- Email: [email protected]
Acoustic absorption increase prediction by placing absorbent material in pieces
Applied Acoustics 113 (2016) 185–192
Contents lists available at ScienceDirect
Applied Acoustics journal homepage: www.elsevier.com/locate/apacoust
Acoustic absorption increase prediction by placing absorbent material in pieces David Caballol a,⇑, Alvaro P. Raposo b a b
Department of Architectural Constructions and Assessment, Technical University of Madrid, Avda. Juan de Herrera, 6, 28040 Madrid, Spain Department of Applied Mathematics, Technical University of Madrid, Avda. Juan de Herrera, 6, 28040 Madrid, Spain
a r t i c l e
i n f o
Article history: Received 8 June 2016 Received in revised form 23 June 2016 Accepted 27 June 2016 Available online 4 July 2016 Keywords: Absorption Building materials Materials in patches Sound-absorbent material
a b s t r a c t We present the results of tests performed in a reverberation room measuring reverberation time with different acoustic absorbent materials in different layouts, compared to those results where the same material was placed as a single piece. With the analysis of the obtained data, a regression model is established in order to predict, for certain frequencies, the improvement produced in the reverberation time of a room, using the same amount of material by placing it in pieces separated from each other, instead of in one piece. The analysis also prove the sturdiness of the model. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction One of the strategies employed to limit the reverberating noise in the interior of an enclosure is to fit it acoustically including a certain amount of absorbent material in one or more surfaces. These acoustic conditioning strategies are very commonly used because of calculation simplicity (performance prediction) and its simple execution. Many of the European acoustic regulations include limitations of reverberation time regarding the use and the volume of the enclosure or premises. Other countries replace the requirement by one on the calculated amount of applied equivalent absorption area [1,2]. The vast majority of these regulations base their prescriptions on the well known Sabine’s equation, published in 1900 [3]:
T ¼ 0:16
V ½s; A
ð1Þ
where T is the reverberation time, in seconds, of a room with volume V in cubic meters and A is the equivalent absorption area in square meters. The equivalent absorption area gathers all the acoustic absorption capabilities of the room, for it is computed as the addition of the absorbent surfaces areas, Si , each one multiplied by the P absorption coefficient of the material of that surface: A ¼ i Si ai . The limitations of Sabine’s formula have been clearly recognized [4] and several improved formulations have been ⇑ Corresponding author. E-mail address: [email protected] (D. Caballol). http://dx.doi.org/10.1016/j.apacoust.2016.06.023 0003-682X/Ó 2016 Elsevier Ltd. All rights reserved.
proposed. The most celebrated ones are slight modifications of Eq. (1), like the one proposed by Eyring [5] or that by Millington [6], both of which consider different forms of introducing the absorption coefficient within the equations. There are also attempts to modelize the reverberation time with non diffuse sound field, like Fitzroy’s equation [7] or Arau-Puchades’s equation [8]. It is well known that all these formulations consider only two properties of each absorbent surface, namely its absorption coefficient, which is an intrinsic property of the material, and the surface area. Therefore they cannot distinguish the effect of different layouts of the same amount of absorbent material. However, an extra absorption when the absorbent material is placed in patches rather than in one piece has been widely documented [9–13] and it has been attributed to lack of diffusion and to the so called edge effect. In the former case some research studies [14,15] have attempted to quantify the diffusion based on the dispersion coefficient and absorption coefficient of the walls. It is also known, in the latter case, and extensively studied [11,12,16], that the edge effect increases the measured absorption in a reverberation chamber due to the extra surface area that may be present because of the thickness of the sample under test. Other researchers have studied a similar issue [10,9,17,18] without finding a simply method that can predict the improvement produced in the time of reverberation in a room, using the same amount of absorbent material by placing it in separated pieces. All proposed research methods to give estimates of the acoustical properties of porous type absorbers are complex for practical use.
186
D. Caballol, A.P. Raposo / Applied Acoustics 113 (2016) 185–192
Nevertheless placing the absorbent material in patches is a quite common situation in construction works, see Fig. 1, and notwithstanding the explanations offered in the references, not a simple account of the influence of these effects in the reverberation time has been reported. It is the goal of this paper to provide such a model, when feasible, and to explain the reason when it is not. To that end we have measured reverberation times in a reverberation chamber with test samples in different layouts: in one piece as well as in several patches with a variety of separation gaps among them. In Section 2 we give a detailed account of the experimental procedure, while in Section 3 we report the results of the measurements followed by their statistical analysis which allow us to provide, when suitable, a simple equation to relate the reverberation time with the layout of the patches. Finally, in Section 4 we enumerate the conclusions of our analysis. 2. Method We have chosen to measure the reverberation time under the same conditions specified by the ISO 354 standard [19] for the measurement of the absorption coefficient of an absorbent material. The method described in this standard measures the mean reverberation time in the reverberation room with and without the test sample. The equivalent sound absorption area is calculated from these reverberation time periods through Sabine’s equation, Eq. (1), and then the absorption coefficient. The testing conditions prescribe a specific reverberation room size and shape, with controlled temperature and humidity. The testing sample must have an area between 10 m2 and 12 m2 and must be rectangular in shape with a width-to-length ratio between 0.7 and 1. We have measured the reverberation time in bands of thirds of octaves of three different materials with similar thickness and in different positions. The tested materials have been:
of 10 m2 (in one piece), increasing the gross area through the separation surface between pieces in the different positions tested. As the pieces are separated, for each case, we compare the space occupied by the absorbent material and the gaps left when the planks are gradually distanced. In this way, different results are obtained for 100% of the occupied surface (all planks together in a single piece), 86%, 75%, 51% and 37% of the area occupied by the material. This ratio of net to gross area is the variable that we refer to as occupation throughout the paper. The test samples are initially rectangular in shape, with a width/length ratio of 0.7 and are placed in such a way that every part is more than 1 m away from the edges of the reverberating room. This condition varies as the separation distance between the pieces increases, but in all cases a separation of at least 0:75 m is maintained (see Fig. 2). In all cases, test sample pieces were allowed to reach a balance between the temperature and the relative humidity of the reverberating room before the tests. The relative humidity of the chamber ranged from 38% to 39% during the tests, and the temperature between 19.9 and 20.6 °C. The interrupted noise signal method was used for measuring the reverberation time and the sound fall curves were measured from equivalent levels (using linear average) with integration times that vary between 20 ms for the third octave bands of frequency 100, 125 and 160 Hz and 10 ms for the other frequency bands. Readings were made in all cases, in third octave bands, as specified in the ISO 266 standard [20]. From the layout of the experiment one may expect different results at different frequency bands. Since the gaps between patches of the material range from 10 cm to less than 1 m, medium and low frequencies waves are expected not to notice the different layouts of the absorbent materials due to their long wavelengths. On the contrary, the effect should be noticeable in the high
Material 1 (M1): non-woven polyester fiber 30 mm thick, in rigid planks with dimensions 1000 500 30 mm and 30 kg=m3 density. Material 2 (M2): 30 mm thick rock wool, in rigid planks with dimensions 1000 600 30 mm and 100 kg=m3 density. Material 3 (M3): melamine foam 30 mm thick, in rigid planks with dimensions 1000 500 30 mm and 10 kg=m3 density. Firstly, the reverberation time with the materials in one piece is obtained, and then with the materials distributed in pieces with given separations among them. The samples used have a net area
Fig. 1. Suspended ceiling with discontinuous absorbent material.
Fig. 2. Test in reverberating room.
187
D. Caballol, A.P. Raposo / Applied Acoustics 113 (2016) 185–192
frequency bands. Another effect that should be kept in mind is the area effect for frequencies of 500 Hz or below, because the size of the patches (of the order of 1 m) is similar to the wavelengths of those frequencies. However, Kawai and Meotoiwa [21] have shown that with regard to the area effect, the layout of the absorbent material in a single or several pieces is irrelevant. Therefore, since the net area of absorbent material is kept the same all along the experiment, the results we are measuring should be the consequence of the edge effect rather than the area effect.
3. Results and discussion The results obtained in the tests carried out in the reverberation room are given as reverberation time for each frequency, each material and each level of occupation. The data are presented in Table 1. For the sake of completeness, the global values of T mid 4001250 and T mid 4002500 are shown in Table 2. In order to analyze these data we have performed several statistical analysis on them with the main question being to determine whether the measured reverberation time depends on the occupation level of the absorbent material and, in the affirmative case, to estimate a model of this dependence. At the same time, examining if the frequency and material variables are factors to be considered in this response is also analyzed. Data in Table 1 are the result of a complete factorial design with three factors: the frequency, with eighteen levels (from 100 Hz to 5000 Hz); the material, with three levels (M1, M2 and M3) and the occupation, with five levels (37%, 51%, 75%, 86% and 100%). In each of the 18 3 5 ¼ 270 slots we have one measured datum, as shown in the aforementioned table. The data can be best viewed in Fig. 3, which shows a chart with the reverberation time T in relation to the occupation level, separated by the frequency and the material. The statistical analysis has been split into three steps. In the first one, the global effect of each of the three factors is evaluated, and the results point that the frequency is the most relevant factor in explaining the reverberation time, while the effects of the material or the occupation are hardly distinguishable from the random or non-controlled variability. Therefore the analysis must proceed by studying each frequency on its own, which is the second step. The outcome of it forces us to consider only high frequencies, and forget about the low and medium ones, for the effect of the material nor the occupation are traceable. In the third step we
try, and succeed, to build a model for the dependence of the reverberation time on the occupation variable. 3.1. First step: global analysis of variance In this first approach to data we consider the whole Table 1 which is a complete factorial design with three factors, as mentioned. The statistical tool which informs us of the relative importance of each factor in explaining the variability of data is the analysis of variance. In this step we have considered the three factors as quantitative variables, for our only concern at this stage is to measure their relative importance in explaining the data. The result of the analysis of variance performed on the data is shown in Table 3, which contains the sums of squares associated with each of the factors as well as the variability not explained by them. Unfortunately, although the results in Table 3 lead to interesting conclusions, we cannot trust them because the diagnosis of the model reveals that the hypothesis of the analysis of variance are not fulfilled. Fig. 4 shows a scatter plot of the residuals and a normal Q–Q plot to test normality. The scatter plot points a failure of the homoscedasticity hypothesis and the second plot a failure of the normality hypothesis of the residuals. Therefore we are lead to a transformation of the data in order to get reliable results. A transformation which has worked fine is to get the inverses of the reverberation times. Since we are only concerned, at this stage, on the relative importance of each factor, it is irrelevant if we analyze the variable T or its inverse 1=T. Now, the result of the analysis of variance on the inverse of the data of Table 1 is summarized in Table 4. The diagnosis of this second analysis of variance can be seen in Fig. 5, which shows an improvement in homoscedasticity of the residuals and, most remarkable, of the normality of their distribution. A Kolmogorov–Smirnov test of normality of the residuals gives the value D ¼ 0:0443 for the statistic, which means a p-value of 0.6651 for the test. The results derived from this analysis are, thus, reliable. In Table 4 we read that the three factors have a statistically significant contribution to explain the variability of the data, for the three of them give tiny critical values of the F-test. However, by looking at the column of sums of squares we see that the frequency explains on its own 92.5% of the total variability, while the material accounts for 2.5% and the occupation for 0.5% but, in contrast, the variability not explained by the model raises to 4.5%. Therefore, the effects of both the material and the occupation are masked by the variability due to other factors (random or not under control in the actual experiment).
Table 1 Reverberation time in third octave bands, in seconds. Freq. (Hz)
100 125 160 200 250 315 400 500 630 800 1000 1250 1600 2000 2500 3150 4000 5000
Material 1
Material 2
Material 3
100%
86%
75%
51%
37%
100%
86%
75%
51%
37%
100%
86%
75%
51%
37%
3.29 4.41 3.20 3.26 3.34 2.95 2.71 2.68 2.56 3.09 2.83 2.75 2.55 2.37 2.20 2.01 1.70 1.41
3.42 4.01 3.76 2.81 4.06 3.29 3.28 2.68 2.73 2.73 2.63 2.51 2.44 2.30 2.10 1.89 1.68 1.41
3.17 4.14 4.14 2.74 3.58 2.91 2.97 2.80 2.78 2.68 2.55 2.50 2.42 2.26 2.08 1.94 1.70 1.40
3.56 3.77 3.61 2.92 3.59 3.17 3.25 2.48 2.33 2.54 2.42 2.40 2.31 2.22 2.07 1.86 1.69 1.39
3.25 3.70 3.39 3.30 2.97 3.57 2.89 2.80 2.70 2.72 2.63 2.43 2.49 2.26 2.13 1.91 1.66 1.35
4.18 4.55 3.81 2.73 3.66 2.69 2.39 2.25 2.03 2.45 2.12 2.14 2.15 2.08 2.02 1.85 1.60 1.33
3.06 4.19 4.19 2.68 3.16 2.87 2.40 2.12 2.15 2.17 2.16 2.20 2.06 2.02 1.91 1.76 1.59 1.32
3.54 4.20 3.75 3.38 3.30 3.31 2.72 2.16 2.19 2.20 2.12 2.07 2.01 2.03 1.92 1.73 1.60 1.37
3.51 4.06 3.25 2.78 4.24 2.62 2.89 2.48 1.95 2.04 1.90 1.95 1.92 1.92 1.79 1.76 1.57 1.31
3.59 3.92 4.01 2.54 3.06 2.64 2.35 2.17 1.83 1.99 1.98 2.01 1.96 1.86 1.81 1.65 1.52 1.24
3.66 4.20 3.40 3.93 3.83 3.18 2.86 2.58 2.41 2.37 2.43 2.48 2.26 2.26 2.08 1.88 1.66 1.44
4.22 4.33 4.13 3.22 3.64 3.08 2.60 2.63 2.45 2.30 2.29 2.22 2.24 2.12 2.00 1.82 1.61 1.34
3.72 4.22 3.94 3.73 3.39 3.37 2.58 2.47 2.40 2.40 2.15 2.13 2.17 2.10 1.99 1.80 1.61 1.33
4.62 5.40 4.12 3.05 3.33 3.11 2.69 2.61 2.51 2.12 2.20 2.09 2.03 2.02 1.93 1.75 1.58 1.29
3.41 4.59 4.24 3.94 2.93 2.91 2.91 2.46 2.21 2.19 1.98 2.07 2.00 2.03 1.98 1.82 1.60 1.35
188
D. Caballol, A.P. Raposo / Applied Acoustics 113 (2016) 185–192
Table 2 Reverberation time, global indexes, in seconds. Global index
Material 1
Material 2
Material 3
100%
86%
75%
51%
37%
100%
86%
75%
51%
37%
100%
86%
75%
51%
37%
T mid;4001250 T mid;4002500
2.77 2.64
2.76 2.60
2.71 2.56
2.57 2.45
2.70 2.56
2.23 2.18
2.20 2.13
2.24 2.16
2.20 2.09
2.06 2.00
2.52 2.41
2.42 2.32
2.36 2.27
2.37 2.24
2.30 2.20
0.4 0.6 0.8 1.0
3150
4000
5000 5 4
M1 M2 M3
3 2
1000
1250
1600
2000
2500
315
400
500
630
800
Reverberation time (seconds)
5 4 3 2
5 4 3 2
100
125
160
200
250
5 4 3 2 0.4 0.6 0.8 1.0
0.4 0.6 0.8 1.0
0.4 0.6 0.8 1.0
Occupation Fig. 3. Reverberation time in relation to occupation, separated by frequency and by material.
Table 3 Anova table for the complete factorial design of Table 1. Factor
Sum of squares
Deg. of freedom
Variance
F value
p-value
Frequency Material Occupation Residuals
159.12 3.02 0.56 16.52
17 2 4 246
9.36 1.51 0.14 0.07
139.2 22.5 2.1
0.0000 0.0000 0.0844
Total
179.22
269
0.67
The conclusion of this analysis is, thus, that in order to distinguish the effect of the occupation, we must study the data in each frequency separated from the rest. This is step two. 3.2. Second step: analysis of variance in each frequency We now consider the data in Table 1 for each frequency and, therefore, we study eighteen factorial designs with two factors each, namely the material and the occupation. We have performed an analysis of variance in each of them to try to discern the role of
each factor in explaining the variability observed. The results are summarized in Table 5, where sums of squares and p-values of the F-test of each factor are shown. It is worth mentioning that the diagnosis of all the eighteen analysis of variance have proven right, both in homoscedasticity and normality of the residues. In each case, in addition to the standard diagnosis graphics, a Shapiro–Wilk test has been applied to the residues, with a resulting p-value greater than 0.5 in all cases. Thus, all the results are reliable. In Table 5 it is interesting to compare the sums of squares associated with the occupation and the residuals. For frequencies
189
15 10 5 0
Residuals quantiles
−10
−5
10 5 0 −5 −10
Standardized residuals
15
D. Caballol, A.P. Raposo / Applied Acoustics 113 (2016) 185–192
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
−3
−2
Fitted values
−1
0
1
2
3
Theoretical quantiles
Fig. 4. Diagnosis plots of the analysis of variance performed on the data of Table 1. The left figure is a scatter plot of the residuals with respect to the fitted values, which shows an inadmissible heterogeneity of the variance. The right figure is a normal Q–Q plot of the residuals, which shows a lack of normality.
Table 4 Anova table for the complete factorial design of the inverse of data in Table 1. Factor
Sum of squares
Deg. of freedom
Variance
F value
p-value
Frequency Material Occupation Residuals
4.252 0.116 0.023 0.209
17 2 4 246
0.25010 0.05790 0.00573 0.00085
293.7 68.0 6.7
0.0000 0.0000 0.0000
Total
4.600
269
0.67
The conclusion, since we are interested in the effect of the occupation in the measured reverberation time, is that we are forced to restrict our study to 800 Hz or greater frequencies. In the case of 5000 Hz we also perform the next step in order to confirm the explanation above. 3.3. Third step: Analysis of covariance In the third step we look for a model that numerically explains the dependence of reverberation time on the variable occupation. In order to give the model predictive power, we consider now the occupation as a quantitative, continuous variable, so the model will take on the form of a regression model. But we cannot forget the effect of the material, as Table 5 proves that the latter is more
50 0 −100
−50
Residuals quantiles
50 0 −50 −100
Standardized residuals
100
100
up to 630 Hz the sum of squares of the residuals is greater than that of the occupation, or very similar in the case of 250 Hz. Hence, the p-values of the occupation are high, say greater than 0.05, so the occupation has not a statistically significant role in explaining the variability. From 800 Hz to 4000 Hz the sum of squares of the occupation is greater than that of the residuals and the p-values are small enough to consider that the factor affects the data. Finally, the last frequency under study, 5000 Hz, gives an unexpected result, as the three sums of squares are quite similar, leading to not so small p-values. In this case a revisit to Fig. 3 shows that in 5000 Hz there is almost no difference in the response by varying either the material nor the occupation level, so the statistical analysis seems right when it points that these factor do not affect significatively the data.
0.2
0.3
0.4
0.5
0.6
Fitted values
0.7
0.8
−3
−2
−1
0
1
2
3
Theoretical quantiles
Fig. 5. Diagnosis plots of the analysis of variance performed on the inverse of the data of Table 1. The left figure is a scatter plot of the residuals with respect to the fitted values, which shows an admissible lack of homoscedasticity. The right figure is a normal Q–Q plot of the residuals, which is very close to normality.
190
D. Caballol, A.P. Raposo / Applied Acoustics 113 (2016) 185–192
Table 5 Results of the analysis of variance with two factors (material and occupation) in each of the eighteen frequencies. The sums of squares (s.s.) and p-values are shown. Frequency
s.s. (material)
s.s. (occupation)
s.s. (residuals)
p-value (material)
p-value (occupation)
100 125 160 200 250 315 400 500 630 800 1000 1250 1600 2000 2500 3150 4000 5000
0.8748 0.7632 0.2996 1.5363 0.0187 0.3644 0.5631 0.5366 0.8871 0.9609 0.8241 0.5249 0.4728 0.2273 0.1279 0.0756 0.0314 0.0152
0.4475 0.2580 0.6202 0.5053 1.0211 0.1242 0.1401 0.0054 0.1156 0.2835 0.1691 0.1883 0.0958 0.0711 0.0468 0.0299 0.0062 0.0118
1.2313 1.2958 0.8472 0.8684 0.9998 0.6090 0.4188 0.1718 0.1533 0.0675 0.0803 0.0399 0.0253 0.0110 0.0104 0.0125 0.0029 0.0118
0.1170 0.1570 0.2980 0.0170 0.9280 0.1530 0.0331 0.0035 0.0005 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0365
0.5980 0.8050 0.2990 0.3950 0.1810 0.7980 0.6314 0.9911 0.2871 0.0058 0.0399 0.0040 0.0080 0.0014 0.0046 0.0288 0.0372 0.1889
important than the former. Nevertheless, our aim is to find a way to predict the effect of the occupation regardless the material and, to this end, we have used the techniques of the analysis of covariance on the data of each of the frequencies starting from 800 Hz. For each frequency we have adjusted three different models to account for the reverberation time as a function of the material and the occupation. The first model is given by the equation
if the inclusion of more parameters is justified with the improvement of the model. To answer this question the appropriate tests are two which involve the quantities Si ; i ¼ 1; 2; 3 and, together, are known as analysis of covariance [22,] [23, pp. 179–184] [24, pp. 75–77]. The homogeneity test measures the difference in the sums of squares of the residuals between models 1 and 3, that is, the one with no effect of the material considered and that with the effect considered in full. The statistic is
T ¼ l þ bO þ e;
F hom ¼
ð2Þ
9 S1 S3 ; 4 S3
ð5Þ
where T is the reverberation time and O is the occupation variable. The parameters of this model are l, an intercept term, and b, the slope. The unknown error e is assumed to be distributed normally, with zero mean and variance r2 . In this first model there is no effect of the material. It is just a regression model of the reverberation time in terms of the occupation, making use of the fifteen available data. The second model is introduced in Eq. (3), where the symbols are as before and ai represents the effect of the material Mi.
and it is distributed as an F random variable with 4 and 9 degrees of freedom. The greater the value of F hom the greater the difference between models 1 and 3 and, thus, more statistical evidence of the material being a necessary explanatory variable. The equality of slopes test measures the difference in the sums of squares of the residuals between models 2 and 3, that is, the difference of considering the same slope for the regression line or a different slope for each level of the material. The statistic is
T ¼ l þ ai þ bO þ e;
F slope ¼
i ¼ 1; 2; 3:
ð3Þ
In this model the effect of the material is accounted for only by means of the parameters ai which, of course, add to zero. But the slope b, which measures the effect of the occupation variable, is the same regardless the material. Finally, the third model considers the effect of the material not only by the parameters ai , but also by introducing different slopes for each material, as the following equation shows:
T ¼ l þ ai þ bi O þ e;
i ¼ 1; 2; 3:
ð4Þ
In this case, each of the three models (for i ¼ 1; 2; 3 respectively) is estimated with the five data values of the corresponding material. Once we have estimated all the models we have proceeded to compute the sums of squares of the residuals in each case. Then, for each frequency under study, we call S1 the sum of squares of the fifteen residuals of the first model and the same applies for S2 in the second model. Eq. (4) gives, in fact, three models for each frequency, one per material. We have computed the sums of squares of the five residuals in each case and added the three to get S3 , the sum of squares of the residuals related to this last model, which is to be compared with S1 and S2 . Obviously, in all cases it has to be true that S1 > S2 > S3 , for the more parameters are included in a model, the better the adjustment of the model and, thus, the smaller the squares of the residuals. But the question is
9 S2 S3 ; 2 S3
ð6Þ
and it is distributed as an F random variable with 2 and 9 degrees of freedom. The greater the value of F slope the greater the difference between models 2 and 3 and, thus, more statistical evidence of the necessity of three different slopes to account for the effect of the occupation. The results of these two tests in each frequency are recorded in Table 6. The results of Table 6 are similar from 800 up to 4000 Hz. On one hand, the homogeneity test is failed (small p-value), so the statistical evidence is to consider the effect of the material in the model, as we expected. On the other hand,
Table 6 Results of the homogeneity test and the equality of slopes test for each of the frequencies studied. The statistics F hom and F slope are defined in Eqs. (5) and (6), and the rest of symbols within them. Frequency
S1
S2
S3
F hom
p-value
F slope
p-value
800 1000 1250 1600 2000 2500 3150 4000 5000
1.1003 0.9261 0.5992 0.5194 0.2448 0.1541 0.0970 0.0351 0.0280
0.1370 0.0999 0.0726 0.0455 0.0168 0.0258 0.0211 0.0036 0.0127
0.1265 0.0912 0.0649 0.0332 0.0122 0.0175 0.0191 0.0029 0.0123
17.32 20.60 18.51 32.96 42.96 17.59 9.15 25.17 2.88
0.0003 0.0001 0.0002 0.0000 0.0000 0.0003 0.0031 0.0000 0.0866
0.37 0.43 0.53 1.67 1.70 2.14 0.47 1.06 0.14
0.6982 0.6618 0.6044 0.2421 0.2369 0.1742 0.6413 0.3845 0.8738
191
D. Caballol, A.P. Raposo / Applied Acoustics 113 (2016) 185–192 Table 7 Slopes, b, of the regression models for the reverberation time as a function of the occupation. Also the standard error, sb is given and the residuals variance s2R . Frequency
b
sb
s2R
800 1000 1250 1600 2000 2500 3150 4000 5000
0.52 0.44 0.44 0.31 0.29 0.20 0.16 0.08 0.12
0.13 0.11 0.09 0.07 0.04 0.05 0.05 0.02 0.04
0.11 0.10 0.08 0.06 0.04 0.05 0.04 0.02 0.03
the equality of slopes test gives small values of the statistic F slope (high values of p), so there is no evidence of the slopes of the regression lines being different for each material. In the case of 5000 Hz the result is that the regression line can be only one for all the data, with no effect of the material. Therefore we arrive at the conclusion that the model in Eq. (3) is the appropriate one for these data. This model is prescribed by the parameters l and ai , which give the intercept of the line, and is different for each material, the parameter b, the slope of the line, which has proven to be independent of the material and, finally, the variance of the error, which is estimated by the residual variance. In Table 7 we show the estimation for the slope b, with its standard error, and the residual variance, s2R , for each frequency. This table is the main result of this paper. As it has been said, the regression lines depend on the material, but only in the intercept parameter. However our aim is to give the model of Eq. (3) a form which is useful regardless the material. We want to substitute the parameters l and ai by an easily accessible property of each material. To that end, consider Eq. (3) with O ¼ 1, that is, with the layout of the material in a single piece, and denote by T 1 the reverberation time in this particular case. We can regard T 1 as well as the value obtained from the sound absorption coefficient given within the manufacturer technical specifications for each material and the use of Sabine’s equation, Eq. (1), and therefore, we can regard T 1 as included in the deterministic part of the model, so we get T 1 ¼ l þ ai þ b. Solving for l þ ai and substituting again in Eq. (3) we get the following form of the deterministic part of the relation between reverberation time and occupation
T ¼ T 1 þ bðO 1Þ;
ð7Þ
where the slope b, the same as before, is given in Table 7. Eq. (7) allows to predict, for high frequencies and in similar conditions to those of our experiment, the improvement in the reverberation time starting from the value T 1 as long as the occupation is decreased. 4. Conclusions Using standard testing in the reverberation room, the reverberation time in thirds of octave bands with three different absorbent materials with similar thickness and with different layouts have been obtained. A statistically significant relation has been found between the reverberation time variable in the third of octave bands from 800 to 5000 Hz, which can be predicted with Eq. (7), using the values of the parameter b given in Table 7. The applicability of this equation is limited by the constrains of our experiment conditions: porous materials without peaky absorption characteristics, similar test conditions (in particular, no backing air layers), and net to gross area ratio ranging in between 37% and 100%. If one wishes to use a rule of thumb, they can observe that the values of the slope b for each of the three frequencies in the octave
Table 8 Averaged slopes, b, among the three values of Table 7 in each octave band. These values are only approximate and given as a rule of thumb for the application of Eq. (7). Octave band
b
1000 2000 4000
0.47 0.27 0.12
band of 1000 Hz are compatible with each other, if one takes into account their respective standard errors. Therefore, as an approximation, the average of these three values can be considered as a common value for the frequencies in this octave band. The same applies to the values of the three frequencies in the octave band of 2000 Hz, and almost in the 4000 Hz octave band. For the sake of giving such rule of thumb for Eq. (7) we give here the mentioned averaged values of the slope in each octave band (see Table 8). For the lower frequencies it has been proved that the ratio between net and gross area of the material (variable occupation) is not a statistically significant variable that can serve to explain the reverberation time. The cause of this lack of precision in the prediction at low and medium frequencies is that variability is very high due to other factors and therefore, the effect of occupation in these frequency bands is masked. This result coincides with our expectations about medium and low frequencies on the base of their long wavelength which prevent these waves of noticing the different layouts of the materials. The robustness of the proposed model has been statistically demonstrated and therefore, the above equation is considered a simple and reliable method to predict the improvement produced in the reverberation time in a room of the characteristics tested, using the same amount of material but placed in pieces separated from each other instead of in one piece. Acknowledgements Thanks to the staff and to the School of Engineering and Telecommunication Systems of the Technical University of Madrid, as well as to the Department of Architectural Constructions of the School of Building Engineering of Madrid. Thanks are due too to an anonymous referee for insightful comments which have helped to greatly improve the paper. References [1] Rasmussen B, Brunskog J, Hoffmeyer D. Reverberation time in classrooms – comparison of regulations and classification criteria in the Nordic countries. In: Proceedings of joint Baltic-Nordic acoustics meeting BNAM2012. p. 1–6. [2] COST Action TU0901, Copenhagen. Towards a common framework in building acoustics throughout Europe; 2013. [3] Sabine WC. Collected papers on acoustics. Cambridge: Harvard University Press; 1923. p. 3–68 [Ch. Reverberation]. [4] Alton F, Pohlmann K. Reverberation. Master handbook of acoustics. 5th ed. McGraw-Hill; 2009. [5] Eyring C. Reverberation time in dead rooms. J Acoust Soc Am 1930;1:217–41. [6] Millington G. A modified formula for reverberation. J Acoust Soc Am 1932;4:69. [7] Fitzroy D. Reverberation formula which seems to be more accurate with nonuniform distribution of absorption. J Acoust Soc Am 1959;31:893. [8] Arau-Puchades H. An improved reverberation formula. Acustica 1988;65:163–80. [9] Chrisler V. Dependence of sound absorption upon the area and distribution of the absorbent material. J Res Natl Bur Stand 1934;13:169–87. [10] Cook R. Absorption of sound by patches of absorbent material. J Acoust Soc Am 1957;29:324–9. [11] Bruijin A. Calculation of edge effect of sound absorbing structures. PhD thesis. Delft, Holland; 1967. [12] Guicking D. Theoretical evaluation of the edge effect of an absorbing strip of a pressure-release boundary. Acustica 1990;70:66–75. [13] Trevor J, D’Antonio P. Acoustics absorbers and diffusers. New York: Taylor & Francis; 2009. p. 388–93 [Ch. Hybrid surfaces].
192
D. Caballol, A.P. Raposo / Applied Acoustics 113 (2016) 185–192
[14] Hanyu A. Theoretical framework for quantitatively characterizing sound field diffusion based on scattering coefficient and absorption coefficient of walls. J Acoust Soc Am 2010;128:1140–8. [15] Omoto A. Comment on a theoretical framework for quantitatively characterizing sound field diffusion based on scattering coefficient and absorption coefficient of walls. J Acoust Soc Am 2013;133:9. [16] Kawakami F. Deep-well approach for cancelling the edge effect in random incident absorption measurement. J Acoust Soc Jpn 1998;19:327–38. [17] Thomasson S. Theory and experiments of the sound absorption as function of the area. Report TRITA-TAK8201. Stockholm: Dept of Acoustics, Royal Institute of Technology; 1982. [18] Jeong C. Converting Sabine absorption coefficients to random incident absorption coefficients. J Acoust Soc Am 2013;133:3951.
[19] ISO 354:2003, Geneva, Switzerland. Acoustics – measurement of sound absorption in a reverberation room; 2015. [20] ISO 266:1997, Geneva, Switzerland. Acoustics – preferred frequencies; 2013. [21] Kawai Y, Meotoiwa H. Estimation of the area effect of sound absorbent surfaces by using a boundary integral equation. Acoust Sci Technol 2005;26:123–7. [22] Peña D. Regresión y diseño de experimentos. Madrid: Alianza Editorial; 2002. [23] Weisberg S. Applied linear regression. 2nd ed. New York: John Wiley & Sons; 1985. [24] Yan X, Gang Su X. Linear regression analysis. Singapore: World Scientific; 2009.
Contents lists available at ScienceDirect
Applied Acoustics journal homepage: www.elsevier.com/locate/apacoust
Acoustic absorption increase prediction by placing absorbent material in pieces David Caballol a,⇑, Alvaro P. Raposo b a b
Department of Architectural Constructions and Assessment, Technical University of Madrid, Avda. Juan de Herrera, 6, 28040 Madrid, Spain Department of Applied Mathematics, Technical University of Madrid, Avda. Juan de Herrera, 6, 28040 Madrid, Spain
a r t i c l e
i n f o
Article history: Received 8 June 2016 Received in revised form 23 June 2016 Accepted 27 June 2016 Available online 4 July 2016 Keywords: Absorption Building materials Materials in patches Sound-absorbent material
a b s t r a c t We present the results of tests performed in a reverberation room measuring reverberation time with different acoustic absorbent materials in different layouts, compared to those results where the same material was placed as a single piece. With the analysis of the obtained data, a regression model is established in order to predict, for certain frequencies, the improvement produced in the reverberation time of a room, using the same amount of material by placing it in pieces separated from each other, instead of in one piece. The analysis also prove the sturdiness of the model. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction One of the strategies employed to limit the reverberating noise in the interior of an enclosure is to fit it acoustically including a certain amount of absorbent material in one or more surfaces. These acoustic conditioning strategies are very commonly used because of calculation simplicity (performance prediction) and its simple execution. Many of the European acoustic regulations include limitations of reverberation time regarding the use and the volume of the enclosure or premises. Other countries replace the requirement by one on the calculated amount of applied equivalent absorption area [1,2]. The vast majority of these regulations base their prescriptions on the well known Sabine’s equation, published in 1900 [3]:
T ¼ 0:16
V ½s; A
ð1Þ
where T is the reverberation time, in seconds, of a room with volume V in cubic meters and A is the equivalent absorption area in square meters. The equivalent absorption area gathers all the acoustic absorption capabilities of the room, for it is computed as the addition of the absorbent surfaces areas, Si , each one multiplied by the P absorption coefficient of the material of that surface: A ¼ i Si ai . The limitations of Sabine’s formula have been clearly recognized [4] and several improved formulations have been ⇑ Corresponding author. E-mail address: [email protected] (D. Caballol). http://dx.doi.org/10.1016/j.apacoust.2016.06.023 0003-682X/Ó 2016 Elsevier Ltd. All rights reserved.
proposed. The most celebrated ones are slight modifications of Eq. (1), like the one proposed by Eyring [5] or that by Millington [6], both of which consider different forms of introducing the absorption coefficient within the equations. There are also attempts to modelize the reverberation time with non diffuse sound field, like Fitzroy’s equation [7] or Arau-Puchades’s equation [8]. It is well known that all these formulations consider only two properties of each absorbent surface, namely its absorption coefficient, which is an intrinsic property of the material, and the surface area. Therefore they cannot distinguish the effect of different layouts of the same amount of absorbent material. However, an extra absorption when the absorbent material is placed in patches rather than in one piece has been widely documented [9–13] and it has been attributed to lack of diffusion and to the so called edge effect. In the former case some research studies [14,15] have attempted to quantify the diffusion based on the dispersion coefficient and absorption coefficient of the walls. It is also known, in the latter case, and extensively studied [11,12,16], that the edge effect increases the measured absorption in a reverberation chamber due to the extra surface area that may be present because of the thickness of the sample under test. Other researchers have studied a similar issue [10,9,17,18] without finding a simply method that can predict the improvement produced in the time of reverberation in a room, using the same amount of absorbent material by placing it in separated pieces. All proposed research methods to give estimates of the acoustical properties of porous type absorbers are complex for practical use.
186
D. Caballol, A.P. Raposo / Applied Acoustics 113 (2016) 185–192
Nevertheless placing the absorbent material in patches is a quite common situation in construction works, see Fig. 1, and notwithstanding the explanations offered in the references, not a simple account of the influence of these effects in the reverberation time has been reported. It is the goal of this paper to provide such a model, when feasible, and to explain the reason when it is not. To that end we have measured reverberation times in a reverberation chamber with test samples in different layouts: in one piece as well as in several patches with a variety of separation gaps among them. In Section 2 we give a detailed account of the experimental procedure, while in Section 3 we report the results of the measurements followed by their statistical analysis which allow us to provide, when suitable, a simple equation to relate the reverberation time with the layout of the patches. Finally, in Section 4 we enumerate the conclusions of our analysis. 2. Method We have chosen to measure the reverberation time under the same conditions specified by the ISO 354 standard [19] for the measurement of the absorption coefficient of an absorbent material. The method described in this standard measures the mean reverberation time in the reverberation room with and without the test sample. The equivalent sound absorption area is calculated from these reverberation time periods through Sabine’s equation, Eq. (1), and then the absorption coefficient. The testing conditions prescribe a specific reverberation room size and shape, with controlled temperature and humidity. The testing sample must have an area between 10 m2 and 12 m2 and must be rectangular in shape with a width-to-length ratio between 0.7 and 1. We have measured the reverberation time in bands of thirds of octaves of three different materials with similar thickness and in different positions. The tested materials have been:
of 10 m2 (in one piece), increasing the gross area through the separation surface between pieces in the different positions tested. As the pieces are separated, for each case, we compare the space occupied by the absorbent material and the gaps left when the planks are gradually distanced. In this way, different results are obtained for 100% of the occupied surface (all planks together in a single piece), 86%, 75%, 51% and 37% of the area occupied by the material. This ratio of net to gross area is the variable that we refer to as occupation throughout the paper. The test samples are initially rectangular in shape, with a width/length ratio of 0.7 and are placed in such a way that every part is more than 1 m away from the edges of the reverberating room. This condition varies as the separation distance between the pieces increases, but in all cases a separation of at least 0:75 m is maintained (see Fig. 2). In all cases, test sample pieces were allowed to reach a balance between the temperature and the relative humidity of the reverberating room before the tests. The relative humidity of the chamber ranged from 38% to 39% during the tests, and the temperature between 19.9 and 20.6 °C. The interrupted noise signal method was used for measuring the reverberation time and the sound fall curves were measured from equivalent levels (using linear average) with integration times that vary between 20 ms for the third octave bands of frequency 100, 125 and 160 Hz and 10 ms for the other frequency bands. Readings were made in all cases, in third octave bands, as specified in the ISO 266 standard [20]. From the layout of the experiment one may expect different results at different frequency bands. Since the gaps between patches of the material range from 10 cm to less than 1 m, medium and low frequencies waves are expected not to notice the different layouts of the absorbent materials due to their long wavelengths. On the contrary, the effect should be noticeable in the high
Material 1 (M1): non-woven polyester fiber 30 mm thick, in rigid planks with dimensions 1000 500 30 mm and 30 kg=m3 density. Material 2 (M2): 30 mm thick rock wool, in rigid planks with dimensions 1000 600 30 mm and 100 kg=m3 density. Material 3 (M3): melamine foam 30 mm thick, in rigid planks with dimensions 1000 500 30 mm and 10 kg=m3 density. Firstly, the reverberation time with the materials in one piece is obtained, and then with the materials distributed in pieces with given separations among them. The samples used have a net area
Fig. 1. Suspended ceiling with discontinuous absorbent material.
Fig. 2. Test in reverberating room.
187
D. Caballol, A.P. Raposo / Applied Acoustics 113 (2016) 185–192
frequency bands. Another effect that should be kept in mind is the area effect for frequencies of 500 Hz or below, because the size of the patches (of the order of 1 m) is similar to the wavelengths of those frequencies. However, Kawai and Meotoiwa [21] have shown that with regard to the area effect, the layout of the absorbent material in a single or several pieces is irrelevant. Therefore, since the net area of absorbent material is kept the same all along the experiment, the results we are measuring should be the consequence of the edge effect rather than the area effect.
3. Results and discussion The results obtained in the tests carried out in the reverberation room are given as reverberation time for each frequency, each material and each level of occupation. The data are presented in Table 1. For the sake of completeness, the global values of T mid 4001250 and T mid 4002500 are shown in Table 2. In order to analyze these data we have performed several statistical analysis on them with the main question being to determine whether the measured reverberation time depends on the occupation level of the absorbent material and, in the affirmative case, to estimate a model of this dependence. At the same time, examining if the frequency and material variables are factors to be considered in this response is also analyzed. Data in Table 1 are the result of a complete factorial design with three factors: the frequency, with eighteen levels (from 100 Hz to 5000 Hz); the material, with three levels (M1, M2 and M3) and the occupation, with five levels (37%, 51%, 75%, 86% and 100%). In each of the 18 3 5 ¼ 270 slots we have one measured datum, as shown in the aforementioned table. The data can be best viewed in Fig. 3, which shows a chart with the reverberation time T in relation to the occupation level, separated by the frequency and the material. The statistical analysis has been split into three steps. In the first one, the global effect of each of the three factors is evaluated, and the results point that the frequency is the most relevant factor in explaining the reverberation time, while the effects of the material or the occupation are hardly distinguishable from the random or non-controlled variability. Therefore the analysis must proceed by studying each frequency on its own, which is the second step. The outcome of it forces us to consider only high frequencies, and forget about the low and medium ones, for the effect of the material nor the occupation are traceable. In the third step we
try, and succeed, to build a model for the dependence of the reverberation time on the occupation variable. 3.1. First step: global analysis of variance In this first approach to data we consider the whole Table 1 which is a complete factorial design with three factors, as mentioned. The statistical tool which informs us of the relative importance of each factor in explaining the variability of data is the analysis of variance. In this step we have considered the three factors as quantitative variables, for our only concern at this stage is to measure their relative importance in explaining the data. The result of the analysis of variance performed on the data is shown in Table 3, which contains the sums of squares associated with each of the factors as well as the variability not explained by them. Unfortunately, although the results in Table 3 lead to interesting conclusions, we cannot trust them because the diagnosis of the model reveals that the hypothesis of the analysis of variance are not fulfilled. Fig. 4 shows a scatter plot of the residuals and a normal Q–Q plot to test normality. The scatter plot points a failure of the homoscedasticity hypothesis and the second plot a failure of the normality hypothesis of the residuals. Therefore we are lead to a transformation of the data in order to get reliable results. A transformation which has worked fine is to get the inverses of the reverberation times. Since we are only concerned, at this stage, on the relative importance of each factor, it is irrelevant if we analyze the variable T or its inverse 1=T. Now, the result of the analysis of variance on the inverse of the data of Table 1 is summarized in Table 4. The diagnosis of this second analysis of variance can be seen in Fig. 5, which shows an improvement in homoscedasticity of the residuals and, most remarkable, of the normality of their distribution. A Kolmogorov–Smirnov test of normality of the residuals gives the value D ¼ 0:0443 for the statistic, which means a p-value of 0.6651 for the test. The results derived from this analysis are, thus, reliable. In Table 4 we read that the three factors have a statistically significant contribution to explain the variability of the data, for the three of them give tiny critical values of the F-test. However, by looking at the column of sums of squares we see that the frequency explains on its own 92.5% of the total variability, while the material accounts for 2.5% and the occupation for 0.5% but, in contrast, the variability not explained by the model raises to 4.5%. Therefore, the effects of both the material and the occupation are masked by the variability due to other factors (random or not under control in the actual experiment).
Table 1 Reverberation time in third octave bands, in seconds. Freq. (Hz)
100 125 160 200 250 315 400 500 630 800 1000 1250 1600 2000 2500 3150 4000 5000
Material 1
Material 2
Material 3
100%
86%
75%
51%
37%
100%
86%
75%
51%
37%
100%
86%
75%
51%
37%
3.29 4.41 3.20 3.26 3.34 2.95 2.71 2.68 2.56 3.09 2.83 2.75 2.55 2.37 2.20 2.01 1.70 1.41
3.42 4.01 3.76 2.81 4.06 3.29 3.28 2.68 2.73 2.73 2.63 2.51 2.44 2.30 2.10 1.89 1.68 1.41
3.17 4.14 4.14 2.74 3.58 2.91 2.97 2.80 2.78 2.68 2.55 2.50 2.42 2.26 2.08 1.94 1.70 1.40
3.56 3.77 3.61 2.92 3.59 3.17 3.25 2.48 2.33 2.54 2.42 2.40 2.31 2.22 2.07 1.86 1.69 1.39
3.25 3.70 3.39 3.30 2.97 3.57 2.89 2.80 2.70 2.72 2.63 2.43 2.49 2.26 2.13 1.91 1.66 1.35
4.18 4.55 3.81 2.73 3.66 2.69 2.39 2.25 2.03 2.45 2.12 2.14 2.15 2.08 2.02 1.85 1.60 1.33
3.06 4.19 4.19 2.68 3.16 2.87 2.40 2.12 2.15 2.17 2.16 2.20 2.06 2.02 1.91 1.76 1.59 1.32
3.54 4.20 3.75 3.38 3.30 3.31 2.72 2.16 2.19 2.20 2.12 2.07 2.01 2.03 1.92 1.73 1.60 1.37
3.51 4.06 3.25 2.78 4.24 2.62 2.89 2.48 1.95 2.04 1.90 1.95 1.92 1.92 1.79 1.76 1.57 1.31
3.59 3.92 4.01 2.54 3.06 2.64 2.35 2.17 1.83 1.99 1.98 2.01 1.96 1.86 1.81 1.65 1.52 1.24
3.66 4.20 3.40 3.93 3.83 3.18 2.86 2.58 2.41 2.37 2.43 2.48 2.26 2.26 2.08 1.88 1.66 1.44
4.22 4.33 4.13 3.22 3.64 3.08 2.60 2.63 2.45 2.30 2.29 2.22 2.24 2.12 2.00 1.82 1.61 1.34
3.72 4.22 3.94 3.73 3.39 3.37 2.58 2.47 2.40 2.40 2.15 2.13 2.17 2.10 1.99 1.80 1.61 1.33
4.62 5.40 4.12 3.05 3.33 3.11 2.69 2.61 2.51 2.12 2.20 2.09 2.03 2.02 1.93 1.75 1.58 1.29
3.41 4.59 4.24 3.94 2.93 2.91 2.91 2.46 2.21 2.19 1.98 2.07 2.00 2.03 1.98 1.82 1.60 1.35
188
D. Caballol, A.P. Raposo / Applied Acoustics 113 (2016) 185–192
Table 2 Reverberation time, global indexes, in seconds. Global index
Material 1
Material 2
Material 3
100%
86%
75%
51%
37%
100%
86%
75%
51%
37%
100%
86%
75%
51%
37%
T mid;4001250 T mid;4002500
2.77 2.64
2.76 2.60
2.71 2.56
2.57 2.45
2.70 2.56
2.23 2.18
2.20 2.13
2.24 2.16
2.20 2.09
2.06 2.00
2.52 2.41
2.42 2.32
2.36 2.27
2.37 2.24
2.30 2.20
0.4 0.6 0.8 1.0
3150
4000
5000 5 4
M1 M2 M3
3 2
1000
1250
1600
2000
2500
315
400
500
630
800
Reverberation time (seconds)
5 4 3 2
5 4 3 2
100
125
160
200
250
5 4 3 2 0.4 0.6 0.8 1.0
0.4 0.6 0.8 1.0
0.4 0.6 0.8 1.0
Occupation Fig. 3. Reverberation time in relation to occupation, separated by frequency and by material.
Table 3 Anova table for the complete factorial design of Table 1. Factor
Sum of squares
Deg. of freedom
Variance
F value
p-value
Frequency Material Occupation Residuals
159.12 3.02 0.56 16.52
17 2 4 246
9.36 1.51 0.14 0.07
139.2 22.5 2.1
0.0000 0.0000 0.0844
Total
179.22
269
0.67
The conclusion of this analysis is, thus, that in order to distinguish the effect of the occupation, we must study the data in each frequency separated from the rest. This is step two. 3.2. Second step: analysis of variance in each frequency We now consider the data in Table 1 for each frequency and, therefore, we study eighteen factorial designs with two factors each, namely the material and the occupation. We have performed an analysis of variance in each of them to try to discern the role of
each factor in explaining the variability observed. The results are summarized in Table 5, where sums of squares and p-values of the F-test of each factor are shown. It is worth mentioning that the diagnosis of all the eighteen analysis of variance have proven right, both in homoscedasticity and normality of the residues. In each case, in addition to the standard diagnosis graphics, a Shapiro–Wilk test has been applied to the residues, with a resulting p-value greater than 0.5 in all cases. Thus, all the results are reliable. In Table 5 it is interesting to compare the sums of squares associated with the occupation and the residuals. For frequencies
189
15 10 5 0
Residuals quantiles
−10
−5
10 5 0 −5 −10
Standardized residuals
15
D. Caballol, A.P. Raposo / Applied Acoustics 113 (2016) 185–192
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
−3
−2
Fitted values
−1
0
1
2
3
Theoretical quantiles
Fig. 4. Diagnosis plots of the analysis of variance performed on the data of Table 1. The left figure is a scatter plot of the residuals with respect to the fitted values, which shows an inadmissible heterogeneity of the variance. The right figure is a normal Q–Q plot of the residuals, which shows a lack of normality.
Table 4 Anova table for the complete factorial design of the inverse of data in Table 1. Factor
Sum of squares
Deg. of freedom
Variance
F value
p-value
Frequency Material Occupation Residuals
4.252 0.116 0.023 0.209
17 2 4 246
0.25010 0.05790 0.00573 0.00085
293.7 68.0 6.7
0.0000 0.0000 0.0000
Total
4.600
269
0.67
The conclusion, since we are interested in the effect of the occupation in the measured reverberation time, is that we are forced to restrict our study to 800 Hz or greater frequencies. In the case of 5000 Hz we also perform the next step in order to confirm the explanation above. 3.3. Third step: Analysis of covariance In the third step we look for a model that numerically explains the dependence of reverberation time on the variable occupation. In order to give the model predictive power, we consider now the occupation as a quantitative, continuous variable, so the model will take on the form of a regression model. But we cannot forget the effect of the material, as Table 5 proves that the latter is more
50 0 −100
−50
Residuals quantiles
50 0 −50 −100
Standardized residuals
100
100
up to 630 Hz the sum of squares of the residuals is greater than that of the occupation, or very similar in the case of 250 Hz. Hence, the p-values of the occupation are high, say greater than 0.05, so the occupation has not a statistically significant role in explaining the variability. From 800 Hz to 4000 Hz the sum of squares of the occupation is greater than that of the residuals and the p-values are small enough to consider that the factor affects the data. Finally, the last frequency under study, 5000 Hz, gives an unexpected result, as the three sums of squares are quite similar, leading to not so small p-values. In this case a revisit to Fig. 3 shows that in 5000 Hz there is almost no difference in the response by varying either the material nor the occupation level, so the statistical analysis seems right when it points that these factor do not affect significatively the data.
0.2
0.3
0.4
0.5
0.6
Fitted values
0.7
0.8
−3
−2
−1
0
1
2
3
Theoretical quantiles
Fig. 5. Diagnosis plots of the analysis of variance performed on the inverse of the data of Table 1. The left figure is a scatter plot of the residuals with respect to the fitted values, which shows an admissible lack of homoscedasticity. The right figure is a normal Q–Q plot of the residuals, which is very close to normality.
190
D. Caballol, A.P. Raposo / Applied Acoustics 113 (2016) 185–192
Table 5 Results of the analysis of variance with two factors (material and occupation) in each of the eighteen frequencies. The sums of squares (s.s.) and p-values are shown. Frequency
s.s. (material)
s.s. (occupation)
s.s. (residuals)
p-value (material)
p-value (occupation)
100 125 160 200 250 315 400 500 630 800 1000 1250 1600 2000 2500 3150 4000 5000
0.8748 0.7632 0.2996 1.5363 0.0187 0.3644 0.5631 0.5366 0.8871 0.9609 0.8241 0.5249 0.4728 0.2273 0.1279 0.0756 0.0314 0.0152
0.4475 0.2580 0.6202 0.5053 1.0211 0.1242 0.1401 0.0054 0.1156 0.2835 0.1691 0.1883 0.0958 0.0711 0.0468 0.0299 0.0062 0.0118
1.2313 1.2958 0.8472 0.8684 0.9998 0.6090 0.4188 0.1718 0.1533 0.0675 0.0803 0.0399 0.0253 0.0110 0.0104 0.0125 0.0029 0.0118
0.1170 0.1570 0.2980 0.0170 0.9280 0.1530 0.0331 0.0035 0.0005 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0365
0.5980 0.8050 0.2990 0.3950 0.1810 0.7980 0.6314 0.9911 0.2871 0.0058 0.0399 0.0040 0.0080 0.0014 0.0046 0.0288 0.0372 0.1889
important than the former. Nevertheless, our aim is to find a way to predict the effect of the occupation regardless the material and, to this end, we have used the techniques of the analysis of covariance on the data of each of the frequencies starting from 800 Hz. For each frequency we have adjusted three different models to account for the reverberation time as a function of the material and the occupation. The first model is given by the equation
if the inclusion of more parameters is justified with the improvement of the model. To answer this question the appropriate tests are two which involve the quantities Si ; i ¼ 1; 2; 3 and, together, are known as analysis of covariance [22,] [23, pp. 179–184] [24, pp. 75–77]. The homogeneity test measures the difference in the sums of squares of the residuals between models 1 and 3, that is, the one with no effect of the material considered and that with the effect considered in full. The statistic is
T ¼ l þ bO þ e;
F hom ¼
ð2Þ
9 S1 S3 ; 4 S3
ð5Þ
where T is the reverberation time and O is the occupation variable. The parameters of this model are l, an intercept term, and b, the slope. The unknown error e is assumed to be distributed normally, with zero mean and variance r2 . In this first model there is no effect of the material. It is just a regression model of the reverberation time in terms of the occupation, making use of the fifteen available data. The second model is introduced in Eq. (3), where the symbols are as before and ai represents the effect of the material Mi.
and it is distributed as an F random variable with 4 and 9 degrees of freedom. The greater the value of F hom the greater the difference between models 1 and 3 and, thus, more statistical evidence of the material being a necessary explanatory variable. The equality of slopes test measures the difference in the sums of squares of the residuals between models 2 and 3, that is, the difference of considering the same slope for the regression line or a different slope for each level of the material. The statistic is
T ¼ l þ ai þ bO þ e;
F slope ¼
i ¼ 1; 2; 3:
ð3Þ
In this model the effect of the material is accounted for only by means of the parameters ai which, of course, add to zero. But the slope b, which measures the effect of the occupation variable, is the same regardless the material. Finally, the third model considers the effect of the material not only by the parameters ai , but also by introducing different slopes for each material, as the following equation shows:
T ¼ l þ ai þ bi O þ e;
i ¼ 1; 2; 3:
ð4Þ
In this case, each of the three models (for i ¼ 1; 2; 3 respectively) is estimated with the five data values of the corresponding material. Once we have estimated all the models we have proceeded to compute the sums of squares of the residuals in each case. Then, for each frequency under study, we call S1 the sum of squares of the fifteen residuals of the first model and the same applies for S2 in the second model. Eq. (4) gives, in fact, three models for each frequency, one per material. We have computed the sums of squares of the five residuals in each case and added the three to get S3 , the sum of squares of the residuals related to this last model, which is to be compared with S1 and S2 . Obviously, in all cases it has to be true that S1 > S2 > S3 , for the more parameters are included in a model, the better the adjustment of the model and, thus, the smaller the squares of the residuals. But the question is
9 S2 S3 ; 2 S3
ð6Þ
and it is distributed as an F random variable with 2 and 9 degrees of freedom. The greater the value of F slope the greater the difference between models 2 and 3 and, thus, more statistical evidence of the necessity of three different slopes to account for the effect of the occupation. The results of these two tests in each frequency are recorded in Table 6. The results of Table 6 are similar from 800 up to 4000 Hz. On one hand, the homogeneity test is failed (small p-value), so the statistical evidence is to consider the effect of the material in the model, as we expected. On the other hand,
Table 6 Results of the homogeneity test and the equality of slopes test for each of the frequencies studied. The statistics F hom and F slope are defined in Eqs. (5) and (6), and the rest of symbols within them. Frequency
S1
S2
S3
F hom
p-value
F slope
p-value
800 1000 1250 1600 2000 2500 3150 4000 5000
1.1003 0.9261 0.5992 0.5194 0.2448 0.1541 0.0970 0.0351 0.0280
0.1370 0.0999 0.0726 0.0455 0.0168 0.0258 0.0211 0.0036 0.0127
0.1265 0.0912 0.0649 0.0332 0.0122 0.0175 0.0191 0.0029 0.0123
17.32 20.60 18.51 32.96 42.96 17.59 9.15 25.17 2.88
0.0003 0.0001 0.0002 0.0000 0.0000 0.0003 0.0031 0.0000 0.0866
0.37 0.43 0.53 1.67 1.70 2.14 0.47 1.06 0.14
0.6982 0.6618 0.6044 0.2421 0.2369 0.1742 0.6413 0.3845 0.8738
191
D. Caballol, A.P. Raposo / Applied Acoustics 113 (2016) 185–192 Table 7 Slopes, b, of the regression models for the reverberation time as a function of the occupation. Also the standard error, sb is given and the residuals variance s2R . Frequency
b
sb
s2R
800 1000 1250 1600 2000 2500 3150 4000 5000
0.52 0.44 0.44 0.31 0.29 0.20 0.16 0.08 0.12
0.13 0.11 0.09 0.07 0.04 0.05 0.05 0.02 0.04
0.11 0.10 0.08 0.06 0.04 0.05 0.04 0.02 0.03
the equality of slopes test gives small values of the statistic F slope (high values of p), so there is no evidence of the slopes of the regression lines being different for each material. In the case of 5000 Hz the result is that the regression line can be only one for all the data, with no effect of the material. Therefore we arrive at the conclusion that the model in Eq. (3) is the appropriate one for these data. This model is prescribed by the parameters l and ai , which give the intercept of the line, and is different for each material, the parameter b, the slope of the line, which has proven to be independent of the material and, finally, the variance of the error, which is estimated by the residual variance. In Table 7 we show the estimation for the slope b, with its standard error, and the residual variance, s2R , for each frequency. This table is the main result of this paper. As it has been said, the regression lines depend on the material, but only in the intercept parameter. However our aim is to give the model of Eq. (3) a form which is useful regardless the material. We want to substitute the parameters l and ai by an easily accessible property of each material. To that end, consider Eq. (3) with O ¼ 1, that is, with the layout of the material in a single piece, and denote by T 1 the reverberation time in this particular case. We can regard T 1 as well as the value obtained from the sound absorption coefficient given within the manufacturer technical specifications for each material and the use of Sabine’s equation, Eq. (1), and therefore, we can regard T 1 as included in the deterministic part of the model, so we get T 1 ¼ l þ ai þ b. Solving for l þ ai and substituting again in Eq. (3) we get the following form of the deterministic part of the relation between reverberation time and occupation
T ¼ T 1 þ bðO 1Þ;
ð7Þ
where the slope b, the same as before, is given in Table 7. Eq. (7) allows to predict, for high frequencies and in similar conditions to those of our experiment, the improvement in the reverberation time starting from the value T 1 as long as the occupation is decreased. 4. Conclusions Using standard testing in the reverberation room, the reverberation time in thirds of octave bands with three different absorbent materials with similar thickness and with different layouts have been obtained. A statistically significant relation has been found between the reverberation time variable in the third of octave bands from 800 to 5000 Hz, which can be predicted with Eq. (7), using the values of the parameter b given in Table 7. The applicability of this equation is limited by the constrains of our experiment conditions: porous materials without peaky absorption characteristics, similar test conditions (in particular, no backing air layers), and net to gross area ratio ranging in between 37% and 100%. If one wishes to use a rule of thumb, they can observe that the values of the slope b for each of the three frequencies in the octave
Table 8 Averaged slopes, b, among the three values of Table 7 in each octave band. These values are only approximate and given as a rule of thumb for the application of Eq. (7). Octave band
b
1000 2000 4000
0.47 0.27 0.12
band of 1000 Hz are compatible with each other, if one takes into account their respective standard errors. Therefore, as an approximation, the average of these three values can be considered as a common value for the frequencies in this octave band. The same applies to the values of the three frequencies in the octave band of 2000 Hz, and almost in the 4000 Hz octave band. For the sake of giving such rule of thumb for Eq. (7) we give here the mentioned averaged values of the slope in each octave band (see Table 8). For the lower frequencies it has been proved that the ratio between net and gross area of the material (variable occupation) is not a statistically significant variable that can serve to explain the reverberation time. The cause of this lack of precision in the prediction at low and medium frequencies is that variability is very high due to other factors and therefore, the effect of occupation in these frequency bands is masked. This result coincides with our expectations about medium and low frequencies on the base of their long wavelength which prevent these waves of noticing the different layouts of the materials. The robustness of the proposed model has been statistically demonstrated and therefore, the above equation is considered a simple and reliable method to predict the improvement produced in the reverberation time in a room of the characteristics tested, using the same amount of material but placed in pieces separated from each other instead of in one piece. Acknowledgements Thanks to the staff and to the School of Engineering and Telecommunication Systems of the Technical University of Madrid, as well as to the Department of Architectural Constructions of the School of Building Engineering of Madrid. Thanks are due too to an anonymous referee for insightful comments which have helped to greatly improve the paper. References [1] Rasmussen B, Brunskog J, Hoffmeyer D. Reverberation time in classrooms – comparison of regulations and classification criteria in the Nordic countries. In: Proceedings of joint Baltic-Nordic acoustics meeting BNAM2012. p. 1–6. [2] COST Action TU0901, Copenhagen. Towards a common framework in building acoustics throughout Europe; 2013. [3] Sabine WC. Collected papers on acoustics. Cambridge: Harvard University Press; 1923. p. 3–68 [Ch. Reverberation]. [4] Alton F, Pohlmann K. Reverberation. Master handbook of acoustics. 5th ed. McGraw-Hill; 2009. [5] Eyring C. Reverberation time in dead rooms. J Acoust Soc Am 1930;1:217–41. [6] Millington G. A modified formula for reverberation. J Acoust Soc Am 1932;4:69. [7] Fitzroy D. Reverberation formula which seems to be more accurate with nonuniform distribution of absorption. J Acoust Soc Am 1959;31:893. [8] Arau-Puchades H. An improved reverberation formula. Acustica 1988;65:163–80. [9] Chrisler V. Dependence of sound absorption upon the area and distribution of the absorbent material. J Res Natl Bur Stand 1934;13:169–87. [10] Cook R. Absorption of sound by patches of absorbent material. J Acoust Soc Am 1957;29:324–9. [11] Bruijin A. Calculation of edge effect of sound absorbing structures. PhD thesis. Delft, Holland; 1967. [12] Guicking D. Theoretical evaluation of the edge effect of an absorbing strip of a pressure-release boundary. Acustica 1990;70:66–75. [13] Trevor J, D’Antonio P. Acoustics absorbers and diffusers. New York: Taylor & Francis; 2009. p. 388–93 [Ch. Hybrid surfaces].
192
D. Caballol, A.P. Raposo / Applied Acoustics 113 (2016) 185–192
[14] Hanyu A. Theoretical framework for quantitatively characterizing sound field diffusion based on scattering coefficient and absorption coefficient of walls. J Acoust Soc Am 2010;128:1140–8. [15] Omoto A. Comment on a theoretical framework for quantitatively characterizing sound field diffusion based on scattering coefficient and absorption coefficient of walls. J Acoust Soc Am 2013;133:9. [16] Kawakami F. Deep-well approach for cancelling the edge effect in random incident absorption measurement. J Acoust Soc Jpn 1998;19:327–38. [17] Thomasson S. Theory and experiments of the sound absorption as function of the area. Report TRITA-TAK8201. Stockholm: Dept of Acoustics, Royal Institute of Technology; 1982. [18] Jeong C. Converting Sabine absorption coefficients to random incident absorption coefficients. J Acoust Soc Am 2013;133:3951.
[19] ISO 354:2003, Geneva, Switzerland. Acoustics – measurement of sound absorption in a reverberation room; 2015. [20] ISO 266:1997, Geneva, Switzerland. Acoustics – preferred frequencies; 2013. [21] Kawai Y, Meotoiwa H. Estimation of the area effect of sound absorbent surfaces by using a boundary integral equation. Acoust Sci Technol 2005;26:123–7. [22] Peña D. Regresión y diseño de experimentos. Madrid: Alianza Editorial; 2002. [23] Weisberg S. Applied linear regression. 2nd ed. New York: John Wiley & Sons; 1985. [24] Yan X, Gang Su X. Linear regression analysis. Singapore: World Scientific; 2009.