l\xaf=4V/S$,^{2,6,40,41} and the stepwise exponential energy decay, changing by $1\u2212\alpha \xaf$ after a specular reflection. The Eyring formula uses

The Eyring formula is designed to give more accurate RT estimates in “dead,” i.e., highly absorptive, rooms (although Eyring does not specify a particular threshold for a room to be “dead” or “live”^{2}) than Sabine's formula and consistently gives lower RT values for the same absorption coefficient.^{2} If the room's surfaces are perfectly absorptive, i.e., $\alpha \xaf=1$, Sabine's formula gives a non-zero number, whereas Eyring's results in $ln\u2009(0)$, which in literature is interpreted as $T60(f)=0$ s.^{42}

However, this common view of Eyring's formula accuracy for high sound absorption is contested.^{38,40} Eyring's theory assumes that the sound decay in a room is a discrete process, which could be described by a probability of the interaction between the sound particle and the surface. For such assumptions to be justified, the sound field is required to be homogeneous and isotropic.^{38} It is discussed in literature that the combination of such assumptions and prerequisites does not result in more accurate RT estimations when $\alpha \xaf$ is high,^{37,38,40} but rather returns correct predictions for small values of average absorptivity [$\alpha \xaf<0.5$ (Refs. 39 and 42)].

## III. SOUND-ABSORPTION-COEFFICIENT CALIBRATION

In this study, we show that the classical equations by Sabine and Eyring can achieve sufficient accuracy of RT predictions, even when the assumptions of homogeneous and isotropic sound field are not met. This is accomplished by reducing the inaccuracy in the estimation of the average absorptivity $\alpha \xaf$.

Several studies exist that aim at calibrating room acoustics simulations to obtain parameter values matching the measurement results.^{43–46} However, applying the calibration to Sabine's or Eyring's models is rare,^{45} as most of the research focuses on simulations in software, such as ODEON^{47–49} or CATT-Acoustic.^{50–52} Also, in the majority of the studies, there is no record of collecting the atmospheric data during measurements.

In the present study, the measurements are performed in an environment that allows changing the amount and distribution of sound absorption in the room (described in more detail in Sec. V). Hence, the $\alpha \xaf$ calibration is conducted with a similarly adjustable case in mind.

Here, we assume that the change of the total absorption within the space $\alpha \xaf$ is directly proportional and linearly related to the amount of absorbing material. The model for the calibrated $\alpha \xaf$ is

where *n* is the number of fixed portions of absorbing material added to the space (in our case, one variable acoustics panel changing its state from reflective to absorptive, cf. Sec. V). The adopted approach consists of a base absorption coefficient $\beta X$ representing the most reflective configuration of the measured space, i.e., with the smallest possible $\alpha \xaf$. The change of absorptivity resulting from the addition of absorbing material is symbolized by a step value $\Delta X$.

Here, the subscript $X\u2208{S,E,T}$ denotes the particular formula to be calibrated. Subscript $S$ represents Sabine's model and $E$ Eyring's method. Additionally, subscript $T$ denotes absorption values that are based on laboratory measurements or the literature. Since neither Sabine's nor Eyring's model considers absorption distribution, the change in $\alpha \xaf$ is also considered as distribution-independent.

To verify that the predictions by Sabine's and Eyring's models can come close to measurement results, the average absorptivity $\alpha \xaf$ is first fitted to match $\alpha \u0302$ based on the obtained values of RT. Stemming directly from the relation between the sound-absorption coefficient and the RT values ($T60\u223c1/\alpha \xaf$), the problem of optimizing $\alpha \xafc,\u200aX$ is of nonlinear nature. Here, error minimization is performed on the absolute difference between the measured $\alpha \u0302$ for combination *k* with certain absorption conditions (the minuend) and an $\alpha \xaf$ estimated from either of the equations for the same combination (the subtrahend). The total number of analyzed absorption combinations is denoted as *K*.

In the case of Sabine's equation, the problem is formulated as

whereas for Eyring's formula, it is defined as

Assuming that the distribution of the measured RTs is not normal, the absolute values are not squared. This avoids amplifying the effect of outliers on the calibration, making the process more robust.

## IV. AIR-ABSORPTION COMPENSATION

This section presents the formulas to theoretically determine the air-absorption coefficient *m*. The applicability of a pure-tone *m* to determine the absorption in full-octave bands and its further use in the air-absorption compensation procedure is discussed as well.

### A. Attenuation of sound in air

To account for both the absorbing properties of the enclosure's surfaces and the decay caused by sound propagating through air, Eq. (1) is extended with a second exponential,^{32,34}

This results in the following changes in Eq. (2):

Equivalent corrections are also made to Sabine's and Eyring's formulas. When the air absorption is omitted, i.e., $m=0$, we write $T60(f,0)=T60(f)$.

The value of the air-absorption coefficient *m* depends on the frequency of sound and the atmospheric conditions. It is derived from the attenuation of sound in the air, $\alpha a$, which is expressed in dB/m,

where $e$ denotes Euler's constant ($e=2.71828$).

The attenuation of sound in the air is a function of the relaxation frequencies of oxygen $frO$ and nitrogen $frN$, which are calculated, respectively, from^{53,54}

and

where $pa$ is the ambient atmospheric pressure in kPa, $pr=101.325$ kPa is the reference ambient atmospheric pressure, $Ta$ is the ambient atmospheric temperature in K, $Tr=293.15$ K is the reference ambient atmospheric temperature, and $hv$ is the molar concentration of water vapour, presented as a percentage and dependent on the relative humidity (RH). Based on these quantities, the pure-tone sound-attenuation coefficient for atmospheric absorption for a specific frequency *f* is expressed as

### B. Air-absorption coefficient in octave frequency bands

The air-absorption coefficient obtained with Eqs. (11)–(14) is applicable for estimating the effect of the atmospheric conditions on the decay of pure tones only. However, the measured RT values as well as those estimated with the prediction formulas are usually given for full- or third-octave bands. Therefore, a suitable representation of the air-absorption coefficient *m* for the whole band needs to be determined.

A few approaches estimate the effect that air absorption has on sound decay in the frequency bands. One, introduced by Sisler and Bass,^{55} integrates the power spectral density of the signal over third-octave bands. They showed that when the atmospheric absorption is comparable with the boundary absorption, the differences between pure-tone and band RT values are significant.

An approach following a similar reasoning was adapted by Wenmaekers *et al.*^{34} to calculate the effective air absorption in full-octave bands. A special requirement of this method is the RT value without the air absorption, which is unavailable in many scenarios utilizing air-absorption compensation, e.g., when estimating the room acoustic parameters from scale model measurements. The RT would have to be obtained either through measurement or simulation, and this is an additional source of uncertainty in further calculations.

Considering that there is no agreement in the literature on how to use the air-absorption coefficient *m*,^{34,56,57} a simple experiment was conducted to compare the air-absorption coefficient values obtained with available methods. First, the *m* values for all the pure tones within the frequency range of interest (125 Hz–8 kHz) were calculated for $Ta=298.15$ K, RH of 50%, and $pa=101.325$ kPa, the standard pressure. The obtained values are marked in Fig. 1 with a solid black line and serve as the reference for further calculations, for which the atmospheric condition parameters were fixed at the aforementioned values. The outcomes of the rest of the calculations are assumed to fall close to the reference. The values of the center frequencies of each octave band are highlighted, since often they are used as nominal values of *m.*^{56}

The air-absorption coefficients of center frequencies were compared with the values averaged over a number of pure-tone *m*s from the whole considered range. The amount of pure tones used for averaging changed with each band, spanning from 88 for the 125-Hz band to over 5000 for the 8-kHz band. The results of this experiment are presented in Fig. 1. The differences between the air-absorption coefficients of the center frequencies and obtained with averaging are small, with the averaged *m* being between 8.2% and 8.4% higher than for the center frequencies.

## V. RT MEASUREMENTS

This section presents the measurements conducted and equipment used in this study in the variable acoustics laboratory *Arni* at the Acoustics Lab of Aalto University, Espoo, Finland.^{58}

### A. Measurement space and setup

The *Arni* room is rectangular in shape with dimensions 8.9 m × 6.3 m × 3.6 m (length, width, and height, respectively). The walls and the ceiling of the room are covered with variable acoustics panels made from painted metal sheets and filled with absorptive material. On the front of the panels, rectangular slots are cut out from the surface. The slots can be opened, letting the sound reach the absorptive material inside, or closed, making the surface reflective. The dimensions of a single panel are 0.6 m × 0.4 m × 2.4 m (length, width, and height). The absorptive material is 25 cm thick, allowing the closing mechanism to move behind the front surface of the panel. There are in all 55 panels in the variable acoustics laboratory, including eight on three of the walls, 11 on the fourth wall, and 20 on the ceiling. The panels on the three walls are placed directly on the floor, whereas those on the fourth wall are hanging 63 cm above the floor due to the heating installations situated on that wall. The view of *Arni* and the equipment used in the measurements are presented in Fig. 2.

During the measurements, a 01 dB LS01 omnidirectional loudspeaker served as the sound source. There were a few types of receivers used in the procedure: two G.R.A.S. (Holte, Denmark) 1/2-in. diffuse-field microphones of type 40AG, two G.R.A.S. 1/2-in. free-field microphones of type 46AF, and one Brüel & Kjær (Nærum, Denmark) 1/2-in. diffuse-field microphone of type 4192. A G.R.A.S. power module of type 12AG was used as an amplifier. All the equipment was connected to a measurement laptop via a MOTU UltraLite mk3 Audio Interface. The atmospheric data were gathered using a Testo 174H Mini data logger. The positions of the sound source, the receivers, and the atmospheric data logger are marked in Fig. 3.

The measurement signal was a 3-s-long exponential sine sweep^{59,60} with frequency response spanning from 20 Hz to 20 kHz. The sweep was played five times for each panel configuration with 2 s of silence in between to allow the sound to fully decay, achieving signal-to-noise ratios between 40 and 50 dB. The total number of recorded measurement signals for each panel configuration was 25 (5 sweeps × 5 receivers). The atmospheric data were collected once for each measurement of five sweeps.

In total, $K=$ 5312 panel combinations were measured. This included the scenarios in which all panels are absorptive, all panels are reflective (one combination each); one panel is absorptive, one panel is reflective (55 combinations each); and 2–54 panels are absorptive (100 combinations each). The panel state transition implies a gradual shift of the total absorption in the room by a value proportional to the number of absorptive and reflective panels. The database of the measured RIRs is available online.^{61}

### B. Measurement accuracy

Since the measurement procedure was automated—changing of the panel states, outputting the excitation signal, recording the measurement, and writing to a file were executed using a python script—the process was not continuously monitored. Therefore, non-stationary noise events disturbing the measurement were not caught during the process and had to be detected in a post-processing stage. To identify which of the measured sweeps are free from non-stationary disturbances, the procedure called rule of two (Ro2) was employed.^{62} From the signals free of contamination, i.e., those classified by Ro2 as having high values of Pearson's correlation coefficient, one sweep per receiver position in a combination was used later in the result analysis.

When presenting results, the median of the *T*_{60} values across all combinations with *n* absorptive panels for all receiver positions was used as an estimator of the “main” value of a combination's RT. The results also show the trends in the RT change with the decrease in the room's absorption. The median RT was estimated as

where $n=0,1,2,\u2026,55$ is the number of absorptive panels in a combination, and *n _{k}* is the number of absorptive panels for combination

*k*.

## VI. RESULTS

This section presents the results of RT measurements and compares them with the outcome of estimations obtained with the Sabine and Eyring formulas. The distribution of the obtained RT values is discussed, and the air-absorption compensation is performed. Furthermore, the section analyzes the correction that sound-absorption calibration introduces to the predicted RT.

### A. Air-absorption compensation

The RT values were calculated from the measured RIRs using the DecayFitNet software^{63} due to the achieved robustness of the RT predictions in the presence of the background noise. The correctness of decay estimations was assessed using the mean squared error (MSE) between the predictions and measured RIRs. The median MSE for all combinations and six frequency bands was 3.34 dB with a median absolute deviation of 2.01 dB, showing good agreement between the recorded and estimated decays.

DecayFitNet estimates the decay parameters of the analyzed RIRs, such as the initial energy level and the decay rate, together with the background noise level. Thus, it is possible to resynthesize the energy-decay curve extended beyond the noise floor.^{63} Such a procedure was used in the present research; hence, the RT values presented here were calculated as *T*_{60} times, over the 60-dB decay range, between –5 and –65 dB. The results for the octave bands 250 Hz–8 kHz are shown in Figs. 4(a)–4(f).

Due to the enormous number of measurements, they were conducted over a long period of time (approximately 2 weeks), during which the atmospheric conditions varied considerably. Figure 5 shows the change in RH, ambient temperature, and ambient atmospheric pressure during the entire measurement period. The influence of these changes on the predicted RT values was significant, especially in the 4- and 8-kHz bands, as shown in Figs. 4(e) and 4(f), with the values of median RT with air absorption.

The effect of air absorption adds considerable bias to the measurement results, such as introducing drops to the RT values, when increase is expected. For instance, for conditions with zero and one absorptive panels, the RT values in the 4- and 8-kHz bands, denoted with white circles in Figs. 4(e) and 4(f), are lower than when two or three panels were absorptive. During the time these conditions were measured, the temperature was relatively stable, staying at between 19.5 °C and 19.9 °C. Bigger differences were registered in the atmospheric pressure, which dropped from 101.9 kPa for zero and one absorptive panels to between 100.5 and 100.7 kPa for conditions with two and three absorptive panels. The most prominent discrepancies, however, were observed in RH, which for the first two sets of measurements oscillated around 28.5%, whereas the latter two sets of measurements were performed in RH changing from 40% to 50%. To increase the robustness of the result analysis, the air absorption is compensated.

In the present work, the effect of the air absorption is subtracted from the measurement results according to the following formula:

To choose the *m* used in this formula to be either the *m* of the center frequency of the octave band or the result of averaging of pure-tone coefficients as discussed in Sec. IV B, both of those approaches were used on the measurement results. The RT values compensated with the use of averaging formed a smoother curve with more shallow drops than those compensated with the center frequency *m*. Hence, averaging was chosen as the air-absorption-compensation method in the remainder of this paper.

The results compensated for air absorption are displayed in Figs. 4(a)–4(f) and consistently used in the remainder of this paper. The biggest change in RT values after the compensation is observed at high frequencies, such as in the 4- and 8-kHz bands. However, the same effect, although much less significant, is observed already for RTs over 1 s in the 250-Hz and over 0.6 s in 500-Hz band in Figs. 4(a) and 4(b).

The above changes oppose the common approach of considering air absorption only for spaces with volumes over 200 m^{3} (Refs. 6 and 17) (the nominal volume of *Arni* is 201.9 m^{3}, but when the adjustable panels are in their closed state, the volume is considered to decrease by 0.58 m^{3} per panel). Air-absorption compensation is, therefore, advised when analyzing the results of acoustic measurements. It is especially needed when comparing results of measurements done in different atmospheric conditions, e.g., on different days, or comparing measured and simulated RT data.

### B. Distribution of RT values

The measured values show that the change in the placement of absorptive and reflective panels produces a relatively broad distribution of RT values around the median. Because of *Arni*'s dimensions, the Schroeder frequency reaches as high as 200 Hz when the total absorption within the room is low. This results in a low modal density below that frequency, impeding the measurement of the RT values in the 250-Hz band and possibly causing wider spreads around the median as well as outliers. Similarly, the distribution of the absorptive and reflective panels seems to affect the distribution of the measured RT values. Such an effect appears to be more prominent in 250-Hz to 1-kHz bands, as seen in Figs. 4(a)–4(c). For high frequencies, the distribution is narrower, even though some outliers are spotted in the 8-kHz band in Fig. 4(f).

In the 1–8 kHz bands, most of the results in Fig. 4 lie within $\xb110%$ from the median, whereas in the low frequencies, the majority of RT values are inside $\xb125%$ from the median. The exact numbers specifying the spread of values for each band are presented in Table I, confirming wider distributions in low frequencies.

Frequency . | 250 Hz . | 500 Hz . | 1 kHz . | 2 kHz . | 4 kHz . | 8 kHz . |
---|---|---|---|---|---|---|

±10% of $T\u030360,n$ | 65% | 79% | 94% | 99% | 99% | 99% |

±25% of $T\u030360,n$ | 97% | 98% | 99% | 100% | 100% | 100% |

Frequency . | 250 Hz . | 500 Hz . | 1 kHz . | 2 kHz . | 4 kHz . | 8 kHz . |
---|---|---|---|---|---|---|

±10% of $T\u030360,n$ | 65% | 79% | 94% | 99% | 99% | 99% |

±25% of $T\u030360,n$ | 97% | 98% | 99% | 100% | 100% | 100% |

For the majority of combinations, the distribution of RT values cannot be classified as normal. This justifies the choice of median as an estimator (since the mean value is more susceptible to contamination by outliers, it would prove less robust). An example of a probability distribution plot showing the measured values relative to the median for the 1-kHz frequency band and all panel conditions is presented in Fig. 6.

### C. Predicted RT values

In this work, the sound-absorption coefficients used to obtain the average absorptivity $\alpha \xaf$ in Sabine's and Eyring's formulas were taken either from the data provided by the material's manufacturer (in the case of variable acoustic panels and curtains) or from the literature (in the case of wall and floor materials). Their values are presented in Table II.

Material . | 250 Hz . | 500 Hz . | 1 kHz . | 2 kHz . | 4 kHz . | 8 kHz . |
---|---|---|---|---|---|---|

Panel absorptive (Ref. 64) | 0.86 | 0.77 | 0.66 | 0.45 | 0.38 | 0.42 |

Panel reflective (Ref. 64) | 0.09 | 0.05 | 0.05 | 0.04 | 0.02 | 0.03 |

Wall (Ref. 65) | 0.02 | 0.03 | 0.03 | 0.04 | 0.05 | 0.05 |

Floor (Ref. 65) | 0.02 | 0.03 | 0.03 | 0.03 | 0.02 | 0.03 |

Curtain (Ref. 66) | 0.45 | 0.95 | 0.99 | 0.99 | 0.99 | 0.99 |

Material . | 250 Hz . | 500 Hz . | 1 kHz . | 2 kHz . | 4 kHz . | 8 kHz . |
---|---|---|---|---|---|---|

Panel absorptive (Ref. 64) | 0.86 | 0.77 | 0.66 | 0.45 | 0.38 | 0.42 |

Panel reflective (Ref. 64) | 0.09 | 0.05 | 0.05 | 0.04 | 0.02 | 0.03 |

Wall (Ref. 65) | 0.02 | 0.03 | 0.03 | 0.04 | 0.05 | 0.05 |

Floor (Ref. 65) | 0.02 | 0.03 | 0.03 | 0.03 | 0.02 | 0.03 |

Curtain (Ref. 66) | 0.45 | 0.95 | 0.99 | 0.99 | 0.99 | 0.99 |

The results of calculations using table-based $\alpha \xaf$ and $m=0$ in the Sabine and the Eyring formulas are compared with the measured RT values and are displayed in Fig. 7 with blue and red dots, respectively. They show that both formulas predict the reverberation of the room with a considerable error. The best estimations are obtained in the 250-Hz band, where both models follow the median RT closely (Sabine with a slight offset) below the value of approximately 0.8 s. At 500 Hz and 1 kHz, the low *T*_{60} combinations are estimated correctly by the formulas, but the error increases rapidly when the *T*_{60} is high. The predicted RT values for frequencies 2–8 kHz are too low across all combinations in Fig. 7. The average error percentage across all combinations for both formulas in each octave band is shown in Table III, further confirming the prediction inaccuracy.

Frequency . | 250 Hz . | 500 Hz . | 1 kHz . | 2 kHz . | 4 kHz . | 8 kHz . |
---|---|---|---|---|---|---|

Average error Sabine | 7% | 12% | 14% | 22% | 23% | 34% |

Average error Eyring | 6% | 19% | 21% | 29% | 29% | 40% |

Frequency . | 250 Hz . | 500 Hz . | 1 kHz . | 2 kHz . | 4 kHz . | 8 kHz . |
---|---|---|---|---|---|---|

Average error Sabine | 7% | 12% | 14% | 22% | 23% | 34% |

Average error Eyring | 6% | 19% | 21% | 29% | 29% | 40% |

An important observation is that whilst the results of the measurements are distributed around the median value, the estimations are not. The inability of Sabine's and Eyring's formulas to adapt to a changing absorption distribution is an error inherent to those equations, which was already discussed in literature^{40,67} and is not further examined in the present study.

### D. Calibrated sound-absorption coefficient

The calibration of sound-absorption coefficients was performed according to Eqs. (7) and (8) for Sabine's and Eyring's formulas, respectively. A median absorption for combination *k* and all five receivers was used as $\alpha \u0302k$ with *K* = 5312. The base values of absorptivity in *Arni*, $\beta S$ and $\beta E$, were equivalent to the case when all the panels are in their reflective state, whereas the difference in absorption, $\Delta S$ and $\Delta E$, indicated the change of panel state from reflective to absorptive. Similar calculations were conducted for $\beta T$ and $\Delta T$, following the example given below for the 500-Hz band,

The comparison between the values of $\alpha \xaf$ based on Table II, as well as $\alpha \xafc,\u200aS$ and $\alpha \xafc,\u200aE$, is presented in Fig. 8. The differences between $\beta T,\u2009\beta S$, and $\beta E$ are small for the 250-Hz band, but the table-based numbers are significantly higher than the calibrated ones in all the remaining bands. As $\Delta S$ is consistently higher than $\Delta E$, no such relation is observed between $\beta S$ and $\beta E$.

The results of the calibration for all measurements are presented in Fig. 9. Ideally, both measured and predicted RT values would be equal, forming a diagonal line. However, this is impossible since the measured values are influenced by absorption distribution, which is not accounted for either in Sabine's or Eyring's formulas. Therefore, in Fig. 9, the estimated absorption coefficients create vertical lines that assign several values of measured *T*_{60} for each number predicted with the Sabine or Eyring formulas. However, the stepwise nature of calibrated values does not contribute to an excessively wide distribution of the results, mostly fitting within ±10% of the target diagonal.

The exact proportions of measured RT values not exceeding the 10% and 25% limits are presented in Table IV, which shows that the spread is almost the same for both analyzed formulas, with slight advantage in Eyring's model in the 250-Hz and 1-kHz bands. The results of Table IV also align well with the distribution of measured RT values presented in Table I. The percentage of average error of calibrated predictions relative to the measured RT values shows great improvement compared with the error from before the calibration, cf. Table III.

Frequency . | 250 Hz . | 500 Hz . | 1 kHz . | 2 kHz . | 4 kHz . | 8 kHz . |
---|---|---|---|---|---|---|

Sabine ±10% | 55% | 75% | 89% | 98% | 99% | 98% |

Sabine ±25% | 94% | 99% | 99% | 99% | 99% | 99% |

Average error Sabine | 4% | 3% | 2% | 2% | 1% | 2% |

Eyring ±10% | 56% | 75% | 92% | 98% | 99% | 98% |

Eyring ±25% | 95% | 99% | 99% | 99% | 99% | 99% |

Average error Eyring | 3% | 3% | 2% | 2% | 1% | 2% |

Frequency . | 250 Hz . | 500 Hz . | 1 kHz . | 2 kHz . | 4 kHz . | 8 kHz . |
---|---|---|---|---|---|---|

Sabine ±10% | 55% | 75% | 89% | 98% | 99% | 98% |

Sabine ±25% | 94% | 99% | 99% | 99% | 99% | 99% |

Average error Sabine | 4% | 3% | 2% | 2% | 1% | 2% |

Eyring ±10% | 56% | 75% | 92% | 98% | 99% | 98% |

Eyring ±25% | 95% | 99% | 99% | 99% | 99% | 99% |

Average error Eyring | 3% | 3% | 2% | 2% | 1% | 2% |

It is crucial to note that some amount of error in the predictions is expected, as both formulas are claimed to be accurate only in a diffuse sound field (cf. Sec. II B). The sound field within *Arni*, however, is assumed to not be fully diffused. Factors such as unevenly distributed absorption^{68} (even when all the panels are in the same state, other elements of the interior prevent the uniformity of absorptivity distribution) and the shoebox shape of *Arni*^{69} both indicate lack of isotropy and homogeneity.

Considering that the just noticeable difference (JND) for reverberation perception varies depending on the type of sound^{70–74} and the character of decay,^{75} ranging from around 3% for speech signals^{73} to over 20% for band limited noise and music,^{71,74,76} the calibrated predictions fit within those constraints. This precision is crucial, considering that design decisions for concert halls and auditoriums are made based on the RT values estimated with Sabine's or Eyring's formulas.

## VII. DISCUSSION

There are many possible sources of uncertainty in RT predictions. A few of them are mentioned in this paper, namely, the incorrectly assumed sound-absorption coefficient, the error inherent to the analyzed formulas, and neglect of the effect of the air. The results may be biased also by the uncertainty coming from the measurement equipment, the method used to calculate the RT values from obtained RIRs, and the destructive effect of stationary and non-stationary noise and time variance.^{77,78}

To account, at least partially, for the diffuse reflections in the decay, the modified sound-absorption coefficient based on the diffusion model^{79} was also examined in the present study. Due to its not showing a significant improvement in predictions with Eyring's formula, it was not included in Sec. VI for the sake of clarity of presentation. The outcome of this method is well in agreement with the results presented in the literature for the sound-absorption coefficient values under 0.6.^{79}

The present study shows that when the sound-absorption coefficient is chosen carefully, the traditional Sabine and Eyring RT prediction formulas achieve good accuracy. However, the estimation of the sound-absorption coefficient remains challenging. The current method to measure the absorption of materials is based on the Sabine formula,^{25} which may not agree well with multiple-slope decays observed in reverberation chambers.^{29} This approach creates an error buildup when an inaccurately estimated $\alpha \xaf$ is used to calculate the RT in a room not fulfilling the conditions required by the Eyring and Sabine formulas.

Apart from that, standardized measurements require the use of a specific amount, edge-to-area ratio, and arrangement of absorbing material to achieve significant effect on the acoustics of a reverberation chamber, as well as to eliminate the edge effect that skews the results.^{25} Such placement of acoustic materials is, however, rarely used in acoustic adaptations of concert halls, auditoriums, and other specialized facilities. Thus, the ideal $\alpha \xaf$ in the measured space, understood as a value that will return the measured RT when used in one of the prediction formulas, might be, in fact, significantly different from the one resulting from the values obtained in the laboratory. Therefore, the debate over the accuracy of RT formulas will probably be resolved only with an improved procedure to estimate sound absorption in practical settings.

## VIII. CONCLUSIONS

This study analyzes two of the most popular RT estimation formulas—Sabine's and Eyring's equations—and verifies their accuracy in predicting RT values on a big dataset of measured impulse responses. The results show that, even in a scenario where the absorption distribution in the room varies considerably between measurements, both of the aforementioned models predict the $T60(f)$ to within approximately ±10% precision after the proposed calibration.

Both formulas assume that the sound field in a considered room is homogeneous and isotropic, an unachievable requirement for the vast majority of rooms and a source of error in the calculations. The uncertainty in results may also come from using an absorption coefficient that is not equivalent to the actual absorption in the room. The present work shows that the error resulting from incorrectly assumed absorptivity preponderates the inaccuracy coming from the non-diffuse sound field. We show that by using a calibrated sound-absorption coefficient, a sufficiently accurate RT estimation, within the limits of JND for reverberation perception, is achieved.

The results show that Sabine's and Eyring's formulas display good scalability of predictions with the change of total absorptivity in the room. This means that having correct estimations of the base and step value of the sound-absorption coefficient, the equations return accurate results regardless of the amount and distribution of absorption added or subtracted from the initial acoustic conditions. This can be valuable in case of acoustic adaptation, where results of a few measurements or simulations (base absorption and absorption change) offer a possibility to precisely tune the designs to desired RT values without need for extra computations or additional measurements.

Another important issue, many times neglected in the RT estimation, is the absorption of the air. The results presented in this paper show that the air absorption may introduce a significant bias to the measurement results. Therefore, we emphasize that it is necessary to compensate for the air absorption in RT measurements and simulations. The study also proposes a method to establish the air-absorption coefficient for full frequency bands based on the pure-tone calculations used in standards. Compared with other approaches for air-absorption compensation, the proposed method is shown to not introduce an error dependent on the RT values. The procedures proposed in this paper improve the accuracy of RT predictions in room acoustics.

## ACKNOWLEDGMENTS

This work was supported by the Nordic Sound and Music Computing Network—NordicSMC, NordForsk Project No. 86892.