The Impact of Surface Scattering on Reverberation Time in Di ﬀ erently Shaped Spaces

: In terms of the overall di ﬀ usion sound ﬁeld, the inherent di ﬀ usion state of the space and the scattering coe ﬃ cient of indoor surfaces work jointly to di ﬀ use sound energy. This study has investigated the impact of surface scattering on reverberation time in di ﬀ erently shaped spaces. First, 10 spaces with the same volume but di ﬀ erent shapes were calculated using a computer simulation; Next, two typical spaces were selected to calculate 10 states, in which the volume was multiplied. The calculations results show that the impact of surface scattering on reverberation time in di ﬀ erently shaped spaces follows three laws: in a group of spaces with the same variation pattern, T 20 varies with scattering coe ﬃ cients at a similar rate. In a group of spaces with di ﬀ erent variation patterns, there is a di ﬀ erence of more than 5% in the change rate of T 20 with scattering coe ﬃ cients; in imperfect di ﬀ usion spaces, decay curves vary in accordance with scattering coe ﬃ cients. If the scattering coe ﬃ cients are the same, T 20 varies with spatial shapes; when the volume of rectangular spaces and trapezoidal space ranges from 3000 to 30,000 m 3 , the change rate of T 20 is less than 5%. In the present study, spaces were classiﬁed by the position combination of shape surfaces. On this basis, we proposed and then graded the concept of “morphology di ﬀ usivity”.


Introduction
In the acoustic design of performance buildings, walls and ceilings are usually fitted with diffusion devices to diffuse indoor sound fields. The acoustic reflection, which results from the tilt or concave/convex shape of surfaces, changes the transmission direction of sound energy and the distribution of resonance frequency in indoor spaces. The diffusion of indoor sound fields is considered a very important method of improving acoustic quality [1].
Two parameters describe the reflection state of sound on surfaces, the diffusion coefficients and scattering coefficients. The scattering coefficient refers to the ratio of the non-specular reflection energy to the total reflection energy, while the diffusion coefficient is used to describe the spatial polar distribution of non-specular reflection sound energy [2,3]. The difference between diffusion coefficient and scattering coefficient is that diffusion coefficient describes the spatial distribution of non-specular sound energy, while scattering coefficient describes the energy proportion of non-specular sound energy. Cox et al. [4] found that using the diffusion coefficients in geometric room acoustic models is likely to produce incorrect results. In this paper, the scattering coefficient was used. To evaluate the effect of diffusion and scattering, a key room acoustics parameter is reverberation time (RT), determined by 20 dB decay or 30 dB decay, specifically referred to as T 20 and T 30 , respectively in this paper, whereas RT refers to general concept of reverberation time.
to affect acoustic parameters. However, none of these studies investigated the relationship between spatial shape and surface scattering, or between spatial volume and surface scattering.
It can be seen from the above research that the influence of surface scattering change on the RT is different in different shapes of space. It can be concluded that different spatial forms can "diffuse" the energy of sound field to a certain extent, and the diffusion of sound field is affected by both spatial forms and surface scattering.
Both the scale model test and computer simulation can be used to study scattering. Using the scale model method, although the results can be close to real situations, the number of cases that can be studied is limited due to the modelling complexity. All of the studies mentioned above pertained to a single space, in which there were scattering sources or not; none compared multiple spaces of different shapes. Currently, no existing study on surface scattering has investigated the sensitivity of RT to surface scattering in differently shaped spaces and the extent to which RT varies in accordance with spatial volume.
The present study thus aims to investigate the impact of scattering on RT in differently shaped spaces. Under the conditions of a constant absorption coefficient, RT is also affected by spatial volume.
This study has mainly conducted analyses relating to the following topics: (1) In spaces of different shapes, the way in which RT varies with an increase in the surface scattering coefficient; (2) Under the same surface scattering coefficient, different spatial shape will have different RT; (3) In spaces with different volumes, the way in which RT varies with an increase in the surface scattering coefficient.

General Settings
Kuttruff's [18] study found that in the frequency bands higher than the Schroeder frequency, geometric acoustics can be used to calculate the indoor sound field. In geometric acoustics, the reflection and scattering of sound rays are used to simulate the sound propagation process. This method has a high accuracy at high frequencies where the surface size is much larger than the wavelength.
Billon et al. [19] reported the calculation results of the Eyring formula after adding scattering in different space shapes. The CATT calculation results when the scattering coefficient was set to 0.4 were compared with the measured results. The RT results using the three methods were in good agreement. Both the RT and Sound Pressure Level (SPL) were investigated in cubic rooms by Jing and Xiang [20]. For uniformly distributed absorption coefficients, both the Eyring diffusion and modified diffusion models demonstrated good agreement with the CATT model.
These research results indicated that the computer simulation can to a certain extent reflect the influence of surface scattering on the RT in real space. It has a good reliability to explore the influence of surface scattering on the RT in ordinary spaces using the sound ray method.
Using Odeon, CATT, and Raven software, Shtrepi and Astolfi [21] performed computer simulations of 480 rectangular concert halls and analyzed the relationship between T 30 and scattering. They found that T 30 tended to decrease after diffusers were added to the walls and ceilings. Shtrepi et al. [22] indicated that the input modes of scattering parameters are different in CATT and Odeon. If the same frequency dependent scattering value assumed automatically in Odeon is applied to CATT for simulation with the same surface absorption coefficients, the T 30 value obtained is consistent with Odeon's calculation result. In this study, we used this method to compare the CATT and Odeon in a rectangular space and a specially shaped space (through the inclined surface to eliminate the parallel planes). It is found that with the same absorption coefficient and scattering coefficient, the difference in T 30 calculated by the two software packages is less than 5%. Referring to the international itinerant comparison of computer simulations, Vorländer [23] and Bork [24,25] used 5% as the subjective threshold of relative difference in T 30 . Therefore, the calculation results of the two software packages are consistent.
CATT Acoustic v9 is a room acoustic simulation software package that offers three different cone-tracing algorithms for source-receiver echograms and impulse responses and one audience area mapping algorithm. When the order ≤ n, diffuse and specular reflections are deterministic (n = max split order) and the diffuse reflections are created by actual diffuse ray separation. The n + 1 order reflections are randomly determined by the scattering coefficient. Diffuse reflection is frequency dependent, individually tracing for each 1/1-octave 125-16 kHz [26,27].
In CATT, the sound absorption and scattering coefficients are set for the six octave bands from 125 Hz to 4 kHz. For example, "ABS ceiling = < α125 α250 α500 α1k α2k α4k > L < s125 s250 s500 s1k s2k s4k >". The sound absorption and scattering coefficients are expressed by 1%-99%. The scattering coefficients of different surfaces can be set directly according to the test data or as suggested by the CATT user manual [26,27]. Since the scattering coefficient in the CATT can be set separately for each frequency band, the CATT was used as the calculation tool in this study.
In the simulated calculation, a shoebox-shaped small theater was used as the basic model for calculation and analysis. Its three-dimensional size was 25 × 16 × 7.5 m (L × W × H), and its indoor volume was 3000 m 3 . CATT was used to carry out simulated calculations and to analyze the results. Based on this model, the spatial shape was changed, while the variation in indoor volume was controlled to the extent of ±1%. In this study, the interfacial acoustic absorption coefficient was uniformly distributed. Considering the impact of acoustic absorption on RT, medium acoustic absorption settings were adopted; specifically, the interfacial acoustic absorption coefficient was set to 0.5.
For each space, all of the surfaces' scattering coefficients were simultaneously changed from 1% and 10% to 30%, 50%, 70%, 90%, and 99% at intervals of 20%. However, since the scattering coefficient cannot be 0 and 100% in CATT, the scattering coefficient's minimum value was set to 1% and its maximum value was set to 99%. The purpose of continuous coefficient variation is to cover the possibility of coefficient variations to the maximum extent. This method was used in prior scattering studies [7,28]. During each calculation, the seam scattering coefficients are provided for all of the octave bands. The CATT user manual states that a scattering coefficient of 20% is suitable for a smooth surface, and a scattering coefficient of 90% is suitable for a rough surface [26,27]. Percentages are replaced by decimals in the following text.

Spatial Shape
Based on previous research, spatial diffusion can be increased by tilted walls, but the diffusion field cannot be achieved only by tilted walls. Therefore, in this study, the spatial shape was adjusted by tilting different surfaces in a rectangular space. The decay curves in the different spaces varied according to the different tilting wall methods [15]. The RT transformed with the changes in the decay curve caused by the modification of the spatial shape.
A computer simulation was performed on nine spaces with the same volume but adjusted shapes. The adjusted shapes included the tilt of a single surface and a combination of the tilts of multiple surfaces (as listed in Table 1). Different adjustment methods could be combined to produce eight spatial shapes. In the simulated calculation, a reverberation chamber was added to stand in for a perfect-diffusion space [29].
As the human ear is most sensitive to sound at a frequency of 2000-4000 Hz, the average RT change rate of the octave (2000 Hz) was used as the evaluation standard in this study [30].
According to the Schroeder frequency formula, the frequency of the rectangular space in this study is 100 Hz; therefore, 2000 Hz was selected as 20 times this value, which was within the geometric acoustics application range [18].
Note: Capital letters denote different cases, and lower-case letters denote position changes. s is the single abbreviation, which means that only one right or left side wall is inclined. d is the double abbreviation, which means that both the right and left side walls are inclined. r is the rear wall abbreviation, which means that the rear wall is inclined. c is the ceiling abbreviation, which means that the ceiling is inclined. Thus, d × r × c means that the side walls, rear wall, and ceiling are inclined simultaneously.
One sound source point and six receiving points were set up in each space. Figure 1 shows the positions of the sound source point and receiving points in each space. The average calculated values of the six receiving points were compared [31,32].
Appl. Sci. 2020, 10, x FOR PEER REVIEW 5 of 14 As the human ear is most sensitive to sound at a frequency of 2,000-4,000 Hz, the average RT change rate of the octave (2,000 Hz) was used as the evaluation standard in this study [30].

Indoor Volume
Volume is a major factor affecting RT. The studies of Ryu [10], Shtrepi [21], Kuttruff [15], Wang [28], Schroeder [33], all pertain to a single space; no studies have been conducted to investigate spaces with the same shape but different volumes. When spatial volume increases and the acoustic absorption coefficient is constant, indoor RT also increases. In this case, the impact of spatial scattering on RT remains unclear. Here, the same space was scaled up to calculate the impact of the surface scattering coefficient on RT when the spatial volume ranged from 3000 to 30,000 m 3 . A spatial volume of 3000 m 3 was used as the reference volume; this was then amplified by a factor of n (n = 1, 2, 3 . . . . . . 10). The aim was to calculate the impact of the scattering coefficient on RT under different volume conditions.

Steps
In the present study, selected spaces were used in a comparative analysis of decay curve and RT. As no threshold standards have been established to judge whether there is any difference in decay curve and RT, it is impossible to assess the relationship between spatial shapes and surface scattering coefficients.
After a space of constant volume is changed, its total surface area and total indoor acoustic absorption also change. According to the Eyring equation, this directly changes RT. A comparison of calculated RT values between different spaces is therefore affected by the change in both spatial shapes and total acoustic absorption. This study used the RT change rate to conduct a comparative analysis of different spaces, eliminating the impact of acoustic absorption.
All of the samples used a scattering coefficient of 0.01 (equal to 1% above) as a reference stage to calculate the RT's change rate when the scattering coefficient was 0.1, 0.3, 0.5, 0.7, 0.9, and 0.99. The influence of the scattering coefficient on the RT was analyzed using the RT's rate. The RT's change rate was calculated using Equation (1).
where, τ denotes the change rate of RT (%); RT s denotes the RT value when the scattering coefficient is s. RT (0.01) denotes the RT value when the scattering coefficient is 0.01.
The threshold of relative difference in RT was used as a standard to evaluate whether there was any impact on RT. A survey of Western music and language, carried out with Western respondents, showed that the threshold of relative difference in RT was 5-10% [34][35][36]. A survey of Chinese folk music, conducted with Chinese respondents, showed that the threshold of relative difference in RT was 25-30% [37]. In an international itinerant comparison of computer simulations, Vorländer [23] and Bork [24,25] used 5% as the subjective threshold of relative difference in T 30 . The 5% threshold of relative difference in RT was used to judge whether the scattering coefficient affected RT significantly.

Calculation Parameters
In previous studies, T 30 has been used as the subject of study [8,11,14,15,21], although T 20 was used in the actual performance space test. Three typical spaces were selected to calculate T 20 and T 30 through CATT. A comparative analysis showed that, in relation to the same space, the trend and quantity of variation with the scattering coefficient were both less than 5% (as shown in Figure 2. Hence, their change rates could be considered the same. To compare the simulated values with measured values in the future, T 20 was used as the subject of this study.
In previous studies, T30 has been used as the subject of study [8,11,14,15,21], although T20 was used in the actual performance space test. Three typical spaces were selected to calculate T20 and T30 through CATT. A comparative analysis showed that, in relation to the same space, the trend and quantity of variation with the scattering coefficient were both less than 5% (as shown in Figure 2. Hence, their change rates could be considered the same. To compare the simulated values with measured values in the future, T20 was used as the subject of this study.

The Decay Curve of Sound Energy
The RT was calculated using the sound energy decay curves. In this section, two spatial shapes (a rectangular space and a reverberation chamber: six faces of the rectangular space were parallel to each other and there was no parallel plane in the reverberation chamber) were used to calculate the RT. Specifically, the RT was calculated when the surface scattering coefficient was 0.01, 0.3, and 0.7. Figure 3 shows the decay curve related to the rectangular space. The decay curve changed (specifically, the slope of the curve increased) as the surface scattering coefficient increased. This indicates that RT was shortened with the increase in the surface scattering coefficient. The decay curve related to the reverberation chamber shows that the slope of the curve remained unchanged under different values of the surface scattering coefficient (as shown in Figure 4). Evidently, variations in the surface scattering coefficient did not affect the RT variation trend in the reverberation chamber. The reverberation chamber's surface was covered by arc diffusers that effectively diffused the reverberation chamber's sound field. Therefore, the decay curve did not change when the surface scattering coefficient increased.

The Decay Curve of Sound Energy
The RT was calculated using the sound energy decay curves. In this section, two spatial shapes (a rectangular space and a reverberation chamber: six faces of the rectangular space were parallel to each other and there was no parallel plane in the reverberation chamber) were used to calculate the RT. Specifically, the RT was calculated when the surface scattering coefficient was 0.01, 0.3, and 0.7. Figure 3 shows the decay curve related to the rectangular space. The decay curve changed (specifically, the slope of the curve increased) as the surface scattering coefficient increased. This indicates that RT was shortened with the increase in the surface scattering coefficient. The decay curve related to the reverberation chamber shows that the slope of the curve remained unchanged under different values of the surface scattering coefficient (as shown in Figure 4). Evidently, variations in the surface scattering coefficient did not affect the RT variation trend in the reverberation chamber. The reverberation chamber's surface was covered by arc diffusers that effectively diffused the reverberation chamber's sound field. Therefore, the decay curve did not change when the surface scattering coefficient increased.    4. The Impact of Spatial Shape on the T20 Change Rate Figure 5 presents the calculation results, with the abscissa indicating the surface scattering coefficient and the ordinate indicating the T20 change rate. The curve in Figure 5 shows how the T20 change rate varied as the surface scattering coefficient increased. The ordinate value was consistent with the just noticeable difference (JND) value of T20 at a regular interval of 5% [23][24][25]30]. As shown in Figure 5, T20 tended to decrease as the scattering coefficient increased in differently shaped spaces. This variation trend is consistent with existing study results [8,11,14,16,21]. However, the decreasing trend was different across different spatial shapes.
The T20 decay curves in Figure 5 can be roughly classified into four groups. Within each group, the decay curves shared a similar variation trend; the variation trend of the decay curves was obviously different across groups. The more surfaces were spatially changed, the lower the RT change rate from increased surface scattering became.
In accordance with the space adjustment described above, the rectangular space was adjusted in three ways. First, the rectangular space remained unchanged, and was marked Space A. Second, the walls or ceiling were tilted separately; specifically, a single side wall was tilted, both side walls were tilted, the rear wall was tilted, and the ceiling was tilted. Spaces B, C, D, E, and F were obtained in this way. Third, both walls and the ceiling were adjusted at same time. Specifically, a single side wall and the ceiling were adjusted; both side walls and the ceiling were adjusted; and rear walls and  Figure 5 presents the calculation results, with the abscissa indicating the surface scattering coefficient and the ordinate indicating the T 20 change rate. The curve in Figure 5 shows how the T 20 change rate varied as the surface scattering coefficient increased. The ordinate value was consistent with the just noticeable difference (JND) value of T 20 at a regular interval of 5% [23][24][25]30]. As shown in Figure 5, T 20 tended to decrease as the scattering coefficient increased in differently shaped spaces. This variation trend is consistent with existing study results [8,11,14,16,21]. However, the decreasing trend was different across different spatial shapes.

The Impact of Spatial Shape on the T 20 Change Rate
Appl. Sci. 2020, 10, x FOR PEER REVIEW 9 of 14 a ceiling were added to both side walls. Spaces G, H, and I were obtained in this way. In each group of spaces, the curves of the T20 change rate shared a similar trend; the maximum difference in the change rate was less than 3% and the difference in the change rate between groups was approximately 5%. Evidently, each group of spaces shared a consistent change-rate trend, while the change-rate trend was obviously different across groups. T20 varied most significantly in relation to the scattering coefficient in Space A. T20 change rate tended to decrease evenly; it decreased most significantly (by 25%) when the scattering coefficient was 0.9. In Spaces B, C, E, and F, the T20 change rate was approximately the same under four states, and the maximum decrease in the T20 change rate was approximately 20%. In Spaces D, G, H, and I, the T20 change rate was approximately the same under three states, and the maximum decrease in the T20 change rate was approximately 20%. In the reverberation chamber, the T20 change rate with the scattering coefficient was less than 3%. The T 20 decay curves in Figure 5 can be roughly classified into four groups. Within each group, the decay curves shared a similar variation trend; the variation trend of the decay curves was obviously different across groups. The more surfaces were spatially changed, the lower the RT change rate from increased surface scattering became.
In accordance with the space adjustment described above, the rectangular space was adjusted in three ways. First, the rectangular space remained unchanged, and was marked Space A. Second, the walls or ceiling were tilted separately; specifically, a single side wall was tilted, both side walls were tilted, the rear wall was tilted, and the ceiling was tilted. Spaces B, C, D, E, and F were obtained in this way. Third, both walls and the ceiling were adjusted at same time. Specifically, a single side wall and the ceiling were adjusted; both side walls and the ceiling were adjusted; and rear walls and a ceiling were added to both side walls. Spaces G, H, and I were obtained in this way. In each group of spaces, the curves of the T 20 change rate shared a similar trend; the maximum difference in the change rate was less than 3% and the difference in the change rate between groups was approximately 5%. Evidently, each group of spaces shared a consistent change-rate trend, while the change-rate trend was obviously different across groups.
T 20 varied most significantly in relation to the scattering coefficient in Space A. T 20 change rate tended to decrease evenly; it decreased most significantly (by 25%) when the scattering coefficient was 0.9. In Spaces B, C, E, and F, the T 20 change rate was approximately the same under four states, and the maximum decrease in the T 20 change rate was approximately 20%. In Spaces D, G, H, and I, the T 20 change rate was approximately the same under three states, and the maximum decrease in the T 20 change rate was approximately 20%. In the reverberation chamber, the T 20 change rate with the scattering coefficient was less than 3%.
The curves of the T 20 change rate shown in Figure 5 presented a dual-slope trend, with 0.3 as the demarcation point. When the scattering coefficient ranged between 0.01 and 0.3, T 20 presented a stage of rapid decrease; the maximum change to T 20 (17%) occurred in the rectangular space. The minimum change to T 20 was 8%. Both were greater than 5%. When the scattering coefficient ranged between 0.3 and 0.99, the T 20 change tended to be smooth, while the T 20 change rate was always lower than 5%.

Impact of the Scattering Coefficient on T 20
Four spaces were selected to compare T 20 values. The selected spaces included the perfect diffusion reverberation chamber, the rectangular space, a space with adjusted walls, and a space with adjustments to both walls and ceiling. In the four spaces, the surface scattering had three states (the surface scattering coefficient was 0.01, 0.3 and 0.7). Figure 6 shows the calculation results. As shown in Figure 6, the T20 varied in accordance with the scattering coefficient across the four spaces. As the scattering coefficient increased, the T20 decreased in three spaces (apart from the reverberation chamber); the T20 change range was different across the four spaces. Space A had the largest change range. The change ranges of Spaces B and G were smaller than Space A. When the scattering coefficient was low (0.01), the T20 differed most among the four spaces. Specifically, the T20 As shown in Figure 6, the T 20 varied in accordance with the scattering coefficient across the four spaces. As the scattering coefficient increased, the T 20 decreased in three spaces (apart from the reverberation chamber); the T 20 change range was different across the four spaces. Space A had the largest change range. The change ranges of Spaces B and G were smaller than Space A. When the scattering coefficient was low (0.01), the T 20 differed most among the four spaces. Specifically, the T 20 in Space A was approximately 1.23 times that in the reverberation chamber, while the T 20 in Spaces B and G were approximately 1.17 times that in the reverberation chamber. As the scattering coefficient increased, the difference in T 20 across the four spaces gradually decreased. When the scattering coefficient was 0.3, the T 20 in Spaces A, B and G were approximately 1.05 times that in the reverberation chamber. When the scattering coefficient increased to 0.7, the T 20 was basically the same across the four spaces.
The above analysis shows that, under the same surface scattering coefficient, T 20 varied in accordance with spatial shapes. When the surface scattering coefficient was low, the T 20 varied significantly. The inherent diffusivity of each space obviously affected the T 20 . The more parallel surfaces existed within a space, the lower the diffusivity of that space, and the more significantly the T 20 varied. As the surface scattering coefficient increased, the inherent diffusivity of each space affected the T 20 less significantly, and the T 20 in all other spaces tended to be consistent with that in the reverberation chamber.

The Impact of Spatial Volume on the T 20 Change Rate
Typical Spaces A and C were selected to be scaled up. While their spatial shapes remained unchanged, their volume gradually increased by a factor of 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, and 10.0, thus ranging from 3000 to 30,000 m 3 . During the calculation process, the average acoustic absorption coefficient was 0.5.
The T 20 change curve related to Space A basically coincided with that related to Space C. The maximum T 20 change rate related to Space A was 0.23, and the difference in the maximum T 20 change rate was 4% between different volumes. The maximum T 20 change rate related to Space C was 13.5%. Except for one or two numerical values, the difference in the maximum T 20 change rate was 4% between different volumes. Evidently, the impact of the surface scattering coefficient on T 20 was weakly correlated with spatial volume when the spatial volume ranged between 3000 and 30,000 m 3 (as shown in Figure 7).

Definition of Morphology Diffusivity
Figures 2-7 show that as the surface scattering coefficient increased, the T 20 tended to decrease. During this process, the slope of the curve changed constantly until the curve became horizontal, presenting a dual-slope trend overall. However, the sensitivity of T 20 to the surface scattering coefficient varied across different spatial shapes. The maximum decay decreased as the tilt area increased. For the standard reverberation chamber, the surface scattering coefficient had almost no effect on T 20 . As yet, no standard has been established to measure spatial diffusivity. For this reason, it is necessary to introduce the concept of "Morphology Diffusivity".
Kuttruff [15] calculated RT by increasing the scattering coefficient, showing that, as the scattering coefficient increased, the decay of sound energy was approximated to the linear decay assumed by the Eyring equation. In a single space study, Wang [28] calculated T 30 under a scattering coefficient of 0; the difference between this T 30 and the T 30 calculated using the Eyring equation became the sensitivity index of the scattering coefficient.
For spaces with the same volume but different shapes, the variation trend of T 20 with the scattering coefficient was different if the acoustic absorption coefficient remained the same. The difference in the variation of T 20 may have been due to differences in the diffusivity of spatial shapes. The results in Figure 6 show that the T 20 change rate was associated with the composition of surface tilt. Spatial surfaces were divided into two groups, including walls and ceilings. When a single group of surfaces changed (e.g., walls only or ceilings only were adjusted), the T 20 change rate was the same, and no perfect diffusion sound fields were generated. When both groups of surfaces changed, the T 20 change rate was the same, but differed significantly from the T 20 change rate for the first group of surfaces. Likewise, no perfect diffusion sound fields were generated. When both walls and ceilings were tilted, and diffusion shapes were added to their surfaces, perfect diffusion sound fields were generated in the spaces. When only spatial volume was changed, the T 20 variation trend remained unchanged. Hence, morphology diffusivity can be defined as the inherent degree of diffusion of different spatial shapes.
Appl. Sci. 2020, 10, x FOR PEER REVIEW 11 of 14 How the T20 of Space A varied in accordance with the scattering coefficient How the T20 of Space C varied in accordance with the scattering coefficient

Definition of Morphology Diffusivity
Figures. [2][3][4][5][6][7] show that as the surface scattering coefficient increased, the T20 tended to decrease. During this process, the slope of the curve changed constantly until the curve became horizontal, presenting a dual-slope trend overall. However, the sensitivity of T20 to the surface scattering coefficient varied across different spatial shapes. The maximum decay decreased as the tilt area increased. For the standard reverberation chamber, the surface scattering coefficient had almost no effect on T20. As yet, no standard has been established to measure spatial diffusivity. For this reason, it is necessary to introduce the concept of "Morphology Diffusivity." Kuttruff [15] calculated RT by increasing the scattering coefficient, showing that, as the scattering coefficient increased, the decay of sound energy was approximated to the linear decay assumed by the Eyring equation. In a single space study, Wang [28] calculated T30 under a scattering coefficient of 0; the difference between this T30 and the T30 calculated using the Eyring equation became the sensitivity index of the scattering coefficient.
For spaces with the same volume but different shapes, the variation trend of T20 with the scattering coefficient was different if the acoustic absorption coefficient remained the same. The difference in the variation of T20 may have been due to differences in the diffusivity of spatial shapes. The results in Figure 6 show that the T20 change rate was associated with the composition of surface tilt. Spatial surfaces were divided into two groups, including walls and ceilings. When a single group of surfaces changed (e.g., walls only or ceilings only were adjusted), the T20 change rate was the same, and no perfect diffusion sound fields were generated. When both groups of surfaces changed, the T20 change rate was the same, but differed significantly from the T20 change rate for the first group of surfaces. Likewise, no perfect diffusion sound fields were generated. When both walls and ceilings were tilted, and diffusion shapes were added to their surfaces, perfect diffusion sound fields were generated in the spaces. When only spatial volume was changed, the T20 variation trend remained unchanged. Hence, morphology diffusivity can be defined as the inherent degree of diffusion of different spatial shapes.
Morphology diffusivity can be divided into four grades by the change amplitude of T20 (as listed in Table 2). Grade I stands for the morphology diffusivity of rectangular spaces. Rectangular Morphology diffusivity can be divided into four grades by the change amplitude of T 20 (as listed in Table 2). Grade I stands for the morphology diffusivity of rectangular spaces. Rectangular spaces are reference spaces. However, the six planes of each rectangular space are three groups of parallel planes, so rectangular spaces are imperfect diffusion spaces. After the walls or ceilings of rectangular spaces were tilted, the morphology diffusivity of the new spaces became superior to that of the original rectangular spaces. The morphology diffusivity of such spaces was defined as Grade II. After both the walls and ceilings of rectangular spaces were tilted, the morphology diffusivity of the new spaces was further improved. The morphology diffusivity of such spaces was defined as Grade III. After diffusers were added to both the walls and the ceilings of reverberation chambers, perfect diffusion sound fields were generated. The morphology diffusivity of the reverberation chambers was then defined as Grade IV, and the reverberation chambers were perfect diffusion spaces. Morphology diffusivity was classified into four grades, ranging from I to IV. Grade I represented the lowest level of morphology diffusivity, while IV indicated the highest level of morphology diffusivity.
While T 20 decreased as the scattering coefficient increased, there were two sections of different slopes in the change rate curve. As the scattering coefficient increased, T 20 first decreased rapidly, with a change rate greater than 5%; then, T 20 varied smoothly, with a change rate less than 5%. Hence, the inflection point of the slope was determined to be the critical scattering coefficient. For the samples used in this study, the critical scattering coefficient of the reverberation chamber was 0.01, and the critical scattering coefficients of the other spaces were 0.3. This confirmed the results of an earlier investigation by Wang [28] and Lam [38].

Conclusions
This study investigated the impact of surface scattering on T 20 in differently shaped spaces. A rectangular space was used as the reference space model. On this basis, spatial shapes were adjusted.
The T 20 was affected by both the inherent morphology diffusivity and surface scattering coefficient. When the morphology diffusivity was high, reflections were reflected in different ways, and the surface scattering coefficient only slightly affected the T 20 . For example, the T 20 in the reverberation chamber did not vary with the surface scattering coefficient due to the complex shape. However, the surface scattering coefficient significantly affected the T 20 when the morphology diffusivity was low. For example, in the rectangular space, the T 20 change rate was 25% when the surface scattering coefficient changed. The morphology diffusivity could be re-graded by adjusting the quantity and positions of the tilted walls and ceilings. Therefore, the morphology diffusivity grades accurately described the aforementioned analysis.
For imperfect diffusion spaces, such as a rectangular space, there is a turning point in each curve of the change rate of the T 20 . The scattering coefficient at the inflection point is the critical scattering coefficient. For spaces such as the rectangular space, the critical scattering coefficients were all 0.3; in the reverberation chamber, the critical scattering coefficient was 0.01. Evidently, no critical scattering coefficients exist in perfect diffusion spaces, although they do exist in imperfect diffusion spaces. The results of this study show that the critical scattering coefficients of imperfect diffusion spaces are approximately 0.3.
The morphology diffusivity of space is proposed. The morphology diffusivity of space can be assigned according to the combination of different surfaces in space. Each morphology diffusivity value corresponds to a diffusion degree level, and different levels correspond to different maximum change rates of the T 20 . For example, when the wall or ceiling is inclined separately, it corresponds to Grade II; when the wall and ceiling are inclined at the same time, it corresponds to Grade III. The RT's accuracy can be improved according to the morphology diffusivity and the scattering coefficient in the RT calculations of performance spaces.
In spaces with different volumes, the impact of the surface scattering coefficient on T 20 is unaffected by the size of spatial volume; specifically, inherent morphology diffusivity is not affected when spatial volume ranges between 3000 and 30,000 m 3 .
In this study, only the uniform arrangement of sound absorption coefficients was assessed; the uneven distribution of sound absorption coefficients will be researched in the future. Funding: The support from the UKRI (EP/X123456/1) is acknowledged.

Conflicts of Interest:
The authors declare no conflict of interest.