Approximate Theory of Reverberation in Rectangular Rooms with Specular and Diffuse Reflections

Approximate theory of reverberation in rectangular rooms with specular and diffuse reflections

Tetsuya Sakumaa) Graduate School of Frontier Sciences, The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8563, Japan (Received 22 August 2011; revised 26 July 2012; accepted 26 July 2012) First, an approximate theory of reverberation in rectangular rooms is formulated as a specular reflection field based on the image source method. In the formulation, image sources are divided into axial, tangential, and oblique groups, which chiefly contribute to the corresponding groups of normal modes in wave acoustics. Consequently, the total energy decay consists of seven kinds of exponential decay curves. Second, considering surface scattering on walls with scattering coefficients, an integrated reverberation theory for nondiffuse field is developed, where the total field is divided into specular and diffuse reflection fields. The specular reflection field is simply formulated by substituting specular absorption coefficients, while the diffuse reflection field is assumed to be a perfectly diffuse field, of which energy is supplied from the specular reflection field at each reflection. Finally, a theoretical case study demonstrates how surface scattering affects the energy decay in rectangular rooms with changing the aspect ratio and the absorption distribution. VC 2012 Acoustical Society of America. [http://dx.doi.org/10.1121/1.4747001] PACS number(s): 43.55.Br, 43.20.Fn [LMW] Pages: 2325–2336

I. INTRODUCTION the effect strongly depends on room shapes and absorption distribution.15–17 In a room with regular shape and nonuni- In the field of room acoustics, reverberation is one of form absorption, the effect of surface scattering can be great, the most important aspects for judging acoustical qualities of and conversely, with irregular shape and uniform absorption, every kind of room, thus it is to be quantified, predicted, and it can be small or negligible. Recently, as a measure of the controlled. For this reason, a variety of theories of reverbera- efficiency of surface scattering, the scattering coefficient has tion in rooms have been proposed, originated from Sabine’s been widely used for geometric acoustic simulation.18 Its formula1 on the assumption of perfectly diffuse field, modi- measurement method was already standardized in ISO fied by Eyring,2 Norris,3 Millington,4 Sette,5 Kuttruff,6,7 and 17497-1,19 and computational methods are also available for others. The first two classical formulas are still widely used predicting it.20,21 However, up to now there exists no rever- for room acoustics design, although they cannot take into beration theory including the scattering coefficient, which account room shapes and absorption distribution.8 Actually, must be based on some theory for nondiffuse field, although the classical formulas often underestimate the reverberation some research has been concerned with sound fields includ- time of usual rooms, most of which have rectangular shapes ing both specular and diffuse reflections.22–24 and nonuniform absorption distribution. In this paper, first, an approximate theory of reverbera- For rectangular rooms, an empirical formula was first tion in rectangular rooms is formulated as a specular reflec- proposed by Fitzroy,9 and followed by Pujolle,10 Hirata,11 tion field based on the image source method by modifying Arau-Puchades,12 Nilsson,13, Neubauer,14 and others. Fitz- Hirata’s theory. Second, considering surface scattering on roy’s formula empirically assumes the arithmetic mean of walls with scattering coefficients, an integrated reverberation reverberation times of the three orthogonal directions, while theory for nondiffuse field is developed, where the total field Arau-Puchades’s formula assumes the geometric mean with is divided into specular and diffuse reflection fields. Finally, a theoretical consideration. Both of the formulas are simple a theoretical case study is presented to demonstrate how sur- to calculate, but do not work adequately in various situations. face scattering affects the energy decay of nondiffuse fields Hirata derived a reverberation theory based on the image in rectangular rooms, with changing the aspect ratio and the source method, decomposing the sound field into one-, two-, absorption distribution. and three-dimensional fields with frequency dependence. At first sight, this approach seems to correspond to the normal mode theory based on wave acoustics; however, some II. REVERBERATION OF SPECULAR REFLECTION discrepancy is seen due to a misunderstanding in the theoret- FIELD IN RECTANGULAR ROOMS ical development. A. Specular field of image sources Generally, surface scattering tends to heighten field dif- fuseness, resulting in a decrease in reverberation time, but As illustrated in Fig. 1, a regular arrangement of image sources is determined for a point source in a rectangular room. When the energy density of the reflection field is dis- a)Author to whom correspondence should be addressed. Electronic mail: cussed with space average for both source and receiver, a [email protected] continuous distribution of image sources is considered with

J. Acoust. Soc. Am. 132 (4), October 2012 0001-4966/2012/132(4)/2325/12/$30.00 VC 2012 Acoustical Society of America 2325 surface area S ¼ 2(LxLy þ LyLz þ LzLx). Furthermore, the area-weighted arithmetic mean of absorption coefﬁcients for the three directions is used as represented by

r r r LyLza~x þ LzLxa~y þ LxLya~z aob ¼ ; (3) LyLz þ LzLx þ LxLy

r where a~xðy;zÞ is the geometric mean of random-incidence absorption coefﬁcients in each direction according to Eq. (2). Consequently, Eq. (1) is approximated by ð 1 2 S W r=lob 4p dr EobðtÞ¼ 2 ð1 aobÞ ct 4pcr !V W cA^ ¼ exp ob t ; (4) cA^ob V

with Aôb ¼ Aob=4; Aob ¼ SaEob and aEob ¼lnð1 aobÞ: For a general expression, the quantity A^ is defined as the room absorption factor taking into account the ratio of incident in- tensity to energy density, thus giving A^/V ¼ aE=l throughout this paper. Equation (4) is regarded as a first approximation for the FIG. 1. Image sources of a rectangular room. specular field of oblique sources, in a similar form to Eyr- ing’s formula using Millington’s average absorption coeffi- a representative receiving point at the center of the room. For cient for each pair of parallel walls. However, it includes the image sources at equal distances from the receiving point, the contributions of axial and tangential sources that are more number of sources, distance attenuation, and wall absorption apt to deviate from the average decay. In the following, one-, are estimated on the condition that only specular reflections two-, and three-dimensional specular fields are considered occur throughout the paths. If the real source of the sound by dividing image sources into axial, tangential, and oblique power W is stopped at t ¼ 0 in the steady state, the average groups, which chiefly contribute to the corresponding groups energy density of the three-dimensional specular field is formu- of normal modes in wave acoustics. lated in spherical coordinates, as follows (t 0): ð ð ð 2p p 1 W B. Specular fields of axial sources S ~ rsinhcosu=Lx E ðtÞ¼ 2 ð1axÞ 0 0 ct 4pcr Consider the image sources near the x axis in the angular 2 ranges within 6hy and within 6hz to the positive and negative rsinhsinu=Ly rcosh=Lz r sinhdrdhdu ð1a~yÞ ð1a~zÞ ; x directions, as illustrated in Fig. 2.Thex-axial sources lead LxLyLz to normal incidence to the x-directional walls (in yz plane), (1) and grazing incidence to the y-andz-directional walls. If the where c is the sound speed, L is the length of each side, off-axial path differences from an x-axial source to the room x(y,z) in y and z directions are, respectively, within 1/4 wavelength and a~xðy;zÞ is the absorption coefficient averaged by the geometric mean of two reflection coefficients in each direction. (p/2 in phase), the image source chiefly contributes to x-axial Because of alternate reflections between the parallel surfa- modes in wave acoustics. For far x-axial sources at small ces, the geometric average is used, represented by angles of hy(z), the maximum path differences are expressed rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi by Dy(z) Ly(z) sin hy(z) Ly(z)hy(z), thus the critical angles are generally defined as þ a~xðy;zÞ ¼ 1 1 axðy;zÞ 1 axðy;zÞ ; (2)

6 where axðy;zÞ are the absorption coefficients of two parallel walls. Generally, Eq. (1) is composed of direction-dependent multiple exponentials, and numerical integration is required to evaluate the exact energy decay.8 In a similar way to the classical reverberation theory, a single exponential decay function is roughly assumed, where the mean free path for oblique sources lob is approximately equal to the theoretical value for the three-dimensional perfectly diffuse field lr ¼ 4 V/S (see the first section of the 25 Appendix), with the room volume V ¼ LxLyLz, and the total FIG. 2. Axial image sources for the x axis in the xy plane.

2326 J. Acoust. Soc. Am., Vol. 132, No. 4, October 2012 Tetsuya Sakuma: Theory of reverberation in rectangular rooms pc h ¼ ; (5) C. Specular fields of tangential sources xðy;zÞ 2xL xðy;zÞ In a similar way to axial sources, consider the image sources near the xy plane in the angular range within 6h to with the angular frequency x. z the xy plane, which lead to random incidence to the x- and y- At the critical angles h , the frequency of normal reflec- y(z) directional walls, and grazing incidence to the z-directional tions on the x-directional walls is expressed by n c/L ,and ax x walls (in xy plane). Again, if the off-tangential path differ- those of grazing reflections on the y-andz-directional walls ence from an xy-tangential source to the room in z direction by n ¼ c sin h /L ch /L . In the angular ranges, axy(z) y(z) y(z) y(z) y(z) is within 1/4 wavelength (p/2 in phase), the image source the average reflection frequency for the x direction, n n , ax ax chiefly contributes to the xy-tangential modes. Thus, the crit- and those for the y and z directions, n n =2. By lim- axyðzÞ axyðzÞ ical angles by Eq. (5) are in common with axial and tangen- iting the angular ranges of integration in Eq. (1), the energy tial sources. density of the one-dimensional specular field from x-axial In the angular range, the average frequency of reflec- sources is formulated as follows: ð tions for the x- and y-directional walls is ntxy c=lxy, with 1 the two-dimensional mean free path l ¼ pL L /2(L þ L ) S 4W n r=Lx g rhy=2Ly xy x y x y EaxðtÞ¼ ð1 a~xÞ ð1 a~yÞ (see the first section of the Appendix), and the average fre- c ct quency of grazing reflections for the z-directional walls, h = 2ð2hyÞð2hzÞdr ð1 a~gÞr z 2Lz n n =2 ch =2L . Similar to Eq. (6), the energy density z 4p tz tz z z ð of the two-dimensional specular field from xy-tangential 4W pc2L 1 x r=lx sources is formulated as follows: ¼ 2 2 ð1 aaxÞ dr c 2x V ct ð ! 1 2 ^ S 2W r r=lxy g rhz=2Lz 2pð2hzÞdr 4W pc Lx cAax E ðtÞ¼ ð1 a~ Þ ð1 a~ Þ ¼ exp t (6) txy c xy z 4pV c 2x2VA^ V ct ð ax 2W pcL L 1 ¼ x y ð1 a Þr=lxy dr where the one-dimensional mean free path l ¼ L , and the 2 txy x x c 2xV ct ! contribution of axial sources to the one-dimensional field is 2W pcL L cA^ assumed quadruple to that of oblique sources to the three- ¼ x y exp txy t ; ^ dimensional field due to the coherence in the two off-axial c 2xVAtxy V directions, in accordance with the ratio of normalization fac- (9) tors in the mode theory. Regarding wall absorption, where the contribution of tangential sources to the two- Aâx ¼ 2Aax; Aax ¼ 2LyLzaEax; aEax ¼lnð1aaxÞ; dimensional field is assumed double that of oblique sources to n g eaxy g eaxz the three-dimensional field due to the coherence in the off- aax ¼ 1ð1a~xÞð1a~yÞ ð1a~z Þ ; (7) tangential direction. Regarding wall absorption, A^txy ¼ Atxy=p; naxyðzÞ pcL x Atxy ¼ 2ðLx þ LyÞLzaEtxy, aEtxy ¼ln(1 atxy), eaxyðzÞ ¼ 2 ; (8) nax 4xLyðzÞ r g etz atxy ¼ 1 ð1 a~xyÞð1 a~z Þ ; (10) n where a~x is the normal-incidence absorption coefficient of g L a~r þ L a~r the x-directional walls, and a~yðzÞ are the grazing-incidence r y x x y a~xy ¼ ; (11) absorption coefficients of the y- and z-directional walls, con- Ly þ Lx sidered as the geometric mean for parallel walls according to 2 Eq. (2). In the strict sense, the above-presented normal and ntz p cLxLy etz ¼ 2 ; (12) grazing incidence correspond to incidence from the angular ntxy 8xðLx þ LyÞLz ranges of axial sources with frequency dependence; accord- r ingly, a database of directional absorption coefficients is where a~xy is the area-weighted arithmetic mean of random- needed. Practically, rough estimates of the normal- and incidence values for the two directions, considering alternate grazing-incidence coefficients are available on the assumption reflections by Eq. (2), and atxy is the total average absorption of local reaction (see the second section of the Appendix). The coefficient for the two-dimensional field, with etz the ratio of total average absorption coefficient for the one-dimensional average reflection frequency of the z direction to the x and y field, aax, is given by taking into account the reflection fre- directions. Note that Eq. (10) is again not valid if one of the g quency in each direction, with eaxy(z) the ratios of average z-directional walls has perfect absorption yielding a~z ¼ 1, reflection frequency of the y and z to the x direction. because the contribution of xy-tangential sources perfectly In Eq. (7), the geometric mean weighted with average vanishes in Eq. (9). reflection frequency is given, for the reason that the reflection frequency in each direction is almost independent for axial sources. However, it is not valid for an extreme case in D. Reverberation of total specular field which that either grazing-incidence absorption coefficient g a~yðzÞ ¼ 1, because the contribution of x-axial sources per- Equation (1) includes the contributions of axial and fectly vanishes in Eq. (6), even if one of the y- and z-direc- tangential sources, and Eq. (9) also includes that of axial tional walls has perfect absorption. sources. Summing up Eqs. (1), (6), and (9), excluding the

J. Acoust. Soc. Am., Vol. 132, No. 4, October 2012 Tetsuya Sakuma: Theory of reverberation in rectangular rooms 2327 above-presented duplicate contributions, the total energy which is consistent with Eq. (5). Thus it can be stated that density of specular fields in the room is expressed by the angular ranges for axial and tangential image sources " ! approximately correspond to axial and tangential modes in W c cA^ ESðtÞ¼ ob exp ob t wavenumber space. c A^ V It should be noted that the present theory includes three ob ! reasonable modifications to the original theory by Hirata,11 on X 2c cA^ þ txy exp txy t the critical angles, the averaging of absorption coefficients, A^ V xy txy !# and the ratios of normalization factors. In the original theory, X 4c cA^ different angular ranges were derived from an ambiguous dis- þ ax exp ax t ; (13) cussion, resulting in some discrepancy with the expression ^ V x Aax based on the normal mode theory. Regarding absorption coefficients, alternate reflections between parallel walls were not where the proportion of sources in each group is denoted by considered, and arithmetic averages weighted with reflection pcS pc2L frequency were given regardless of one- or two-dimensional c 1 ; 28 ob ¼ þ 2 field, thus finally leading to different decay rates. 4xV8x V 2 In the modified theory, geometric averages are used pcLxLy cðLx þ LyÞ pc Lx ctxy ¼ 1 ; cax ¼ ; with regard to independence of reflection frequencies. How- 2xV xL L 2x2V x y ever, if it is applied to extreme cases where one of the walls has perfect absorption, all of the one-dimensional fields and with the total edge length L ¼ 4(Lx þ Ly þ Lz). Conse- quently, the reverberation of the total specular field is com- a two-dimensional field that is tangential to the absorptive posed of seven kinds of exponential decay, arising from one wall vanish completely. Therefore the modified theory is oblique, three tangential, and three axial source groups. Each not applicable to such cases, specifically with perfect absorp- decay rate in decibels per unit time is given by tion for grazing incidence. If the original theory is applied to the cases, none of the one- and the two-dimensional S ^ Dobðtxy;axÞ ¼ 10log10e cAobðtxy;axÞ=V; (14) fields vanish, even a one-dimensional field that is normal to the absorptive wall. It is not realistic from a different aspect, with the relation so the original theory is also not applicable to such cases, specifically those with perfect absorption for normal min ðDS ; DS ; DS Þ < min ðDS ; DS ; DS Þ < DS : (15) ax ay az txy tyz tzx ob incidence. Equation (13) entirely corresponds to the approximate expression derived from the normal mode theory in wave 26,27 III. REVERBERATION IN RECTANGULAR ROOMS acoustics. The critical angles for axial and tangential image WITH SURFACE SCATTERING sources can be interpreted in view of normal mode distribution in wavenumber space, as illustrated in Fig. 3. Supposing the A. Specular and diffuse fields with surface scattering ranges dominated by axial modes to be within the middle to the The scattering coefficient of a diffuse surface is defined adjacent oblique modes, the critical angles are determined by as the ratio of nonspecularly reflected energy to total reflected p energy.18 By introducing the scattering coefficients of wall ¼ k sin hyðzÞ khyðzÞ; (16) 2LyðzÞ surfaces, sound energy propagating from image sources can be divided into specular and diffuse components at each reflection. One possible assumption is that the specular reflection field is composed of arriving components that are never scattered at all reflections from image sources to a receiving point. On that assumption, the energy density of the specular field is simply given by replacing all values related to absorption coefficient with those related to specular absorption coefficient b ¼ a þ (1 a)s, with s the scattering coefficient. Accordingly, Eq. (13) is modified by substituting all kinds of ~ ^ ^ b, bE, B,andb for a~, aE, A,andA, as follows: W c cB^ ESðtÞ¼ ob exp ob t c Bôb V X 2c cB^ þ txyexp txy t ^ xy Btxy V # X 4c cB^ þ ax exp ax t ; (17) B^ V FIG. 3. Axial modes in the xy plane in wavenumber space. x ax

2328 J. Acoust. Soc. Am., Vol. 132, No. 4, October 2012 Tetsuya Sakuma: Theory of reverberation in rectangular rooms ð ð 1 r 0 0 0 thus each decay rate is also modified as D W r =lob ðrr Þ=lr sEobdr dr EobðtÞ¼ ð1 bobÞ ð1 arÞ c ct ð0 lob V S ^ 1 Dob txy;ax ¼ 10log10e cBobðtxy;axÞ=V: (18) W dr ð Þ ¼ l ½ð1 a Þr=lr ð1 b Þr=lob c ob r ob V "#ct ! As illustrated in Fig. 4, it is assumed that a part of the W 1 cA^r 1 cBôb specular energy is transformed into diffuse energy at each ¼ l exp t exp t ; c ob A^ V B^ V refection, and after the transition, the diffuse energy is r ob decayed by perfectly diffuse reflections and finally arrives to (21) a receiving point by random incidence. With respect to an ar- ^ ^ ^ bitrary distance r0 from an image source, the rate of scattered with lob ¼ Sob=ðBob ArÞ, and the mean free path for energy in a very small path is given by the diffuse field lr. In the specular reflection path before ^ = ; the transition, the energy is decayed with Bob ¼ Bob 4 Bob 0 0 ¼ SbEob; bEob ¼ln(1 bob) ¼ aEob þ sEob, while in the dif- dr =lob sEobdr lim 1 ð1 sobÞ ¼ lim 1 exp fuse reflection path after the transition, the energy is decayed dr0!0 dr0!0 l ob ^ with Ar ¼ Ar=4, Ar ¼ SaEr, aEr ¼ln(1 ar), where ar is the s dr0 S^ dr0 ! Eob ¼ ob ; area-weighted mean of random-incidence absorption coeffi- l V ob cients. The relations lob lr and aob ar hold for oblique ^ (19) sources, thus resulting in Bôb Ar and 0 lob 1. with S^ ¼ S =4; S ¼ Ss , and s ¼ln (1 s ). ob ob ob Eob Eob ob B. Diffuse fields of axial and tangential sources The average scattering coefficient for the three-dimensional specular field is given by Considering axial image sources as mentioned previously, and referring to Eqs. (6) and (21), the energy density r r r r r r LyLzð1 a~xÞs~x þ LzLrð1 a~yÞs~y þ LxLyð1 a~zÞs~z of the diffuse field from x-axial sources is formulated as sob ¼ r r r ; follows: LyLzð1 a~xÞþLzLxð1 a~yÞþLxLyð1 a~zÞ ð ð (20) 1 r W n 0 g 0 ED ðtÞ¼ ð1b~ Þr =Lx ð1b~ Þr hy=2Ly ax c x y r ct 0 where s~ is the geometrical mean of random-incidence 0 xðy;zÞ g 0 0 s dr ~ r hz=2Lz ðrr Þ=lr Eax scattering coefficients of two parallel walls, considering ð1b Þ ð1arÞ z l alternate reflections by Eq. (2), and s is assumed the arith- x ob 2ð2h Þð2h Þdr metic mean weighted with area and reflection coefficient for y z the three directions. Estimating the energy scattered at r0 and 4pðV ð 2 1 r W pc L 0 0 decayed before and after the transition, and subsequently ¼ x ð1b Þr =lx ð1a Þðrr Þ=lr 0 2 ax r integrating it with respect to r throughout the path, the c 2x V ct 0 0 energy density of the diffuse field for oblique sources is for- Sâxdr dr mulated as follows: V V " ! 2 W pc Lx 1 cA^r ¼ 2 lax exp t c 2x V A^r V 1 cB^ exp ax t ; (22) Bâx V

^ ^ ^ with lax ¼ Sax=ðBax ArÞ, where the contribution of axial sources to diffuse ﬁeld is equal to that of oblique sources due to random incidence to the room. Regarding wall scattering, ^ Sax ¼ Sax=2, Sax ¼ 2LyLzsEax, sEax ¼ln(1 sax), and similar to Eq. (7),

n g eaxy g eaxz sax ¼ 1 ð1 s~xÞð1 s~yÞ ð1 s~z Þ ; (23) n with s~x the normal-incidence scattering coefficient of the g x-directional walls and s~yðzÞ the grazing-incidence scattering coefficients of the y- and z-directional walls, considered as the geometric mean for parallel walls according to Eq. (2). Again, because the above-presented normal- and grazing- incidence coefficients are the averages within the frequency- dependent angular range, a database of directional scattering FIG. 4. Transition from specular to diffuse reflections. coefficients is needed, which can be obtained by the

J. Acoust. Soc. Am., Vol. 132, No. 4, October 2012 Tetsuya Sakuma: Theory of reverberation in rectangular rooms 2329 free field measurement18 or numerical computation.20 If the one-dimensional specular field decays more rapidly than the ^ ^ diffuse field ðBax > ArÞ; lax 0. Otherwise, the sign of the inequality is changed, where the decay of the diffuse field is obeyed by the one-dimensional specular field, and a flutter echo is liable to occur. In the singular case that Bâx ¼ A^r, Eq. (22) can be transformed into ! 2 ^ ^ D W pc Lx sEax 1 þ Arct=V cAr EaxðtÞ¼ 2 exp t : (24) c 2x V bEax A^r V

Similar to absorption coefficient, it should be noted that Eq. (23) is not valid for an extreme case in which either g grazing-incidence scattering coefficient s~yðzÞ ¼ 1, because the specular component of x-axial sources perfectly vanishes in Eq. (17), even if one of the y- and z-directional walls has perfect diffusion. In a similar way to axial sources, the energy density of the diffuse field from xy-tangential sources is formulated as follows: ð ð FIG. 5. Energy decay of specular, diffuse, and total fields for case 1b, with a 1 r scattering coefficient of 0.1: (a) 500 Hz, (b) 2000 Hz. Abbreviations: Specu- W r 0 g 0 D ~ r =lxy ~ r hz=2Lz EtxyðtÞ¼ ð1 bxyÞ ð1 bz Þ lar/diffuse fields in subtotal (S/D), of oblique sources (Sob/Dob), of xy-tan- c ct 0 gential sources (Stxy/Dtxy), of x-axial sources (Sax/Dax). 0 0 s dr 2pð2h Þdr ð1 a Þðrr Þ=lr Etxy z r l 4pV ð ð xy ! 1 r ^ ^ W pcL L 0 0 W pcLxLy sEtxy 1 þ Arct=V cAr x y r =lxy ðrr Þ=lr ED ðtÞ¼ exp t : ¼ ð1 btxyÞ ð1 arÞ txy ^ c 2xV ct 0 c 2xV bEtxy Ar V S^ dr0 dr (28) txy V V " ! ^ Note that Eq. (26) is again not valid if one of the z-direc- W pcLxLy 1 cAr g ¼ ltxy exp t tional walls has perfect diffusion with yielding s~z ¼ 1, c 2xV A^r V because the specular component of xy-tangential sources 1 cB^ perfectly vanishes in Eq. (17). exp txy t ; (25) B^txy V

^ ^ ^ with ltxy ¼ Stxy=ðBtxy ArÞ. Regarding wall scattering, ^ Stxy ¼ Stxy=p, Stxy ¼ 2(Lx þ Ly)LzsEtxy, sEtxy ¼ln(1 stxy), and similarly to Eqs. (10) and (11),

r g etz stxy ¼ 1 ð1 s~xyÞð1 s~z Þ ; (26) ~r ~r ~r ~r r Lyð1 axÞsx þ Lxð1 ayÞsy s~xy ¼ r r ; (27) Lyð1 a~xÞþLxð1 a~yÞ assuming the arithmetic mean weighted with area and reﬂec- tion coefﬁcient for the two directions. Again, ltxy 0if B^txy > A^r, otherwise changing the sign of the inequality. In the singular case that B^txy ¼ A^r, Eq. (25) can be transformed into

TABLE I. Dimensions and absorption coefﬁcients of rectangular rooms.

Case Lx (m) Ly (m) Lz (m) ax ay az ax/Lx: ay/Ly: az/Lz

1a 10 10 10 0.35 0.35 0.35 1: 1: 1 1b 10 10 10 0.15 0.30 0.60 1: 2: 4 2a 20 10 5 0.30 0.30 0.30 1: 2: 4 FIG. 6. Energy decay of specular, diffuse, and total ﬁelds for case 2b, with a 2b 20 10 5 0.10 0.20 0.40 1: 4: 16 scattering coefﬁcient of 0.1: (a) 500 Hz, (b) 2000 Hz. Abbreviations: See Fig. 5.

2330 J. Acoust. Soc. Am., Vol. 132, No. 4, October 2012 Tetsuya Sakuma: Theory of reverberation in rectangular rooms C. Reverberation of total diffuse field Similar to the total specular ﬁeld, summing up Eqs. (21), (22), and (25) and excluding the duplicate contributions of axial and tangential sources, the total energy density of diffuse ﬁelds in the room is expressed by

( "# ! "# ! W 1 cA^ 1 cB^ X 1 cA^ 1 cB^ EDðtÞ¼ c l exp r t exp ob t þ c l exp r t exp txy t ob ob ^ ^ txy txy ^ ^ c Ar V Bob V xy Ar V Btxy V "#) ! X 1 cA^ 1 cB^ þ c l exp r t exp ax t ; (29) ax ax ^ ^ x Ar V Bax V

where cob(txy,ax) are the proportions of sources as denoted for It is confirmed that if all wall surfaces are perfectly D Eq. (13).Att ¼ 0, the total energy density in the steady state diffuse with s ! 1, sE/bE ! 1, thus resulting in E ð0Þ is expressed by ! W=cA^r. !The reverberation of the total diffuse field is also appa- X X rently composed of seven kinds of decay, but double expo- W sEob sEtxy sEax EDð0Þ¼ c þ c þ c : nential decay. With the decay rate of the three-dimensional ^ ob txy ax cAr bEob xy bEtxy x bEax D ^ perfectly diffuse field, Dr ¼ 10log10e cAr=V each rate is (30) given by

^ ^ 1 exp½ Bobðtxy;axÞ Ar ct=V DD ðtÞ¼DD : (31) obðtxy;axÞ r ^ ^ ^ ^ 1 Ar=Bobðtxy;axÞ exp½ Bobðtxy;axÞ Ar ct=V

In Eq. (31), just after stopping the source (t ! 0), D. Integrated reverberation in rectangular rooms DD ðtÞ!0; that is, all seven groups of sources do obðtxy;axÞ On account that specular and diffuse reflection fields are not immediately start to decay. In the late decay (t ), !1 superposed, the reverberation of an actual room is generally DD ðtÞ!DD DS for oblique sources, while DD ðtÞ! ob r ob txyðaxÞ observed as the overall energy decay. Adding the energy den- min DD; DS for axial and tangential sources, where the r txyðaxÞ sity of specular and diffuse fields by Eqs. (17) and (29),the decay rates of the specular fields DS are given by Eq. (18). txyðaxÞ overall energy density in a rectangular room is expressed by

" ! # X X W l cA^ 1l cB^ 2l cB^ 4l cB^ EðtÞ¼ r exp r t þc ob exp ob t þ c txy exp txy t þ c ax exp ax t ; ^ ob ^ txy ^ ax ^ c Ar V Bob V xy Btxy V x Bax V (32)

P P S D with lr ¼ coblob þ xyctxyltxy þ xcaxlax. Dobðtxy;axÞ, and one rate for the diffuse ﬁeld Dr . The relation by Consequently, the reverberation of the room is apparently Eq. (15) holds among the former seven rates, thus the total D S S S composed of eight kinds of exponential decay, of which decay decay rate approaches min (Dr ; Dax; Day;Daz) in the late decay. rates correspond to seven rates for the specular ﬁelds At t ¼ 0, the overall energy density is expressed by

"# X X W A^ þ S^ 2A^ þ S^ 4A^ þ S^ Eð0Þ¼ c r ob þ c r txy þ c r ax : (33) ^ ob ^ txy ^ ax ^ cAr Bob xy Btxy x Bax

J. Acoust. Soc. Am., Vol. 132, No. 4, October 2012 Tetsuya Sakuma: Theory of reverberation in rectangular rooms 2331 Considering actual rooms in the mid and high frequency which depends on the average scattering and absorption range, oblique sources are usually dominant in the steady coefficients for the one-dimensional field, with the conven- state, thus roughly giving tional average absorption coefficient.

W aEr þ sEob W IV. THEORETICAL CASE STUDY Eð0Þ ; (34) ^ b ^ cAr Eob cAr A. Conditions of rectangular rooms where the ratio of specular to diffuse field is aEr/sEob. As a case study in rectangular rooms, the influences of Accordingly, the energy density ratio of diffuse to total field room aspect ratio, absorption distribution, and surface scat- in decibels is given by tering on reverberation are investigated by the present theory. As shown in Table I, four types of rooms are given, all of which have a volume of 1000 m3 and an absorption D aEr R ð0Þ10log10 1 þ ; (35) area of 210 m2, additionally assuming that absorption and sEob scattering coefficients have no dependence on incidence angle, and the latter has an uniform value for all surfaces. which depends on the average scattering coefficient for Based on the two classical theories, Sabine’s reverberation oblique sources with the conventional average absorption D time is 0.77 s for all cases, whereas Eyring’s is 0.62 s for coefficient. In the late decay (t !1), if Dr is the minimum ^ ^ cases 1a and 1b, and 0.65 s for cases 2a and 2b. In the fol- rate (Ar Baxðy;zÞ), the energy of the specular field vanishes D lowing, energy density levels relative to L ¼ 10log (W/c) earlier than the diffuse field, thus R (t) ! 0. Otherwise, the 0 10 are calculated for the specular and the diffuse fields of axial, contribution of one of axial source groups lingers with the tangential, and oblique sources in each and subtotal, and for minimum rate among DS , and the energy is balanced axðy;zÞ the total field. between the specular and the diffuse fields as follows:

! B. Results and discussion ^ ^ Ar Baxðy;zÞ RDðtÞ!10log 1 þ 4 ; (36) Figures 5 and 6 show the energy decay curve of each 10 S^ axðy;zÞ component, calculated for cases 1b and 2b with a scattering coefﬁcient of 0.1, at 500 and 2000 Hz. It is seen that the

FIG. 7. Energy decay of specular, diffuse, and total ﬁelds at 500 Hz, with changing the scattering coefﬁcient from 0.05 to 0.8: (a) Case 1a, (b) case 1b, (c) case 2a, (d) case 2b.

2332 J. Acoust. Soc. Am., Vol. 132, No. 4, October 2012 Tetsuya Sakuma: Theory of reverberation in rectangular rooms FIG. 8. Transient proportion of specular and diffuse to the total energy at 500 Hz, with changing the scattering coefficient from 0.05 to 0.8: (a) Case 1a, (b) case 1b, (c) case 2a, (d) case 2b. decay curve of the specular field for every source group is decay, but it has a remarkably affect in the other cases. It is straight, but not for the diffuse field. Regarding frequency seen that with increasing the scattering coefficient the specu- dependence, the energy density levels of axial and tangential lar field decays rapidly, which suppresses the curvature of sources are relatively higher at the lower frequency, thus the the total decay. However, the change of scattering coefficient curvature of the total decay is more remarkable, especially from 0.4 to 0.8 hardly affects the total decay in all cases. in case 2b. Figure 8 shows the transient proportion of specular and Figure 7 shows the energy decay curves calculated at diffuse energy to total energy in the decay curves of Fig. 7. 500 Hz for all cases, uniformly changing the scattering coef- It is seen that at the beginning of the curves the balance ficient from 0.05 to 0.8. In case 1a with cubic shape and uni- between specular and diffuse energy depends on the scatter- form absorption, surface scattering does not affect the total ing coefficient, but not on the cases under the condition of an

J. Acoust. Soc. Am., Vol. 132, No. 4, October 2012 Tetsuya Sakuma: Theory of reverberation in rectangular rooms 2333 FIG. 9. Reverberation time T30 at 500 and 2000 Hz, with changing the scattering coefficient from 0.05 to 0.8: (a) Case 1a, (b) case 1b, (c) case 2a, (d) case 2b. Four types of lines represent theoretical values by Sabine, Eyring, Fitzroy, and Arau-Puchades. equal absorption area. Regarding the components of the study demonstrated the following effects of surface scatter- specular energy, it is observed that the three-dimensional ing on energy decay in rectangular rooms: (a) With an irreg- field is dominant at the beginning, thus validating the ular aspect ratio or nonuniform absorption distribution, the approximation by Eq. (35). In the late decay, the ratio of dif- curvature of the total decay occurs if the scattering coeffi- fuse energy converges to a nonzero value with increasing the cient is low; (b) in the steady state, the balance between ratio of one-dimensional specular energy, if the scattering specular and diffuse energy is determined by the absorption coefficient is below some value that depends on the cases. and scattering coefficients; (c) in the late decay, the diffuse This tendency can be confirmed by Eq. (36), which specifies field or one of one-dimensional specular fields is dominant the above-mentioned condition as A^r > Bâx. depending on the scattering coefficient. Figure 9 shows the reverberation time T30 in the decay However, the present theory has some problems to be range from 5to35 dB at 500 and 2000 Hz for all cases, further investigated for practical use. First of all, the theory with changing the scattering coefficient from 0.05 to 0.8. For is based on an approximation for specular fields that assumes reference, the theoretical values by Sabine, Eyring, Fitzroy, single exponential decay for each group of image sources. In and Arau-Puchades are indicated regardless of scattering the strict sense, each source can have a different decay rate, coefficient. In all cases, the calculated values are almost which leads to a somewhat curvature in each group.29,30 equal to Eyring’s if the scattering coefficient is above 0.4. Consequently, some discrepancy from the true decay will On the other hand, with the lowest scattering coefficient of occur in rooms with extremely irregular aspect ratio or non- 0.05, increases of 5% to 30% to Eyring’s are observed in uniform absorption distribution. Regarding absorption and cases 1b and 2a, while drastic increases of two to three times scattering coefficients, the theory involves normal- and in case 2b, which are considerably beyond Fitzroy’s. Up to grazing-incidence values in addition to random-incidence now the present theory does not take into account additional values. For the normal and grazing incidence, the critical scattering from the edges of a room, which can occur due to angle changes depending on the frequency and the size of a a difference of absorption coefficients between adjacent wall, which makes it troublesome to evaluate the values. A walls. Generally, the effect of edge scattering can be greater rough estimation for absorption coefficients, presented in the at lower frequencies, which will suppress reverberation of second section of the Appendix, may be a practical way to specular fields to some extent. Therefore the calculated be tested. Furthermore, it should be noted that the theory is reverberation time may be rather overestimated in rooms not applicable for an extreme case in which one of walls has with very low surface scattering. perfect absorption or diffusion. A different problem may arise from the fact that the theory takes into account surface V. CONCLUSIONS scattering on the walls, but not considering edge scattering around the walls. The influence of edge scattering should not An approximate theory of reverberation in rectangular be negligible at lower frequencies, thus some way is needed rooms with specular and diffuse reflections was newly devel- to include it. oped based on the image source method by introducing scat- As mentioned previously, the scope of the present tering coefficients of wall surfaces. First, it was described theory should be intricately limited by many factors. Future that the specular reflection field is decomposed into one-, work on validating the theory by experimental measurement two-, and three-dimensional fields with frequency depend- and computational simulation, and also by comparing with ence, and its formulation corresponds to the approximate other theories in various kinds of rooms is recommended. expression by the normal mode theory in wave acoustics. Second, an integrated theory for specular and diffuse fields ACKNOWLEDGMENT was formulated with scattering coefficients, which expresses that the total energy decay is apparently composed of seven This project has been funded by the Grant-in-Aid Scien- kinds of exponential decay for specular fields and one for the tific Research from Japan Society for the Promotion of Sci- three-dimensional diffuse field. Finally, a theoretical case ence (No. 21360275).

2334 J. Acoust. Soc. Am., Vol. 132, No. 4, October 2012 Tetsuya Sakuma: Theory of reverberation in rectangular rooms ð ð APPENDIX 1 2p p cS nxyz ¼ ðnx þ ny þ nzÞ sin hdhdu ¼ ; (A1) 4p 0 0 4V 1. Approximation of mean free paths for specular fields

Although there exist well-known arguments on the mean with nx ¼ c|sin h cos u|/Lx, ny ¼ c|sin h sin u|/Ly, and 25,31,32 free path, the inverse of mean reflection frequency with nz ¼ c|cos h|/Lz, giving the mean free path for the specular respect to sound speed is considered as the value for a specu- field as equal to the value for the perfectly diffuse field, lar field of image sources. The three-dimensional mean reflec- lr ¼ 4 V/S. By excluding axial and tangential sources, Eq. tion frequency for all image sources is represented by (A1) is modified for oblique sources as follows:

X X 2pð2h Þ 2ð2h Þð2h Þ n ðn þ n Þ z þ ðn þ n þ n Þ y z 2cL xyz txy tz 4p ax axy axz 4p 1 xy x xS nob ¼ X X nxyz ; (A2) 2pð2h Þ 2ð2h Þð2h Þ pcS 1 z þ y z 1 4xV xy 4p x 4p

with hx(y z) the critical angles, nax, naxy z for x-axial sources, 4rn cos h , ð Þ a ¼ ; (A6) and n , n for xy-tangential sources (see Secs. II B and h 2 2 txy tz ðrn cos h þ 1Þ þðxn cos hÞ II C). Thus, the mean free path for oblique sources is approximately given by where rn and xn are the resistance and the reactance of nor- malized impedance for normal incidence. With statistical 4V pcS=x averaging, the random-incidence coefficient is given by l < l ; (A3) ob r ð ð S 2cL=x p=2 p=2 ar ¼ ah cos h sin hdh cos h sin hdh for which a further approximation, lob lr, can be assumed 0 0 for a relatively large room at mid and high frequencies. 8r r2 x2 x n 1 n n tan1 n In the same way, the two-dimensional mean reflection ¼ 2 2 þ 2 2 rn þ xn xnðrn þ xnÞ 1 þ rn frequencies are modified for tangential sources except axial rn 2 2 sources, as follows (in xy plane): 2 2 ln½ð1 þ rnÞ þ xn ; (A7) rn þ xn 33 2hy 2hx which was first derived by Paris. n ðn þ n Þ ðn þ n Þ xy ax axy 2p ay ayx 2p In the same way, limiting the incidence angles from 0 to ntxy ¼ 2hy 2hx h , that is a critical angle defined by Eq. (3), the quasi-nor- 1 c 2p 2p mal-incidence coefficient for the corresponding axial sources pc 1 is expressed by 1 2x Lx þ Ly 2 2 n ; (A4) 8rn r x xy n n c Lx þ Ly an ¼ 2 2 2 1 cos hc þ 2 2 1 ðrn þ xnÞ sin hc xnðrn þ xnÞ 2x LxLy 1 xnð1 cos hcÞ tan 2 ð1 þ rnÞð1 þ rn cos hcÞþxn cos hc with the original value, nxy ¼ c=lxy ¼ 2cðLx þ LyÞ=pLxLy. "#) Thus, the mean free paths for tangential sources except axial r ð1 þ r Þ2 þ x2 n ln n n : sources are approximately given by 2 2 2 2 rn þ xn ð1 þ rn cos hcÞ þðxn cos hcÞ (A8) pLxLy pcðLx þ LyÞ=2x ltxy < lxy; (A5) 2ðLx þ LyÞpc=x For a small angle of hc, Eq. (A8) is almost equal to the true normal-incidence coefficient, for which a further approximation, ltxy lxy, can be assumed as well. 4rn an a0 ¼ ; (A9) 2 2 ðrn þ 1Þ þ xn 2. Estimation of normal- and grazing-incidence because the angle dependence is fairly weak near the normal absorption coefficients direction in Eq. (A6). Consequently, in the present reverber- If local reaction is assumed on wall surfaces, the direc- ation theory, true normal-incidence coefficients can be used tional absorption coefficient is theoretically given by regardless of the critical angle.

J. Acoust. Soc. Am., Vol. 132, No. 4, October 2012 Tetsuya Sakuma: Theory of reverberation in rectangular rooms 2335 On the other hand, limiting the incidence angles from p/2–hc to p/2, the grazing-incidence coefﬁcient for the corresponding axial or tangential sources is expressed by

hi 8r r2 x2 x sin h r n n n 1 n c n 2 2 (A10) ag ¼ 2 2 2 sin hc þ 2 2 tan 2 2 ln ð1þrn sin hcÞ þðxn sin hcÞ : ðrn þ xnÞ sin hc xnðrn þ xnÞ 1þrn sin hc rn þ xn

For a small angle of hc that satisﬁes jxnjhc 1 þ rnhc, Eq. (A10) is approximated by

2 2 2 8rn ½2rn þðrn þ xnÞhchc 2 2 ag ln½ð1 þ rnhcÞ þðxnhcÞ ; (A11) 2 2 2 ðrn þ xnÞhc 1 þ rnhc

15 € and furthermore, if a real impedance is assumed (xn ¼ 0), B.-I. L. Dalenback, “Room acoustic prediction based on a unified treat- ment of diffuse and specular reflection,” J. Acoust. Soc. Am. 100, 899– 909 (1996). 8 1 2 16 ag 1 þ lnð1 þ rnhcÞ ; (A12) Y. W. Lam, “The dependence of diffusion parameters in a room acoustics rnhc 1 þ rnhc rnhc prediction model on auditorium sizes and shapes,” J. Acoust. Soc. Am. 100, 2193–2203 (1996). 17L. M. Wang and J. Rathsam, “The influence of absorption factors on the which results in ag 8=rnhc for rnhc 1, and ag ð8=3Þ h h 2 h sensitivity of a virtual room’s sound field to scattering coefficients,” Appl. rn c 4ðrn cÞ for rn c 1. Certainly it is difficult to evalu- Acoust. 69, 1249–1257 (2008). ate a grazing-incidence coefficient that strongly depends on 18M. Vorl€ander and E. Mommertz, “Definition and measurement of random- the critical angle, whereas Eq. (A12) may be useful for a incidence scattering coefficients,” Appl. Acoust. 60, 187–200 (2000). 19 rough estimate from a measured absorption coefficient. ISO 17497-1:2004: “Acoustics—Measurement of the sound scattering properties of surfaces, Part 1: Measurement of the random-incidence scattering coefficient in a reverberation room” (2004). 1W. C. Sabine, Collected Papers on Acoustics (Harvard University Press, 20Y. Kosaka and T. Sakuma, “Numerical examination on the scattering coef- Cambridge, 1922), Chap. 1. ficients of architectural surfaces using the boundary element method,” 2C. F. Eyring, “Reverberation time in ‘dead’ rooms,” J. Acoust. Soc. Am. Acoust. Sci. & Tech. 26, 136–144 (2005). 1, 217–241 (1930). 21J.-J. Embrechts, L. de Geetere, G. Vermeir, M. Vorl€ander, and T. Sakuma, 3R. F. Norris, “A derivation of the reverberation formula,” in Architec- “Calculation of the random-incidence scattering coefficients of a sine- tural Acoustics, edited by V. O. Knudsen (Wiley, New York, 1932), shaped surface,” Acust. Acta Acust. 92, 593–603 (2006). 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2336 J. Acoust. Soc. Am., Vol. 132, No. 4, October 2012 Tetsuya Sakuma: Theory of reverberation in rectangular rooms