Development of an optimised, standard-compliant procedure to calculate sound transmission loss: design of transmission rooms

Applied Acoustics 63 (2002) 1003–1029 www.elsevier.com/locate/apacoust

Development of an optimised, standardcompliant procedure to calculate sound transmission loss: design of transmission rooms Christos I. Papadopoulos* Department of Naval Architecture and Marine Engineering, NTUA, Iroon Polytechneiou 9, 15773, Zografou, Greece Received 16 July 2001; received in revised form 20 December 2001; accepted 15 January 2002

Abstract A numerical procedure to estimate the transmission loss of sound insulating structures is proposed based upon the technology of acoustic measurements and standards. A virtual laboratory (VL), namely, a numerical representation of a real laboratory consisting of two reverberation rooms meeting certain sound field quality criteria is designed. VL is to be used for the numerical simulation of standardised measurements under predefined, controlled, acoustic conditions. In this paper, the design and optimisation of VL is investigated. The geometry of the transmission rooms is designed following first principles, in order for diffuse field conditions and sufficiently smooth primary mode distribution in the low frequency to be achieved. A finite element-based optimisation procedure, introduced by the author in previous work, is extended to arbitrarily shaped rooms. It is used to predict the appropriate local geometric modifications so as for improved mode distribution and smoother sound pressure fluctuations of the transmission rooms in the low-frequency range to be achieved and lowfrequency measurement reproducibility and accuracy to be increased. Steady-state acoustic response analysis is performed in order to quantify the acoustic field quality of the virtual transmission rooms in the frequency range of measurements. A method to calculate the total absorption, A, of the receiving room is introduced by simulation of the reverberation time measurement procedure using Transient acoustic response analysis. The acoustic performance of VL is overall considered and is shown to meet in a sufficient degree, relative laboratory measurement standards in the frequency range of 100
704 Hz. # 2002 Elsevier Science Ltd. All rights reserved.

* Tel.: +30-1-7721114; fax: +30-1-7721117. E-mail address: [email protected] 0003-682X/02/$ - see front matter # 2002 Elsevier Science Ltd. All rights reserved. PII: S0003-682X(02)00005-1

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1. Introduction In order for laboratory acoustic measurements to be performed and reliable data about the acoustic performance of sound insulating elements to be obtained, national or international recommendations concerning laboratory measurements should be followed [1–3]. A typical laboratory for the measurement of sound insulating properties in the frequency range of 100H2
5 kHz consists of two independent reverberation rooms, which share a common interface [1,3]. The geometry of both rooms is such as for perfectly diffuse-field acoustic conditions to be generated. The test specimen, whose transmission loss (TL), is to be measured, is placed along the common interface. A sound field is generated in the source room by application of a corner loudspeaker placed opposite the test-specimen area. TL is equivalent to the sound reduction index R [1] given by:

R ¼ L1  L2 þ 10 log

S A

ð1Þ

where L1 and L2 are the average sound pressure levels in dB of the source and receiving room respectively, S is the test-specimen area in m2 and A is the total absorption of the receiving room in Sabins. The theoretical background and practical guidelines to efficiently design a laboratory for the measurement of sound insulation of building elements as well as the acoustic field quality criteria and geometric characteristics that must be met, have been extensively dealt with in the literature. More specifically, the relative national and international recommendations show aspects of the facilities, installation and measurement procedure [1–5], while experts’ texts focus on the methodology to design rooms where certain quality criteria of their acoustic field are fulfilled [6–10]. Lately, the problem of unreliability in performing acoustic measurements in the low frequency range due to the existence of unwanted, multiple resonating phenomena and the strong coupling of the dynamic performance of the measurement rooms and that of the test-specimens has been addressed [11–13]. Serious work has been done in the field of analytical and numerical prediction of sound transmission through simple and complex insulating structures. In most of the cases simple or complex plate-like structural problems have been studied and analytical or semi-empirical formulas have been derived. They can be categorised in Helmholtz-type analytical calculations [12,14–22], where the sound propagation and the sound-structure interaction is tackled by application of the wave equation with appropriate velocity matching across the air-structure interface, in statistical energy analysis methods [17,23–25] which can be used in a wider frequency range but include simplifications, which prevent from understanding the local spectral sound and vibration fluctuations, and numerical methods utilising finite element analysis techniques [26–30]. However, simple, rectangular measurement rooms and structures have mainly been modelled and the level difference between the measurement rooms has in most of the cases been calculated [for calculation of the correction term

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10 log(S/A) of Eq. (1), which takes into account the sound reflections inside the receiving room (RR), additional knowledge of the total absorption A of RR with respect to frequency is required]. Thus, the extracted results cannot be appropriately compared with TL data of sound insulating materials and structures obtained at authorised acoustic laboratories. This is mainly due to the special geometric design of common transmission suites and the standardised conditions of the acoustic field that hold at the time of laboratory experiments, which lead to specific acoustic loading of the test-specimen under measurement. Such acoustic loading cannot be simulated in the case of simple rectangular transmission rooms being employed. Furthermore, techniques that use ideally generated diffuse acoustic field in order for estimation of TL to be achieved [21,30], may sufficiently lead to room-independent values of TL but they cannot be practicably applied to common rooms with finite dimensions where the existence of standing sound waves leads to only near-diffuse field conditions being attainable especially in the low frequency range. The basic concept of this paper is the design of a virtual laboratory (VL) in order to create an optimised calculation environment for the estimation of TL of sound insulating structures which provides at the same time standard-compliant acoustic conditions as those encountered in common measurement practice. Specially designed models of measurement rooms, the acoustic field quality of which resembles that of typical laboratories, but with increased efficiency in low-frequency measurements are utilised and numerical calculation of the acoustic response of various structures can be more appropriately performed. In this manner, the extracted results at higher frequencies ( > 150 Hz) are analogous to those measured in the case of real laboratories while at lower frequencies more realistic estimation of the sound insulating properties can be obtained. The measured low-frequency TL depends strongly on the mode distribution of the transmission rooms. Low-frequency TL measurement is generally hampered by the presence of various resonance phenomena at the rooms’ mode frequencies [11]. The measured TL is in that case affected by such phenomena and becomes strongly dependent on the employed measurement configuration. Hence, the obtained results are less accurate and reproducible between laboratories with different low-frequency dynamic behaviour [12]. ISO 140 and ELOT 370 recommend that the mode frequencies in the low frequency range should be as smoothly distributed as possible [1,3], in order for maximum decoupling of the measurement rooms’ dynamics and the vibrational behaviour of the test-specimens to be achieved. Generally, if certain modes coincide within a frequency range, certain sound peaks are observed in the neighbourhood of those modes, due to modal superposition, while the sound pressure is significantly lower further afield. The generated sound field is therefore less diffuse and if measurements are performed at such an acoustic field, the test-specimen is unevenly loaded in the frequency domain, and thus the obtained results become strongly dependant on the employed measurement configuration. The design of VL takes into account up-to-date regulations and state-of-the-art numerical techniques in order for efficient simulation of laboratory conditions to be achieved. The principle dimensions of the two reverberation rooms are primarily selected so as for mode distribution in the low frequency range to be smooth

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[1,3,25,31–33] and for the acoustic field inside the two rooms to be sufficiently diffuse [6–10], satisfying at the same time geometric limitations set by ISO [1] or ELOT [3]. Optimal mode distribution in the low frequency range cannot be sufficiently attained by mere selection of the principle dimensions and geometry of the rooms [11,13,31–33]. However, such a distribution is shown to be eventually attainable for rooms of arbitrary geometry by application of an extension of the mode redistribution method [13] in both transmission rooms. This method is shown to appropriately affect the low frequency modes of the rooms by application of local geometric modifications to their walls. After the optimised geometry of the two rooms has been determined, numerical calculations are performed to verify the low-frequency mode distribution and the sound pressure diffusivity inside the transmission rooms throughout the frequency range of measurements. In order to calculate TL the actual total absorption A in Sabins of the receiving room is also needed. For that, numerical simulation of the reverberation time measurements is performed [4,34] and A is indirectly estimated by utilising the Sabin formula (see for example Refs. [9,10]).

2. VL room design in compliance with national/international standards The main national and international recommendations concerning the acoustic performance of the transmission rooms [1,3] are hereby summarized:  even distribution of the mode frequencies of the rooms, especially in the low frequency range. This would allow increased TL measurement reproducibility between laboratories at low frequencies;  diffuse-field sound conditions inside the transmission rooms throughout the frequency range of measurements; Many practical reasons prevent the engineer from being able to explicitly select such dimensions so as for mode distribution in the low frequency range to be smooth and for the generated sound field to be diffuse. This is due to the difficulty to simultaneously control in an efficient manner several mode frequencies [13] and to the fact that sound diffusion is hard to quantify and its effectiveness is hard to prove [35]. Diffusion can be defined as a measure of the directional distribution of sound energy arriving at one point. A diffuse sound field is such that at any position, sound waves are incident from all directions with equal intensities and random phase relations [36]. In order to achieve both goals, trial-and-error techniques have been used accompanied by the following general rules:  The basic principle dimensions of the transmission rooms must be selected in such a way as to avoid mode coincidence in narrow frequency ranges. For that to be achieved, the ratio of the principle dimensions must not be small integers [31]. Dimension ratios such as 2n=3 or 5n=3 (n=1, 2, 3, . . .) should be preferably chosen [25].

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 Geometric symmetries, such as parallel surfaces, should be avoided. Such symmetries usually lead to uneven mode distribution and symmetrically-patterned less-diffuse acoustic field [25].  Convex surfaces cause the acoustic field to be more diffuse by producing multiple reflections toward different directions, while concave surfaces focus the acoustic waves leading to local non-uniformities of the acoustic field. Thus, convex rather than concave surfaces should be preferred [7,8].  Regions in the vicinity of sharp-angled intersecting walls lie within acoustic shadows and may give rise to local acoustic resonances; therefore, such formations should be avoided [6,8].  Highly reflective walls should be preferred so as to allow for multiple sound reflections and lead to acceptable near-diffuse acoustic field conditions [3]. Apart from limitations, which stem from the acoustic field quality requirements inside the rooms, certain geometric limitations do also apply, due to the recommended relative dimensions and volumes of the transmission rooms [1,3]. Briefly described, the volume of each room should be at least 50 m3 while the two volumes should differ by at least 10%. The test specimen opening should be at least 10 m2 and the reverberation time of both rooms should be limited to values less than 2 seconds. Taking into account all these considerations and after extensive numerical experiments, the principle dimensions of the rooms as well as their shape and volume are chosen, in order for primarily smooth mode distribution and sufficient sound diffusivity to be achieved. A 3-D room of arbitrary shape can be uniquely determined by specifying its principle dimensions. The final geometry of the transmission rooms is illustrated in Fig. 1. Additional geometric information is presented in Table 1. The final rooms have a volume difference of 10.8%, share a plane interface with area of 12 m2 and have relatively different shapes. Symmetries as well as sharp-angled intersecting walls are avoided and some walls are designed as convex surfaces. The respective design recommendations by ISO are therefore satisfied. The mode distribution of the transmission rooms in the low frequency range is accounted for by performing numerical calculations. Finite element models of the two rooms are constructed. Second order, 3-D, 20-node, hexahedral elements[37] are selected for the modelling, to increase the calculation accuracy. The mesh size is selected such as to accurately predict the modes and corresponding mode shapes in the low frequency range 0
125 Hz (convergence studies have been performed by the author in Ref. [13]). Eigenvalue analysis [37] is used for the extraction of the mode frequencies of the rooms. The analysis results are presented in Table 2 (‘starting Table 1 Geometric data about the two reverberation rooms of the virtual laboratory

3

Total volume (m ) Total area (m2) Total air mass (kg)

Source room

Receiving room

61.619 96.016 73.943

54.962 88.942 65.954

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Fig. 1. Geometric design of the transmission rooms. Location of sound sources and measurement positions throughout the virtual measurement procedure.

mode frequencies’ column) and Fig. 2 (-^- line). Even after a thorough investigation of the geometric properties of the rooms, multiple resonances do exist in the low frequency range, detracting from the rooms’ suitability. It is evident that although satisfactory to start with, the low-order mode frequencies of the transmission rooms need to be redistributed in order to avoid problematic multi-resonant frequency ranges. Such ranges, for example, can be observed at frequencies of 80, 89, 99 and 104 Hz for the source room and 82, 105 and 114 Hz for the RR as it is shown in Fig. 2. The effect of uneven mode distribution upon the generated sound field can be calculated by performing steady-state acoustic response analysis.

3. Steady-state acoustic response of the untreated rooms Steady-state analysis [37] is performed to calculate the acoustic field distribution inside the transmission rooms when harmonic acoustic load is applied. The three-

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Table 2 Starting, optimal and achieved mode distribution of the transmission rooms in the low frequency range. Calculation of objective function F for each case [Eq. (3)] Source room

Receiving room

Mode number

Mode number

Mode frequencies, Hz Starting

Optimal [Eq. (2)]

Final

7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

70.357 77.715 79.864 83.032 88.929 89.922 97.86 99.727 102.92 106.03 107.42 111.35 112.11 116.4 118.09 120.83 122 123.11

69.742 75.131 80.021 84.486 88.583 92.361 95.859 99.109 102.138 104.971 107.626 110.122 112.473 114.695 116.790 118.778 120.664 122.458

69.668 75.706 79.680 84.366 88.523 91.413 96.190 99.538 102.47 105.27 107.92 109.87 111.94 114.86 116.48 118.93 120.61 122.58

F

20.896

5.391

0

Mode Frequencies, Hz Starting

Optimal [Eq. (2)]

Final

7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

72.873 80.23 82.463 88.679 91.399 96.11 100.1 105.01 105.44 110.31 113.07 114.03 116.18 120.12 122.47 124.27 125.43 126.67

73.275 78.808 83.823 88.398 92.593 96.460 100.038 103.362 106.460 109.357 112.076 114.623 117.026 119.295 121.440 123.473 125.403 127.237

72.937 79.345 83.588 88.708 92.424 96.440 100.05 103.44 106.00 109.43 112.37 114.78 116.17 119.46 121.74 123.94 125.63 126.70

F

14.374

0

5.235

dimensional wave equation governs the propagation of the sound waves. The boundary conditions over the room surfaces as well as the magnitude and the direction of the acoustic load must be applied in order to calculate the space and frequency distribution of the acoustic field. ISO recommends that the rooms’ surfaces should be highly reflective to ensure multiple reflections and therefore satisfactory diffusivity of the acoustic field. For that to be achieved, boundary conditions of partial reflection are applied to the rooms’ walls. A realistic absorption coefficient of 0.05 is chosen for every boundary surface and remains constant in the frequency range of interest. For the determination of the numerical equivalent to the actual absorption coefficient, the concept of complex impedance is facilitated [5,10,37]. The sound source is applied to each room in such a way as to simulate a corner loudspeaker, placed in the wall opposite to the test-specimen and pointing towards the centre of each room. This is achieved by simultaneous application of three inphase inward sinusoidal acceleration vectors acting toward the three principle axes and with direction toward the interior of each room. The sound source is applied to the upper left corner of SR and to the upper right corner of RR as they are seen from the test-specimen area (see Fig. 1).

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Fig. 2. The -^- line is the mode distribution of the non-optimised transmission rooms in the low frequency range. The smooth grey line corresponds to optimal mode distribution. Problematic frequency ranges of both rooms are depicted.

Due to the low absorption coefficient used to model the rooms’ walls, the generated acoustic field inside the rooms is modal. That is, due to small amount of total damping, intense response peaks are expected at resonant frequencies. This can be seen in Figs. 3 and 4, which show the transfer function of the average sound pressure lever (SPL) of both transmission rooms as a frequency sweep with a 1-Hz-interval

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Fig. 3. Source room: the 1-Hz-interval dashed line shows the transfer function of the average SPL of the non-optimised room while the solid line shows the analogous transfer function of the optimised room. The circled maxima correspond to mode frequencies of the non-optimised room. The smooth dashed and solid lines correspond to the mean (fitted) SPL of the non-optimised and optimised room respectively.

over the frequency range of 50
130 Hz (dashed -^- line). The problematic frequencies corresponding to (a) modal superposition of neighbouring modes and (b) high-energy single modes are depicted in those figures (circled peaks). Such frequencies are for example those of 70, 78, 83, 89, 100, 103, 111 Hz for the source room and of 68 73, 82, 100, 105, 114 Hz for the receiving room. Certain maxima of SPL are observed at those frequencies. The problem is hereby briefly explained: Uneven mode distribution leads to multiple resonant frequencies in narrow frequency ranges. The sound pressure response at those frequencies is more likely to be high due to modal superposition (viz. excitation frequencies of 89, 99, 112 Hz— Fig. 3, of the source room and of 82, 105, 113 Hz—Fig. 4, of the receiving room), while its much lower at non-resonating frequencies (viz. excitation frequencies of 74, 93, 114 Hz—Fig. 3 and 76, 85, 108 Hz—Fig. 4). Taking into account that the SPL difference of the transmission rooms leads to calculation of TL, uneven mode distribution and eventually intensely fluctuating SPL in both rooms might lead to significant TL estimation fluctuations when measured at different transmission suites. Redistribution of the low-order mode frequencies would cause the acoustic field to be smoother with less intense SPL fluctuations, namely, more suitable for measurements. The measured TL in this case is less dependent on the measurement rooms. The effect of mode redistribution upon the virtually measured TL is a subject of future work. The mode redistribution method [13] is used for optimisation of the

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Fig. 4. Receiving room: the 1-Hz-interval dashed line shows the transfer function of the average SPL of the non-optimised room while the solid line shows the analogous transfer function of the optimised room. The circled maxima correspond to mode frequencies of the non-optimised room. The smooth dashed and solid lines correspond to the mean (fitted) SPL of the non-optimised and optimised room respectively.

mode distribution of both rooms. The implementation of the method is analytically presented in the next paragraph.

4. Mode redistribution In this section, an extension of the mode redistribution method presented in Ref. [13] to non-rectangular, arbitrarily shaped rooms is performed. The mode frequencies and mode shapes of a room are a function of its geometric boundaries. It has been shown [13] that appropriate local changes in geometry can lead to smoother mode distribution in the low frequency range. This is achieved by means of an optimisation procedure for which the independent variables array and the objective function must be formed. The transmission rooms share a common interface of 43 m2 where the test specimen is to be placed (Fig. 1). No local geometric modifications can therefore be applied along that interface. Therefore, if 9 variable local geometric modifications per each of the five remaining room walls are selected, they lead to a total of 45 per room. These constitute the independent variable array X(i) of the optimisation procedure. In order for the modifications to be applied, the nodes of the finite element model are changed in a such way as to create spatial irregularities over the wall

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surfaces acting either as cavities, if the modifications lead to increased air volume, or as obstacles to the sound waves propagation, if the modifications lead to decreased air volume (see for example the optimised rooms in Fig. 6 or Ref. [13]). The objective function, which will lead to optimal mode distribution, can be defined by using non-linear regression models of the growth family [13]. The MMF (Morgan, Mercer and Florin) non-linear regression model [38] provides a best fit to the mode distribution. Its mathematical expression is presented below: fðiÞ ¼

a b þ c id b þ id

ð2Þ

where i is the mode order and a, b, c and d are fitting constants. For this particular case the MMF model constants have the values a=0.005336, b=11.09372, c=228.99833 and d=0.809669 for SR and a=0.0080576, b=10.06214, c=209.5282 and d=0.866657 for RR. In Fig. 2 it has been shown graphically that the grey smooth fitted lines correspond to the ideal mode distribution each room should have in order to be acoustically optimised. Frequencies below 70 Hz are out of the measurement capability of VL. Usually, at low frequencies the modal density is low and low-frequency acoustic modes carry substantial amount of energy, and it is difficult for them to be simultaneously redistributed (it would take vast changes of the rooms geometry) . On the other hand the mode distribution at frequencies above 125 Hz is quite dense, each mode carries less energy and the energy distribution is reasonably satisfactory [13]. The geometric modifications previously discussed must be therefore applied to modify the mode frequencies of the rooms in the range of 70
125 Hz in such a way as to correspond to the optimal values suggested by Eq. 2 and Fig. 2. Therefore, with i varying from 7 to 24, M(i) being the mode frequencies of the room and f(i) the corresponding optimal mode frequencies, we can define the norm between the optimal and non-optimal mode distribution[13]. This norm, is the objective function F of the optimisation problem and it can be expressed in terms of i, M(i) and f(i) as in Eq. (3):

F¼

24  X   fðiÞ  MðiÞ 

ð3Þ

i¼7

Minimization of the objective function F would cause the sum of the absolute values of the deviation of each mode frequency from its respective optimum value to be minimized, leading to rooms with evenly distributed acoustic modes [13]. Table 2 apart from the mode frequencies of the untreated rooms also shows the analogous of the optimised rooms compared with the optimal ones. The starting values of the objective function can be calculated using Eq. (3) for each case and are 20.896 for the source and 14.374 for the receiving room. The respective achieved objective function after the optimisation procedure has converged is 5.391 and 5.235 respectively. Fig. 5 illustrates the starting vs. achieved mode distribution where the -- line denotes the achieved mode distribution of the optimised rooms and the -^- line the

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Fig. 5. Starting vs. achieved mode distribution of the transmission rooms. The -^- line is the mode distribution of the non-optimised room, the -- is the achieved distribution of the optimised room and the grey smooth line is the desired optimum.

corresponding distribution of the untreated ones. The grey smooth line corresponds to optimal mode distribution. It can be seen that sufficiently smooth mode distribution has been achieved for both rooms in the frequency range under consideration. The FE models of the two rooms, after the optimisation procedure has converged, are illustrated in Fig. 6. The total volume of the modified source room has increased to 61.6 from 61 m3 while the volume of the modified receiving room has reduced to 54.5 from 55 m3. Generally the geometry of several test-specimens might affect the modal distribution of the transmission rooms. However for planar test-specimens (such as sound

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Fig. 6. The final geometry of the transmission rooms after the optimising local geometric modifications have been applied.

insulating sheets, doors, windows and numerous common sound insulating structures) the geometry of the transmission rooms is slightly affected by the test-specimen installation. Therefore, the room modes do remain unaffected as well. It is also well known that low or intermediate absorption coefficients of the test-specimen would have minor effect on the room’s mode frequencies. It can therefore be argued that, in the case of common planar sound insulating surfaces with low or intermediate sound absorption coefficients the achieved mode redistribution will be insignificantly affected. The effect of mode redistribution upon the spectral sound pressure distribution of the rooms will be considered in the next section.

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5. Verification of the acoustic field quality 5.1. Introduction The quality of the sound field of a room can be determined on a basis of its spatial, frequency and time dependence of the sound pressure distribution. The recommended sound field quality of the two reverberation rooms has been analysed in detail in Refs. [1,3,6–9,36]. Generally, in order for diffuseness to be obtained, the spatial sound distribution inside the rooms should exhibit small fluctuations and the spatially averaged SPL curve with respect to frequency should be smooth. The combination of these two conditions could characterise an acoustic field as sufficiently diffuse. Practically, in order to prove the existence of diffuse field conditions inside the rooms the following issues should be accounted for:

 The spatial sound distribution of the acoustic field inside the two rooms should be uniform, namely, no major local non-uniformities should be observed except those at locations near the sound source or near highly reflecting surfaces.  The sound generated in the transmission rooms should be steady and have a continuous spectrum in the frequency range considered. The sound spectrum should not exhibit differences in level greater than 6 dB between adjacent onethird-octave bands.  The transfer function of the spatially averaged sound pressure level in the low-frequency range should be smooth and intense fluctuations should be avoided. This would be a means of verifying the effect of mode redistribution upon the sound field quality of the rooms.  The decay rate of the acoustic field, after the sound source has been stopped, should be monotonic and smooth.  The reverberation time of the rooms should have negligible frequency and spatial dependence.

Steady-state and transient finite element analyses are performed to verify the acoustic field quality of the proposed reverberation rooms. For these analyses and in order to perform accurate calculations at excitation frequencies up to 704 Hz, finer mesh has been selected. More specifically, second-order, 20-node hexahedral elements with characteristic length of 0.1 m have been selected so as to provide with at least 6 finite element nodes per acoustic wavelength [37] in the frequency range under consideration. In the following paragraphs, the spatial sound distribution and the sound pressure spectrum at steady-state conditions will be calculated by performing steady-state analysis. The decay rate of the acoustic field and the reverberation time in the frequency range of measurements will be calculated by performing Transient analysis and by simulating the reverberation time measurements proposed by ISO [4,34].

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5.2. Steady-state response analysis By utilising steady-state analysis, the average sound pressure level response in the frequency domain and the spatial distribution of the sound field in any given excitation frequency can be predicted. The same source positions and boundary conditions as in Section 3 are applied to both optimised transmission rooms. The transfer function of their sound pressure level with respect to excitation frequency averaged over space and one-third octave frequency bands is presented in Fig. 7. The spatial averaging has been conducted over every finite element node inside the two rooms using the following formula: SPL ¼ 10  log10

p21 þ p22 þ þ p2n n  p2ref

ð4Þ

where p1, p2 . . . pn are the sound pressure at n points inside the room and pref =20 mPa is the reference sound pressure used for the logarithmic calculations. In the frequency range of 70
704 Hz the rooms are shown to provide sufficiently low fluctuation of the one-third octave band average sound pressure level. In order to verify the effectiveness of the mode redistribution method the acoustic response of the optimised rooms (see again Figs. 3 and 4, solid line) have been presented in contradistinction to that of the untreated rooms (Figs. 3 and 4, dashed line and

Fig. 7. Transfer function of the 1/3 octave band average SPL of the optimised transmission rooms with respect to excitation frequency. Fluctuation of the SPL of adjacent one-third-octave bands is shown to be sufficiently low in the frequency range of measurements.

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remarks in Section 3). It can be seen that the improvement on the mode distribution has a positive influence on the spectral sound pressure distribution in the frequency range of interest. In the source room for example (Fig. 3), the local maximum of the average sound pressure at 82.5 Hz, namely, between modes 9 and 10 (see also Table 2), has been divided to two less intense maxima having frequencies of 80 and 84 Hz. The same is true for frequencies of 89, 100 and 112 Hz. For the same room, the local minimum of 74 Hz for the unmodified case has been divided into two neighbouring minima having frequencies of 72 and 77 Hz. In the unmodified case, the frequency of 74 Hz is in the middle of two neighbouring frequency ranges with concentrated modes (compare with results of Table 2). That is, no modes act near that frequency, therefore the sound pressure is low. The same is true for the frequencies of 85 and 93 Hz. The improved distribution of the mode frequencies spreads more evenly the sound energy to neighbouring frequency ranges and smoothes those global minima. The same can be observed for the receiving room (Fig. 4 and Table 2). Local maxima (as those at frequencies of 68, 82, 91, 105 Hz) and minima (such as those at frequencies of 76, 86, 103, 109 Hz) were made less intense leading to a smoother sound field distribution. In order to derive a quantitative result of the mode redistribution effect, the mean average deviation of the 1-Hz-interval spatially averaged SPL from the fitted (mean) SPL in the frequency range of optimisation is calculated. In Figs. 3 and 4, the dashed smooth lines correspond to the fitted SPL of the untreated rooms and the solid to the analogous of the optimised rooms. The average SPL fluctuation, SPLf, can be calculated as the average of the absolute deviation between 1-Hz-interval and mean (fitted) SPL in the optimisation frequency range: SPLf ¼

1 55

125  X SPLi

ð1Hzinterval Þ

 -SPLiðfittedÞ 

ð5Þ

i¼70 Hz

These values are 2.79 and 2.16 dB for the untreated and optimised SR respectively while for the RR the respective values are 2.53 and 1.88 dB. A significant decrease of the envelope enclosing the 1-Hz-interval SPL curve of both rooms can, therefore, be observed for both rooms in the frequency range of optimisation. Summarizing, the mode redistribution method has resulted in reverberation rooms where multiple resonating phenomena in the low frequency range were made less intense. The number of local minima and maxima has increased and their effect upon the sound energy distribution has been reduced, leading to a decrease of the SPL fluctuation in the frequency range of interest. Namely, the redistribution of the acoustic modes of the measurement rooms in the low frequency range affected positively the distribution of the total acoustic energy in the same range and consequently the diffuseness of their acoustic field as it will be shown thereinafter. The spatial distribution of the acoustic field inside the transmission rooms is illustrated in Fig. 8 at a representative excitation frequency of 300 Hz. The sound pressure over a cross section is presented in order to clarify the sound distribution in the interior of the rooms. The spatial distribution of the sound field can be utilised in

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Fig. 8. A cross section showing the spatial sound pressure distribution of the transmission rooms. A typical example at excitation frequency of 300 Hz.

order for the diffusivity of a reverberation room to be characterised in a quantitative manner. The standard deviation, , of the acoustic pressure field which is related to spatial variance can be used [39]. If N points having sound pressure level SPLi are considered inside a room, the standard deviation at a discrete frequency f can be calculated by: 2ð f Þ ¼

N  2 1 X SPLði; f Þ  SPLð f Þ N  1 i¼1

ð6Þ

A frequency band response of  can be calculated by logarithmically averaging SPL over the frequency band of interest. Standard deviation of the one-third-octave band sound pressure levels less than 1.5 dB is considered sufficient to characterise a

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sound field as adequately diffuse [39]. The spatial averaging of sound pressure of the transmission rooms has been conducted over 2400 nodes of the finite element model lying in an imaginary, orthogonal hexahedral volume in the interior of the rooms and being 1.5 m away from the sound source and 1.0 m away from any reflecting wall. Fig. 9 shows the value of  for both rooms over the frequency range of measurements. It can be observed that apart from a small deviation in the frequency range of 90–140 Hz, the desired value of 1.5 dB is achieved for both rooms. Generally, in the low frequency range the excitation is intensely modal and the involved modes are dominant, whereas at higher frequencies the total sound variation inside the source room is decreased leading to more smoothly spatial sound distribution. This can be attributed to greater number of modes being excited at higher frequencies, whereas the energy of each mode decreases as mode order increases. Therefore the variation becomes more local and less intense. In conclusion, the

Fig. 9. Transmission rooms: descriptor  ðfÞ as a function of frequency for the 1=3 octave bands.

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spatial sound pressure distribution of the two rooms can be characterised as sufficient in the frequency range of measurements. ISO [1] states that, in low frequency measurements, when using a single sound source it should be operated in at least two positions so as to partly compensate the lack of diffusivity in small rooms and attain excitation of greater number of loworder room modes. However, the sound field inside the designed transmission rooms has been shown to provide enough diffusivity in the low frequency range. In order to deeper clarify the acoustic loading of a candidate test-specimen in the low frequency range when measured using the proposed transmission rooms, the acoustic response of the source room is calculated when the sound source operates in an alternative location (Fig. 1; the loudspeaker is placed on the lower left corner of the source room as it is seen from the test-specimen area). Fig. 10 shows the spatial sound distribution over the test-specimen area for the standard and alternative sound source operations at frequencies of 70, 80, 90 and 100 Hz. It can be seen that the pattern of the sound field between the two operating positions shows increasing resemblance as frequency increases. In the case of excitation frequency of 100 Hz only minor differences can be observed. The situation is further improving at greater excitation frequencies due to greater number of lower-energy modes being excited and more diffusivity of the sound filed being in that way attained. It can therefore be argued that at frequencies greater than 100 Hz there exists no need of sound sources being operated in different locations since the resulting sound distribution over the testspecimen area is equivalent due to increasing diffusivity of the source room sound field. 5.3. Transient response analysis In order to further examine the diffuseness of the acoustic field, the acoustic response of the transmission rooms at transient conditions should also be verified. This is accomplished by simulating the reverberation time measurements as described in [34]. Harmonic acoustic load is applied to both rooms in 15 discrete excitation frequencies (lying in the frequency range from 70 to 704 Hz) at time t=0 and it remains constant until steady-state conditions have been reached. The order of magnitude for settling time of rooms as those studied is typically below 1 s and in the vicinity of 0.5 s. After steady-state conditions have been established, the sound excitation source is abruptly stopped. Simulation is continued until the average sound pressure level in the rooms is well below 35 dB. In order to calculate the sound pressure spectral density of the applied load, the Fourier transform F of each wave tone is employed. The Fourier transform of a single sine wave being instantly applied at t=0s is provided by the following equation [40]:   fi ðtÞ ¼ uðtÞ sin !0;i t ! Fi ð!Þ ¼ Fið fi ðtÞÞ ¼

!0;i ; i ¼ 1; 2; :::; 15  !2

!20; i

ð7Þ

In order to predict the total spectral loading of the rooms, the superposition principle is used:

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Fig. 10. Source room: spatial SPL distribution over the test-specimen area for excitation frequencies of 70, 80, 90 and 100 Hz when the sound source is operated in two different positions (as it is shown in Fig. 1).





(

Fð!Þ ¼ f ðtÞ ¼ F

15 X i¼1

) fi ðtÞ ¼ F

15 X Fi ð!Þ

ð8Þ

i¼1

For the test-case examined (in Fig. 11) the dashed line shows the spectral content of a single tone at 70 Hz, while the superimposed spectral density of the 15 discrete harmonic loads is calculated as a summation over the frequency range of interest (solid line). Should a single excitation had been applied, the signal would have been

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Fig. 11. Application of a superposition of harmonic loading at discrete excitation frequencies and calculation of the spectral density of the applied load.

as that in the lower right window of the same figure. Actually this is the summation of the 15 single tones, had they been applied separately. Note that the signal exhibits a pseudo-random nature, simulating thus the white-noise excitation used in experiments. It can be seen that minimum loading of 82.7 dB is in that way imposed on the rooms over the entire frequency range of interest. Figs. 12 and 13 show the response of the rooms to the suggested load. The acoustic field is developing for 0.5 s and the average sound is decaying after the sound source has been ceased. Both rooms are shown to provide with an almost straight-line decay curve over the frequency range of interest, which corresponds to exponential decay when projected to natural scale. Based on Eq. (1), in order for calculation of TL to be performed, the total absorption, A, of the receiving room is needed. A, can be estimated by calculating the reverberation time T60 of the receiving room and utilising the Sabine formula: A¼

0:160 V T60

ð9Þ

The calculation of T60 is performed based upon the recommendations of the ISO 3382 standard. T60 is derived from 5 to 35 dB of the sound decay curve of the receiving room. It can be seen that there exist a small deviation of T60 between different excitation frequencies. The two extreme values of T60 are T60, min =0.89 s in

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the case of f=90 Hz and T60, max =1.06 s in the case of f=500 Hz. The corresponding total absorption Amin and Amax can, therefore, be estimated as 9.88 and 8.29 Sabine respectively. The TL correction term 10 log(S/A) min/max [Eq. (1)] can now be calculated for the two extreme values of A and is 0.84 and 1.61 dB respectively. Fluctuation of A over the frequency of interest can therefore introduce an uncertainty to the calculated TL, bounded in the interval of  0.385 dB. However such an uncertainty level is negligible. Finally, the independence of the decay time, as well as, rise time from the tones’ excitation frequency supports the assumption that there exist adequate spectral decoupling between the actions of switching on and off the sound source and the spectral content of the various single tones applied to the room in the frequency range of measurements. If the values of Table 1 and the design absorption coefficient a of 0.05 are used to calculate the total absorption of the receiving room a value of approximately A=(total wall area)  a=4.9 would arise. However, such a calculation would take no account of the modal dynamic behaviour of the room. Especially in the low-frequency range, the pattern of the developed low-frequency sound field is mastered by the low-order, high-energy mode shapes of the room. Therefore, at each frequency, the actual absorption coefficient should be calculated by taking into account the sound field pattern and the development and reflections of sound inside the room.

Fig. 12. Transient response of the optimised SR at several excitation frequencies in accordance with ISO 3382 [34]. Calculation of the decay curve with respect to excitation frequency.

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Fig. 13. Transient response of the optimised RR at several excitation frequencies in accordance with ISO 3382 [34] standard. Calculation of the reverberation time of RR.

Hence, by the methodology introduced in this section, the effective rather than the design total absorption of the room can be more efficiently calculated. In order to examine the spatial dependence of the reverberation time, the decay curve at several points in the interior of the two rooms is also calculated. These points are selected so as to meet the ISO standards. ISO recommends a minimum of five microphone positions to be used in each room. These points should be spaced uniformly and be distributed within the maximum permitted space throughout each room. Generally there must be a minimum separating distance of 0.7 m between microphone positions, of 0.7 m between any microphone position and room boundaries and of 1.0 m between any microphone position and the sound source or the test-specimen area. The selected microphone positions inside the two rooms have been shown in Fig. 1. Calculations of reverberation time were performed at each of the 5 microphone positions per room and for harmonic loading in the frequency range of 70 to 708 Hz. Fig. 14 presents the spatial dependence of the reverberation time compared to the spatially averaged reverberation time of both transmission rooms in a representative excitation frequency (150 Hz). It can be seen that the reverberation time at each location bears strong resemblance to the spatially averaged reverberation time. The decay rate can be approximated by a straight line, the maximum sound at steady-state conditions is approximately the same for each of the 5 locations and the reverberation time can be characterised as spatially independent.

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Fig. 14. The decay curve at 5 different locations inside the transmission rooms compared to the spatially averaged decay curve for excitation frequency of 150 Hz.

Accounting for the transient response calculations, the following observations can be put down to bullets:  the total absorption, A, as a function of frequency, of the receiving room can be efficiently calculated with negligible calculation error;  the decay curve in the frequency range of measurements is reasonably smooth. A straight line can approximate the decay curve of both rooms;  the reverberation time remains practically constant in the frequency range of interest;  the reverberation time shows negligible spatial dependence.

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If these observations are compared with the respective recommendations (see Section 5.1), the sound field can be characterised as being sufficiently diffuse. The calculated actual total absorption of the receiving room can be used in conjunction with Eq. (1) to calculate the TL of a test specimen if the average sound pressure level in both the source and receiving room (being a function of frequency) and the area of the specimen are known. Installation of a high absorbing test-specimen would vary the reverberation time of the receiving room and consequently additional error to the predicted TL would be introduced. Generally, for test specimens with low absorbing behaviour their absorption coefficient can be compared to those of common reflecting painted walls (a=0.03–0.05). Minor change in the reverberation characteristics of the receiving room would then be introduced by such test-specimen configuration. In the case of test-specimens having one of their two sides covered with sound absorbing material, this side should face the source room [1]. In that way no effect on the RT of the receiving room would be observed. If, however, a test specimen with both absorbing sides is to be measured, the reverberation characteristics of the receiving room should be recalculated using the same methodology as that proposed in this section.

6. Conclusions In order to obtain numerical results, which comply with ISO recommendations and are comparable to commonly performed laboratory acoustic measurements, the procedure proposed by ISO should be precisely followed. This paper has dealt with the optimised design and modelling of a virtual laboratory for the estimation of sound transmission loss of sound insulating structures. In order for accurate numerical measurements to be performed the general theoretical aspects of sound field quality have been followed. The transmission rooms have been designed using first principles, in order for diffuse field conditions and preliminarily even mode distribution in the low frequency to be achieved. In order to further smoothen lowfrequency SPL fluctuations, application of the mode redistribution method has been performed. The finite element-based optimisation procedure has shown adequacy in converging to an optimised, even mode distribution in the low frequency range. The corresponding sound energy distribution has shown to be smoother and the multiple resonance phenomena to be less intense. Steady-state and transient analyses have been performed to verify the rooms’ sound field quality. The two rooms have shown to provide satisfactory diffuse sound field in the frequency range of calculations. The local geometric modifications upon the room walls apart from redistributing the low-frequency modes also provide geometric irregularities, which improve the diffuseness of the acoustic field. The actual absorption of the receiving room has been efficiently calculated by simulating the reverberation time measurements. This has been accomplished by performing transient acoustic response analysis. The decay curve of the sound energy in both rooms is shown to be smooth (straight-line decay) and the frequency and spatial dependence of the reverberation time of both rooms is shown to be negligible in the frequency range of interest. Therefore, the relative ISO

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recommendations are reasonably met and numerical calculations of the transmission loss of test specimen models can be performed in this virtual laboratory.

Acknowledgements The author is indebted to Professor J.P. Ioannidis of National Technical University of Athens, Greece, for his supervision and his valuable assistance throughout this work. The author would also like to acknowledge helpful discussions with Mr. N. Xiros from National Technical University of Athens, Greece and the financial support by the IKY foundation (Greek State Scholarship Foundation).

References [1] ISO 140-I,III. Acoustics—measurement of sound insulation in buildings and of building elements. ISO, 1995. [2] ISO/DIS 15186–3, Acoustics—measurement of sound insulation in buildings and of building elements using sound intensity, ISO, 2000. [3] ELOT 370-I/IX (Hellenic Standard). Acoustics—measurement of sound insulation of building elements. ELOT, 1978 (in Greek). [4] Goydke H. New international standards for building and room acoustics. Applied Acoustics 1997; 52:185–96. [5] ELOT EN 20354 (Hellenic standard). Acoustics—measurement of sound absorption in a reverberation room. ELOT, 1996 (in Greek). [6] Alton EF. The master handbook of acoustics. McGraw-Hill, 1994. [7] Ando Y, Noson D. Music and concert hall acoustics. Academic Press, 1997. [8] Cavanaugh WJ, Wilkes JA. Architectural acoustics. John Wiley & Sons Inc., 1999. [9] Harris CM. Handbook of acoustical measurements and noise control. McGraw-Hill, 1991. [10] Beranek LL, Ver IL. Noise and vibration control engineering. John Wiley & Sons Inc., 1992. [11] Fuchs HN, Zha X, Pommerer M. Qualifying freefield and reverberation rooms for frequencies below 100 Hz. Applied Acoustics 2000;59:303–22. [12] Osipov P, Mees G, Vermeir X. Low-frequency airborne transmission through single partitions in buildings. Applied Acoustics 1997;52:273–88. [13] Papadopoulos CI. Redistribution of the low frequency acoustic modes of a room: a finite element— based optimisation method. Applied Acoustics 2001;62(11):1267–85. [14] Gomperts MC. Radiation from rigid baffled, rectangular plates with general boundary conditions. Acustica 1974;30:321–6. [15] Guy RW. The transmission of airborne sound through a finite panel, air gap, panel and cavity configuration—a steady state analysis. Acustica 1981;49:323–33. [16] Trochidis A, Kalaroutis A. Sound transmission through double partitions with cavity absorption. Journal of Sound and Vibration 1986;107:321–7. [17] Norton MP. Fundamentals of noise and vibration analysis for engineers. Cambridge: Cambridge University Press, 1989. [18] Cervera F, et al. Sound intensity in the near field above a vibrating flat plate. Noise Control Eng J 1997;45(5):193–9. [19] Fujiwara K, Hothersall DC, Kim C. Noise barriers with reactive surfaces. Applied Acoustics 1998; 53:255–72. [20] Tang WC, Zheng H, Ng CF. Low frequency sound transmission through close-fitting finite sandwich panels. Applied Acoustics 1998;55:13–30.

C.I. Papadopoulos / Applied Acoustics 63 (2002) 1003–1029

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[21] Kang H-Ju. Prediction of sound transmission loss through multilayered panels by using Gaussian distribution of directional incident energy. J Acoust Soc Am 2000;107:1414–20. [22] Tadeu A, Mateus D. Sound transmission through single, double and triple glazing. Experimental Evaluation, Applied Acoustics 2001;62:307–25. [23] Craik RJM. Sound transmission through parallel plates coupled along a line. Applied Acoustics 1997;49:353–72. [24] Craik RJM, Smith RS. Sound transmission through double leaf lightweight partitions. Part I: airborne sound. Applied Acoustics 1996;61:223–45. [25] Skarlatos D. Applied acoustics. Athens: ION publications, 1998 (in Greek).. [26] Hansen CH. Effect of size on the sound transmission loss of both heavily and lightly damped orthogonal panels. Proc. of Int. Conf. on Noise Control Engineering, Sydney, Australia, Dec 1991. [27] McClure CK. Predicting wall panel transmission loss with ABAQUS. ABAQUS Users’ Conference, Newport, RI., 1992; pp.389–400. [28] Haroum MA, Yland CV, Elsanadeby HM. Finite element modelling of structural re-inforced concrete grid walls. Proc. of 7th Int. Conf. on Civil and Structural Engineering, Oxford, England, September 1999. [29] Maluski S, Gibbs BM. Application of a finite-element model to low-frequency sound insulation in dwellings. J Acoust Soc Am 2000;108:1741–51. [30] Sgard FC. A numerical model for the low-frequency diffuse field sound transmission loss of double wall sound barriers with elastic porous lining. J Acoust Soc Am 2000;108:2865–72. [31] Louden MM. Dimension-ratios of rectangular rooms with good distribution of eigentones. Acoustica 1971;24:101–4. [32] Sepmeyer LW. Computed frequency and angular distribution of the normal modes of vibration in rectangular rooms. Journal of the Acoustical Society of America 1965;37:413–23. [33] Bonello OJ. A new criterion for the distribution of normal room modes. J Audio Eng Soc 1981;29: 597–606. [34] ISO 3382. Acoustics—measurement of reverberation time in Auditoria. ISO, 1997. [35] Haan C, Fricke FR. An evaluation of the importance of surface diffusivity in concert halls. Applied Acoustics 1997;51:53–69. [36] Skarlatos D, Kostakis D. A statistical approach for the diffusion of sound by reflecting surfaces in large enclosures. Applied Acoustics 1996;47:67–81. [37] Abaqus. Abaqus standard reference manual, HKS/ABAQUS Corp., 1997. [38] SPSS. SPSS for Windows, base system user’s guide release 10.0. SPSS Inc. 1998. [39] Nelisse H, Nicolas J. Characterisation of a diffuse field in a reverberant room. J Acous Soc Am 1997; 101(6):3517–24. [40] Nagrath IJ. Control systems engineerin, 2nd ed. Wiley Eastern Limited, 1982.