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Prediction of reverberation time using the residual minimization method
Applied Acoustics 106 (2016) 42–50
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Applied Acoustics journal homepage: www.elsevier.com/locate/apacoust
Prediction of reverberation time using the residual minimization method Artur Nowos´wiat ⇑, Marcelina Olechowska, Jan S´lusarek Faculty of Civil Engineering, Silesian University of Technology, Gliwice, Poland
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 9 August 2015 Received in revised form 4 December 2015 Accepted 28 December 2015
Keywords: Reverberation time Prediction of reverberation time Residual minimization method Sabine
In many practical situations the assumption of sound field dispersion needed for the application of the Sabine’s theory is not fulfilled. In general, sound field is sufficiently dispersed if there are no large differences in the dimensions of the room, limiting partitions are not parallel, or the sound absorbing material is uniformly distributed. In practice, very few of these requirements are satisfied. As a result, a number of other formulas describing reverberation time have been created, for example Fitzroy’s or Neubauer’s formulas. However, these methods in many cases differ significantly from the actual measurements. The paper presents a method used to estimate reverberation time as well as its applicability potential involving laboratory models and auditorium rooms. The proposed method can be classified into a group of learning methods and involves the use of statistical methods which allow for approximation with the use of the least squares method. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction
T EYR ¼
In many practical applications we must be able to predict the reverberation time in rooms. In the present paper we propose the estimation method of reverberation time based on the analysis of mathematical statistics. As the initial step in this method, the difference between the actual measurement and the recognized theoretical formulas such as Sabine’s, Eyring’s, Millington’s, Kuttruff’s, Fitzroy’s, Arau’s, Neubaera’s and Pujoll’s formulas should be determined. Sabine described the reverberation phenomenon in the statistical acoustic field theory of the room, and basing on his research results he also provided an empirical formula for the calculation of reverberation time, which has the following form [1]:
T SAB ¼
0:161V S aSAB
½saSAB ¼
n 1X ai Si S i¼1
ð1Þ
where V – volume of the room, S – total internal surface area of the room, aSAB – average sound absorption coefficient, ai – sound absorption coefficient of the i-th partition limiting the room, Si – surface area of this partition. A modified determination method of reverberation time was suggested by Norris [2] and Eyring [3]. Developing the calculation concept of the sound absorption coefficient formulated by Sabine, they introduced a logarithmic dependence for the average coefficient a in the denominator.
⇑ Corresponding author. E-mail address: [email protected] (A. Nowos´wiat). http://dx.doi.org/10.1016/j.apacoust.2015.12.024 0003-682X/Ó 2015 Elsevier Ltd. All rights reserved.
0:161V ½s; S aEYR
aEYR ¼ lnð1 aSAB Þ
ð2Þ
Another formula was presented by Millington [4] and Sette [5]. The provided model differs from the previously described formulas in the applied determination method of the average sound absorption coefficient. Millington suggested calculating the coefficient aMIL as the geometric mean.
T MIL ¼
0:161V ½s; S aMIL
aMIL ¼
n 1X Si ln ð1 ai Þ S i¼1
ð3Þ
Kuttruff on the other hand [6] suggested a statistical distribution of sound, taking into account the Gaussian random variable as well as the Rayleigh’s probability. Basing on the above, he created the definition of the function of mean free path
c2 ¼ ðl2 l2 Þ=l2 as a variation of probability. To calculate c2 , he used the Monte Carlo simulation method. Kuttruff introduced two important changes to the Eyring’s equation. The first involved the shape of the room, while the other involved the distribution of absorbent material. He also introduced a correction used to determine the average sound absorption coefficient, which yielded the following equation:
T KUT ¼
0:161V ½s; S aKUT
aKUT ¼ ln ð1 aSAB Þ 1 þ
c2 2
ln ð1 aSAB Þ
ð4Þ The formulas developed by Fitzroy [7] allow for uneven distribution of sound absorbing materials as well as sound absorbing x – a y – a z ). systems in the room (a
A. Nowos´wiat et al. / Applied Acoustics 106 (2016) 42–50
Sx Sy Sz ¼ Tx þ Ty þ T z; S S S
T FIT
T x ¼ 0:161V ; Sa
where
FIT;x
T y ¼ 0:161V ; Sa FIT;y
½s
ð5Þ
T z ¼ 0:161V ,aFIT;x ¼ lnð1 ax Þ; Sa FIT;z
aFIT;y ¼ lnð1 ay Þ; aFIT;z ¼ lnð1 az Þwhere Sx, Sy, Sz – pairs of opposite surfaces of the walls, ax ; ay ; az – average reverberant sound absorption coefficients of the material on the respective wall pairs. Arau-Puchades [8] suggested an improved equation in which he assumes that the reverberation time of the interior is determined as a geometric weighted average of three reverberation times derived from orthogonal directions (x, y, z). He also assumes that the decay of reverberation time is hyperbolic. The absorption coefficients used in his formula are understood as the mean absorption for each pair of the opposite walls. Simultaneous sound reflections are taking place between these surfaces, and therefore the decay of sound should be considered in three directions. Arau-Puchades determines the reverberation time of the interior in the following way:
T ARAU ¼
0:161V 0:161V 0:161V SaARAU;x SaARAU;y SaARAU;z Sy X
Sx X
Sz X
ð6Þ
where aARAU;x ¼ ln ð1 ax Þ; aARAU;y ¼ lnð1 ay Þ; aARAU;z ¼ lnð1 az Þ. Neubauer and Kostek [9] presented the modification of Fitzroy’s equation, dividing the Kuttruf’s correction section into two parts. One part reflects the floor and ceiling surfaces, while the second part takes into account the impact of the remaining walls.
T NEU ¼
0:45V S2
lw
aCF
þ
hðl þ wÞ ;
aWW
½s
ð7Þ
The Neubauer equation examines the division of acoustic field into two parts, treating the determined absorption coefficients as an adjustment to the Eyring and Kuttruff’s formula:
qCF ðqCF qÞS2CF ; aWW ðqSÞ2 q ðq qÞS2WW ¼ lnð1 aSAB Þ þ WW WW 2 ðqSÞ
aCF ¼ lnð1 aSAB Þ þ
where l, w, h – the length, width and height of the room, aCF – the average effective sound absorption coefficient of ceiling and floor, aWW – the average effective sound absorption coefficient of side partitions, q ¼ 1 a – reflectance coefficient, SCF – the surface of ceiling and floor, SWW – the surface area of side walls. Pujolle [10] proposed another determination method of the mean free path lm ; taking into account the dimensions of the room. He presented two formulas to determine lm: lm ¼
1 6
T PUJ ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 2 2 2 2 2 2 2 2 4 L2 þ l þ L2 þ h þ h þ l and lm ¼ pffiffiffiffi L2 l þ L2 h þ h l :
p
0:04lm
aEYR
½s
ð8Þ
where L, h, l – length, width and height of the room, In many available papers the authors compare their prediction methods of reverberation time with the values obtained by means of most commonly applied formulas. The results of analytical calculations are often compared with computer simulations and actual measurements. The present paper analyzes a method developed by the authors, which can be used to determine the differences in the formulas described above from the actual measurement, with reference to the said formulas and the measurement. Ultimately, the method is based on the search of K(f) correction for the Sabine’s formula. The first reason explaining the choice of
43
the Sabine’s model was its simplicity. If the described below residual minimization method had been applied for more complex models, they would probably have been very difficult to apply. The same approach was employed with respect to the application of perturbation methods and the new algebra of perturbation numbers [11] for the estimation of reverberation time, where the Sabine’s formula was used. Another reason involved its common application. In the work [12] the usability of Sabine’s formula was tested for rooms having complicated shapes. The authors of the work [13] analyzed the calculation methods of reverberation time is spacious rooms (atrium). They found in effect of their research studies that the values of reverberation time obtained on the basis of Sabine’s formula were comparable to the average value obtained from the measurement carried out for four receivers. Another example where the Sabine’s model is commonly applied is the standard [14] and the analysis of this model by Prof. Gerretsen [15]. 2. Residual minimization method MMR The proposed method depends on the choice of rooms to be investigated. We do not want to suggest here any specific criteria for such a selection, since at the present stage of works it would be premature. However, we can suggest that a preliminary classification should be applied for the procedure presented below:
Auditorium rooms Classrooms Sacral rooms Etc.
Additionally, for the analysis of rooms from a particular classification, rooms having similar shapes should be selected. The selected rooms can have different coefficient of sound absorption, yet we suggest a separate investigation for rooms with poor soundproofing a < 0:2 and a separate one for well soundproofed rooms a > 0:2. For each room, the measurements and calculations are carried out with the use of commonly accepted theoretical methods. Then, the minimum difference is determined from among the differences R1 ¼ T p T Sab ; R2 ¼ T p T Eyr ; . . ., Rn ¼ T p T sym (referred to as residues), where Tp – measured reverberation time, TSab, TEyr, . . ., Tsym – reverberation times calculated with the use of the theoretical methods described in Chapter I and using computer simulation. The applied theoretical methods to a different extent allow for a complicated geometry of the room, non-uniform distribution of sound-absorbing materials, etc. Therefore, the differences R were being determined between the measurement and each of the theoretical equations described in Chapter 1, whereby the lowest difference could be found. For different rooms, the minimum difference can be determined by means of a different theoretical model. Such a minimal difference (hence the name of the method) is applied in point 1 of the method described below. The correcting element found in this way can be applied in the Sabine’s method. The following definitions are applied: Definition of reverberation time function. We define the function of reverberation time on the set F ¼ f125; 250; 500; 1000; 2000; 4000g; T : F ! Rþ , where the point Tðf Þ 2 Rþ is assigned to any point f 2 F.
Definition of correction function. We define the function K : F ! Rþ on the set F ¼ f125; 250; 500; 1000; 2000; 4000g, where the point K : F ! Rþ is assigned to any point f 2 F.
A. Nowos´wiat et al. / Applied Acoustics 106 (2016) 42–50
44
Definition of product function. The product of the functions K and Tsab having numerical values and common domain F is defined as a function whose value at each point f of the set F is equal to the product of the functions K and Tsab at this point, i.e. the function defined by the formula T Sabine sk ðf Þ ¼ Kðf Þ T Sab ðf Þ for f 2 F: Definition of quotient function. The quotient of the functions Tsr, and Tt having numerical values and common domain F is defined as a function whose value at each point f of the set F is equal to the quotient of the functions Tsr,Sab and Tt at this point. Algorithm of the MMR method
Sab
1. Using the least squares method, we determined the polynomial T t ðf Þ approximating the points ðf k ; Tðf k ÞÞ which were defined in line with the following procedure. (a) If the whole set of residues is positive or negative for the frequency fk, then Tðf k Þ is calculated by means of this theoretical method, which corresponds to the residue
Rmin ¼ minfjR1 j; jR2 j; . . . ; jRn jg (b) If there are residues of different characters for the frequency fk, then T
ðf ÞþT
ðf Þ
Tðf k Þ ¼ theoretical;A k 2 theoretical;B k : The reverberation time T theoretical;A ðf k Þ is calculated using the theoretical method corresponding to the residue Rmin specified as the smallest value among all positive residues, whereas T theoretical;B ðf k Þ is calculated using the theoretical method corresponding to the residue specified as the largest value among all negative residues.
built. The model tests were carried out using two variants. In the first variant, the rooms diffused the sound equally. In the second variant, the sound absorbing elements were fixed locally (mineral wool 10 cm thick), disturbing in that way the uniformity of sound dispersion. Fig. 1a–e present the rooms pertaining to the said variants with the indicated absorbers used for the second variant. In all the work the size of the room has been determined by means of x_y_z (width–length–height). The models of rooms were built from 22 mm thick OSB panels of the dimensions 1,25 2,5 m. The rooms were small, simpleshaped and characterized by a changeable surface limiting the interior. In the second variant, we assumed that 13% of the surface limiting the interior would be covered with mineral wool plates (product of the density of 80 kg/m3 and 10 cm thick) arranged symmetrically on the walls of the test room (Fig. 1a–e). The values of sound absorption coefficients a of the materials used in the laboratory tests are presented in Table 1. The studies on reverberation time were conducted at six measuring points located inside the room. The measurements were carried out at points 1–6 located 0.35 m away from the walls and placed at the distance of 0.6 or 1.2 m from the floor of the room. The omnidirectional sound source was located in the middle of the room at the height of 0.8 m from the floor of the interior. The measurements of reverberation time were carried out using the method of intermittent noise. The interior was excited by means of broadband noise. We obtained approximately pink spectrum of the steady-state reverberant sound for the range covering 1/3rd octave bands of the center frequencies of 50–5000 Hz. In order to obtain high accuracy results and to minimize the effect
2. We defined the points that would be used to determine the correction function K.
T t ð125Þ T t ð250Þ T t ð500Þ ; K 250 ¼ ; K 500 ¼ ; T sr;Sab ð125Þ T sr;Sab ð250Þ T sr;Sab ð500Þ T t ð1000Þ T t ð2000Þ T t ð4000Þ ; K 2000 ¼ ; K 4000 ¼ : K 1000 ¼ T sr;Sab ð1000Þ T sr;Sab ð2000Þ T sr;Sab ð4000Þ
K 125 ¼
where T sr;Sab ðf k Þ is the average Sabine’s reverberation time for the frequencies fk determined in all premises/rooms. The rooms should be of similar shape, similar limiting surfaces and volume. The more rooms we analyze the more accurate results we obtain. T t ð125Þ, . . ., T t ð4000Þ are the values determined in point 1 of the function T t ðf Þ. 3. Using the least squares method, we determined the corrective function Kðf Þ. The obtained function is a polynomial approximating the points specified in point 2. 4. We determined, as a frequency function, the corrected Sabine’s reverberation time described by the following equation T Sabine sk ðf Þ ¼ Kðf Þ T sab ðf Þ, where T Sabine sk ðf Þ – adjusted Sabine’s reverberation time as a frequency function. T sab ðf Þ – Sabine’s reverberation time as a frequency function. Kðf Þ – corrective function determined in point 3. 3. Experimental procedures 3.1. Laboratory research The experimental studies were carried out in a laboratory where the models of rectangular rooms of changeable size were
Fig. 1. Location of soundproofing material in model spaces/rooms. 1 presents the room of the dimensions 1a – 250_250_125 cm, 1b – 375_250_125 cm, 1c – 500_250_125 cm, 1d – 250_250_250 cm, 1e – 500_250_250 cm.
A. Nowos´wiat et al. / Applied Acoustics 106 (2016) 42–50 Table 1 Summary of sound absorption coefficients a of the materials used in the laboratory research. Type of material
OSB plates Mineral wool panel
Sound absorption coefficient a Frequency (Hz) 125
250
500
1000
2000
4000
0.07 0.75
0.07 0.95
0.12 0.95
0.11 0.90
0.12 0.90
0.15 0.75
45
repeated three times at each point. The Figs. 2–4 show the test rooms. The acoustic computer simulations of the investigated phenomenon were carried out using the ODEON software.
4. Results 4.1. Analysis
of random excitation signal, the measurement was repeated four times at each point. The computer simulation of room acoustics was carried out using the program ODEON. In order to determine the reverberation time of the room, a geometric model of the interior was created and relevant material data were assigned to all surfaces. The sound absorption coefficients a of the applied materials were previously determined as presented in Table 1. The simulation was performed for the same settings of sound source and measuring points as in the actual room. 3.2. Testing eeal auditorium rooms The research studies involved the auditorium halls of the Silesian University of Technology, Faculty of Civil Engineering. The building was made as a monolithic frame construction. The interior finishing of walls and ceilings made of cement and lime plasters is covered with paint. The floor is made of terrazzo. The ceilings in the auditoriums are made of a plate-beam system based on a framework. Curtain walls are probably constructed of hollow brick. The assembly halls have an amphitheater seating arrangement for listeners (wooden benches). The smallest room subjected to tests was the auditorium I with the capacity of about 300 m3. The volume of the auditorium III was about 840 m3 and that of the auditorium II – 423 m3. The measurement of reverberation time in all tested rooms was carried out according to the precise method described in the ISO Standard (ISO 3382-2:2010). Using the precise method, two sound source positions and eight microphone positions. The measurement was
The analysis was to demonstrate the applicability of the MMR method. The graphs present the Sabine’s reverberation time corrected by the MMR method, the measurement of reverberation time, reverberation time determined with the Sabine’s method and the reverberation time calculated with the ODEON software. The analysis involved five variants of the room without sound absorbing material and 5 variants with the application of sound absorbing material (Fig. 1). Moreover, the analysis for three lecture theaters was carried out. First, we presented the results involving the rooms shown in Fig. 1a–e in which no absorbing elements were used. Then, the reverberation time was calculated for each room, using Sabin’s, Eyring’s, Millington’s, Kuttruff’s, Fitzroy’s, Arau’s, Neubaera’s and Pujolle’s methods as well as the standard ISO and Odeon. Then, the differences R ¼ T p T m were determined, where T m was the reverberation time calculated by means of the theoretical methods. Basing on the above, the remaining functions were determined using the MMR method. The values of the functions are presented in Table 2. In Table 2 an approximating polynomial Tðf Þ assumes for the frequency 125–4000 Hz the same values as the values of the function Tðf Þ at these points. This is because the polynomial which passes exactly the same points as Tðf Þ has been determined by means of the method of least squares. The final results for the investigated cases are illustrated on the graphs in Fig. 5. The next experiments involved the rooms in which sound absorbing materials were applied. The installation method as well as the amount of mineral wool panels are presented in Fig. 1. As before, the differences R ¼ T p T m were determined.
Fig. 2. Distribution of measurement points in the auditorium/halls (I).
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A. Nowos´wiat et al. / Applied Acoustics 106 (2016) 42–50
Fig. 3. Distribution of measurement points in the auditorium/halls (II).
Fig. 4. Distribution of measurement points in the auditorium/halls (III).
Then, the remaining functions were determined using the MMR method, and their values are shown in Table 3. The final results for the investigated cases are illustrated on the graphs in Fig. 6. At the end of the series of experiments, measurements were carried out in three auditoriums having similar geometry, similar average sound absorption but significantly different cubic volume. The results, as in the case of the laboratory measurements, are summarized in Table 4 and illustrated in the graphs 7. The final results for the considered cases are illustrated on the graphs below (see Fig. 7).
Since the presented method is correcting the Sabine’s theoretical model as to the uncertainties resulting from the room’s geometry or from non-uniform distribution of the acoustic field, it was possible, therefore, to examine during the verification of the results the same rooms as during the determination process of K(f) function. We can also assume that the determined K(f) function will be satisfactorily correcting the Sabine’s model for another room of the geometry similar to that in the investigated rooms and of the cubature lower than the largest investigated room and higher than the smallest room. Such an example will be presented in the following Chapter involving the discussion of results.
A. Nowos´wiat et al. / Applied Acoustics 106 (2016) 42–50 Table 2 The values of the functions used in the MMR method for rooms without the absorbing materials.
Table 3 The values of the functions used in the MMR method for rooms with the absorbing materials.
Frequency
Tðf Þ T t ðf Þ T sr;Sab ðf Þ Kðf Þ
47
Frequency
125
250
500
1000
2000
4000
0.906 0.906 0.86 1.0535
1.149 1.149 0.87 1.3169
0.696 0.696 0.52 1.3641
0.732 0.732 0.55 1.3163
0.646 0.646 0.53 1.2947
0.514 0.514 0.41 1.4205
Tðf Þ T t ðf Þ T sr;Sab ðf Þ Kðf Þ
125
250
500
1000
2000
4000
0.517 0.518 0.39 1.325
0.317 0.319 0.33 0.9865
0.238 0.254 0.27 0.9027
0.221 0.250 0.29 0.9117
0.251 0.257 0.28 0.9163
0.215 0.226 0.27 0.8841
Fig. 5. Experimental and analytical predictions of reverberation time. The presented results involve the premises/rooms of a diffused sound field of the dimensions L_W_H (in centimeters), where L – length, W – width, H – height of the room. The results are given as a function of frequency for the octave within the range 125 Hz–4 kHz.
A. Nowos´wiat et al. / Applied Acoustics 106 (2016) 42–50
48
Fig. 6. Experimental and analytical predictions of reverberation time. The presented results involve the premises/rooms of a disturbed sound field of the dimensions L_W_H (in centimeters), where L – length, W – width, H – height of the room. The results are given as a function of frequency for the octave within the range 125 Hz–4 kHz.
Table 4 The values of the functions used in the MMR method for the tested auditoriums/halls. Frequency
Tðf Þ T t ðf Þ T sr;Sab ðf Þ Kðf Þ
125
250
500
1000
2000
4000
2.40 2.39 3.54 0.68
2.08 2.08 3.51 0.57
2.01 2.06 3.75 0.58
2.21 2.12 3.24 0.64
1.99 2.05 3.00 0.68
1.65 1.62 2.5 0.65
5. Discussion The present paper involves the estimation of the reverberation time with the use of the elaborated MMR method. Fig. 5 shows the reverberation time as a frequency function for the Sabine’s
method, the Sabine’s methods corrected by the MMR method, Odeon simulation and measurement results. We can observe that with respect to the rooms with the diffused sound field, the described here estimation method of reverberation time corrects the already known formulas. We can therefore obtain the reverberation time as a frequency function close to the measurement results. The estimation inaccuracy may be due to the measurement inaccuracy for such small spaces/rooms. Fig. 6 shows the reverberation time as a frequency function for the rooms with the applied absorbing material (see Fig. 1). Also in this case, the residual minimization method proves to be quite efficient. Actually, in all cases the Sabine_sk chart is similar to the measurement and not worse than the results of the simulation performed with the use of ODEON software. The method was finally verified by the measurements carried out in
A. Nowos´wiat et al. / Applied Acoustics 106 (2016) 42–50
49
Fig. 8. Experimental and analytical predictions of reverberation time.
Fig. 7. Experimental and analytical predictions of reverberation time. The presented results involve the three assembly halls of various size I – 300 m3, II – 423 m3, III – 840 m3. The results are given as a function of frequency for the octave within the range 125 Hz–4 kHz.
three auditoriums. Also in that case the results could be generally regarded as satisfactory. However, it should be emphasized that they are not as clear as in the laboratory tests. It results from the fact that the method is actually the correction of Sabine’s method, and therefore it is burdened with similar errors. Such errors as the insufficient knowledge involving the exact values of sound absorption coefficients for room insulation materials, uneven distribution of the sound field have impact on the final results. Another estimation error of reverberation time in the auditoriums/halls involved the choice of the rooms in terms of their size and cubature, which varied significantly. The smallest room volume was 300 m3 and the biggest one was 840 m3. It is worth noting, however, that even
with such a selection, the results were evaluated fairly well. We can take for granted that the selection of premises/rooms of a similar size will yield results close to the measurements. To confirm this hypothesis, another auditorium room of the Civil Engineering Department was examined which is the same in terms of cubature and shape as the auditorium room III. The results are presented in Fig. 8 The differences between the Sabine’s model and the measurement are effected principally by the lack of precise data involving sound absorbing parameters of the partition walls limiting the room (tabular values for plaster and flooring were accepted), but still the corrected model yields interesting results. We can speculate here that if the sound absorption parameters had been known, the corrected results would have been even more satisfying. The reverberation time algorithm for MMR will rank among learning processes, which improves its performance basing on the past experience. The method is based on statistical learning where more measured objects are being added. It is important to ensure in this method that for a learning process we acquire the knowledge of the experimental data involving so called ‘‘similar” objects i.e. having a similar shape, volume, and dispersion of sound in the room. Learning, to paraphrase Herbert Simon’s idea dated 1983, is seen as a change in the adjusted formula enabling in the future the determination of more accurate results (similar to the measurement), acting on objects of a similar nature. The MMR method as a learning process is used because the existing models of reverberation time are not very reliable. It may be noted that if the new measurements of reverberation time are acquired in the future, the whole algorithm can be repeated again, thereby acquiring the correcting function K, which will provide even greater approximation to the actual measurement. The described method can be applied principally to correct inaccuracies involving the estimation of reverberation time by means of Sabine’s model effected by: non-uniform distribution of sound absorbing panels, the influence of room geometry on the difference between the measurement and the calculations, or the lack of knowledge of the acoustic absorption of sound absorbing materials. The present work does not terminate the investigation studies on the subject. Works are being carried out currently on the application of perturbation numbers (new algebra) to assess the approximation uncertainty of reverberation time.
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References [1] Sabine WC. Collected papers on acoustics. Cambridge (MA): Harvard University Press; 1922. [2] Norris RF, Andree CA. An instrumental method of reverberation measurement. J. Acoust. Soc. Am. 1930;1(3):366–72. [3] Eyring CF. Reverberation time in ‘‘dead” rooms. J. Acoust. Soc. Am. 1930;1: 217–41. [4] Millington G. A modified formula for reverberation. J. Acoust. Soc. Am. 1932;4: 69–82. [5] Sette WJ. A new reverberation time formula. J. Acoust. Soc. Am. 1933;4:193–210. [6] Kuttruff H. Room acoustics. 5th ed. London: Spon Press; 2009. ISBN 10: 0-20387637-7. [7] Fitzroy D. Reverberation formula which seems to be more accurate with nonuniform distribution of absorption. J. Acoust. Soc. Am. 1959;31:893–7. [8] Arau-Puchades H. An improved reverberation formula. Acustica 1988;65: 163–80.
[9] Neubauer RO, Kostek B. Prediction of the reverberation time in rectangular rooms with non-uniformly distributed sound absorption. Arch. Acoust. 2000;26(3):183–201. [10] Pujolle J. Nouvelle formule pour la durée de réverbération (New formula for the length of time of reverberation). Rev d’Acoust 1975;19:107–13. [11] Skrzypczyk J. Perturbation methods for acoustic systems with interval parameters. Arch. Acoust. 2008;33(4 supplement):165–70. [12] Iordache V, Catalina T, Cucu B-M. Experimental investigation of the reverberation time inside a complex geometry indoor space. Roman. J. Acoust. Vibr. 2013;X(2):109–14. [13] Kang J, Yap PL, Meng Y, Chen B. Acoustics in large atrium spaces. In: 14th International Congress on Sound and Vibration; 2007. [14] EN 12354-6. Building acoustics – estimation of acoustic performance of buildings from the performance of products – Part 6: Sound absorption in enclosed spaces; 2003. [15] Gerretsen E. Measurement and prediction in building acoustics: A good match in standardization. Conference on Noise Control; 2004, Gdynia.
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Applied Acoustics journal homepage: www.elsevier.com/locate/apacoust
Prediction of reverberation time using the residual minimization method Artur Nowos´wiat ⇑, Marcelina Olechowska, Jan S´lusarek Faculty of Civil Engineering, Silesian University of Technology, Gliwice, Poland
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 9 August 2015 Received in revised form 4 December 2015 Accepted 28 December 2015
Keywords: Reverberation time Prediction of reverberation time Residual minimization method Sabine
In many practical situations the assumption of sound field dispersion needed for the application of the Sabine’s theory is not fulfilled. In general, sound field is sufficiently dispersed if there are no large differences in the dimensions of the room, limiting partitions are not parallel, or the sound absorbing material is uniformly distributed. In practice, very few of these requirements are satisfied. As a result, a number of other formulas describing reverberation time have been created, for example Fitzroy’s or Neubauer’s formulas. However, these methods in many cases differ significantly from the actual measurements. The paper presents a method used to estimate reverberation time as well as its applicability potential involving laboratory models and auditorium rooms. The proposed method can be classified into a group of learning methods and involves the use of statistical methods which allow for approximation with the use of the least squares method. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction
T EYR ¼
In many practical applications we must be able to predict the reverberation time in rooms. In the present paper we propose the estimation method of reverberation time based on the analysis of mathematical statistics. As the initial step in this method, the difference between the actual measurement and the recognized theoretical formulas such as Sabine’s, Eyring’s, Millington’s, Kuttruff’s, Fitzroy’s, Arau’s, Neubaera’s and Pujoll’s formulas should be determined. Sabine described the reverberation phenomenon in the statistical acoustic field theory of the room, and basing on his research results he also provided an empirical formula for the calculation of reverberation time, which has the following form [1]:
T SAB ¼
0:161V S aSAB
½saSAB ¼
n 1X ai Si S i¼1
ð1Þ
where V – volume of the room, S – total internal surface area of the room, aSAB – average sound absorption coefficient, ai – sound absorption coefficient of the i-th partition limiting the room, Si – surface area of this partition. A modified determination method of reverberation time was suggested by Norris [2] and Eyring [3]. Developing the calculation concept of the sound absorption coefficient formulated by Sabine, they introduced a logarithmic dependence for the average coefficient a in the denominator.
⇑ Corresponding author. E-mail address: [email protected] (A. Nowos´wiat). http://dx.doi.org/10.1016/j.apacoust.2015.12.024 0003-682X/Ó 2015 Elsevier Ltd. All rights reserved.
0:161V ½s; S aEYR
aEYR ¼ lnð1 aSAB Þ
ð2Þ
Another formula was presented by Millington [4] and Sette [5]. The provided model differs from the previously described formulas in the applied determination method of the average sound absorption coefficient. Millington suggested calculating the coefficient aMIL as the geometric mean.
T MIL ¼
0:161V ½s; S aMIL
aMIL ¼
n 1X Si ln ð1 ai Þ S i¼1
ð3Þ
Kuttruff on the other hand [6] suggested a statistical distribution of sound, taking into account the Gaussian random variable as well as the Rayleigh’s probability. Basing on the above, he created the definition of the function of mean free path
c2 ¼ ðl2 l2 Þ=l2 as a variation of probability. To calculate c2 , he used the Monte Carlo simulation method. Kuttruff introduced two important changes to the Eyring’s equation. The first involved the shape of the room, while the other involved the distribution of absorbent material. He also introduced a correction used to determine the average sound absorption coefficient, which yielded the following equation:
T KUT ¼
0:161V ½s; S aKUT
aKUT ¼ ln ð1 aSAB Þ 1 þ
c2 2
ln ð1 aSAB Þ
ð4Þ The formulas developed by Fitzroy [7] allow for uneven distribution of sound absorbing materials as well as sound absorbing x – a y – a z ). systems in the room (a
A. Nowos´wiat et al. / Applied Acoustics 106 (2016) 42–50
Sx Sy Sz ¼ Tx þ Ty þ T z; S S S
T FIT
T x ¼ 0:161V ; Sa
where
FIT;x
T y ¼ 0:161V ; Sa FIT;y
½s
ð5Þ
T z ¼ 0:161V ,aFIT;x ¼ lnð1 ax Þ; Sa FIT;z
aFIT;y ¼ lnð1 ay Þ; aFIT;z ¼ lnð1 az Þwhere Sx, Sy, Sz – pairs of opposite surfaces of the walls, ax ; ay ; az – average reverberant sound absorption coefficients of the material on the respective wall pairs. Arau-Puchades [8] suggested an improved equation in which he assumes that the reverberation time of the interior is determined as a geometric weighted average of three reverberation times derived from orthogonal directions (x, y, z). He also assumes that the decay of reverberation time is hyperbolic. The absorption coefficients used in his formula are understood as the mean absorption for each pair of the opposite walls. Simultaneous sound reflections are taking place between these surfaces, and therefore the decay of sound should be considered in three directions. Arau-Puchades determines the reverberation time of the interior in the following way:
T ARAU ¼
0:161V 0:161V 0:161V SaARAU;x SaARAU;y SaARAU;z Sy X
Sx X
Sz X
ð6Þ
where aARAU;x ¼ ln ð1 ax Þ; aARAU;y ¼ lnð1 ay Þ; aARAU;z ¼ lnð1 az Þ. Neubauer and Kostek [9] presented the modification of Fitzroy’s equation, dividing the Kuttruf’s correction section into two parts. One part reflects the floor and ceiling surfaces, while the second part takes into account the impact of the remaining walls.
T NEU ¼
0:45V S2
lw
aCF
þ
hðl þ wÞ ;
aWW
½s
ð7Þ
The Neubauer equation examines the division of acoustic field into two parts, treating the determined absorption coefficients as an adjustment to the Eyring and Kuttruff’s formula:
qCF ðqCF qÞS2CF ; aWW ðqSÞ2 q ðq qÞS2WW ¼ lnð1 aSAB Þ þ WW WW 2 ðqSÞ
aCF ¼ lnð1 aSAB Þ þ
where l, w, h – the length, width and height of the room, aCF – the average effective sound absorption coefficient of ceiling and floor, aWW – the average effective sound absorption coefficient of side partitions, q ¼ 1 a – reflectance coefficient, SCF – the surface of ceiling and floor, SWW – the surface area of side walls. Pujolle [10] proposed another determination method of the mean free path lm ; taking into account the dimensions of the room. He presented two formulas to determine lm: lm ¼
1 6
T PUJ ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 2 2 2 2 2 2 2 2 4 L2 þ l þ L2 þ h þ h þ l and lm ¼ pffiffiffiffi L2 l þ L2 h þ h l :
p
0:04lm
aEYR
½s
ð8Þ
where L, h, l – length, width and height of the room, In many available papers the authors compare their prediction methods of reverberation time with the values obtained by means of most commonly applied formulas. The results of analytical calculations are often compared with computer simulations and actual measurements. The present paper analyzes a method developed by the authors, which can be used to determine the differences in the formulas described above from the actual measurement, with reference to the said formulas and the measurement. Ultimately, the method is based on the search of K(f) correction for the Sabine’s formula. The first reason explaining the choice of
43
the Sabine’s model was its simplicity. If the described below residual minimization method had been applied for more complex models, they would probably have been very difficult to apply. The same approach was employed with respect to the application of perturbation methods and the new algebra of perturbation numbers [11] for the estimation of reverberation time, where the Sabine’s formula was used. Another reason involved its common application. In the work [12] the usability of Sabine’s formula was tested for rooms having complicated shapes. The authors of the work [13] analyzed the calculation methods of reverberation time is spacious rooms (atrium). They found in effect of their research studies that the values of reverberation time obtained on the basis of Sabine’s formula were comparable to the average value obtained from the measurement carried out for four receivers. Another example where the Sabine’s model is commonly applied is the standard [14] and the analysis of this model by Prof. Gerretsen [15]. 2. Residual minimization method MMR The proposed method depends on the choice of rooms to be investigated. We do not want to suggest here any specific criteria for such a selection, since at the present stage of works it would be premature. However, we can suggest that a preliminary classification should be applied for the procedure presented below:
Auditorium rooms Classrooms Sacral rooms Etc.
Additionally, for the analysis of rooms from a particular classification, rooms having similar shapes should be selected. The selected rooms can have different coefficient of sound absorption, yet we suggest a separate investigation for rooms with poor soundproofing a < 0:2 and a separate one for well soundproofed rooms a > 0:2. For each room, the measurements and calculations are carried out with the use of commonly accepted theoretical methods. Then, the minimum difference is determined from among the differences R1 ¼ T p T Sab ; R2 ¼ T p T Eyr ; . . ., Rn ¼ T p T sym (referred to as residues), where Tp – measured reverberation time, TSab, TEyr, . . ., Tsym – reverberation times calculated with the use of the theoretical methods described in Chapter I and using computer simulation. The applied theoretical methods to a different extent allow for a complicated geometry of the room, non-uniform distribution of sound-absorbing materials, etc. Therefore, the differences R were being determined between the measurement and each of the theoretical equations described in Chapter 1, whereby the lowest difference could be found. For different rooms, the minimum difference can be determined by means of a different theoretical model. Such a minimal difference (hence the name of the method) is applied in point 1 of the method described below. The correcting element found in this way can be applied in the Sabine’s method. The following definitions are applied: Definition of reverberation time function. We define the function of reverberation time on the set F ¼ f125; 250; 500; 1000; 2000; 4000g; T : F ! Rþ , where the point Tðf Þ 2 Rþ is assigned to any point f 2 F.
Definition of correction function. We define the function K : F ! Rþ on the set F ¼ f125; 250; 500; 1000; 2000; 4000g, where the point K : F ! Rþ is assigned to any point f 2 F.
A. Nowos´wiat et al. / Applied Acoustics 106 (2016) 42–50
44
Definition of product function. The product of the functions K and Tsab having numerical values and common domain F is defined as a function whose value at each point f of the set F is equal to the product of the functions K and Tsab at this point, i.e. the function defined by the formula T Sabine sk ðf Þ ¼ Kðf Þ T Sab ðf Þ for f 2 F: Definition of quotient function. The quotient of the functions Tsr, and Tt having numerical values and common domain F is defined as a function whose value at each point f of the set F is equal to the quotient of the functions Tsr,Sab and Tt at this point. Algorithm of the MMR method
Sab
1. Using the least squares method, we determined the polynomial T t ðf Þ approximating the points ðf k ; Tðf k ÞÞ which were defined in line with the following procedure. (a) If the whole set of residues is positive or negative for the frequency fk, then Tðf k Þ is calculated by means of this theoretical method, which corresponds to the residue
Rmin ¼ minfjR1 j; jR2 j; . . . ; jRn jg (b) If there are residues of different characters for the frequency fk, then T
ðf ÞþT
ðf Þ
Tðf k Þ ¼ theoretical;A k 2 theoretical;B k : The reverberation time T theoretical;A ðf k Þ is calculated using the theoretical method corresponding to the residue Rmin specified as the smallest value among all positive residues, whereas T theoretical;B ðf k Þ is calculated using the theoretical method corresponding to the residue specified as the largest value among all negative residues.
built. The model tests were carried out using two variants. In the first variant, the rooms diffused the sound equally. In the second variant, the sound absorbing elements were fixed locally (mineral wool 10 cm thick), disturbing in that way the uniformity of sound dispersion. Fig. 1a–e present the rooms pertaining to the said variants with the indicated absorbers used for the second variant. In all the work the size of the room has been determined by means of x_y_z (width–length–height). The models of rooms were built from 22 mm thick OSB panels of the dimensions 1,25 2,5 m. The rooms were small, simpleshaped and characterized by a changeable surface limiting the interior. In the second variant, we assumed that 13% of the surface limiting the interior would be covered with mineral wool plates (product of the density of 80 kg/m3 and 10 cm thick) arranged symmetrically on the walls of the test room (Fig. 1a–e). The values of sound absorption coefficients a of the materials used in the laboratory tests are presented in Table 1. The studies on reverberation time were conducted at six measuring points located inside the room. The measurements were carried out at points 1–6 located 0.35 m away from the walls and placed at the distance of 0.6 or 1.2 m from the floor of the room. The omnidirectional sound source was located in the middle of the room at the height of 0.8 m from the floor of the interior. The measurements of reverberation time were carried out using the method of intermittent noise. The interior was excited by means of broadband noise. We obtained approximately pink spectrum of the steady-state reverberant sound for the range covering 1/3rd octave bands of the center frequencies of 50–5000 Hz. In order to obtain high accuracy results and to minimize the effect
2. We defined the points that would be used to determine the correction function K.
T t ð125Þ T t ð250Þ T t ð500Þ ; K 250 ¼ ; K 500 ¼ ; T sr;Sab ð125Þ T sr;Sab ð250Þ T sr;Sab ð500Þ T t ð1000Þ T t ð2000Þ T t ð4000Þ ; K 2000 ¼ ; K 4000 ¼ : K 1000 ¼ T sr;Sab ð1000Þ T sr;Sab ð2000Þ T sr;Sab ð4000Þ
K 125 ¼
where T sr;Sab ðf k Þ is the average Sabine’s reverberation time for the frequencies fk determined in all premises/rooms. The rooms should be of similar shape, similar limiting surfaces and volume. The more rooms we analyze the more accurate results we obtain. T t ð125Þ, . . ., T t ð4000Þ are the values determined in point 1 of the function T t ðf Þ. 3. Using the least squares method, we determined the corrective function Kðf Þ. The obtained function is a polynomial approximating the points specified in point 2. 4. We determined, as a frequency function, the corrected Sabine’s reverberation time described by the following equation T Sabine sk ðf Þ ¼ Kðf Þ T sab ðf Þ, where T Sabine sk ðf Þ – adjusted Sabine’s reverberation time as a frequency function. T sab ðf Þ – Sabine’s reverberation time as a frequency function. Kðf Þ – corrective function determined in point 3. 3. Experimental procedures 3.1. Laboratory research The experimental studies were carried out in a laboratory where the models of rectangular rooms of changeable size were
Fig. 1. Location of soundproofing material in model spaces/rooms. 1 presents the room of the dimensions 1a – 250_250_125 cm, 1b – 375_250_125 cm, 1c – 500_250_125 cm, 1d – 250_250_250 cm, 1e – 500_250_250 cm.
A. Nowos´wiat et al. / Applied Acoustics 106 (2016) 42–50 Table 1 Summary of sound absorption coefficients a of the materials used in the laboratory research. Type of material
OSB plates Mineral wool panel
Sound absorption coefficient a Frequency (Hz) 125
250
500
1000
2000
4000
0.07 0.75
0.07 0.95
0.12 0.95
0.11 0.90
0.12 0.90
0.15 0.75
45
repeated three times at each point. The Figs. 2–4 show the test rooms. The acoustic computer simulations of the investigated phenomenon were carried out using the ODEON software.
4. Results 4.1. Analysis
of random excitation signal, the measurement was repeated four times at each point. The computer simulation of room acoustics was carried out using the program ODEON. In order to determine the reverberation time of the room, a geometric model of the interior was created and relevant material data were assigned to all surfaces. The sound absorption coefficients a of the applied materials were previously determined as presented in Table 1. The simulation was performed for the same settings of sound source and measuring points as in the actual room. 3.2. Testing eeal auditorium rooms The research studies involved the auditorium halls of the Silesian University of Technology, Faculty of Civil Engineering. The building was made as a monolithic frame construction. The interior finishing of walls and ceilings made of cement and lime plasters is covered with paint. The floor is made of terrazzo. The ceilings in the auditoriums are made of a plate-beam system based on a framework. Curtain walls are probably constructed of hollow brick. The assembly halls have an amphitheater seating arrangement for listeners (wooden benches). The smallest room subjected to tests was the auditorium I with the capacity of about 300 m3. The volume of the auditorium III was about 840 m3 and that of the auditorium II – 423 m3. The measurement of reverberation time in all tested rooms was carried out according to the precise method described in the ISO Standard (ISO 3382-2:2010). Using the precise method, two sound source positions and eight microphone positions. The measurement was
The analysis was to demonstrate the applicability of the MMR method. The graphs present the Sabine’s reverberation time corrected by the MMR method, the measurement of reverberation time, reverberation time determined with the Sabine’s method and the reverberation time calculated with the ODEON software. The analysis involved five variants of the room without sound absorbing material and 5 variants with the application of sound absorbing material (Fig. 1). Moreover, the analysis for three lecture theaters was carried out. First, we presented the results involving the rooms shown in Fig. 1a–e in which no absorbing elements were used. Then, the reverberation time was calculated for each room, using Sabin’s, Eyring’s, Millington’s, Kuttruff’s, Fitzroy’s, Arau’s, Neubaera’s and Pujolle’s methods as well as the standard ISO and Odeon. Then, the differences R ¼ T p T m were determined, where T m was the reverberation time calculated by means of the theoretical methods. Basing on the above, the remaining functions were determined using the MMR method. The values of the functions are presented in Table 2. In Table 2 an approximating polynomial Tðf Þ assumes for the frequency 125–4000 Hz the same values as the values of the function Tðf Þ at these points. This is because the polynomial which passes exactly the same points as Tðf Þ has been determined by means of the method of least squares. The final results for the investigated cases are illustrated on the graphs in Fig. 5. The next experiments involved the rooms in which sound absorbing materials were applied. The installation method as well as the amount of mineral wool panels are presented in Fig. 1. As before, the differences R ¼ T p T m were determined.
Fig. 2. Distribution of measurement points in the auditorium/halls (I).
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A. Nowos´wiat et al. / Applied Acoustics 106 (2016) 42–50
Fig. 3. Distribution of measurement points in the auditorium/halls (II).
Fig. 4. Distribution of measurement points in the auditorium/halls (III).
Then, the remaining functions were determined using the MMR method, and their values are shown in Table 3. The final results for the investigated cases are illustrated on the graphs in Fig. 6. At the end of the series of experiments, measurements were carried out in three auditoriums having similar geometry, similar average sound absorption but significantly different cubic volume. The results, as in the case of the laboratory measurements, are summarized in Table 4 and illustrated in the graphs 7. The final results for the considered cases are illustrated on the graphs below (see Fig. 7).
Since the presented method is correcting the Sabine’s theoretical model as to the uncertainties resulting from the room’s geometry or from non-uniform distribution of the acoustic field, it was possible, therefore, to examine during the verification of the results the same rooms as during the determination process of K(f) function. We can also assume that the determined K(f) function will be satisfactorily correcting the Sabine’s model for another room of the geometry similar to that in the investigated rooms and of the cubature lower than the largest investigated room and higher than the smallest room. Such an example will be presented in the following Chapter involving the discussion of results.
A. Nowos´wiat et al. / Applied Acoustics 106 (2016) 42–50 Table 2 The values of the functions used in the MMR method for rooms without the absorbing materials.
Table 3 The values of the functions used in the MMR method for rooms with the absorbing materials.
Frequency
Tðf Þ T t ðf Þ T sr;Sab ðf Þ Kðf Þ
47
Frequency
125
250
500
1000
2000
4000
0.906 0.906 0.86 1.0535
1.149 1.149 0.87 1.3169
0.696 0.696 0.52 1.3641
0.732 0.732 0.55 1.3163
0.646 0.646 0.53 1.2947
0.514 0.514 0.41 1.4205
Tðf Þ T t ðf Þ T sr;Sab ðf Þ Kðf Þ
125
250
500
1000
2000
4000
0.517 0.518 0.39 1.325
0.317 0.319 0.33 0.9865
0.238 0.254 0.27 0.9027
0.221 0.250 0.29 0.9117
0.251 0.257 0.28 0.9163
0.215 0.226 0.27 0.8841
Fig. 5. Experimental and analytical predictions of reverberation time. The presented results involve the premises/rooms of a diffused sound field of the dimensions L_W_H (in centimeters), where L – length, W – width, H – height of the room. The results are given as a function of frequency for the octave within the range 125 Hz–4 kHz.
A. Nowos´wiat et al. / Applied Acoustics 106 (2016) 42–50
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Fig. 6. Experimental and analytical predictions of reverberation time. The presented results involve the premises/rooms of a disturbed sound field of the dimensions L_W_H (in centimeters), where L – length, W – width, H – height of the room. The results are given as a function of frequency for the octave within the range 125 Hz–4 kHz.
Table 4 The values of the functions used in the MMR method for the tested auditoriums/halls. Frequency
Tðf Þ T t ðf Þ T sr;Sab ðf Þ Kðf Þ
125
250
500
1000
2000
4000
2.40 2.39 3.54 0.68
2.08 2.08 3.51 0.57
2.01 2.06 3.75 0.58
2.21 2.12 3.24 0.64
1.99 2.05 3.00 0.68
1.65 1.62 2.5 0.65
5. Discussion The present paper involves the estimation of the reverberation time with the use of the elaborated MMR method. Fig. 5 shows the reverberation time as a frequency function for the Sabine’s
method, the Sabine’s methods corrected by the MMR method, Odeon simulation and measurement results. We can observe that with respect to the rooms with the diffused sound field, the described here estimation method of reverberation time corrects the already known formulas. We can therefore obtain the reverberation time as a frequency function close to the measurement results. The estimation inaccuracy may be due to the measurement inaccuracy for such small spaces/rooms. Fig. 6 shows the reverberation time as a frequency function for the rooms with the applied absorbing material (see Fig. 1). Also in this case, the residual minimization method proves to be quite efficient. Actually, in all cases the Sabine_sk chart is similar to the measurement and not worse than the results of the simulation performed with the use of ODEON software. The method was finally verified by the measurements carried out in
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49
Fig. 8. Experimental and analytical predictions of reverberation time.
Fig. 7. Experimental and analytical predictions of reverberation time. The presented results involve the three assembly halls of various size I – 300 m3, II – 423 m3, III – 840 m3. The results are given as a function of frequency for the octave within the range 125 Hz–4 kHz.
three auditoriums. Also in that case the results could be generally regarded as satisfactory. However, it should be emphasized that they are not as clear as in the laboratory tests. It results from the fact that the method is actually the correction of Sabine’s method, and therefore it is burdened with similar errors. Such errors as the insufficient knowledge involving the exact values of sound absorption coefficients for room insulation materials, uneven distribution of the sound field have impact on the final results. Another estimation error of reverberation time in the auditoriums/halls involved the choice of the rooms in terms of their size and cubature, which varied significantly. The smallest room volume was 300 m3 and the biggest one was 840 m3. It is worth noting, however, that even
with such a selection, the results were evaluated fairly well. We can take for granted that the selection of premises/rooms of a similar size will yield results close to the measurements. To confirm this hypothesis, another auditorium room of the Civil Engineering Department was examined which is the same in terms of cubature and shape as the auditorium room III. The results are presented in Fig. 8 The differences between the Sabine’s model and the measurement are effected principally by the lack of precise data involving sound absorbing parameters of the partition walls limiting the room (tabular values for plaster and flooring were accepted), but still the corrected model yields interesting results. We can speculate here that if the sound absorption parameters had been known, the corrected results would have been even more satisfying. The reverberation time algorithm for MMR will rank among learning processes, which improves its performance basing on the past experience. The method is based on statistical learning where more measured objects are being added. It is important to ensure in this method that for a learning process we acquire the knowledge of the experimental data involving so called ‘‘similar” objects i.e. having a similar shape, volume, and dispersion of sound in the room. Learning, to paraphrase Herbert Simon’s idea dated 1983, is seen as a change in the adjusted formula enabling in the future the determination of more accurate results (similar to the measurement), acting on objects of a similar nature. The MMR method as a learning process is used because the existing models of reverberation time are not very reliable. It may be noted that if the new measurements of reverberation time are acquired in the future, the whole algorithm can be repeated again, thereby acquiring the correcting function K, which will provide even greater approximation to the actual measurement. The described method can be applied principally to correct inaccuracies involving the estimation of reverberation time by means of Sabine’s model effected by: non-uniform distribution of sound absorbing panels, the influence of room geometry on the difference between the measurement and the calculations, or the lack of knowledge of the acoustic absorption of sound absorbing materials. The present work does not terminate the investigation studies on the subject. Works are being carried out currently on the application of perturbation numbers (new algebra) to assess the approximation uncertainty of reverberation time.
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