Derivation of the Sabine Equation: Conservation of Energy

UIUC Physics 406 Acoustical Physics of Music Derivation of the Sabine Equation: Conservation of Energy

Consider a large room of volume V = HWL (m3) with perfectly reflecting walls, filled with  a uniform, steady-state (i.e. equilibrium) acoustic energy density wrtfa ,, at given frequency f (Hz) within the volume V of the room. Uniform energy density means that a given time t:  3 waa r,, t f w t , f constant (SI units: Joules/m ). The large room also has a small opening of area A (m2) in it, as shown in the figure below:

H V A   I rt,, f ac   nAˆˆ,  An

In the steady-state, the rate of acoustical energy Wa input e.g. by a point sound source within the large room equals the rate at which acoustical energy is “leaking” out of the room through the hole of area A, i.e. the acoustical power input by the sound source in the room into the room = the acoustical power leaving the room through the hole of area A. In this idealized model of a room with perfectly reflecting walls, the hole of area A thus represents absorption of sound in a real room with finite reflectivity walls, i.e. walls that have some absorption associated with them.

Suppose at time t = 0 the sound source in the room {located far from the hole} is turned off. Since the sound energy density is uniform in the room, the sound energy contained in the room  Wtf,,,, wrtfdrwtf  33  drwtfV  , will thus decrease with time, since aaVV a a acoustical energy is (slowly) leaking out of the room through the opening of area A.

The instantaneous acoustical power at the frequency f passing through the hole of area A is the instantaneous time-rate of change of the acoustic energy in the room:

Wtf, Ptf,  a a t

However, the instantaneous acoustical power loss at the frequency f associated with the flux    of acoustic energy passing through the hole of area A is also Ptf,,, I  rtf  dA where aaA    I rt,, f is the instantaneous 3-D vector sound intensity at the point r at frequency f (SI units: a  Watts/m2) and dA dAnˆ is a infinitesimal vector area element associated with the hole of area A, and nˆ is the outward-pointing unit normal to the hole of area A, as shown in the above figure. -1- Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2002 - 2017. All rights reserved.

UIUC Physics 406 Acoustical Physics of Music Thus: Wtf,  P t,,,,,,cos fa I r t f dA I  r t f ndAˆ I  t f t dA aaaat AA  A where cos t is the instantaneous direction cosine , and thus t is the instantaneous 3-D   opening angle between the two vectors Ia tf, and AAn ˆ (as shown in the above figure), and   Iaart,, f nˆ  I  r , f cos  t. In the steady-state, the magnitude of the 3-D vector sound  intensity is constant in time at any given point r inside the volume V of the room, and on/at the opening of the hole of area A, however the direction of the 3-D vector sound intensity at any  given point r fluctuates randomly from one moment to the next. At/on the surface of the opening of the hole of area A, the direction of the 3-D vector sound intensity associated with energy leaking out of the room of volume V is such that the direction of the 3-D sound intensity points randomly from moment-to-moment in the forward-going hemisphere, i.e. is contained within a solid angle d associated only with the forward half of 4 steradians (since sound energy is leaking out of the room – sound energy is not coming into the room from the outside).

We are not interested in following the instantaneous, moment-to-moment/short-time scale   fluctuations in the 3-D vector sound intensity Ia rt,, f , but we are interested in the mean power loss, time-averaged over these moment-to-moment fluctuations. For randomly fluctuating   direction in Ia rt,, f , the mean power loss through the hole of area A, time-averaging over such moment-to-moment fluctuations is:

 Wtf,   Ptf,,,cosa I r tf dA I r f t dA at AA a hole a hole

The random, fluctuating moment-to-moment direction in the 3-D vector sound intensity  I rtfIrf,, , cos  t means that the fluctuating, moment-to-moment cost is a hole a hole  constant random at the hole opening of area A. What this means physically is that the probability density distribution ddP cos cos 1 2 is flat/uniform in the cos variable, as shown in the figure below: 뼀 dP cos 

d cos ½

cos  1 0 +1

UIUC Physics 406 Acoustical Physics of Music Since probability P is (always) conserved, and defining x cos , we must have:

1111ddxPPcos  11 11 1 dcos dxdxdxx 2 1 1111  1 ddxcos  2 2 2 2

However, for our physical situation here, only the forward half of this probability distribution is occupied 0cos1 - {sound energy is leaking out of the hole, not into it}. Thus, the time- averaged value of cost , for a flat random distribution in cos over the forward hemisphere is:

111dP cos 111111 costddxdxx cos  cos  cos  cos  2 000  0 d cos  2 2 2 2 4 and hence:  Wtf,   Ptf,,,,cosa I rtf dA I rf t dA aaaAA t 11 1  I rf , dA  I r , f dA  I r , f A 44AAa a hole 4 a hole

A clarificational note: In the steady-state, note that the time interval tavg needed for averaging over the moment-to-moment fluctuations in the instantaneous direction of the 3-D sound   intensity Ia rt,, f is much less than the characteristic time constant W associated with sound energy leaking out of the room of volume V through the hole of area A, i.e. tavg W .

For a large room, we {can safely} assume that the nature of sound propagation is very similar to that in “free air” – i.e. the great outdoors. Then the instantaneous 3-D vector sound intensity    3 Ia rt,, f is related to the instantaneous scalar acoustic energy density wrtfa ,, (Joules/m ) by    the relation Iaart,, f cwrt  ,, f where c = velocity vector associated with propagation of  sound in free-air with c  344 m/s at NTP. Thus, from the above discussion on averaging out random, moment-to-moment fluctuations in the direction of 3-D sound intensity at the hole of area A, we see that the time-averaged version of this relation also holds:

   Iaart,, f cwrt ,, f  Iartfcwrtf hole,, a hole ,,

Note further that in the steady-state, at time t the acoustic energy Wtfa , contained within the  room of volume V is related to the {uniform} acoustic energy density wrtfa ,, by:

 Wtf,,,, wrtfdrwtf  33  drwtfV  , aaVV a a

This relation also holds for time-averaged quantities:

 Wtf,,,, wrtfdr33 wtf dr wtfV , aaVV a a

UIUC Physics 406 Acoustical Physics of Music Since the acoustic energy leaking out of the hole comes from inside the room, by energy conservation, we see that:

 Wtfa ,   11  Ptf,,,,,,, IrtfdA IrtfA cwrtfA aA a a hole a hole t 44 cA  Wtf , 4V a

 Wtfa , cA 1 or:  Wtfaac,,  W tf tV4 W 4V where we have defined the characteristic time constant   (SI units: seconds). W cA

 Wtfa , 1 The equation  Wtfa , is a linear, first-order homogeneous differential t W equation {known as the diffusion, or heat equation} which, for our situation/our initial conditions (at t = 0) has the well-known solution of the form:

o t W Wtfaa,  W f e

o where Wfa  is the time-averaged value of the acoustic energy contained in the room at the

ooW 1 frequency f at time t = 0. Thus, at timet W : WftaWa,  W fe  W a fe i.e. the {time-averaged} acoustic energy at frequency f decreases to 1/e = 1/2.7183 = 0.3679 of its initial value in a time interval t W .

For a large room, since Iaaaft,,, cw ft cW ft V, we can equivalently rewrite the solution for the time-averaged acoustic energy in terms of the time-averaged sound

o t W o intensity as: Itfaa,  I f e where Ia f is the time-averaged sound intensity at the frequency f at time t = 0, and instead ask: how long does it take for the time-averaged sound intensity to decay to one-millionth (106) of its initial value, i.e. what is the reverberation time T60? This occurs when:

ooT60 W 6 IaatT60 ,10 f I fe  I a f i.e. this occurs when eT60 W 106 . Take the natural log of both sides of this relation:

T60 W 6 T60 W 6 lne  ln 10 . But ln eT 60 W . Thus: T60 W ln 10 or:

6 4V T60 W ln 10 and since W  , we thus find that the reverberation time T60 is: cA

4ln10 6 664VVV T60 W ln 10  ln 10      cA c A A  

UIUC Physics 406 Acoustical Physics of Music The numerical value of this “universal” constant,  is:

6 4ln10 4 13.8155 55.262     0.1611sm   0.049 s ft cmsms343 343

VV Thus, the Sabine equation is: T0.161 ( metric units ) 0.049 ( english units ) . 60 AA

6.0 We also see that: T60 WWln 10  13.8155 .

Similarly, we can also show that the reverberation time T30 , defined as the time it takes for the time-averaged sound intensity to decay to one-thousandth (103) of its initial value is given by:

3.0 1 TT30WWln 10  6.9078  2 60

How do we physically measure/determine the reverberation time T60 (and/or T30 )?

Method I:

Note that since:

ooT60 W 6 IaatT60 ,10 f I fe  I a f

ooT30 W 3 IaatT30 ,10 f I fe  I a f

Then, from the above {time-averaged} sound intensity level formulae:

SILtTf60,10log 10 ItTfao  60 , I

SIL t T30,10log f 10 Iac t  T 30 , f I o

12 2 where Io 10 Watts m is the (time-averaged) reference sound intensity (at the threshold of human hearing).

At time t  0 we have:

SIL t0, f 10log10  Iac t  0, f I o

o  10log10 Ifao I

UIUC Physics 406 Acoustical Physics of Music

Thus, the t  0 to tT 60 difference in {time-averaged} sound intensity levels is:

SIL t T60,,0, f SIL t  T 60 f  SIL t  f

 10log10ItTaoao 60 , f I 10log 10  It 0, f I

6 oo  10log10 10Ifao I 10log 10  If ao I

6 o o 10log10 10 10log 10 Ifao I  10log10 IaofI

6  10log10  10 60 dB

Likewise, the t  0 to tT 30 difference in {time-averaged} sound intensity levels is:

SIL t T30,,0, f SIL t  T 30 f  SIL t  f

 10log10ItTaoao 30 , f I 10log 10  It 0, f I

3 oo  10log10 10Ifao I 10log 10  If ao I

3 o o 10log10 10 10log 10 Ifao I  10log10 IaofI

3  10log10  10 30 dB

Now, for a large room (i.e. in a “free field” situation), the sound intensity level (SIL) will be within ~ 0.1 dB of the sound pressure level (SPL) over the human auditory range (~ 20 Hz – 20 KHz). Thus, using an accurately-calibrated SPL meter, technically speaking, the reverberation time Tttt60  2  1 is the measured time interval corresponding to a decrease of:

SPL t T60 ,60 f dB measured from SPL t t1,5 f dB to

SPL t t2 ,65 f dB referenced to SPL t 0, f .

Similarly, technically speaking, the reverberation time Tttt30  2  1 is the measured time interval corresponding to a decrease of:

SPL t T30 ,30 f dB measured from SPL t t1,5 f dB to

SPL t t2,35 f dB referenced to SPL t 0, f .

For example, if SPL t0, f 100 dB , then SPL t t11,,95 f SPL t  t f dB and SPL t t2 ,35 f dB , whereas SPL t t2,65 f dB , thus:

SPL t T60, f SPL t  t 2 , f  SPL t  t 1 , f 35 dB95 dB 60 dB and: tt11 

SPL t T30, f SPL t  t 2 , f  SPL t  t 1 , f 65 dB95 dB 30 dB -6- Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2002 - 2017. All rights reserved.

UIUC Physics 406 Acoustical Physics of Music

These relations are shown in the figure below:

SPL t, f Sound Pressure Level from Sound Source 100 dB Ambient Noise Level aka Noise Floor, 95 dB e.g. 55 dB SPL t, f 30 dB Tttt   65 dB 30 2 1

SPL t, f 60 dB

TtttT60  2  1 2 30 35 dB

t  0 tt11 t tt 2 tt 2

Note that the ambient noise level (aka noise floor) of the room may (often) be such that it is significantly above SPL t t2 ,35 f dB (e.g. 55 dB ), hence measuring the T60 reverberation time will not be possible, whereas measuring the T30 reverberation time will be possible (as long as the noise floor of the room is below SPL t t2,65 f dB ). The T60 reverberation time can then be calculated from the measured T30 reverberation time using the simple relation TT60 2 30 .

Note that one can also simply increase the sound power input to the room, such that with e.g.

SPL t0, f 120 dB , then: SPL t t11, f SPL t  t , f 115 dB ,

SPL t t2 ,55 f dB , and SPL t t2,85 f dB , thus:

SPL t T60, f SPL t  t 2 , f  SPL t  t 1 , f 55 dB115 dB  60 dB and: tt11 

SPL t T30, f SPL t  t 2 , f  SPL t  t 1 , f 85 dB115 dB  30 dB

Note further that there is nothing sacred about usingT60 or T30 ; We can also define other reverberation times: TTTT10, 20 , 40 , 50 ... which correspond to relative decreases of SIL′s/SPL′s of 10, 20, 40, 50 ... dB respectively… The general relation for an arbitrary definition of the xx 10 reverberation time is: Txx W ln 10 , with: SPL t Txx , f xx dB .

UIUC Physics 406 Acoustical Physics of Music Method II:

oooWW Note that since: IaWt , f Ife a  1 eIf a   0.3679 If a we can use a true sound intensity meter (e.g. Bruel & Kjaer 2260E) to measure the time interval

t WW 0  that the time-averaged sound intensity falls to 1/e = 0.3679 of its initial (t = 0) value. We can then subsequently calculate T60 and T30 using the relations:

6.0 3.0 1 T60 WWln 10  13.8155 and: TT30WWln 10  6.9078  2 60

1 However, note that if the typical T60 reverberation time is ~ 1.4 sec, then TT30 2 60 ~0.7sec and W ~ 0.1sec ! Hence, experimentally it is much better to (directly) measure T60 and/or T30 if the time resolution of the measurement,  tW ~ ~ 0.1sec .

Note further that the following is also a difficult quantity to (accurately) measure:

SIL tWW,,0,, f SIL t  f  SIL t  f  SPL  t  W f

 10log10ItfIaW , o 10log 10  ItfI a 0, o

oo  10log10 1eIao  f I 10log 10  I ao f I

o o  10log10 1eIfI 10log 10 ao   10log10 IaofI

 10log10  1edB 4.343

However, additionally note that one could very easily carry out a least-squares fit of the SIL(t)

(or SPL(t)) data to a decaying exponential in order to obtain an accurate determination of W .

Method III:

For a large room (i.e. in a “free field” situation), the (time-averaged) sound intensity Ia tf, is related to the time-averaged square of the instantaneous over-pressure by the relation:

2 Iaotf,, p tf c

3 where o 1.204 kg m is the density of air and cms 343 is the longitudinal speed of propagation of sound in air {at NTP}.

o t W 22t W Then since: Itfaa,  I f e , then: ptf,0, pt f e

For a large room (i.e. in a “free field” situation), the time-average square of the over-pressure amplitude is related to the over-pressure amplitude by:

22ttWW1 2 ptf,0, pt f e 2 peo

UIUC Physics 406 Acoustical Physics of Music

tt p  p where po is the over-pressure amplitude at t = 0, and ptf,0,cos pt  f  e  po  te  is the instantaneous over-pressure, and  p is the characteristic time constant associated with the exponential decay of the over-pressure amplitude with time.

Now, time-averaging over the rapid oscillations associated with costft cos 2 , but not the slow-decay of the exponential, we have:

222222222ttt p pp1 ptf,coscos poo te p te 2 pe o

Comparing this result with the above, we see that a relation exists between the two time constants W and  p :

Wp 2 or:  pW 2

Hence, we obtain the following relations:

6.0 T60 WWppln 10  13.8155  13.8155  2  6.90776  and: 3.0 1 TT30 WWppln 10  6.9078  6.9078  2  3.45388  2 60

We have developed for the UIUC Physics 406 (and Physics 193) POM course a method which has enabled us to determine the over-pressure decay time constant  p to high accuracy by: a.) 24-bit digital recording the time-dependent signal output from a (reference) pressure mic situated somewhere in the large room, stimulated by either a single frequency, or white/pink noise emanating from a sound source located somewhere in the large room. b.) offline analyzing the pressure mic’s time-dependent 24-bit digital signal data (*.wav format), using digital filtering techniques to window around the single frequency (in order to reject ambient noise), or use e.g. digital filters to window pink/white noise signals in 31 1/3-octave bands across the full audio spectrum, calculating the standard deviations  p t of the over- pressure signals (which are linearly proportional to the over-pressure amplitudes) of the filtered signals in a short, running time window t (with pW21tf  ), and then:

o t  p c.) carrying out least-squares fits to decaying exponentials ppte associated with each of the windowed standard deviations, e.g. in the first ½ second (second ½ second) time interval – often there is more than one time constant, e.g. if the ceiling and floor of the large room are more absorptive than the walls of the room…

If interested, please see/read e.g. Serin Yoon’s Fall 2012, Nathan Oliveira and Frank Horger’s Fall 2010 and also Eric Egner’s Fall 2007 UIUC 193 POM Final Reports, available on-line at the following URL:

http://courses.physics.illinois.edu/phys193/193_student_projects.html