Acoustic design of small rectangular rooms: Normal frequency statistics

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Applied Acoustics 69 (2008) 1356–1360 www.elsevier.com/locate/apacoust

Technical Note

Acoustic design of small rectangular rooms: Normal frequency statistics M.A. Błaszak Adam Mickiewicz University, Institute of Acoustics, Umultowska 85, 61-614 Poznan, Poland Received 30 March 2007; received in revised form 18 October 2007; accepted 19 October 2007

Abstract The subject of discrete modes in small rectangular rooms has been considered. A new procedure for selecting optimum geometric proportions of rooms has been proposed, taking into account the eigenfrequencies up to the Schroeder frequency and considering also the surface averaged sound absorption coefficient (a) of a given room. This new procedure leads to a series of plots describing the geometric proportions of small rectangular rooms corresponding to the smoothest frequency response for different absorption conditions. When taking a into account, the range of the acceptable dimension ratios X:Y has proved relatively wide, so the standard deviation calculated for the distances between subsequent modes does not exceed 1.5 (as in Bolt’s work). However, the range of the acceptable dimension ratios decreases with decreasing a and for mean absorption coefficient lower or equal 0.3 there are only a few of the ratios for which a uniform distribution of eigenmodes is obtained. Ó 2007 Elsevier Ltd. All rights reserved. Keywords: Room acoustics; Schroeder frequency; Eigenfrequency

1. Introduction Each enclosure behaves as a filter and therefore it affects all signals generating inside it by changing their spectra. A standing wave can significantly strengthen the perceived level of sound corresponding to particular frequency components of noise, generated for example by various devices (such as projector or computer). General nature of resonances has been considered by many authors who tried to calculate the size of an enclosure that will maximize uniformity of the frequency response. At low frequencies the modal sound field is distinctly non-uniform, dominated by discrete room modes. This non-uniformity have also an influence the music and speech perception, causing changes in the sound spectra and making them unnatural. Certain low frequency components that agree with the natural frequencies of an enclosure may be intensified as much as by 20 dB [9]. Because of the fact that the frequencies below 100 Hz can cause E-mail address: [email protected] 0003-682X/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.apacoust.2007.10.005

unpleasant (in perception) acoustical effects, a large variety of enclosures with ‘‘poor acoustic climate’’ was investigated beyond the standard low frequency limit [6]. The nature of resonance in large rooms has been also recognized to be an important factor in determining the acoustics of music halls. Many organists maintain that certain enclosures (churches, concert halls) posses a ‘‘sympathetic note’’. These enclosures tend to reinforce tones within a restricted region of pitch [9]. However, the subject of this paper is limited to the effects of standing waves in small rectangular enclosures. 2. The method of analysis The eigenfrequencies of a room with the dimensions Dx, Dy and Dz can be calculated from Eq. (1) sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2  2  2ffi c nx ny nz fnx ;ny ;nz ¼ þ þ ; 2 Dx Dy Dz nx ; ny ; nz ¼ 0; 1; 2 ; . . . ;

ð1Þ

M.A. Błaszak / Applied Acoustics 69 (2008) 1356–1360

where c is the sound speed [m/s] and integers nx, ny, nz represent the order of the modes. According to Bolt [2–5], the mean square of the deviations (w) of the actual distances between subsequent modes from the mean value in a certain frequency range is a measure of the regularity of distribution of the eigenmodes on the frequency scale (2) Pn1 2 i¼1 ei w¼ þ 1; ð2Þ ðn  1Þd2 where d the mean distance between subsequent eigenmodes [Hz], ei – the deviation from the mean value [Hz], i – the index denoting the pair of modes whose distance is considered and n – the number of modes. The higher the value of w, the larger the fluctuations in the frequency spacing and the sound transmission irregularity over the specified frequency range [1]. However, of particular interest is the problem of frequency range that should be taken for analysis. In this paper modal analysis has been performed in the frequency range up to the Schroeder frequency. Schroeder [10,12] derived the critical frequency fsch representing the limit between the frequency range with well-separated and that with strongly-overlapping resonances (the limit of diffuse and non-diffuse sound field). In 1962 Schroeder and Kutruff [12] defined this frequency as (3) rffiffiffiffiffiffiffi T 60 fsch ¼ 2000 ; ð3Þ V where, T60 is the reverberation time and V is the volume of considered enclosure. This frequency corresponds to the case of three overlapping normal modes. Assuming that the time of reverberation can be obtained from Eyring formula (4) T 60 ¼

0:16V ; S lnð1  aÞ

ð4Þ

where S is the total surface of the enclosure [m2] and a – surface averaged sound absorption coefficient. According to Eqs. (3) and (4) the range of frequency of the eigenmodes taken for consideration depends on the total area limiting the enclosure and the surface averaged sound absorption coefficient. Therefore, the range of the mode frequencies taken for analysis is not the same as in Bolt’s work, as with changing  a the upper bound frequency changes, while the lower bound energy is determined by the lowest mode frequency in a given enclosure. As the curves change with  a, they are presented for enclosures of different size but a constant  a. According to Bolt’s work, the lower boundary of the region corresponds approximately to the lowest normal frequency in the room. This was exactly the lowest normal frequency in a cube, but in rooms with other proportions there might be a few lower normal frequencies [5]. It should be noticed that w does not bring complete information on the character of this non-uniformity. To get this information an additional parameter should be introduced. Let X be defined as follows (5)

Pn1 X¼

i¼1 ðjei j

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2

 CÞ ; ðn  1ÞC2

where pffiffiffiffiffiffiffiffiffiffiffiffi C ¼ w  1:

ð5Þ

ð6Þ

The higher the value of X, the larger the ‘‘gaps’’ in the characteristics (the irregularity increases). Knowing the accurate frequencies of subsequent eigenmodes of a given room, it is possible to calculate the value of X. 3. Results 3.1. The uniform distribution Fig. 1 presents the sizes of rooms in which the value of w is, after Bolt, lower or equal 1.5 (the height is normalised to 1). In the calculations w = 1 stands for fully uniform distribution of eigenmodes on the frequency scale. The higher the mean absorption coefficient the broader the acceptable area of X:Y dimension ratios. The values of w are different for two enclosures of the same geometrical size but different a, because the Schroeder frequency depends on the total acoustic absorption of a room related to the time of reverberation T60. Therefore w assumes different values for rooms of the same geometric proportions but equipped with different acoustic lining materials. Hence, a comparison of the geometrical sizes of rooms makes sense only if the rooms have the surfaces of the same mean absorbing properties. The values of w for the geometric proportions assumed by Bolt as one of the best (1:1.5:2.5), are 3.28 ða ¼ 0:2Þ or 1:66ða ¼ 0:6Þ so they are not optimum if the mode distribution analysis is made in the procedure proposed. Table 1 presents the least mean square values of the deviations of the actual distances between subsequent modes from the mean value for different mean absorption coefficients. For rectangular enclosures (z = 1, x = 1–3, y = 1–3) characterized by a < 0:6 one of the dimension ratios that give the lowest value of w is 1:1.2:1.4. For the dimension ratios giving the smoothest frequency response for the frequencies up to the Schroeder’s one (see Fig. 1), parameter X has been calculated and the results for selected surface averaged sound absorption coefficient are presented in Fig. 2. The dark areas correspond to the dimension ratios for which parameter: w is lower or equal 1.5 and X is lower than 100. It should be noticed that for a ¼ 0:3 there are just two dimension ratios fulfilling above conditions (1:1.22:1.44 and 1:1.23:1.46). 3.2. The random distribution In analogy to the above calculations, the geometric proportions were calculated for which the distribution of eigenmodes in the frequency scale was the least advantageous (very random distribution) and the values of w was equal to or higher than two (see Fig. 3).

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Fig. 1. The dark areas correspond to the dimension ratios giving the smoothest frequency response (w 6 1.5) for the frequencies up to Schroeder’s one in small rectangular rooms. The figure is drawn for three values of the surface averaged sound absorption coefficient ða ¼ 0:3; 0:4; 0:5 and 0:6Þ.

Table 1 The dimension ratios giving the smoothest frequency response for the frequencies up to the Schroeder’s one: (a) in small rectangular rooms (z = 1, x = 1–3, y = 1–3) and (b) in a rectangular oblong room (z = 1, x = 1–3, y = 3–5) as a function of the surface averaged sound absorption coefficient (a)  a

0.1

0.2

0.3

0.4

0.5

0.6

(a) w1 1:X1:Y1 w2 1:X2:Y2 w3 1:X3:Y3

2.30 1:1.2:1.4 2.44 1:1.2:1.5 2.50 1:1.4:1.8

1.84 1:1.2:1.4 1.96 1:1.2:1.5 1.96 1:1.2:1.6

1.48 1:1.1:1.4 1.52 1:1.2:1.4 1.55 1:1.2:1.5

1.39 1:1.2:1.4 1.40 1:1.4:1.9 1.42 1:1.6:2.1

1.19 1:1.2:1.4 1.24 1:1.2:1.5 1.30 1:1.2:1.6

1.06 1:2.3:3.0 1.11 1:2.4:3.0 1.18 1:2.2:2.8

(b) w1 1:X1:Y1 w2 1:X2:Y2 w3 1:X3:Y3

3.04 1:2.9:4.9 3.09 1:2.7:4.9 3.11 1:2.7:4.8

2.44 1:1.8:4.4 2.45 1:2.5:4.6 2.45 1:2.9:4.9

1.98 1:2.7:4.9 1.99 1:2.9:4.9 2.06 1:3.0:5.0

1.46 1:3.0:5.0 1.58 1:2.8:5.0 1.59 1:2.9:5.0

1.40 1:2.0:4.9 1.41 1:2.0:5.0 1.50 1:3.0:5.0

1.42 1:2.3:4.3 1.43 1:2.4:4.5 1.44 1:1.4:5.0

From the point of view of the wave theory, the rooms of the least advantageous frequency distribution are a cube and a cubicoid with a square base. For these rooms

it is impossible to achieve low values of w even on using large amounts of strongly sound-absorbing materials, see Fig. 4.

M.A. Błaszak / Applied Acoustics 69 (2008) 1356–1360

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Fig. 2. The dark areas correspond to the dimension ratios for which parameter w is lower or equal 1.5 and X is lower than 100. The figure is drawn for two values of the surface averaged sound absorption coefficient ða ¼ 0:4 and 0:5Þ.

Fig. 3. The dark areas correspond to the dimension ratios giving the very random frequency response (w > 2) for the frequencies up to the Schroeder’s one in small rectangular rooms. The figure is drawn for three values of the surface averaged sound absorption coefficient ða ¼ 0:3; 0:4; 0:5 and 0:6Þ.

1360

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Fig. 4. The mean square of the deviations (w) of the actual distances between subsequent modes from the mean value in a certain frequency range (up to the Schroeder frequency) as a function of the surface averaged sound absorption coefficient for a cube and a few other proportions of rooms with a square base.

4. Practical applications in classrooms The problem of mode distribution in classrooms is of particular importance as these rooms are designed to be used for learning and verbal communication or music teaching take place there a few hours a day. Often classrooms are equipped with computers, projectors, air-conditioning or other noise producing devices. Amplification of particular frequency components from the spectra of these devices may lead to unpleasant acoustic effects causing fatigue of the hearing system, in the listeners in the antinode of the standing wave [1]. Therefore, design of acoustic adaptation in the already built rooms should be performed taking into regard the optimisation of the reverberation time and that of the modal characterisation of the room. Performing calculations for each room it is possible to find the surface averaged sound absorption coefficient for which the value of w is optimum or lower than 1.5. It should be noted that not always a low value of w can be achieved using the acoustic adaptation for which  a 6 0:7. The steady-state frequency response of an enclosure is irregular ‘‘in nature’’, but is has been shown that the irregularity decreases with increasing absorption coefficient. Unfortunately, taking into regard the type of materials for acoustic adaptation of rooms, it is neither cheap or simple to achieve a high absorption coefficient in the range of the lowest frequencies. 5. Conclusions The problem of discrete modes in small rectangular rooms has been considered. Assuming the Schroeder frequency as the upper frequency limit, a statistical fluctuation in frequency spacing was calculated. This new

procedure permitted obtaining a series of plots describing the geometric proportions of small rectangular rooms corresponding to the smoothest frequency response for different surface averaged sound absorption coefficient ( a) values. The calculations indicate that for rectangular enclosures, but not for oblong rooms (corridors), characterized by a < 0:6 one of the dimension ratios that give the lowest value of w is 1:1.2:1.4. When taking a into account, the range of the acceptable dimension ratios X:Y has proved relatively wide, so the standard deviation calculated for the distances between subsequent modes does not exceed 1.5 (as in Bolt’s work). However, the range of the acceptable dimension ratios decreases with decreasing a and for a 6 0:3 there are only a few of the ratios for which a uniform distribution of eigenmodes is obtained. The results have shown that analysis of the distribution of eigenmodes should be performed taking into regard not only the geometrical dimensions of the room but also its mean acoustic absorption. Hence, the acoustic design of a room should be based on the volume of a given enclosure, the acoustic adaptation materials [7,8,11] to optimise the reverberation time and then the geometric size of the room to optimise the distribution of eigenmodes. The higher the value of a the lower fsch and the greater the set of geometric dimensions for which the root-mean-square spacing is acceptable.

References [1] Błaszak MA. A new method for analysis of the modal characteristics of small rectangular rooms. Arch Acoust 2006;31(4):273–81. [2] Bolt RH. Frequency distribution of eigentones in a three-dimensional continuum. J Acoust Soc Am 1939;10:228. [3] Bolt RH. Normal frequency spacing statistics. J Acoust Soc Am 1947;19:79. [4] Bolt RH. Normal modes of vibration in room acoustics: angular distribution theory. J Acoust Soc Am 1939;11:74. [5] Bolt RH. Note on normal frequency statistics for rectangular rooms. J Acoust Soc Am 1946;17(101A):130–3. [6] Fuchs HV, Zha X, Zhou X, Drotleff H. Creating low-noise environments in communication rooms. Appl Acoust 2001;62: 1375–96. [7] Fuchs HV, Zha X, Pommerer M. Qualifying freefield and reverberation rooms for frequencies below 100 Hz. Appl Acoust 2000;4:303–22. [8] Harris CM, Molloy CT. The theory of sound absorptive materials. J Acoust Soc Am 1952;24:1–7. [9] Knudsen VO. Resonance in small rooms. J Acoust Soc Am 1932;4(1A):20–37. [10] Kylliainen M. Standard deviations in field measurements of impact sound insulation. Joint Baltic-Nordic acoustics meeting 2004; Marie˚ land. hamn, A [11] Morse PM, Bolt RH, Brown RL. Acoustic impedance and sound absorption. J Acoust Soc Am 1940;12:217–27. [12] Schroeder MR, Kutruff H. On frequency response curves in room. J Acoust Soc Am 1962;34(1):76–80.