investigation studies on the application of reverberation time · investigation studies on the...

ARCHIVES OF ACOUSTICS

Vol. 41, No. 1, pp. 15–26 (2016)

Copyright c© 2016 by PAN – IPPT

DOI: 10.1515/aoa-2016-0002

Investigation Studies on the Application of Reverberation Time

Artur NOWOŚWIAT, Marcelina OLECHOWSKA

Faculty of Civil Engineering, Silesian University of TechnologyAkademicka 5, 44-100 Gliwice, Poland; e-mail: {artur.nowoswiat, marcelina.olechowska}@polsl.pl

(received February 12, 2015; accepted November 25, 2015)

The paper presents the research studies carried out on the reverberation time of rooms, in terms oftheoretical aspects and applicability potentials. Over the last century a very large number of scientistshave been attempting to work out models describing the reverberation time in enclosed rooms. They havealso been trying to apply these models for the description of various acoustic parameters of the interior,i.e. the intelligibility of speech, clarity, articulation, etc. In fact, all these models are based on the Sabine’sstatistical method. The paper presents the work of the scientists working on this problem, together withprospective applicability potentials. Such a review may be helpful for researchers, designers or architectsinvolved in the discussed subject.

Keywords: reverberation time; research studies; Sabine.

1. Introduction

In the early twentieth century Wallace ClementSabine developed a statistical model based on the sta-tistical theory of sound field. It assumed that the den-sity of sound energy at any point of the interior wasequal, and the probability of incidence of the soundwave was the same in all directions. The model uses theconcepts of mean free path length of sound waves andaverage waveform time. According to the statisticaltheory, the decay process of sound energy in the room,after turning off the stationary sound source, is of ex-ponential nature, and the parameter of such a decayis expressed by a constant quantity, i.e. reverberation(T – reverberation time). That method was appliedto formulate the generally known Sabine and Eyringformulas, which are now used to determine the rever-beration time of the interior with the known volumeand sound absorption of the room. In the statisticaltheory of sound field in a room, Sabine described thephenomenon of reverberation, and also, basing on theresults of his research, he provided the empirical (pri-mary) formula to calculate reverberation time, whichhas the following form (Sabine, 1922):

TSAB =0.161V

SαSAB[s], αSAB =

1

S

n∑

i=1

αiSi. (1)

Taking into account sound isolation effected by trans-fer in the air, we obtain the following equation:

TSAB =0.161V

SαSAB + 4mV[s], (2)

where TSAB – reverberation time acc. Sabine [s], V –volume of the room, S – total internal surface area ofthe room [m2], αSAB – average absorption coefficient,m – air sound absorption coefficient [Np·m−1].Many empirical equations have been formulated for

rectangular rooms. The first one was offered in thework of (Fitzroy, 1959), then (Pujolle, 1975; Hi-rata, 1979), but also (Neubauer, Kostek, 2001),or (Arau-Puchades, 2005) and others. The said sci-entists presented different approach to their empiricalequations. Fritzoy assumed the arithmetic average ofreverberation time in three orthogonal directions. Onthe other hand the Arau-Puchades formula favours thegeometric average. Both mentioned equations are sim-ple in their assumptions but they do not prove effi-cient in more complicated rooms. Hirata based his re-verberation theory on the distribution of acoustic fieldin one-, two- and three-dimensional areas dependanton frequency. In other words, the increase of scatter-ing field results in the decrease of reverberation time.Yet the said effect strongly depends on the shape andsound absorption of the room (Lam, 1996; Wang,Rathsam, 2008). Measurement methods which allowfor a scattered field are presented in the Standard

16 Archives of Acoustics – Volume 41, Number 1, 2016

ISO 17497-1. Unfortunately, there are not many the-ories in room acoustics which allow for uneven scat-tering of sound in a room. Such an approach was pro-posed by Sakuma (2012). In his work he proposed,first of all, that an approximate theory of reverber-ation in rectangular rooms should be formulated asa specular reflection field based on the image sourcemethod by modifying Hirata’s theory. Secondly, con-sidering surface scattering on walls with scattering co-efficients, an integrated reverberation theory for non-diffuse field should be developed, where the total fieldis divided into specular and diffuse reflection fields. Fi-nally, he obtained a theoretical case study demonstrat-ing how surface scattering affects the energy decay ofnondiffuse fields in rectangular rooms, when changingthe aspect ratio and the absorption distribution.Reverberation time is also widely applied for the

estimation of other acoustic parameters.The results of many research studies indicate

(Plomb et al., 1980; Houthast, Steeneken, 1984;1985; Ozimek, Rutkowski, 1985) that apart fromthe level of speech signal or the interference level ofamplitude-time structure of sound reflection, the in-telligibility of speech in a room is highly dependent onreverberation time. On the other hand, Galbrun andKitapci (2014) investigated the accuracy of speechtransmission index as dependent on reverberation timeand on the signal-noise ratio.Many research studies demonstrate that by the ap-

plication of reverberation time we can quickly esti-mate various parameters or indexes describing roomacoustics. Such an approach was pursued by Lam. Inhis work (Lam, 1999), using only reverberation time,he worked out a method for the estimation of Deut-lichkeit, clarity and center time. In the same wayNowoświat and Olechowska (2016), using the sta-tistical analysis, worked out the estimation method ofSTI, applying for that purpose reverberation time andBistafa and Bradley (2000) plotted STI values ver-sus reverberation time for unamplified speech in class-rooms.At present, when designing lecture rooms, concert

halls, auditoriums, etc., reverberation time is definitelytaken into account. Aretz and Orlowski (2009)demonstrated in their measurements that for a defi-nite volume of a room, sound strength can be decreasedby means of an appropriate alteration of reverberationtime. But on the other hand, to ensure proper recep-tion of chamber music, reverberation time cannot bedecreased freely. Therefore, they suggested that rever-beration time and sound strength should be balanced.Basing on the least squares method, they worked outa regression model.Reverberation time is also important in shaping

the acoustics of sacral interiors. For example,Berardi(2012) and earlier Engel and Kosała (2007) appliedreverberation time among other things for a single in-

dicator rating of such interiors. Wide application ofreverberation time in shaping the acoustics of interiorsinstigated the authors to describe in one paper moreor less known models and theories involving it.

2. Theoretical models of reverberation time

The reverberation time of a room can be calculatedusing the empirical formulas described in the worksof (Arau-Puchades, 1988; Eyring, 1930; Fitzroy,1959;Millington, 1932; Neubauer, Kostek, 2001;Pujolle, 1975; Sabine, 1922; Sette, 1933). Equa-tion (1) for the prediction of reverberation time showsthat the reverberation time of a room can be deter-mined basing on the interior volume V and on itssound absorption A. The Sabine’s formula is very sim-ple, but it is also burdened with some limitations. Theformula gives wrong results for completely soundproofrooms (αSAB = 1). Instead of the expected T = 0,the value of reverberation time different from zero isobtained for the room interior. Therefore, the formuladerived by Sabine is designed for poorly soundproofrooms (αSAB < 0.2) with evenly distributed sound ab-sorption. This formula has historical significance, andwith the development of science, it has given rise toseveral transformations which are free from such lim-itations. Additionally, a modification has been intro-duced into the formulas for rooms of the volume largerthan 200 m3 which allows for the effect of air absorp-tion (PN EN 12354-6, 2005). A modified determina-tion method of reverberation time was proposed byNorris and Eyring (Eyring, 1930; Norris, Andree,1930). When they were expanding the calculation con-cept for sound absorption coefficient given by Sabine(1922), they introduced a logarithmic dependence forthe average coefficient α into the denominator. Theypresented the converted Sabine’s formula which is de-void of the limitation mentioned above and takes thefollowing form:

TEYR =0.161V

SαEYR[s], αEYR = − ln (1− αSAB), (3)

Knudsen modified the Eyring’s formula by introducingthe isolation of sound in the air into Eq. (3) (Knudsen,1929):

TEYR =0.161V

SαEYR + 4mV[s].

The formula introduced by Eyring, in contrast tothe Sabine’s formula, can be used for all values of themean sound absorption coefficient. If the mean soundabsorption coefficient is small (αSAB < 0.2), the devel-opment of the denominator in the expression (3) intoTaylor’s sequence is equal to − ln(1 − αSAB) = αSAB,and then the formula takes the form of Sabine’s for-mula. In such a case, the relative difference ∆T of theinterior reverberation time calculated with the use of

A. Nowoświat, M. Olechowska – Investigation Studies on the Application of Reverberation Time 17

the above formulas does not exceed 9%. The differ-ence is brought about by omitting further terms of thesequence in the Sabine’s formula. The value of ∆T in-creases with the rise of αSAB, and thus the scope ofapplicability of the Sabine’s formula is reduced to thevalues of the coefficient (αSAB < 0.2):

∆T ∼=0.16 VSαSAB

− 0.16 V−S(1−αSAB)

0.16 VSαSAB

· 100%

=ln (1− αSAB) + αSAB

ln (1− αSAB)· 100%

=−0.22 + 0.2

0.22· 100% = 9%.

Another formula was presented byMillington (1932)and Sette (1933). Their formula differs from thepreviously described formulas in the determinationmethod of the average sound absorption coefficient.In the Sabine’s formula, the coefficient αSAB is de-termined as the arithmetic mean, whereas Millingtonsuggested the calculation of the coefficient αMIL as thegeometric mean. The Millington’s formula cannot beused in perfectly absorbing rooms, since in such a caseindeterminacy is obtained in the denominator:

TMIL =0.161 V

SαMIL[s],

αMIL = − 1

S

n∑

i=1

Si ln (1− αi).

(4)

Kuttruff (2009), on the other hand, suggested inhis work a statistical distribution of sound, tak-ing into account the Gaussian random variable andRayleigh probability. Basing on that concept, he cre-ated a definition of the function of mean free pathγ2 =

(l2 − l

2)/l

2as a variation of probability. To

calculate γ2 he applied the Monte Carlo simulationmethod. Kuttruff introduced two important changes tothe Eyring equation. The first one involved the shapeof the room, while the other one involved the distri-bution of the absorbing material. He also introduceda correction in determining the average sound absorp-tion coefficient, which yielded the following equation:

TKUT =0.161V

SαKUT[s],

αKUT = − ln (1− αSAB)

(1 +

γ2

2ln (1− αSAB)

),

(5)

where γ2 is the mean free path.All the formulas presented above are applied to de-

termine the reverberation time in rooms where the dis-tribution and acoustic properties of the used materialsare uniform in all directions. It means that the absorp-tion of the opposite plane pairs (pairs of walls, ceiling

and floor) limiting the room is approximately equal. Inthe further part of the paper we present three formulas(Fitzroy’s, Arau’s and Neubauer’s) used to predict thereverberation time of a room, which take into accountthe propagation of sound in three orthogonal directionsx, y, z. The formulas developed by Fitzroy (Fitzroy,1959) or (Arau-Puchades, 2005) take into accountthe uneven distribution of sound absorbing materi-als and systems in the room (αx 6= αy 6= αz). Thus,isotropic field condition is not fulfilled, which leads tothe increase of reverberation time of the interior deter-mined with the application of the previously discussedformulas (Sabine, Eyring, Kuttruff and Millington).Therefore, the Fitzroy’s formula should be used, as itoffers values that are more consistent with the valuesobtained in the measurement. The Fitzroy’s equationhas the following form:

TFIT =

(Sx

S

)· Tx +

(Sy

S

)· Ty +

(Sz

S

)· Tz [s], (6)

where

Tx =0.161 V

SαFIT,x, Tx =

0.161 V

SαFIT,y, Tz =

0.161 V

SαFIT,z,

αFIT,x = − ln (1− αx), αFIT,y = − ln (1− αy),

αFIT,z = − ln (1− αz),

where Sx, Sy, Sz – surfaces of the opposite pairs ofwalls [m2], αx, αy, αz – average reverberation soundabsorption coefficients of the material on the respectivepairs of walls.Arau-Puchades (2005) proposed an improved

equation which assumed that the reverberation timeof the interior could be determined as a geometricweighted average of three reverberation times obtainedfrom the orthogonal directions (x, y, z). He also as-sumed that the decay of the reverberation time was ofhyperbolic nature. The absorption coefficients used inhis formula are the mean absorption values for eachpair of the opposite walls. Simultaneous sound reflec-tions are formed between these surfaces, and thereforethe decay of sound should be considered in three di-rections. Arau-Puchades determines the reverberationtime of the interior in the following way:

TARAU =

[0.161V

SαARAU,x

]Sx/S

·[

0.161V

SαARAU,y

]Sy/S

·[

0.161V

SαARAU,z

]Sz/S

, (7)

where

αARAU,x = − ln (1− αx) , αARAU,y = − ln (1− αy) ,

αARAU,z = − ln (1− αz)

Arau-Puchades together with Berardi (2013)modified the non-symmetrical model TARAU. Know-ing the expressions for the direct and diffuse sound


pressure, it is possible to obtain a formula for thereverberation radius. This comes from equaling thesound intensity in the direct field in Idirect = Qw/4πr2,where Q is the source directivity, w is the sound powerof the source and r is the distance from the sourceto the receiver. And the reflected sound intensity ina diffuse field in Idiffuse = 4w/A, where A is theequivalent absorption A = [−S ln(1− α) + 4mV ], αis the absorption coefficient, and m is the sound ab-sorption coefficient of the air. We obtain in effect:rHD = ((0.01/π)(V/T ))

1/2.From the equation TARAU, it is possible to calculate

the reverberation radius in each direction, as in theexpression rHD: r2HNDi · 16π = Ai, i = x, y, z.In the last equation the reverberation radius in each

direction increases as the equivalent absorption in thatdirection rises. From this, it is possible to calculatethe reverberation time as a product of the terms ofreverberation for the facing surfaces:

T = (0.01V/πr2HNDx)Sx/S · (0.01V/πr2HNDy)Sy/S

· (0.01V/πr2HNDz)Sz/S , (8)

T = (0.01V/πr2HND),

whererHND =

[rSx/SHNDx · rSy/S

HNDy · rSz/SHNDz

]

with the reverberation radii for a non-diffuse soundfield rHND obtained as:

rHND = ((0.01/π)(V/T ))1/2

,

rHND = (A/16π)1/2 ,

whereA = ASx/S

x · ASy/Sy · ASz/S

z .

Neubauer, Kostek (2001) presented a modifica-tion of the Fitzroy’s equation, dividing appropriatelythe Kuttruff’s correction element into two parts. Onepart reflects the surfaces of the floor and ceiling, whilethe second part takes into account the impact of theother walls.

TNEU =0.45V

S2

(lw

αCF+h (l + w)

αWW

)[s]. (9)

In the Neubauer’s equation, the sound field is di-vided into two parts, where the determined absorptioncoefficients are treated as an adjustment to the Eyringand Kuttruff’s formula:

αCF = − ln (1− αSAB) +ρCF (ρCF − ρ)S2

CF

(ρS)2 ,

αWW = − ln (1− αSAB) +ρWW (ρWW − ρ)S2

WW

(ρS)2,

where l, w, h – length, width and height of theroom [m], αCF – average effective sound absorption co-efficient of the ceiling and floor, αWW – average effec-tive sound absorption coefficient of the side partitions,

ρ = 1− α – reflection coefficient, SCF – surface of theceiling and floor, SWW surface of the side walls [m].Pujolle (1975) proposed another determination

method of the mean free path lm taking into accountthe dimensions of the room. He presented two formulasto determine lm:

lm =1

6

(√L2 + l2 +

√L2 + h2 +

√h2 + l2

)

orlm =

1√π

(L2l2 + L2h2 + h2l2

)1/4.

The Pujolle reverberation time can be determined ac-cording to the formula:

TPUJ =0.04lmαEYR

[s], (10)

where L, h, l – length, height and width of theroom [m].According to Skrzypczyk (2008) and Winkler-

Skalna (2008) reverberation time can be also es-timated using the perturbation numbers. They pre-sented the application of a new algebraic system forthe determination of reverberation time. They in-troduced the perturbations of parameters into theSabine’s equation. Assuming that all the parameters(VPER, SPER, αSAB,PER) in Sabine’s formula un-dergo 2ε-perturbation, we obtain:

TSAB,PER =0.161VPER

4mVPER + SPERαSAB,PER, (11)

where

VPER = V0 + ε1V1 + ε2V2,

SPER = S0 + ε1S1 + ε2S2,

αSAB,PER = αSAB,0 + ε1αSAB,1 + ε2αSAB,2,

where VPER, SPER, αSAB,PER – room volume, totalsurface area of the partitions limiting the room, ab-sorption coefficient (parameters with perturbations),εi – small parameter, i = 1, 2, respectively.The inverse of the 2-perturbation number:

ζ−1 = (x, y, z)−1

=

(1

x, − y

x2, − z

x2

), x 6= 0.

Thus, by using the inverse of 2ε-number, the reverber-ation time has the following form:

TSAB,PER =0.161VPER

4mV + SαSAB,0− 0.161V S

(4mV + SαSAB,0)2

· (ε1αSAB,1 + ε2αSAB,2)

= TSAB,0 + ε1TSAB,1 + ε2TSAB,2,

where TSAB,0 – reverberation time without perturba-tion [s], TSAB,i – perturbation components of the re-verberation time [s].


The reverberation time of a room can be also deter-mined using the recommendations of the standard (PNEN 12354-6, 2005). The said standard reads that theroom reverberation time is dependent on the volumeand on the total equivalent area of the sound-absorbinginterior (total acoustic absorption), as well as on thefraction of the objects:

T =55.3

c0

V (1− ψ)

A, (12)

where c0 – speed of sound in the air, A – equivalentarea of the sound-absorbing interior, referred to as theacoustic absorption of the interior, ψ – fraction of ob-jects defining the volume of the objects in the room.The total acoustic absorption can be calculated as

follows:

A =

n∑

i=1

αs,iSi +

o∑

j=1

Aob,j +

p∑

k=1

αs,kSk +Air ,

where n – number of surfaces i, o – number of objectsj, p – number of k – system objects.The equivalent surface area for air absorption is

determined basing on the formula Aair = 4mV (1−ψ).This standard reads that the air absorption may beomitted if the room has the volume of less than 200 m3

and the calculations are carried out to 1000 Hz octaveband as the highest frequency. The fraction of objectswas determined from the following equation:

ψ =

o∑j=1

Vobj,j +p∑

k=1

Vobj,k

V,

where Vobj – volume of hard objects.Due to some limitations, this method can be ap-

plied for rooms of regular-shape (none of the interiordimensions can be five times greater than the others),with a small number of objects which are also char-acterized by uniform sound distribution. If the aboveconditions are not met, the obtained values of roomreverberation time can differ from the estimated ones.

3. Investigation studies on reverberation time

Using the commonly available technical data in-volving room acoustics, the verification of the useful-ness of the theoretical models of reverberation timewas carried out in terms of their applicability to differ-ent rooms. In the available articles, the authors com-pare their new methods for the prediction of reverbera-tion time with the most commonly applied formulas ofSabine, Eyring, Millington-Sette and Fitzroy. The re-sults of the analytical calculations are often comparedwith the values obtained from computer simulationsand actual measurements. The world literature offersmany publications helpful to verify the usefulness oftheoretical models used for the estimation of reverber-ation time.

In the work (Neubaer, Kostek, 2001), the au-thors presented a new formula for the calculation ofreverberation time and compared it to other existingformulas by Sabine, Eyring, Millington-Sette, Fitzroy,Tohyama (1986), Arau as well as to a new formula(Neubauer) and model incorporated in the Europeanstandard (Annex D of the proposition of EN 12354-6, 2003) along with the measurements. First, the re-searchers analyzed two extreme cases of rooms havingregular shapes. In the first room (alive) they used ma-terials of low absorption (α = 0.02), while in the secondone (dead), the average absorption coefficient of thesurface was (α = 0.95). Then, the authors examinedthe room with the unevenly distributed acoustic ab-sorption. In that case, the floor and ceiling were char-acterized by higher absorption coefficient (α = 0.43),while the other areas had low absorption (α = 0.02). Ineffect of the carried out research studies, the authorsarrived at the following conclusions:1) the reverberation time obtained from the standardformula, by Tohyama and Fitzroy, is significantlydifferent from the measured values;

2) when the room has a low absorption coefficient,there are no significant differences between themeasured values, except for the formula includedin the standard (Annex D of the proposition ofEN 12354-6, 2003) and Tohyama’s formula;

3) the new formula is more effective in determiningthe reverberation time than the classical formulas.In the next work (Kang, Neubaer, 2001), the

authors compare the analytical formulas and com-puter simulations performed by means of the CATT-Acoustic software (CM) and radiosity model (RM).They analyzed two rectangular rooms in which thelength and width equaled 10 [m], while the height wasvariable – 8 or 3 [m]. They examined 8 cases involv-ing the distribution of absorbing material on the sur-faces limiting the room. For the analysis they used twodifferent absorbing surfaces, with the absorption coef-ficient (α = 0.08) for the surfaces marked with graycolor, and with the absorption coefficient (α = 0.05)for the remaining ones. In the present paper the au-thors arrived at the following conclusions:1) for the room with a constant absorption coefficient(α = 0.05) on all limiting surfaces, they receivedsimilar values using the classical formulas (case 1);

2) for the remaining cases they received high rever-beration times for the Fitzroy’s, Arau’s and stan-dard formulas;

3) the Fitzroy’s formula had the highest result, andthe Millington’s formula the lowest;

4) the values obtained from the calculations per-formed with the use of the standard formula andArau’s formulas are similar;

5) they also observed that the values obtained fromCM were higher than those from RM;


6) the radiosity model generates the average rever-beration time close to the values obtained fromthe Fitzroy’s and Kuttruff’s formulas.

Another approach to the estimation of reverbera-tion time was presented by Zhang (2005). In his dis-sertation he suggested a new model for the predictionof reverberation time and also defined the deviationbetween the measurement and the expected T value.On the basis of the obtained results Zhang providedthe following conclusions:

1. The Fitzroy’s method reached the greatest devia-tion of 12.9.

2. For high frequencies, the Kuttruff’s equation isapplicable whereas for low frequencies the Arau-Puchade’s formula is applicable.

3. It was also demonstrated that the model (Zhang)had the smallest deviation equal to 0.22.

The estimation of reverberation time has been de-veloped not only on the basis of empirical formulas, butalso on the basis of the commonly known mathematicalmethods, based on finite element methods, perturba-tions or neural networks. Using one of these methods,Skrzypczyk (2008) and Winkler-Skalna (2008)proposed a new determination method of reverberationtime. The authors presented the methods which makeuse of the new perturbation algebra for the analysisof acoustic problems in enclosed rooms. They intro-duced an assumption advocating that all parametersin the Sabine, Eyring and Kuttruff formulas undergo2ε-perturbation. Basing on the above, they presentedmodified classical formulas for the determination ofreverberation time. In his work, Skrzypczyk (2010)presented the results of the reverberation time calcu-lations in Aula Magna and in the Olivertani Churchdetermined according to the Eyring’s formula. Basingon the achieved results, the author provided the fol-lowing conclusions:

1. The results of the calculations made with the useof the Eyring’s and Sabine’s perturbation formu-las coincide for small-sized perturbations.

2. When only one parameter is subjected to per-turbation, the perturbation calculations are com-patible with the actual perturbation in the wholerange.

Despite the fact that the classical formulas havebeen attributed to simple objects and objects withuniform absorption, some authors attempted to finda correlation between the results of reverberation timeaccording to the already mentioned formulas for com-plex objects. In the paper (Iordache et al., 2013),which was to determine the applicability of Sabine’sformula in the rooms of complex shape, the authorsstudied a room consisting of two levels. The height ofeach level was 4.5 m and the volume of the whole roomwas 2926 m3. Basing on the obtained results, they

observed the differences between the value obtainedfrom the classical formula and from the measurement.The value determined from the Sabine’s formula wasby 0.5 s higher than that obtained from the measure-ment. The authors of the publication explained thatthe difference could have been effected by the follow-ing reasons. Firstly, errors in the measurement of thesurfaces limiting the room due to complex shape ofthe room. Secondly, erroneously employed material ab-sorption coefficients. Thirdly, they questioned the useof the Sabine’s formula for that type of room. Summingup the results obtained in the course of the studies,they concluded that the Sabine’s formula should notbe applied to the interiors of complex shape during thedesign phase.As in the publication (Iordache et al., 2013), the

authors (Kang et al., 2007) analyze the calculationmethods of reverberation time in large spaces (atrium).The research found that the reverberation times ob-tained using the Sabine’s formula were comparable tothe average value obtained from the measurement forfour receivers. However, when they compared the in-dividual results at each point, they were different by15% to 23%. They assumed in the analysis conductedin such a way that the classical formula might be usedin the estimation process of average reverberation timefor the entire interior.The research in the field of acoustics did not fo-

cus solely on the errors in the formulas for the esti-mation of reverberation time. The authors analyzedalso the impact of two standard absorption coeffi-cients and one determined from the Millington’s for-mula with respect to the mentioned parameter. Inthe work (Petelj et al., 2012), the authors com-pare the reverberation times obtained from the clas-sical formulas where standard sound absorption coef-ficients were used with the formula given by Milling-ton. In the Millington’s formula they used his soundabsorption coefficients. The measurements were car-ried out in two rooms of irregular shape. The firsttested room had the floor surface of 32.6 m2 and heightof 3.25 m, while the second one the floor surface of14.15 m2 and height of 2.8 m. The research studies inthe first empty room yielded the average value of re-verberation time equal to T = 4 s. In effect of thetheoretical works, they obtained the value T = 3.99 sfrom the Millington’s formula and T = 4.11 s deter-mined on the basis of the Zhang method. The calcu-lations carried out on the basis of the Sabine’s for-mula yielded the value T = 4.88 s. The best estimatewas obtained for the first example of Millington’s for-mula. In the next room they received the average valueof reverberation time of the in situ research equal toT = 1.21 s. From the classical formulas they obtainedthe value of reverberation time for Sabine T = 1.26 s,Millington T = 1.15 s and Zhang T = 1.20 s, respec-tively.


The authors presented the following conclusion: theMillington’s formula yielded results with small devia-tions, but only when the average reverberation timewas determined for the frequency range from 125 Hzto 4 kHz.Also the works (Dance, Shield, 1999; 2000) com-

pare the formulas which apply the Millington and stan-dard absorption coefficients. In the first publication,the authors analyzed the application of the standardand Millington’s absorption coefficients and comparedthe applied classical formulas. Research studies werealso carried out in a recording studio and concert hallin order to estimate the accuracy of the reverberationtime with the use of Millington’s formula and his ab-sorption coefficients. Then, the obtained values werecompared with those obtained on the basis of Sabine’sand Eyring’s formulas. Having analyzed the results ofthe measurements and the prediction of reverberationtime based on the Sabine’s, Eyring’s and Millington’sformulas in the recording studio, they found that theformula suggested by Millington was as accurate asthe classical formulas. Afterwards, the authors deter-mined the errors yielded by the formulas. The errorfor the Millington’s formula was 10.9% for all fre-quencies, while for the Sabine’s formula 12.8% andEyring’s 20%. Then, they conducted a study in a con-cert hall. Summing up the results of their works, theyobserved significant differences between the obtainedresults. They found that the classical formulas werebecoming less accurate for rooms with unevenly dis-tributed field. In the second paper, the Sabine’s andMillington’s formulas were used to analyze which onewas more useful. The research involved an experimen-tal room (4.54× 2.73× 2.4 m) with differently situatedsound-absorbing material.The authors compared the results of the calcula-

tions made with the use of different formulas of rever-beration time to the values obtained by means of themeasurement. The authors drew the following conclu-sions:

1. For the case 0 they obtained (by means ofSabine’s, Eyring’s, Millington’s and Arau’s formu-las) the reverberation time values with the errorwithin the range of 6%.

2. The Arau’s formula for the cases 1–5 had the errorof 8.3% in comparison to other methods which hadthe errors of 42%, 31% and 33% for the Eyring’s,Sabine’s and Millington’s formulas respectively.

3. The above results illustrate the applicability lim-its of the classical methods, except for the Arau’sformula.

Lawrence presented in his dissertation (Lawren-ce, 2006) a problem involving the estimation of soundabsorption coefficient basing on the Sabine’s andEyring’s formulas. The author outlined the probleminvolving the determination of the coefficient α based

on the Sabine’s formula which implied that the absorp-tion exceeding 100% was possible. Referring to otherscientists, he presented the problems related to themeasurement of the absorption coefficient. The pub-lication also cites Hodgson’s (1993) statement whichreads that the Eyring’s formula is more accurate toestimate the reverberation time and to determine theabsorption coefficient. Despite many discussions andvarious studies presented in papers, none of these pub-lications leads to a solution, but only underlines thesignificance of the problem.The problem involving the determination of the ab-

sorption coefficient was also referred to by Beranek(2006). Having analyzed the examples, he found thatthe Sabine’s formula could be used to determine the re-verberation time in the room for which the coefficientα had been earlier determined in a similar location.A considerable part of the publication presented

the analysis involving the impact of the extent of roomfilling as well as the impact of the unevenly distributedabsorbing material. In another study (Passero, Zan-nin, 2010), the authors attempted to show and ver-ify the similarity of procedures for the determinationof reverberation time in a classroom. In the publica-tion Passero, Zannin (2010) presented the statisticalanalysis of data obtained from the comparison of allapplied methods. Basing on the data included in thatstudy, the authors presented the following conclusions:1. Lack of differences between the measurementmethods, which was also confirmed by the As-tolfi’s publication (Astolfi et al., 2008).

2. The results of the study show that the Eyring’sformula is the most similar to the two measure-ments and to the computer simulation of all themodels used, while the Arau’s formula is the leastsimilar.

3. The reverberation time obtained from theSabine’s formula was longer than that calculatedaccording to Eyring, which was also reported inthe work (Bistafa, Bradley, 2000).The work (Wilmshurst, Thompson, 2012) shows

the reverberation time prediction methods for rectan-gular rooms with unevenly distributed absorbing ma-terial. The authors suggested a different way of deter-mining the reverberation time using the Statistical En-ergy Analysis method (SEA) and compared the clas-sical methods with the computer simulations carriedout with the use of CATT-Acoustic software. The SEAmodel calculates the total potential energy of sound inconnection with the time function, and on that basisa theoretical graph of sound decay is obtained. In theanalyzed room of the dimensions 10× 9× 8 m, theyplaced absorbing materials on different surfaces limit-ing the interiors and checked the usefulness of the usedcalculation methods. Basing on the values obtainedfrom all methods, the authors presented the followingconclusions:


1. The SEA model yielded values close to those ofCATT-Acoustic and Arau.

2. In the case of a non-homogeneous room (ceiling,floor and side walls are lined with absorbing mate-rial), the Fitzroy’s formula and SEAmodel yieldedthe highest value of reverberation time.

Also I. Rossell and I. Arnet studied a similar prob-lem, taking into account the audience in the Hall(Rossell, Arnet, 2002). The authors presented intheir publication the results of reverberation time forthree cases of hall filling (in an empty classroom, with50% filling and filled in 100%). Basing on the resultsobtained for all determination methods of reverbera-tion time, the authors summarized the work with thefollowing conclusions:

1. For the first situation (no audience) the obtainedreverberation time values are comparable, withthe exception of Fitzroy’s equation. In that casewe have a diffused field in which the classical the-ories yield good results.

2. For the room with 50% filling, the results obtainedfrom the theoretical calculations differ from themeasurement values.

3. The values obtained from the Arau’s formulaproved to be the most accurate in estimating thereverberation time. Summarizing the analyses ofthe material collected from the literature, it canbe concluded that the current formulas used tocalculate the reverberation time in a room havelarge deviations from the actual values, and inparticular for cuboid objects with non-diffusive(dispersed) sound field. The objects of that typeare most frequently met in practice. For that typeof objects, the acoustic properties of a room orabsorption values applied in the design processshould be more accurate than the values havinga high degree of approximation.

In recent times, many publications involve roomacoustics by means of computer simulations. In theauthors’ opinion two works can be quoted as the mostinteresting. Berardi (2014) presented in his paper theresults of computer simulations involving the acous-tics of box-shaped churches. During the simulation hewas changing the ratios of width, length and height ofthe buildings. Basically, he was analyzing the depen-dence between the selected acoustic parameters andthe shape of the object. He presented equations de-scribing the acoustics of the simulated churches andthen he verified the obtained results with the measure-ments carried out in five Italian churches.In the work of Bustamante,Girón, Zamarreno

(2014) simulation techniques have been implementedto study the sound fields of a multi-configurable per-formance enclosure by creating computer acoustic 3D-models for each room configuration.

The digital models have been tuned by means of aniterative fitting procedure that uses the reverberationtimes measured on site for unoccupied conditions withthe orchestra shell on the stage.The most interesting works quoted here by the au-

thors contain references to other sources which are alsoworth attention.

4. Case study

For the needs of the case study in an anechoic room,a cuboid room of the floor dimensions 2.5× 5.0 m andheight 2.5 m made of OSB was built. Then, reverbera-tion time was measured for such a structure in two vari-ants. The first variant involved an empty room builtfrom the boards having the sound absorption indexlower than 0.2. The second variant involved the sameroom but with disturbed acoustic field in the form ofmineral wool placed on the partitions limiting the room(see Fig. 1).

Fig. 1. Location of sound absorbingmaterial in the model room.

The investigation of reverberation time consistedin measurements and theoretical calculations with theapplication of equations described in this paper. Theresults are presented on the diagrams below.

Fig. 2. Measured and calculated reverberation times in theanalyzed variants.

Basing on such a simple analysis, we can see adifference between the measurement and theoretical


equations. The results are different for the room withuniformly distributed sound absorption index and dif-ferent for the rooms with sound absorption elements.This simple case study confirms the observations ofmany researchers which have been presented in theprevious sections of this paper.Case studies have been also carried out by many

other researchers as for example McMinn (1996).In his work he investigated among other things theMusikvereinssaal Hall in Vienna which is one of thebiggest concert halls in the world (see Fig. 3).

Fig. 3. Grosser Musikvereinssaal Hall(McMinn, 1996).

The research results provided by McMinn also con-firm the observations of many researchers quoted inthis paper. It is also worthwhile mentioning anotherwork (Neubauer, Kostek, 2001) in which other caseshave been analyzed, e.g. the work presenting reverber-ation time as a function of the volume of the poorlydamped room, with reverberation times being calcu-lated with the application of different equations.Such an approach was employed to show the vari-

ation of the obtained results.Interesting research studies were published by

Kraszewski (2012). In his paper he presented the-oretical calculations for long rooms. He observed in hisconclusions that the reverberation time was influencednot only by the mean absorption index but also bythe location of the absorbing material or the place atwhich the reverberation time was calculated by meansof the diffusion equation.

5. Application of reverberation time

for the description of room acoustics

As it has been already mentioned in the Introduc-tion of the present paper, reverberation time can influ-ence other parameters used to design rooms of varioustypes. In this section we will only signal the results ofsome selected works.

Lam (1999) provided the approximations of certainparameters by means of reverberation time, such as:Definition (Deutlichkeit): D50 = 1− e

− 0.69T60 ;

Clarity: C80 = 10 log(e

1.1T60 − 1

);

Centre Time: Tc = T60

13.8 ;

Level: L = 10 log(

16π102T60

0.161T

).

The authors of this article Nowoświat andOlechowska (2016) provided the approximation ofspeech transmission index by means of reverberationtime: STI = A ln+B, where A = −0.2078,B = 0.6488.Aretz and Orlowski (2009) indicated in their

measurements that sound strength can be lowered bymeans of an appropriate shaping of reverberation time.On the other hand, to ensure proper reception of cham-ber music, reverberation time cannot be lowered freely.Therefore, they suggested that the reverberation timeand sound strength should be balanced. Basing on theleast squares method, they worked out a regressionmodel

Greflected = 10 · log10 ((31200RT/V ) · b)

= 10 · log10(31200RT/V ) + b.

Then, the authors state that between Greflected and10 · log10(31200RT/V ) we can observe high correlationr = 0.94 for b = −3.3 dB with the standard deviationof 1.1 dB.

6. Conclusion

The review of the literature shows that the existingand widely used formulas for the determination of re-verberation time do not allow us to predict accuratelythe reverberation time in the interiors filled with au-dience (currently, too large differences have been ob-served as compared to the actual values, measuredwhile determining the reverberation time). Designersinvolved in shaping the acoustic properties of roomsare looking forward to the development of a formulawhich would more precisely determine the predictionof reverberation time. This paper reviews the avail-able literature in the field concerned with the assess-ment and verification of the usefulness of the appliedmodels designed to determine reverberation time of aroom.The volume of the present publication indicates

that a great number of researchers are still workingon the estimation of reverberation time. A lot of themcreate new formulas whereas others verify them. Withrespect to many works, it is not possible to decide ex-plicitly which formula is the best. To provide an exam-ple, we can quote the research carried out by Passeroand Zannin (2010) in which the reverberation time ina classroom was analyzed. They used for that purposethe calculations with the formulas of Sabine, Eyring


and Arau-Puchades. Their work demonstrated, as inthe case of Bistafa (Bistafa, Bradley, 2000), thatthe reverberation time calculated with the Sabine for-mula was slightly longer than that calculated with theEyring formula. And it was considerably longer whenthe Arau-Puchades formula was applied. The obtainedreverberation time by means of Eyring formula wassimilar to the measurement and computer simulationin ODEON. In variant I the calculations by means ofSabine and Eyring formula yield the values relativelyclose to the measurement. The result provided by Kut-truff formula yields a considerably shorter reverbera-tion time, and using the Arau-Puchades formula, itis considerably longer. It differs slightly in variant IIwhere the tested room is damped. In that case, theformula of Arau-Puchades yields results which are notconsiderably different from other results. Much highervariation in the obtained results can be observed inthe analysis of Grosser Musikvereinssaal Hall, wherethe Albine formula yields shorter reverberation timethan that of Arau-Puchades but longer than the mea-surement. A novel approach was presented by Lam whodescribed the basic parameters of room acoustics, suchas Deutlichkeit, Clarity, Centre Time, Level by meansof reverberation time.In the same way the authors (Nowoświat, Ole-

chowska, 2016) estimated the speech transmission in-dex by means of reverberation time.The presented research studies carried out by dif-

ferent research teams have prompted us to develop ourown model based on the statistical theory and trendfunction. The model developed by the authors of thiswork will be described in the next publication and itwill be based on “learning” algorithms and trend func-tions (Nowoświat et al., 2016).

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