Reverberation time in an almost-two-dimensional diffuse field

Journal of Sound and Vibration (1986) 111(3),391-398

REVERBERATION TIME IN AN ALMOST-TWO-DIMENSIONAL DIFFUSE FIELD M.

TOHYAMA AND

A.

SUZUKI

Musashino Electrical Communication Laboratories, Nippon Telegraph and Telephone Corporation, 3-9-11, Midoricho, Musashino-shi, Tokyo, 180 Japan (Received 27 June 1985, and in revised form 7 December 1985) A derivation for the reverberation time (RT) formula based on an almost-twadimensional diffuse field theory is presented. The RT formula is derived from wave theory. The formula is applicable for the RT of a room where two-dimensional waves are dominant. In such rooms, generally, the RT cannot be estimated accurately by using conventional reverberation theory. RT results measured in a special reverberation-variable room which follow the almost-two-dimensional diffuse field theory are presented.

I. INTRODUCTION

It is difficult to estimate the reverberation time (RT) in a room accurately. For example, suppose that the floor and ceiling of a room are covered by absorbing materials, while the side walls, which are perpendicular to the floor, are acoustically hard. In such rooms, the frequency characteristics of the RT cannot be estimated by classical reverberation theory. Schroeder and Hackman [1] and Fujiwara [2] used an integral equation, which describes the reverberation process, in order to calculate the RT. With this equation account is taken of the location of the absorptive materials, which is not considered in classical reverberation theory. The frequency characteristics, however, cannot be derived from the integral equation, because it does not contain a frequency parameter. Hirata [3] recently proposed an almost-two-dimensional diffuse field theory for the RT frequency characteristics when the room conditions are those mentioned in the first paragraph. This theory was derived on the basis of geometric acoustics. In this paper, the authors show that the almost-two-dimensional diffuse field theory can also be based on wave theory. Measurement results which follow the theory are presented. The measurements were made in a special reverberation-variable room. 2. RT IN TWO-DIMENSIONAL DIFFUSE FIELDS

Kosten [4] described the mean free path in a two-dimensional diffuse field. The mean free path is given by

mxy = 1TSxy/ L xy (m),

(1)

where mxy is the mean free path in a two-dimensional diffuse field, Sxy is the area occupied by the two-dimensional diffuse field, and Lx)' is the circumference of the two-dimensional diffuse field. Therefore, the RT becomes [5] RTxy = 0'1285xy/ axyLx)' 391 0022-460X/86/24039t+08 $03.00/0

(s),

© 1986 Academic

(2) Press Inc. (London) Limited

392

M. TOHYAMA AND A. SUZUKI

where R Txy is the reverberation time in the two-dimensional diffuse field, a xy = -In (1- aXY ) ' and ax y is the averaged absorption coefficient of the walls in the twodimensional diffuse field. The RT as given by equation (2) cannot adequately describe the frequency characteristics. 3. RT IN THE ALMOST TWO-DIMENSIONAL DIFFUSE FIELD

3.1. AVERAGE NUMBER OF REFLECTIONS PER UNIT TIME Batchelder [6] described the average number of reflections that a sound wave undergoes at the walls in a rectangular room. The average number of reflections depends on the direction cosines of the sound wave. The average number of reflections of an oblique wave in a rectangular room, v, is given by (3)

where Lx, Ly, L z are the respective lengths of the sides of the rectangular room (m), V is the volume of the room (rrr'), c is the sound speed in air (rn/s), a, f3, 'Yare direction cosines, a = kx / ko, {3 = ky/ s«; 'Y = k z / ko, k o is the wave number constant of the oblique wave, fo is the frequency of the oblique wave, k~ = k~ + k; + k;, p is the spatial distribution of the pressure in the oblique wave, P = cos (kx,X) cos (kyY) cos (k,z), and V vy, v, are the respective average numbers of reflections which the sound wave undergoes at the x-, yo, or z-walls respectively (z-walls are located on z = 0 and z == L, in rectangular coordinates, where the origin is located at one corner; x- and y-side walls are also located in a similar manner). X ,

3.2. NUMBER OF REFLECTIONS AND THE WAVE NUMBER CONSTANT The average number of reflections that an oblique wave undergoes at z-walls, v., is given by (4)

Thus, when is,

lJ,

= r, the

a-component of the wave number constant, k-, becomes

kzr = kotL,/ c (L'm),

k-: that (5)

Then, by introducing (6)

the equation (7)

can be obtained. Therefore, the average number of reflections at the z-walls, IJm in the oblique wave field, where the z components of the wave number constants of the oblique waves range from 0 to k m is given by v,r=r/2

3.3.

(l/s).

(8)

THE RATIO OF THE AVERAGE NUMBER OF REFLECTIONS AT THE z-WALLS TO THAT AT THE OTHER SIDE WALLS

The average number of reflections at the a-walls and that at the other side walls in the almost-xy-two-dimensional diffuse field is described in this section. The number of

RT IN AN ALMOST 2D DIFFUSE FIELD

393

reflections at the z-walls is neglected in the usual xy-two-dimensional diffuse field, which is constructed of xy-tangential waves whose z-components of the wave number constants are zero. As stated in the previous section, the RT frequency characteristics, however, cannot be derived from the conventional two-dimensional diffuse field theory. For use in such cases, an almost-two-dimensional diffuse field theory can be introduced in which reflections at the z-walls are taken into account. This field is assumed to be composed of both tangential waves and oblique waves which are close to tangential waves, called here almost-tangential-waves. The ratio of the average number of reflections at the z-walls to that at the other side walls (x- and y-walIs) for the almost-xy-two-dimensional diffuse field is needed to calculate the RT frequency characteristics. Suppose that the sound field contains the almost-xy-tangential waves whose zcomponents of the wave number constants range from 0 to k zr given by equation (5): that is, there are n, groups of almost-xy-tangential waves in the field, and each group has a different z-component of the wave number constant. n, is given by equations (6) and (7). The average number of reflections at the z-waIls, IIzn is given by equation (8); that is, Vzr == r/2. The average number of reflections at the x- and y-walls in the almost twodimensional diffuse field, lJxyn becomes lJxyr = nrc/ rnxy. Here nr shows that there are n, groups of almost-xy-tangential waves and the average number of reflections in each group of the waves is c] m xy , where rnx y is the mean free path in the xy-two-dimensional diffuse field given by equation (1). Consequently, the ratio defined above, /L, becomes (9)

This ratio contains the frequency as a parameter, thus allowing the frequency characteristics to be derived. 3.4. RT IN THE ALMOST TWO-DIMENSIONAL DIFFUSE FIELD The RT in the almost two-dimensional diffuse field is also given by equation (2). However, axy in the equation must be replaced by the averaged absorption coefficient a~y in the almost two-dimensional diffuse field. The absorption coefficient a~y can be found by using the ratio of the number of reflections given by equation (9). Thus the RT in the almost two-dimensional diffuse field is given by

«r; =0'128Sxy/a~yLxy where the averaged absorption coefficient,

a~y=axy(l-

a~y,

(s),

(10)

is given by

zr zr II )+az( V )=aXy(l-,u)+crz/L. lJxyr + V zr xyr + lJzr

(11)

lJ

Here a z is the averaged absorption coefficient of the z-walls, and a~y = -In (1- a~y). The RT formula (10) corresponds to Hirata's formula [3]; however, this equation was derived here by using wave theory, in a manner different from that of Hirata, whose approach was based on geometric acoustics. Thus, the same RT formula for an almost-twodimensional diffuse field can be derived either from wave theory or geometric acoustics. The RT formula contains frequency as a parameter, and therefore the frequency characteristics of the RT can be derived. It should be noted that the frequency characteristics mentioned here are not those due to the frequency characteristics of the absorption coefficients of the materials used in a room: that is, even if there were no frequency dependency of the absorption coefficients of materials, frequency characteristics of the RT may exist and can be derived, because of the almost-two-dimensionally-diffuse character of the field.

394

M. TOBY AMA AND A. SUZUKI

The averaged absorption coefficient shown by equation (11) contains the absorption coefficient of the z-walls, The ratio of the absorption coefficient of the z-walls to that of the other side walls in the total average absorption coefficient is determined by J.L as given in equation (9). As the frequency increases, f.L decreases; conversely, when the frequency decreases, J.L increases. For high frequency bands the height of the side walls surrounding the two-dimensional diffuse field is acoustically larger, so that the absorption of the z-walls is not significant. For low frequency bands, the height of the side walls is acoustically shorter, and the absorption of the z-walls becomes significant. Fixed plale Movable plate

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RT IN AN ALMOST 2D DIFFUSE FIELD

395

4. EXAMPLES OF CALCULATIONS AND MEASUREMENTS OF RT Measurement results in ordinary listening rooms which can be modeled by the almosttwo-dimensional diffuse theory have been presented in reference [3]. In what follows here examples of the RT in a special room, rather than an ordinary rectangular room, are presented. This room is provided with a lot of scattering obstacles, and was designed as a reverberation-variable room. In some configurations the acoustic field in the room can be modeled by using almost-twa-dimensional diffuse field theory. 4.1. REVERBERATION·VARIABLE ROOM Figure I shows the vertical (Figure lea)) and horizontal (Figure l(b)) aspects of the room. The geometric and other parameters are given in Tables I and 2. The side walls are constructed of 32 rotatable cylinders. One half of the surface area of each cylinder is covered by sound absorbing materials, and the other half is covered by reflective materials. The acoustic conditions in the room can be varied by removing or rotating the wall elements, so that the RT can be changed.

4.2. REVERBERATION TIME CHARACTERISTICS IN THE ROOM Figure 2 shows measurement results for the RT under three different room conditions. In condition A all the walls, the floor and the ceiling are reflective. In condition B all TABLE

1

Geometric parameters in the reverberation-variable room Figure

Section

25·2 m 2 (= L,L,) 20.4m (=2(Lx-i-Ly ) )

Floor SXy Circumference L x y Length L; Width i; Height i; Volume V

6·0m

4·2m 2'875 m 72'45 m3 (=LxLyL z)

2 Other parameters in the reverberation-variable room TABLE

Section

Area (m") Material

Floor

26·22

Side walls (cylinder)

60·65

Ceiling

31000

Outer walls

t Total surface area.

117·87t

Plastic tile, carpet, absorbing mat (cloth and OW 100 mrn, 32 kg/rrr') Absorbing side: cloth and OW 32 kg/rrr' Reflective side: iron plate (1·6 mrn), plywood (2·7 mm x 2), OW (32 kg/rn") Fixed (14 m2 ) : plasterboard (12 mm), plywood (12 mm), plasterboard (12 mrn), Fixed (I m 2 ) : window for air conditioning. Movable (16 m"): iron plate (2·3 mm) RW (50 mm, 80 kg/rn"), RC (300 mm) Sound trap (above the ceiling): OW (50 mm, 32 kg/rn"), plywood (5,5 mm), OW (50 mm, 32 kg/rrr')

396

M. TOHYAMA AND A. SUZUKI

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Frequency (kHz)

Figure 2. Measurement results of the reverberation time under three room conditions. Condition A (0): walls reflective, floor reflective (plastic tile), ceiling reflective (movable parts closed); condition B (e): walls absorptive, floor absorptive (absorbing mat and carpet), ceiling absorptive (movable parts opened); condition C (6): walls reflective, floor absorptive (absorbing mat and carpet), ceiling absorptive (movable parts opened).

the walls, the floor and the ceiling are absorptive. The RT at 500 Hz is 0·65 seconds under condition A, and 0·07 seconds under condition B. The RT under condition C has a different frequency characteristic from those under conditions A and B, as shown in Figure 2. In condition C all the side walls are reflective as in condition A; however, the floor and ceiling are absorptive as in condition B. The RT becomes longer as the frequency increases and above 1 kHz the RT for this condition is virtually the same as that for condition A. It thus is evident that the RT frequency characteristics under condition C cannot be predicted by using conventional reverberation theory. 4.3.

A COMPARISON OF THE NUMERICALLY CALCULATED RT VERSUS EXPERIMENTAL RESULTS

Accordingly the RT frequency characteristics under the condition C were calculated according to equation (10) and the results were compared with those obtained experimentally. The RT formula, equation (10), was deduced here by assuming the geometrical configuration to be that of a rectangular room. The RT formula, however, is similar to the formula derived from geometric acoustics theory in reference [3]. Therefore, it could be assumed that the RT formula should be applicable to non-rectangular rooms in which two-dimensional wave fields are significant. The geometric parameters are obtained approximately in such non-rectangular rooms. Although the reverberant sound field under condition C was actually produced in the special reverberation-variable room, it could be assumed to be composed mainly of two-dimensional waves, parallel to the floor, reflected repeatedly by only the side walls. Other types of two-dimensional waves should not contribute significantly since the reflective side walls are perpendicular to the floor, and both the floor and ceiling are highly absorptive. Furthermore, one-dimensional axial waves should not be significant, because there are no parallel walls in the non-rectangular room and many scattering cylinders are used in the side walls. The absorption coefficients of the side walls needed for the calculation are given in Table 3; those of the ceiling and floor, a., are assumed to be 0·8 on average. In the calculations account is not taken of the frequency dependency of the absorption

397

RT IN AN ALMOST 2D DIFFUSE FIELD

TABLE

3

Average absorption coefficients of reflective side walls ii xy

Frequency (Hz)

c¥Xy

125 250 500 1000 2000 4000 8000

0·19 0·26 0·23 0·23 0·20 0·18 0·24

coefficients of the ceiling and floor, in order to emphasize the frequency characteristics of the RT derived from the almost-twa-dimensional diffuse theory. These a's were estimated approximately according to the RT measured in the conditions A and B shown in Figure 2. Other geometric parameters needed for the calculation are given in Table 1. '·0

3

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8

Frequency (kHz)

Figure 3. Reverberation time calculated by using the almost-two-dimensional diffuse field theory. \/, Calculated by using equation (10), where ct, = 0·8 is the average absorption coefficient of the ceiling and floor, and ctx y is shown in Table 3; 6, measured under condition C similar to Figure 2; 0, calculated by using equation (2) (conventional two-dimensional diffuse field theory); 0, calculated by using three-dimensional diffuse field theory (ct, = 0·8).

Figure 3 shows the RT as calculated ana measured under condition C. The results calculated from the almost-twa-dimensional diffuse field theory given by equation (10) are for the most part in good agreement with the measured results. The RT characteristics cannot be calculated by using conventional reverberation theory (0 or 0 in Figure 3); however, they can be estimated from the almost-two-dimensional-diffuse field theory. 5. CONCLUSION

It has been shown that a reverberation time (RT) formula for an almost-twa-dimensional

diffuse field can be derived from wave theory, as well as from geometric acoustics, as was done previously by Hirata [3]. Some measurement results for the RT in a rather special non-rectangular room have been presented. The measured frequency characteristics of the RT agree with those predicted by the almost-twa-dimensional diffuse theory

398

M. TOHYAMA AND A. SUZUKI

for the most part. Under such almost-two-dimensional diffuse conditions the RT cannot be estimated by conventional reverberation theory. REFERENCES 1. M. R. SCHROEDER and D. HACKMAN 1980 Acustica 45, 269-273. Iterative calculation of reverberation time. 2. K. FUJIWARA 1984 Acustica 54, 266-273. Steady state sound field in an enclosure with diffusely and speculariy reflecting boundaries. 3. y. HIRATA 1982 Journal of Sound and Vibration 84, 509-517. Dependence of the curvature of sound decay curves and absorption distribution on room shapes. 4. C. W. KOSTEN 1960 Acustica 10, 245-250. The mean free path in room acoustics. 5. Y. HIRATA 1979 Acustica 43, 247-252. Geometric acoustics for rectangular rooms. 6. L. BATCHELDER 1964 Journal of the Acoustical Society of America 36, 551-555. Reciprocal of the mean free path.