Confidence in room acoustic design: an empirical approach for classrooms

Jack Harvie-Clark 1 , Weigang Wei 2

Apex Acoustics Ltd, Design Works, William St, Gateshead, NE10 0JP, UK

ABSTRACT

Methods to predict reverberation time range from the simple Sabine or Eyring relation to more elaborate means such as geometric and numerical acoustic modelling. Classrooms often have a very uneven distribution of sound absorption; measured reverberation times can exhibit a much wider range of values between equivalent rooms than prediction methods would generally suggest. The analytical tools can commonly fail to provide what the designer actually needs: a simple and robust method to advise on solutions for sound absorption that achieve the stated reverberation time criteria with confidence, without excessive prudence. This paper presents an empirical approach to room acoustic design for classrooms with free-hanging absorbent rafts, wall panels and furniture. Statistical analysis of the measured reverberation time in over a hundred rooms is used to determine appropriate design points. With this approach, the acoustic designer can not only describe the proposed requirements to the design team, they can also identify with a stated confidence the anticipated compliance rate. The method could be extended to other indicators such as Clarity, C 50 , and Strength, G.

1 jack.harvie-clark@apexacoustics.co.uk

2 weigang.wei@apexacoustics.co.uk

1. INTRODUCTION

Traditionally, practitioners have sought to explain the acoustic response of rooms by proposing theories that may have a wide range of applications. Methods of analysis range from the simple Sabine or Eyring relation to more elaborate analytical compositions to predict reverberation time in spaces where there is an uneven distribution of absorption; see Arau-Puchades [1] for a range of formulae. Geometric room acoustic modelling and finite element methods are computational approaches to analysis of potential room response.

However, these tools can all commonly fail to provide what the designer actually needs: this is a simple robust method to advise the design team on solutions for sound absorption that can comply with the stated performance requirements, typically a reverberation time criterion, with confidence, but without excessive prudence that would incur unnecessary expense. The present work demonstrates how an empirical approach to room acoustic design can be adopted for classrooms with free-hanging absorbent rafts, wall panels and furniture. Statistical analysis of the measured response in over a hundred rooms has been undertaken. A level of confidence is determined for the design points identified.

The reasons for the variation of measured values is discussed. Rather than trying to fit an acoustic theory to the measured response, the approach presented in this paper uses the measured response to empirically determine the optimum design advice, based on the confidence interval that the practitioner wishes to adopt. Thus the acoustic designer can not only describe the proposed requirements to the design team, they can also identify with a stated confidence the anticipated compliance rate. Conversely, it enables a quantitative assessment of the confidence (or chance) of achieving a particular result on the basis of a quantity of absorption in a room.

The room acoustic response of classrooms is notoriously difficult to predict; uncertainty in predicting reverberation time is due to the varying diffuseness of the sound field, such that theoretical models based on a diffuse field are not always accurate. This highlights that fact that reverberation time may not be the optimum indicator for acoustic response in classrooms, according to Harvie-Clark et al [2], for example, and Minelli [3]. Alternative indicators that may be more appropriate include Clarity, C 50 , Unfavourable ratio, U 50 , and Strength, G. Theoretical models that can better predict these indicators have been developed by Nilsson et al [4], although the uncertainty in application to classrooms is unknown. The same approach as used here for reverberation times could also be extended to evaluate uncertainty in other indicators.

The method is extended to identify the potential range of scattering coefficients, again based on confidence intervals, that may be appropriate based on the measured responses. The range of scattering coefficients may have a broader application beyond the immediate scope, and can be used to model the potential range of acoustic response in rooms of this type.

For new build schools and material change of use from non-school building in England, it is a legal requirement that the spaces achieve certain performance standard as specified in Building Bulletin 93 [5] (BB 93). To achieve good speech intelligibility, BB 93 specifies performance requirements in terms of the mid- frequency reverberation time, T mf , which is the arithmetic average of the reverberation times in 500 Hz, 1k Hz and 2k Hz octave bands.

2. TEST SAMPLES

The reverberation time of 131 school rooms were measured, with room volumes ranging from approximately 120 m 3 to more than 300 m 3 . The majority of rooms have volumes between 150 m 3 and 200 m 3 as shown in Figure 1. These school rooms include Classrooms, ICT, Science, Music, Workshop and Others and the

breakdown of each room type is shown in Figure 2. All the rooms included in this review had a soffit height of 3.45 m.

Figure 2: Sample of room types

Figure 1: Sample of room volumes

Absorbing wall panels were installed to most of the rooms and the percentage of the wall absorbing area (in Sabines) to total absorption area ranges from about 10 % to 40 %. Approximately half the schools were primary schools, and half were secondary schools. Typical primary school (Figure 3, Figure 4) and secondary school classrooms (Figure 5, Figure 6) are illustrated. The classrooms all have a ventilation unit within a bulkhead on one side of the soffit. This has a particular type of ceiling tile beneath, and a plasterboard bulkhead. The primary schools have bespoke rafts, typically 2.4 m x 1.2 m, suspended 200 mm from the soffit. Raft sizes vary where they are fitted around and between structural steelwork. Wall panels are generally mounted above 2.1 m, and sometimes extend above the level of the rafts. The secondary schools generally have rafts composed of ceiling tiles, as shown in Figure 5 and Figure 6. These are arranged in one or more blocks, depending on the structural steelwork. The secondary schools have the same bulkhead arrangement to contain the ventilation unit.

m<150m3 1150-200 m3 1200-300 m3. m1 >300m3

2.1 Data collection

At the time of the reverberation time measurement, a detailed measurement with a tape measure was made of the sound absorption in each room, including wall panels and rafts. The products were known, and laboratory sound reduction data was used for each. The room volume was calculated by excluding the volume of the bulkhead containing the ventilation unit. The volume above the suspended rafts was included in the calculation, for both the bespoke rafts and rafts made of ceiling tiles. Textbook vales are ascribed for the plasterboard walls and floor coverings.

Reverberation time was measured in octave bands with Engineering accuracy according to ISO 3382-2 [6], using an omni-directional loud speaker according to ISO 3382-1, [7]. The performance criteria relate to normally furnished rooms, but all rooms were unfurnished at the time of testing; therefore diffusers were used to simulate furniture. The diffusers comprised two pieces of plywood, each approximately 1.2*0.6 m, with a taped hinge; four diffusers were used in each room, and two are just visible in Figure 5.

R 93 (Classrooms m ICT sm Science = Music Workshop m Other

Figure 3: Typical Primary school with 2.4 * 1.2 m rafts and wall panels

Figure 4: Primary school classroom. The underside of the bulkhead can

be seen in the foreground, top left of the picture.

Figure 6: Secondary school classroom showing the bulkhead and single block of ceiling tile raft.

Figure 5: Secondary school with rafts of

ceiling tiles divided by the structural

Diffusers are visible in the foreground

steelwork.

3. EMPIRICAL DATA ANALYSIS

3.1 Potential form of relation between absorption and reverberation time

The Sabine equation (as used in BS EN 12354-6 [8], for example), describes a relation between the reverberation time, room volume and sound absorption. However, due to the spatial distribution of absorbing materials and scattering conditions, the Sabine equation is not always reliable for predicting the reverberation time in school rooms; these can often have an uneven distribution of absorption and lack of diffusing surfaces. In well-damped spaces, the Eyring-Norris equation may be expected to be more accurate. Four experimental functions are tested for reverberation time prediction to investigate potential for a better estimate of performance along with confidence intervals.

3.2 Algorithm and fitting functions

The least mean squared method is used in the data fitting. This method finds the set of parameters that minimizes the squares of the error function (the difference between the predicted reverberation time and the measured reverberation time). The tested functions are shown in Table 1:

Table 1: Functions tested for RT prediction

Tested function Purpose

𝑅𝑇= 𝑝𝑉𝐴 ⁄ Find correlation coefficient for Sabine relation

𝑅𝑇= 𝑎+ 𝑝𝑉𝐴 ⁄ Test the adapted Sabine relation

𝑅𝑇= 𝑉 𝑝𝐴 𝑤 + 𝑞𝐴 𝑐

Check whether the contributions from wall &

ceiling absorption are significantly different

𝑅𝑇= 𝑝 𝑉 𝑆 𝑇 𝑙𝑛(1 −𝛼)

Find correlation coefficient for Eyring relation

Where: RT = Reverberation time / s A = sound absorption, calculated from the surfaces and elements in the room / m 2 Sabines A w , A c =Absorption on walls and absorption on ceiling / floor respectively V = room volume / m 3 α = mean absorption coefficient, ie A/S where S = total surface area a, p = empirical coefficients fitted from the data.

The offset parameter, a, is included as it is observed that the data does not align with the origin. The regressed fitting parameters and their corresponding standard deviations of the difference between the predicted T mf and measured T mf are shown in Table 2.

Table 2: Prediction models and the standard deviations

Regressed parameter values

Tested function Standard deviation

500 Hz 1k Hz 2k Hz

𝑅𝑇= 𝑝𝑉𝐴 ⁄ 0.098 p = 0.154 p = 0.145 p = 0.153

a = 0. 346 ,

a = 0.336 ,

a = 0.372 ,

𝑅𝑇= 𝑎+ 𝑝𝑉𝐴 ⁄ 0.070

p = 0.083

p = 0.069

p = 0.060

p = 1.091

p = 0.902

p = 0.869,

𝑅𝑇= 𝑉 𝑝𝐴 𝑤 + 𝑞𝐴 𝑐

0.097

q = 0.993

q = 1.187

q = 1.110

𝑅𝑇= 𝑝 𝑉 𝑆 𝑇 𝑙𝑛(1 −𝛼) 0.105 p = 0.168 p = 0.162 p = 0.178

The function 𝑅𝑇= 𝑎+ 𝑝𝑉𝐴 ⁄ has the smallest standard deviation when fitting the data, suggesting the least uncertainty to predict the reverberation time. The fitted scatter plots are shown in Figure 7, Figure 8 and Figure 9 for the 500 Hz, 1 kHz and 2 kHz frequency bands respectively.

Figure 7: RT prediction model – 500 Hz

Figure 8: RT prediction model – 1k Hz

Figure 9: RT prediction model – 2k Hz

3.3 Discussion of results

The first notable feature of the regressed parameters is that for the traditional Sabine construct, the constant has a value of 0.154, 0.145, 0.153 for the 500 Hz. 1 kHz. 2 kHz octave bands respectively – all lower than the value of 0.163 that is derived from BS EN 12354-6, based on air at 20 degrees C and hence a speed of sound of 343 m/s. Thus a design based on using a constant of 0.163 is going to include an (unknown) margin of safety, as a greater provision of sound absorption will be indicated than is required.

The illustration of the data in Figure 7, Figure 8 and Figure 9 indicates the best fit straight line does not pass through the origin. Figure 10 shows both the straight lines with and without the constraint of the relation going through the origin. The wide spread of data points on the graph also illustrates why any alternative equation does not result in much more reliable predictions. Considering the wall-mounted (vertical) and soffit / floor positioned (horizontal) sound absorption does not result in any greater correlations – the best-fit parameter values are similar, but most notably there is a different priority in different frequency bands.

Alternative compositions of the data have been considered, but not investigated at this stage. These include:

• Omission from the room volume of the space above rafts (both types)

• Inclusion of some sound absorption for the area between the rafts of ceiling tiles and soffit

These types of data composition have been found to provide better correlations between predictions and measurements with smaller data sets in previous (unpublished) work. These remain as future work to investigate for this data set.

Figure 10: 1 kHz values. Blue line shows RT = 0.145.V/A , yellow line shoes RT = 0.336 + 0.069.V/A

3.4 Correlations for mid-frequency absorption and mid-frequency reverberation time

However, for the purposes of complying with BB 93, for rooms of this type and style there are two common design points – to meet either a reverberation time of 0.6 or 0.8 seconds T mf , being the arithmetic average of the 500 Hz, 1 kHz and 2 kHz octave bands. These are the current reverberation time limits for primary and secondary school classrooms respectively. The performance requirement is described to one decimal place, therefore we can surmise that a value of 0.64 complies with the “less than or equal to” 0.6 seconds, for example.

Analysis of octave band values is not necessarily helpful – we don’t need each of the individual octave bands to achieve the same target, we need the average to achieve the target. For these room types the reverberation time is relatively well balanced between octave bands in these centre frequencies, so the additional constraint may be small. Figure 11 shows the three octave bands superimposed. If for example the 500 Hz band RT was a little longer and the 2 kHz band RT a little shorter on average, that could also meet the requirements. Therefore a mid-frequency analysis will be easier to use in practice, based on the average of the 500 Hz, 1 kHz and 2 kHz octave bands.

3.5 Calculated v.s measured reverberation time

It is useful to compare the reverberation time calculated according to BS EN 12354-6, to offer a more familiar depiction of calculated v.s measured parameters. Figure 12 demonstrates that the relation in EN 12354-6 is on average almost perfect for this data. This is an important result – demonstrating that in these rooms with this quantity of absorption on the walls, and using the diffusers during measurements, there are on average no significant adverse effects from a horizontal sound field that may cause excessively long reverberation times compared to predicted reverberation times [4]. However, this depiction of the data can make it difficult to assess the uncertainty in the prediction, as the data are not normally distributed. The standard deviation of the error (Calculated – Measured T mf ) is 0.092 seconds, or 14.4 %.

Figure 11: Correlations for each of the 500 Hz, 1 kHz and 2 kHz octave bands superimposed

Predicted v.s Measured T mf

1.20

y = 1.0091x R² = 0.9812

1.10

1.00

Measured T mf / s

0.90

0.80

0.70

0.60

0.50

0.40

0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10 1.20

Predicted T mf / s

Figure 12: Calculated T mf based on EN 12354-6, against measured T mf

Measured Tmf, s es 2 oP BR we S § 9 & 6 KR & VIA

4. APPLICATION TO DESIGN FOR BUILDING REGULATIONS COMPLIANCE

The relationship between the mid-frequency reverberation time the mid-frequency sound absorption can be calculated. The arithmetic mean of the sound absorption in the 500 Hz, 1 kHz and 2 kHz octave bands is calculated, and compared with the arithmetic mean of the reverberation times in the same frequency bands, in the same manner as for individual octave bands. This is merely a correlation between input parameters – the sound absorption – and the output indicator, the “mid-frequency reverberation time”. There is no theoretical relationship between smaller frequency band reverberation times and a larger frequency band.

As the data is not normally distributed around the mean values, using the standard deviation to determine confidence intervals does not yield meaningful results. An alternative method to determine an appropriate deviation from the mean values is required.

One method to assess a deviation from the mean is to shift the correlation line until a certain number of data points sit beyond it. If we are comfortable with achieving the performance criterion 90 % of the time, for example, we can shift the correlation line until only 10 % of the data points sit beyond it. This approach is used below.

Figure 13 shows the mid-frequency correlation between V/A mf and T mf . The blue line shows the best fit regression going through the origin. The red line has the same slope but is shifted so that only 10 % of the data lie above it. The equation for the red line is shown on the graph.

Figure 13: Mid-frequency sound absorption and mid-frequency reverberation times

As noted, we are most commonly concerned with designing to meet a requirement of 0.6 or 0.8 seconds, meaning an upper limit of 0.64 or 0.84 seconds to two decimal places. These points are indicated in Figure 13, with the data points shown in Table 3 for a typical classroom with floor area of 55 m 2 and volume of 190 m 3 .

As well as the mid-frequency sound absorption required, the equivalent reverberation time target calculated according to EN 12354-6 is also indicated for these design points, i.e. the reverberation time that would be calculated with this quantity of absorption in the example room using that method.

Table 3: Examples of design points for a room of 55 m 2 area, 190 m 3 volume

T mf / s A mf / m 2 V/A / m EN 12354-6

T / s

0.64 56 3.4 0.54

0.84 40 4.7 0.75

This means that using a conventional design calculation according to EN 12354-6 with a target of 0.54 seconds will achieve a measured result not exceeding 0.64 seconds in 9 out of 10 cases, for example. Reviewing these design points in Figure 13 indicates that all the values that lie above the red line are at lower values of V/A; hence it would appear that designing this way to achieve the 0.8 second criterion carries little risk, as there are in fact no data points where the value of V/A is less than 4.7 m and the measured reverberation time is greater than 0.84 seconds. Considering the 0.6 second criterion, there are two data points (ie rooms) where the value of V/A is less than 3.4 m, but the 0.6 second criterion is not achieved. Based on the sample data, two in about 20 rooms may be expected to fail to meet the 0.6 second requirement.

5. SCATTERING COEFFICIENTS IN ACOUSTIC MODELLING – PRELIMINARY WORK

Having demonstrated that a simple calculation according to EN 12354-6 is, on average, almost exactly correct, but that there remains a significant amount of uncertainty in the outcome for any particular room, what additional information could geometric acoustic modelling provide? There may be a desire to understand other room acoustic parameters such as Strength and Clarity [2], or to understand the spatial variation of acoustic conditions. The sound absorption coefficients for the proposed materials may available and are required to assess the reverberation time; however, different consultants may use different data or make different judgements for scattering coefficients, for which there is much less consensus. Uncertainty in scattering coefficients can have a significant effect on model predictions. There are some approaches such as the Lambert method [9] to estimate the scattering coefficients based on the geometrical dimension of the elements. However, even using this estimation, the predicted reverberation times can include considerable uncertainty due to the uncertainty associated with the scattering coefficients.

Based on the measured reverberation time in 76 classrooms (with room volume approximately 190 m 3 and the measured average 0.6 s T mf ), a simple CATT model is established to find the appropriate scattering coefficients by adjusting the scattering coefficients to match the simulated and measured reverberation times in octave bands. The overall view of the model is shown in Figure 14.

Although the T mf requirements are based on a normally furnished room condition, it is common that when the compliance test is carried out the room is often empty. In this case, the scattering from the walls, floor and ceiling may be the same. Therefore, in this preliminary work, the CATT model scattering coefficients for the walls, floor, and ceiling are assumed to be the same. Other details of the model are shown in Table 4. The appropriate scattering coefficients to achieve 0.6s T mf with 95% confidence are shown in Table 5.

Figure 14: CATT model used to determine the scattering coefficients

Table 4: Details of CATT modelling

Frequency, Hz

Parameter

Comments 500 1000 2000

RT target, s 0.69 0.59 0.57 Average of 76 classrooms

Absorbing areas of

ceiling, m 2 43 35 34 80% of the total absorbing area

Absorbing areas of

walls, m 2 11 9 9 20% of the total absorbing area. Wall

panels are installed above 2 m high

Room volume, m 3 190 Average of 76 classrooms

Floor area, m 2 56

Soffit height, m 3.4

Table 5: Scattering coefficients in acoustic modeling for classrooms

Frequency, Hz

500 1000 2000

Scattering coefficients to achieve

0.6 s T mf with 95% confidence 0.25-0.28 0.50-0.60 0.50-0.65

oon onn

Scattering coefficients for rooms with similar surface conditions as empty classrooms may be similar as the values in Table 5 as well, even though these scattering coefficients are specifically tested based on measurements in classrooms. This is preliminary work at present; it may be developed by enhancing the model to be more representative of the classrooms tested, including furniture, allowing different scattering coefficients on different surfaces, and exploring the range of scattering coefficients that yield a similar range of reverberation times as found in the measurements. This would be most relevant for rooms which exhibit a non-monotonic reverberation decay curve.

6. CONCLUSIONS AND FURTHER WORK

Measurements of 131 school rooms are used to find the experimental approach to predict the reverberation time. It is calculated that the relation 𝑅𝑇= 𝑎+ 𝑝𝑉𝐴 ⁄ has the least error to predict the reverberation time in these school rooms. The parameters a and p for each frequency are fitted based on the measurements. As the data does not follow a normal distribution, a manual method is used to identify appropriate design points to achieve the most common performance requirements. Further statistical work could provide more insight into the uncertainty of achieving particular design goals.

The required absorbing areas to achieve 0.6 s and 0.8 s T mf are identified as an example of the application. The potential range of scattering coefficients to achieve T mf requirements for geometric acoustic simulations are explored by matching the reverberation time of the CATT model and measurements.

The findings are not considered to be readily extendable to other rooms with different geometries, distribution of absorption or surface conditions.

7. ACKNOWLEDGEMENTS

We are grateful to all our colleagues who helped form the ideas and response in this paper.

8. REFERENCES

1. H Arau-Puchades. Sound Pressure Levels in Rooms : A Study of Steady State Intensity, Total Sound Level, Reverberation Distance, a New Discussion of Steady State Intensity and other Experimental Formulae. Building Acoustics, Sept 1, 2012. 2. J Harvie-Clark, N Dobinson, F Larrieu. Use of G and C50 for classroom design . Proc IOA Vol 36 Pt.3 2014. 3. G Minelli, G E Puglisi, A Astolfi, Acoustical parameters for learning in classroom: A review , Building and Environment, Vol 208, 2022, 4. E Nilsson, E Arvidsson. An Energy Model for the Calculation of Room Acoustic Parameters in Rectangular Rooms with Absorbent Ceilings .. Appl. Sci. 2021, 11, 6607. 5. Building Bulletin 93: Acoustic design of schools: performance standards, February 2015 6. BS EN ISO 3382-2:2008. Acoustics. Measurement of room acoustic parameters - Reverberation time in ordinary rooms. 7. BS EN ISO 3382-1:2009. Acoustics. Measurement of room acoustic parameters - Performance spaces 8. BS EN 12354-6:2003 Building acoustics. Estimation of acoustic performance of buildings from the performance of elements Sound absorption in enclosed spaces

9. CATT-A v9.0 User's Manual CATT-Acoustic V9, 2011