Using Sabine’s and sound power relation to study: The effect of diffusers on measured sound absorption

Abstract

The purpose of this study is to evaluate the effect of the number of diffusers placed in the same room, as well as the effect of diffusers with a flat area (8.4, 13.2, and 18 m²) on the calculated sound absorption coefficient in a reverberation room, when an iterative procedure based on the evaluation of the sound power level is applied (this is done by using the sound power equation and substituting with different values of equivalent absorption area on both sides of the transformed equation to reach the correct value for absorption area, which achieves equality on both sides of the equation in order to achieve the goal). The sound absorption coefficient at random incident of three materials having different degrees of absorption—has been measured in a reverberation room according to ISO 345: a highly absorbent material (sponge) a medium absorbent material (rubber) and a low absorbent material (wood).There is a noticeable convergence between the calculated and measured sound absorption coefficient from the usual ISO 354 procedure and from the sound power level equation from ISO 3741,when diffusers are present. The difference in sound power level is close to 1.0 dB depending on the position of the source with the diffuser area. Thanks to the diffusers effect, the difference between the measured and the calculated sound absorption coefficient can be considerably reduced.

Keywords

Sound absorption diffusers sound power estimation iteration equivalent absorption area

Introduction

Benedetto et al.¹ studied the effect of the position of the absorbing materials in the reverberation room, as well as the position of the sound source and the microphone, on the sound absorption coefficient of the materials; they recommended to take more than one value of the reverberation time to be sure the sound field in the room is diffuse enough. Diffusers should be employed in an uneven pattern to improve diffusion. Their diffusers, on the other hand, were large and reflective diffusers placed in the room’s corners. Toyoda et al.² used the ray tracing method (geometrical acoustics) to investigate the influence of sound propagation and room shape on the sound absorption coefficient. They concluded that the sound propagation in the reverberation room strongly decreases when placing a highly absorbent material in the room. Nolan et al.³ used two acoustical parameters to study the effect of volume diffusers on the reverberation room diffusivity; the equivalent sound absorption area and the diffuse field factor. Drabek et al.⁴ indicated that the design of a reverberation room is based on determining the size and shape of the room according to the relevant specifications, and that the difference inside the room is due to the selection of measurement positions for the reverberation time, all due to the non-uniform spread of the sound field in the room. Vercammen⁵ recommended to correct distribution of diffusers in the room, as well as deducting the size of the diffusers from the size of the room, as well as the use of volumetric diffusers instead of the flat ones.

Lautenbach and Vercammen⁶ compared volume diffusers and panels diffusers, finding that volume diffusers increase the sound diffusivity inside the room and making a model on a scale of 1:10 for the real reverberation rooms. Kim and Jeon⁷ found that it is possible to improve the diffusivity, but the only reliable indicator of the diffusivity is the sound absorption of the sample itself. Berardi and Arau-Puchades⁸ mentioned that the use of volume diffusers instead of flat ones spread sound clearly in music halls, improving the acoustic properties of the music rooms and listening. Stephenson et al.⁹ showed that starting from different assumptions on the sound field behavior different reverberation formulas are obtained; no one of them can fully explain the correct propagation of the sound field in the reverberation room.

Cops et al.¹⁰ described the measurement of the absorption coefficient of different types of absorbing materials with totally different absorption performances, taking into account the influence of equipment control, source and microphone positions, diffuser versus diffusivity, the position of the absorbing material within the room, and the change in configuration of the reverberation room. Vercammen¹¹ pointed out the insufficient diffuse sound field in a reverberation chamber when a highly absorbing test sample is inside; the shape of the reverberation room and the placing of diffusers do influence the result. Mee and Vallis¹² present an analysis about the importance of diffusion levels and the effects of diffuser panels on the variations in the sound field in rooms. Lautenbach and Vercammen¹³ mentioned that the equipment of the reverberation room is made of materials with some sound absorption, and the noise signal energy inside the reverberation rooms should be more widespread so that there is no difference when making comparisons using the standard deviation of the reverberation time.

In a recent inter-laboratory test, Scrosati et al.¹⁴ investigated some proposed variants of the ISO 354 procedure to get more reliable measurements of sound absorption coefficient in reverberation rooms.

It is worth that, in general reverberation time can be measured according to three different methods: the interrupted noise method,¹⁵ the integrated impulse response method^15,16 and the energy detection method.¹⁷ In this study, the interrupted noise method has been used.

Objective of study

The goal of this research is to estimate the sound absorption coefficient and explore the effect of diffusers in the standardized procedures for sound absorption (ISO 354) and sound power level (ISO 3740) measurements. The effect of diffusers on the estimated sound absorption coefficient is investigated in this study using the Newton iterative method on the sound power level formula; the calculated sound absorption coefficient is then compared with the measured sound absorption coefficient (using the Sabine formula).

Measurements in the reverberation room were done following these steps:

(1) Reverberation time and sound pressure level measurements in the reverberation room with different diffusers,

(2) Calculation of the effect of the diffusers on the sound power level of a reference sound source (as per ISO 3741),

(3) Calculation of the effect of the diffuser on measured sound absorption coefficient (α_m) for different materials (using the Sabine formula as per ISO 354),

(4) Evaluation of the diffusers’ effect on calculated sound absorption coefficient (α_c) for different materials using iteration process (see equation (10) below and related explanation).

(5) Obtaining the sound absorption coefficient (α_c) from sound power for different materials.

System setup

The measurements were carried out in the reverberation room of total size 160 m³ and total surface area 178 m², with a non-parallel surfaces and unequal room dimensions. Reverberation time and sound pressure level were measured in the reverberation room with different diffusers hanged from the room ceiling with different heights ranged between 0.7 m to 1.2 m. The total flat area of the diffusers in three different cases is, respectively: 8.4 m², 13.2 m², and 18 m², (11 diffusers, each one having area 1 m × 1.2 m, and two diffusers, each one with area 2.4 m × 1 m).

Figure 1 shows the diffusers distribution in the room and the reverberation room dimensions with sample and sound source locations. According to ISO 354 requirements,¹⁷ the sound source height from the room floor is 1.5 m, the spacing between any two microphone positions is 1.8 m, and the source-to-microphone distance is 1.8 m. The sample location is distant from walls by 1 m. The absorbent material had an area of 9.36 m² with dimensions of 3.6 m × 2.6 m. The material was located at two different locations Y and Z in the room, for each sample location the sound source had two locations A and B as well. The distance between the two locations A and B is 2.2 m and it is more than 1.3 m from the nearer room walls, see Figure 1. A reference sound source B&K type 4204 was located on the room floor at two locations A and B; for each location of the sound source, 6 positions for microphone measurements were considered.

Figure 1.

Left: diffusers distribution in the room; right: reverberation room dimensions with sample positions (Y and Z) and sound source (A and B) locations.

Experimental methods

Sound power measurements in the reverberation room

Instrumentations and measurement procedure

Firstly, measurements were carried out to qualify the reverberation room. Two sound source locations, labeled A and B, were chosen to measure the reverberation time T_e in the empty reverberation room without diffusers. Sound source B&K type 4296, precision sound level meter B&K type 2260 with software 7204, and power amplifier B&K type 2716 were used in the tests. Sound level calibrator B&K type 4231 was used to adjust the instruments before beginning the measurements.

A calibration of the reference sound source (RSS) was carried out in the reverberation room according to ISO 6926 and ISO 3741 at the two selected source locations A and B.^18,19 The values of sound power level (L_w) for the reference sound source at the two locations are reported in Table 3. Measurements of the sound pressure level (L_sp) were carried out according to ISO 3741 standard and then the sound power level (L_w) was determined for the reference sound source (RSS) B&K type 4204, at A and B locations, with six microphone positions for each location. These measurements were then repeated with diffusers of area total of 8.4, 13.2, and 18 m² in the reverberation room; the respective sound power is designated as L_w8, L_w13, and L_w18.

Sound power level calculations

ISO 3741 prescribes to calculate the sound power L_w of RSS from the following formula (1):

L_{W} = \bar{L_{p (ST)}} + {10 \log \frac{A}{A_{0}} + 4.34 \frac{A}{S} + 10 \log (1 + \frac{Sc}{8 Vf}) + C_{1} + C_{2} - 6}

(1)

Where;

$\bar{L_{p (ST)}}$ : the average sound pressure level one-third-octave band time averaging over all source and microphone positions, in dB.

A: equivalent sound absorption area of the test room, in m².

A = [\frac{55.26 * V}{C}] [\frac{1}{T}]

(2)

A₀ = 1 m²: reference value

S: the total surface area of the reverberation room, in m²

T: reverberation time of the room, in seconds

c: speed of sound, in m/s

V: volume of the room, in m³

f: mid-band frequency, Hz

C₁, is the reference quantity correction, in decibels, to account for the different reference quantities used to calculate decibel sound pressure level and decibel sound power level, see ISO 3741¹⁹

C₂, is the radiation impedance correction, in decibels, to change the actual sound power relevant for the meteorological conditions at the time and place of the measurement into the sound power under reference meteorological conditions, see ISO 3741¹⁹

- The standard deviation of the measured sound pressure level is:

S_{S} = \sqrt{\frac{1}{N s - 1} \sum_{i = 1}^{Ns} {(L_{p, i} - L_{pm})}^{2}}

(3)

where

N_s: number of combined source-microphone positions

L_pi: the band time-averaged sound pressure level

L_pm: is the arithmetic mean of the band time-averaged sound pressure levels

Table 1, shows the environmental conditions in the reverberation room during measurements.

Table 1.

Environmental conditions in the reverberation room during measurements.

Measured quantity	Temperature	Static pressure	Relative humidity
Measured quantity	t₀ (°C)	p_s (kPa)	H (%)
L _w	22.5 ± 0.5	101.1 ± 0.1	55 ± 1
Α	22 ± 0.5	101.2 ± 0.1	55 ± 1

Uncertainty determination

According to ISO 3741 standards, the uncertainties of the sound power level u(L_W), have been calculated from the total standard deviation (σ_tot) in decibels (dB) as:

u (L_{w}) = σ_{tot}

(4)

The total standard deviation σ_tot can be determined as:

σ_{tot} = \sqrt{σ_{Ro}^{2} + σ_{omc}^{2}}

(5)

σ_Ro is the standard deviation of reproducibility of the method:

σ_{Ro} = \sqrt{{(c_{1} u_{1})}^{2} + {(c_{2} u_{2})}^{2} + \dots + {(c_{n} u_{n})}^{2}}

(6)

and σ_omc is the uncertainty due to the operating and mounting conditions:

σ_{omc} = \sqrt{\frac{1}{N - 1} \sum_{j = 1}^{N} {(L_{p, j} - L_{av})}^{2}}

(7)

where

L_p,j: sound pressure level measured at a position

L_av: arithmetic mean level calculated for all the repetitions

N: number of microphone positions

Sound absorption measurements in the reverberation room

Instrumentations and measurement procedure

According to ISO 354 requirements, the sound source height from room floor was 1.5 m, the spacing between any two microphone positions 1.8 m, the source-microphone distance 1.8 m.¹⁵ The position of the sound-absorbing material in the reverberation room is affected by the degree of diffusion of the sound field and its strength.²⁰ For each sample location the sound source had two locations A and B as well (see Figure 1). The measurements were repeated three times for the different materials presented in Table 2.

Table 2.

Characteristics of the measured materials in the reverberation room.

Material	Density	Thickness	Air-permeability
Material	(kg/m³)	(m)	(cm³/cm²/s)
Wood (W)	582	0.016	0.0
Rubber (R)	50.71	0.027	0.7
Sponge (S)	19.5	0.047	148

Calculations of Sabine and iterative sound absorption coefficients

The reverberation time values measured in the reverberation room, empty and with the samples, are labeled as follows in Section 5:

T_e totally empty room, without diffusers and samples,

T_ef in the empty room with diffusers only,

T_x in the room without diffusers, but with a sample of material,

T_fx in the room with diffusers and a sample of material.

The reverberation time with a sample of material inside the reverberation room is further specified as: T_R for the rubber sample, T_w for the wood sample, and T_s for the sponge sample.

The calculation procedure can be divided into two main steps.

Step 1: From the measured reverberation times with and without the sample, the “measured” equivalent sound absorption area (A_m) and the “measured” sound absorption coefficient (α_m) are calculated using the Sabine formula as follows:

A_{T} = A_{2} - A_{1} = 55.3 V [\frac{1}{C_{2} T_{2}} - \frac{1}{C_{1} T_{1}}] - 4 V (m_{2} - m_{1})

(8)

Where:

T₁ and T₂ are the reverberation time of the room without and with sample, respectively, in s.

c is the speed of sound at temperature t, in °C:

c = (331 + 0.6 t) m / s

(9)

for constant temperature t₀, then c₁ = c₂ = c.

The sound power attenuation coefficient in air, m₁ and m₂, can be calculated from ISO 9613-1.²¹

Then, the measured sound absorption coefficient α_m can be calculated, knowing the sample area S, as α_m = A_m/S.

Step 2: the equivalent absorption area (A_C) is iteratively calculated from formula (10) as follows (see equation (1) for comparison):

10 \lg \frac{A}{A_{0}} = L_{W} - \bar{L_{p (ST)}} - 4.34 \frac{A}{S} - 10 \lg (1 + \frac{SC}{8 Vf} + C_{1} + C_{2} - 6)

(10)

The calculated sound absorption coefficient (α_C) can be calculated knowing the sample area S as before.

This is an iterative calculation, meaning that the procedure tries to make equal the right side and the left side of equation (10) by searching the optimal value of A_c.

Repeatability of reverberation time measurements

The relative standard deviation of the reverberation time T₂₀, measured with the interrupted noise method, can be estimated by the following formula (15):

\frac{ε (T_{20})}{T_{20}} = \sqrt{\frac{2.42 + 3, 59 / N}{f \cdot T_{20}}}

(11)

Where:

Ɛ(T₂₀): is the standard deviation of the reverberation time;

T: is the measured reverberation time;

f: is the center frequency of each one-third-octave band;

N: is the number of decay curves evaluated.

Results and discussion

Diffusers’ effect on the sound power level of RSS

Diffusers’ effect on reverberation time (T)

Figure 2 shows that the average reverberation time (T₈, T₁₃, and T₁₈) curves for the diffuser’s presence of areas 8.4, 13.2, and 18 m². For the diffuser absence case, the reverberation time curves (Te) are convergent with each other after 1250 Hz, while the difference appeared clearly between all curves in the frequency range from 125 Hz up to 500 Hz. The lowest reverberation time appeared when using diffusers with an area of 18 m². The difference between two cases of reverberation time for diffusers absence and presence (T_absence − T_presence) appears at 125 Hz and its values ranges from 0.75 to 1.44 s. At frequencies upper than 125 Hz, the difference ranges from 0 to 0.58 s.

Figure 2.

Average reverberation time with and without diffusers, with different areas.

Diffuser’s effect on sound power level (L_w)

In Figure 3 it is clear the effect of diffusers cases (absence and presence) with different area on the sound power and represented by (L_w18, L_w13, L_w8, and L_we). Also it is clear the obtained sound power was effected by the average sound pressure level measurements for two cases of diffusers.

Figure 3.

Effect of diffusers on sound power in the reverberation room.

It is noticeable that the lowest values represent the average sound power level in the case of 18 m² diffusers (L_w18) and the highest average sound power level values represent the diffusers absence of (L_we). The difference between the highest and lowest values is approximately 1.1 dB. All values have almost the same behavior during the studied frequency range, as a result of the sound diffusivity effect on the average sound pressure level in the room.

Diffuser’s effect on average sound power level in A-weighting (L_w dBA)

The average sound power levels in dBA (using A-weighting table)¹⁷ of the sound source in the reverberation room are shown in Table 3. It highlights the effect of the area of the diffusers on the values of the sound power of the sound source, when the sound source was placed at two locations A and B, with different diffusers area for the two cases with and without diffusers in the room.

Table 3.

Sound power in dBA at source positions A and B with different diffusers area.

Area (m²)	Pos. A	L_w/A (dBA)	Pos. B	L_w/B (dBA)	L_wt (dBA)
18	L _w18/A	89.01	L _w18/B	88.99	89
13.2	L _w13/A	89.43	L _w13/B	89.41	89.42
8.4	L _w8/A	90.43	L _w8/B	89.79	90.12
No. diff.	L _w/A	90.18	L _w/B	89.82	90

The calculations carried out when placing the source in locations A are represented by the sound power level L_w/A in dBA, when the source was placed at locations B as L_w/B in dBA and when calculating the average value of the sound power for the two positions as L_wt in dBA.

Effect of increasing surface area (S) on the sound power

When diffusers are inserted in the room, having areas of 8, 13, and 18 m², the total surface area of the room (S) is increased by the area of used diffusers, assuming that it affects the accompanied sound power as:

For 8 m² diffusers, the total surface area of the room (S = 178 m²) will be increased to S = 186 m², with changes in sound power level represented as L_w8(186). Accordingly, for 13 m² diffusers, the surface area is S = 191 m² and the sound power level is L_w13(191), and for 18 m² diffusers, the surface area is S = 196 m² and the sound power level is L_w18(196). As appear in Figure 4, there is no noticeable difference between the two cases (diffusers area added or not added to the room surface area) on the sound power level. In absence of diffusers the total surface area is (S = 178 m²), and the sound power level is represented as L_we. As appear in Figure 4, there is not a noticeable effect of diffusers area on the sound power of the sound source in the room.

Figure 4.

Effect of change in total room surface area on sound power level.

Diffuser’s effect on the standard deviation (Ss)

The standard deviations in sound pressure level measurements with varied diffuser areas of 8, 13, and 18 m² (Ss₈, Ss₁₃, Ss₁₈, and Sse) in the reverberation room were determined using equation (3). Figure 5 shows how the area of the diffuser in the room affects the standard deviation values. The smallest standard deviation values were obtained with a diffuser area of 18 m², so it can be assumed that in this latter case, good sound diffusion in the room was achieved.

Figure 5.

Standard deviations of sound level in the reverberation room for different diffuser.

Sound absorption measurements in the reverberation room (α)

Diffuser’s effect on reverberation time of reverberation room contained materials

According to ISO 354, two source locations and six reception positions were used, totaling 12 independent source microphone combinations, employing diffusers with an area of 18 m². The sound source is at two locations in the room, and the microphones are at six positions with at least a 1.8 m distance between positions, 2 m from any sound source, 1 m from any room surface, and 1.7 m away from the test specimen.

Two situations (absence and presence of diffusers with areas of 18 m²) were considered testing different materials, in the reverberation room. The values of the reverberation time (T) for each situation and each sample were recorded in the empty reverberation room (T_e) and the room with samples (T_f) in different positions the reverberation time in the room was measured with two cases of diffusers.

Figure 6, shows the difference in the reverberation time for the three different materials in two cases of diffusers [absence (T_e) − presence (T_f)] in the reverberation room, represented as a sponge (T_SDIF), rubber (T_RDIF), and wood (T_wDIF), and for diffusers presence but without materials, it represented as T_eDIF. The highest difference in the reverberation time T_SDIF appears for higher porous material (sponge presence) as a higher sound absorption coefficient, whereas the small difference appears for poor absorption in reverberation time T_eDIF for empty reverberation room case (diffusers presence but without materials).

Figure 6.

Difference in the reverberation time for different materials, with and without diffusers.

Diffuser’s effect on equivalent absorption area of empty reverberation room (A_e)

Average reverberation time measurements were carried out in the reverberation room. In Figure 7, the equivalent sound absorption area of the empty room was calculated in two ways:

A- Using Sabine formula; for the diffuser’s absence the equivalent sound absorption area and the measured sound absorption coefficient of the empty room are represented as A_em and α_em, and when the room contained the diffusers it is represented by A_efm and α_efm.

With many diffusers in the room, multiple reflections occur, and more sound energy is scattered, so the amount of energy absorbed by the specimen increases, resulting in higher sound absorption. This increase in absorption revealed a direct relationship between sound diffusion and sound absorption.^22,23 When diffusers are added, the mean free path (MFP) becomes shorter and therefore the sound waves hit the surfaces more often. This might be partially responsible for the increase in measured sound absorption as diffusers are.²⁴

B- Using equation (10); for the diffusers’ absence case the estimated equivalent sound absorption area and the estimated sound absorption coefficient of the empty room, it represented as A_eC and α_eC, and when the room contained diffusers, it is represented by A_efC and α_efC.

From Figure 7, by comparing the two curves of the effect of the diffuser, we noticed that there is almost no noticeable difference between A_efC and A_efm, meaning that the inferred equation (10) is valid for equivalent sound absorption area calculations.

Figure 7.

The diffusers effect on calculated and Sabine absorption area of the reverberation room wall surfaces.

In general, Figure 8, shows that there is no noticeable difference in the sound absorption coefficient for different cases (α_em, α_ec, α_efm, and α_efc). The difference between the four curves over the whole frequency range does not exceed 0.5% for the absorption of the walls. The values of the sound absorption coefficient of the wall surfaces α_em, α_ec, α_efm, and α_efc are in the range of 0.02–0.09, which is a very low value. So it is possible to calculate the sound absorption coefficient α from equation (10) for all surfaces of the empty room.

Figure 8.

The diffusers effect on calculated and Sabine sound absorption of the reverberation room wall surfaces.

Diffuser’s effect on sound absorption of sponge

Average reverberation time measurements were carried out in the reverberation room when contained sponge material. In Figure 9, the total sound absorption area of the room with the sponge is calculated in two ways:

A- Using Sabine formula; for diffusers absence, the measured total sound absorption area and sound absorption coefficient of the sponge, is represented as A_STm and α_Sm, and when the room contained diffusers it is represented by A_TSfm and α_Sfm.

B- Using equation (10); for diffusers absence, the estimated total sound absorption area and sound absorption coefficient of the sponge, is represented as A_TSC and α_SC, and when the room contained diffusers it is represented by A_TSfC and α_SfC.

From a comparison between the two curves of the effect of the diffuser, the absorption area of the room with the sponge increased in the case of using diffusers than in the absence of diffusers, especially in the range above 250 Hz, as in Figure 9. Also, the difference between A_TSC and A_TSm is larger than the difference between A_TSfC and A_TSfm in the diffusers presence case. It is noticed that there is a better convergence between the values measured by Sabine and those calculated from equation (10) in the case of the diffuser’s presence than in the diffuser’s absence case.

Figure 9.

The diffusers effect on calculated and Sabine absorption area of a sponge.

From Figure 10, It is clear that the curve of the values of the sound absorption coefficient of the sponge from the Sabine formula with diffusers represents the highest values of absorption (α_Sfm > α_Sm). In the diffuser absence case, the values of the calculated sound absorption coefficient of the sponge are lower than in the diffuser presence case (α_SC <α_SfC). Also, the difference of the sound absorption coefficient of sponge without diffusers (α_SC − α_Sm) is larger than the difference in diffusers presence case for sponge (α_SfC − α_Sfm).

Figure 10.

The diffusers effect on calculated and Sabine sound absorption of sponge material.

In the presence of diffusers, there is a stronger convergence between the values measured by Sabine and those calculated from equation (10) than in the lack of diffusers. This is because the sound pressure level in the absence of diffusers is higher than in the presence of diffusers. The calculated equation is based on the sound pressure level that has been measured.

For porous and highly absorbent materials, the effect of the presence of diffusers on sound absorption is evident, especially in the majority of the frequency range from 125 to 6300 Hz; it is weak in the low-frequency range from 125 to 250 Hz where most porous material do have a low absorption.

Diffuser’s effect on sound absorption of rubber

Average reverberation time measurements were carried out in the reverberation room when contained rubber material. In Figure 11, the total sound absorption area of the room with rubber is calculated in two ways:

A- Using Sabine formula; for the diffuser’s absence the measured total sound absorption area and sound absorption coefficient of rubber will be represented as A_TRm and α_Rm. and when the room contained diffusers, it represented by A_TRfm and α_Rfm.

B-Using equation (10); for diffusers absence, the calculated equivalent sound absorption area and sound absorption coefficient of rubber will be represented as A_TRC and α_RC, and when the room contained diffusers, represented by A_TRfC and α_RfC.

from Figure 11, comparing the two curves of the effect of the diffuser, the diffusers absence case the difference between A_TRC and A_TRm is larger than the difference A_TRfC − A_TRfm with the case of diffusers presence, it is indicated that there is a better convergence in diffusers presence case. The absorption area curve increased clearly in the case of using diffusers case than in the case without diffusers, especially in the frequency range 400–2000 Hz.

Figure 11.

The diffusers effect on calculated and Sabine absorption area of a rubber.

In Figure 12, with diffusers case, the values of the sound absorption coefficient of rubber α_Rfm were higher than those without diffusers case α_Rm, and it is clear that the curve of the values of the calculated sound absorption coefficient of the rubber α_RC without diffusers (from equation (10)) represents the highest values of absorption in the frequency range 125–1250 Hz.

Figure 12.

The diffusers effect on calculated and Sabine sound absorption of rubber material.

Additionally, the difference in the sound absorption coefficient of rubber without diffusers (α_RC − α_Rm) is greater than the difference in the diffusers presence for rubber sound absorption (α_RfC − α_Rfm). It is shown that in the presence of diffusers, the measured values by Sabine and the calculated values are closer than in the absence of diffusers. The effect of the presence of diffusers on sound absorption appears, especially in the majority of the frequency range, except for the weak effect in the low-frequency range at 125 and 250 Hz.

Diffuser’s effect on sound absorption of wood

Average reverberation time measurements were carried out in the reverberation room when contained wood material. In Figure 13, the total sound absorption area of the room with wood is calculated in two ways:

A- Using Sabine formula; for diffusers absence, the measured total sound absorption area and sound absorption coefficient of wood will be represented as A_Twm and α_wm, and when the room contained diffusers it represented by A_Twfm and α_wfm.

B- Using equation (10); for diffusers absence the calculated total sound absorption area and the measured sound absorption coefficient of wood, it represented as A_TwC and α_wC, and when the room contained diffusers represented by A_TwfC and α_wfC.

In Figure 13, from comparing the two curves of the effect of the diffuser, (A_Twfm − A_TwfC), we have a little difference between them in the frequency range 315–2000 Hz. The absorption area increased in the case of using diffusers than in the absence of diffusers, especially in the range above 250 Hz.

Figure 13.

The diffusers effect on calculated and Sabine absorption area of a wood.

Also, the difference in (A_TwC − A_Twm) > (A_TwfC − A_Twfm) of the measured absorption area calculated by the Sabine equation is larger than the difference resulting from those derived from equation (10), which arises from the diffusers’ presence. In general, the diffuser’s presence causes a slight increase in the equivalent sound absorption area of the room containing wood in all frequency bands.

Figure 14 shows the effect of the diffusers’ absence and presence α_wm and α_wfm on the values of measured and calculated sound absorption coefficient of wood when the sample is placed inside the room. Also, the difference between α_wC and α_wm values of the sound absorption coefficient of the wood without diffusers is smaller than the difference α_wfC − α_wfm arising from the diffusers’ presence, in the frequency bands higher than 800 Hz.

Figure 14.

The diffusers effect on calculated and Sabine sound absorption of wood material.

Diffusers, in general, have little influence on the sound absorption of poor-absorbing materials. In the presence of diffusers, the values measured by Sabine and those derived from equation (10) have a better convergence than in the absence of diffusers.

The presence of diffusers has an influence on sound absorption in poor absorbent materials, particularly across the majority of the frequency range, except for a weak effect in the low-frequency range from 125 to 250 Hz. The sound absorption coefficient curves of wood are comparable in both cases of diffuser’s presence and absence at frequency bands below 3150 Hz, while they differ at higher frequency bands over 3150 Hz.

The two equations used in the comparison illustrations in Figures 7, 9, 11, and 13 depends on the use of the same equivalent absorption area (A), but one of two equations use (A) values and in the other equation we deduce (A) from it.

The comparison illustrations in Figures 8, 10, 12, and 14 show the effect of the presence of diffusers or not, as well as a comparison to obtain the value of the sound absorption coefficient without the need to measure the reverberation time of the sample, and it is sufficient to measure only the amount of sound levels in the room. The figures showed the compatibility and success of the proposal

Figure 15, shows the relative standard deviation (Ɛ_TSf/T, Ɛ_TRf/T, Ɛ_Twf/T, and Ɛ_Tef/T) of reverberation time measurements in a reverberation room containing diffusers for sponge, rubber, and wood respectively. The relative standard deviation of reverberation time measurements is equal to 0.01. Except for sponge and rubber Ɛ_TSf/T, Ɛ_TRf/T, it increased up to 0.02, that due to the mutual effect between the room characteristics and the highly absorbent material. Relative standard deviation with low values indicates a higher sound field diffusivity.²⁵

Figure 15.

Relative standard deviation (Ɛ_TSf/T, Ɛ_TRf/T, Ɛ_Twf/T, and Ɛ_Tef/T) of reverberation time.

Conclusion

The effect of diffusers influences two key parameters: the equivalent sound absorption area and the sound pressure level. The current study has generated a collection of data that may be used to refine the sound absorption coefficient estimate based on sound power.

The influence of diffusers on sound absorption emerges across the whole frequency range for a porous and highly absorbing materials (sponge), except for the low-frequency range from 125 to 250 Hz.

The presence of diffusers has an effect on sound absorption in median absorbent materials (rubber), especially in the mid-frequency region from 630 to 1250 Hz. At high frequencies, such as after 3150 Hz, there appears to be little difference between the presence and absence of diffusers and their influence on sound absorption in weakly absorbing materials (wood and walls).

An increase in the values of the sound absorption coefficient of porous materials is brought about by the use of diffusing elements at frequencies above 630–800 Hz. For porous material (sponge) in two diffusers case of 18 m², there is a relative difference in the sound absorption coefficient values; in particular, at 1000 Hz it increased from 51% to 81%. As far as poor absorbing materials are concerned, diffusers increase the difference in sound absorption coefficient values from 0% to 21% at 6300 Hz, and for lower frequency bands from 0.12% to 0.14%. therefore, it can be said that, when doing measurements of the sound absorption coefficient of materials in a reverberation room, the use of diffusers is essential, as mentioned in ISO standards.

The presence or absence of diffusers has little influence on low-absorbent materials. In fact, for materials with poor sound absorption, such as walls and wood, there is a clear and noticeable convergence in calculating the values of the equivalent absorption area and the sound absorption coefficient using the Sabine formula and the iterative calculation based on sound power levels.

With the diffusers presence in the room, there is also a noticeable convergence in the values of the absorption area and sound absorption coefficient using the Sabine formula and the iterative calculation based on sound power levels for medium absorbing materials, such as rubber, and high sound absorbing ones, such as a sponge, with exceptions at two or three frequency bands

Footnotes

Acknowledgements

The authors would like to express their gratitude to all the workers who have participated in this research.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Hany A Shawky

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Using Sabine’s and sound power relation to study: The effect of diffusers on measured sound absorption

Abstract

Keywords

Introduction

Objective of study

System setup

Experimental methods

Sound power measurements in the reverberation room

Instrumentations and measurement procedure

Sound power level calculations

Uncertainty determination

Sound absorption measurements in the reverberation room

Instrumentations and measurement procedure

Calculations of Sabine and iterative sound absorption coefficients

Repeatability of reverberation time measurements

Results and discussion

Diffusers’ effect on the sound power level of RSS

Diffusers’ effect on reverberation time (T)

Diffuser’s effect on sound power level (Lw)

Diffuser’s effect on average sound power level in A-weighting (Lw dBA)

Effect of increasing surface area (S) on the sound power

Diffuser’s effect on the standard deviation (Ss)

Sound absorption measurements in the reverberation room (α)

Diffuser’s effect on reverberation time of reverberation room contained materials

Diffuser’s effect on equivalent absorption area of empty reverberation room (Ae)

Diffuser’s effect on sound absorption of sponge

Diffuser’s effect on sound absorption of rubber

Diffuser’s effect on sound absorption of wood

Conclusion

Footnotes

Acknowledgements

Declaration of conflicting interests

Funding

ORCID iD

References

Diffuser’s effect on sound power level (L_w)

Diffuser’s effect on average sound power level in A-weighting (L_w dBA)

Diffuser’s effect on equivalent absorption area of empty reverberation room (A_e)