Theory of Coupled Rooms For: nternal only Report No.: R/50/TCR Prepared by:. N. taey B.., MO Otober 00
.00 Objet.. The objet of this doument is present the theory alulations to estimate the reverberant sound pressure level in a room due to a sound soure operating in an adjaent oupled room..00 ope.. The sope of this doument is limited to a: 3.00 ssumptions. ssumptions;. Theory; 3. Disussion. 3.. For the theory to be valid the following assumptions must be made:.00 Theory. truly diffuse sound field is generated in eah room;. ah room an be onsidered statistially lassial; 3. teady state onditions;. Reverberation follows steady state onditions;.. This theory is based on hapter.3 of Priniples ppliations of Room oustis, Volume by Cremer et. al... Figure shows the general situation where Room is oupled to Room through an open oupling area. Room P Room V 0 0 P V where: Figure
is the ross setional area of the oupling between Room Room (m ), P is the aousti power of a noise soure operated in Room (W), V is the volume of Room (m 3), 0 is the absorption in Room exluding the area (m ), P is the aousti power of a noise soure operated in Room (W), V is the volume of Room (m 3), 0 is the absorption in Room exluding the area (m ), teady state sound pressure level.3. f we first turn off soure P onsider only the energy densities related to soure P, we an say the following:.. The sound power absorbed in eah Room is given by: 0 for Room 0 for Room [] where are the average energy densities due to soure P in Room Room respetively is the speed of sound in air..5. The power transferred between rooms is given by: Room > Room : Room > Room : [].6. quations [] [] lead to the following power balane equations: P 0 + 0 [3] 0 0 [].7. The total absorption in Room Room inluding the oupling area is given by: + 0 0 + respetively. [5].8. ubstituting equations [5] into equations [3] [] leads to:
P J/m 3 [6] P J/m 3 [7].9. Using the same proedure with soure P off soure P on leads to: P J/m 3 [8] P J/m 3 [9].0. Combining equations [6] [9] equations [7] [8] gives the total energy densities in eah room with both P P on: P + P T J/m [0] P + P T J/m [].. The pressure is related to the energy density by the following equations: p rms ρ o where ρ o is the density of the air. Hene, where p o is 0µPa. ρo L 0log db [] po
.. By operating one soure at a time measuring the sound pressure levels in eah room, we an find the absorption areas, 0, 0 as follows: giving giving L 0log 0log db L ( L L) /0 ( 0 ) 0 m [3] L 0log 0log db L ( L L )/0 ( 0 ) 0 m [].3. quations [3] [] show that to inrease the level differene L - L L -L we must inrease the absorption areas respetively... The ratios / / are alled the oupling fators respetively. They indiate the degree of oupling from room to room from room to room. values lie between 0..5. The geometri mean oupling fator is given by, so;, 0 + + 0 [5].6. Low values indiate loosely oupled rooms with large level differenes, while high values indiate tightly oupled rooms with small level differenes. Deay proesses following steady state exitation.7. ssume that only soure P is on that it is swithed off(p 0) at time t0, allowing the energy in both rooms to deay..8. We replae equations [3] [] by the following linear differential equations:
( ( d( V [6] dt ( ( d( + V [7] dt where V, are the volumes of room room (m 3 )..9. ssuming the reverberant deay is omposed of exponential funtions, we an set: t ( 0e, t ( 0e [8] where is the damping onstant indiates the rate of deay of the sound pressure..0. ssuming abine s reverberation equation holds, we have the following relation: ( ) 6ln 0 [9] T where T is abine s reverberation time(s)... ubstituting equations [8] into [6] [7] setting:, 8V [0] 8V where, are the damping onstants of the unoupled rooms inluding the absorption area, we get for t0: ( ) V 0 V 0 0 [] ( ) 0 V 0 + V 0 [].. For the above equations to be valid, we must have: i.e. 0 0 ( ) ( )
( ) ( ) 0.3. olving this determinant leads to a quadrati in as follows: so ( + ) + ( ) ( + ) ± ( + ) ( ), [3].. Considering the extreme ases of equation [3] gives: For (total oupling) we have + 0, i.e. the two rooms should be treated as one with a single damping onstant volume VV +V total absorption area 0 + 0. etting 0(no oupling),, i.e. the two rooms should be onsidered separate with no interation between them..5. For all intermediate values of, the aoustis of one room has an influene on the aoustis of the other the resulting deay proesses in both rooms are ditated by, i.e.: ( e + e [] t t ( e + e [5] t t where, are the initial values of the different exponential deays..6. Using equations [] [] we find: [6].7. We find by substituting equations [6] into equations [] [5] setting t0: 0 ( ) (( )( )) 0 [7]
0 ( ) (( )( )) 0 [8] where 0 0 is given by equations [6] [7] for steady state onditions..8. Using the same proedure, we an find the deay proesses when a soure is operated in room..9. Reverberant deays are generally expressed in db levels, so: ( L( 0log db [9] 0 where o is the energy density at time t0..30. We an modify all of the above equations for the ase of oupling through a partition: τ, 0 + α 0 + α where: τ is the transmission oeffiient of the partition, assumed the same in both diretions. α α are the absorption oeffiients of the partition on the side of room room respetively. 5.00 Disussion 5.. Rooms are often assumed aoustially isolated from adjaent rooms when evaluating their aousti properties. n some irumstanes, this an be a very dangerous assumption. 5.. The easiest way to show this is by taing an example. ssume we have two rooms with the following parameters: V 300m 3, V 800m 3 an open oupling area between them of m, not unusual in publi buildings.
5.3. The reverberation time in room is said to be exessive so is measured with a view to inreasing the absorption, so dereasing the RT to a value of.6ses. 5.. Further, the reverberation time is to be evaluated over a deay of 5 to 35dB. 5.5. Room is measured the RT(-5,-35) is found to be.9ses. Using abine s equation, we find that to redue the RT to.6ses requires an additional 3.5m of absorption. We add this to room find that the RT(-5,-35) has only redued to.3ses. 5.6. The RT(-5,-35) of room was later also found to be.9ses. Before dding bsorption fter dding bsorption 0 0-0 -0-0 -0-30 -30-0 -0-50 -50-60 -0.5 0.0 0.5.0.5.0.5 3.0 Time, ses -60-0.5 0.0 0.5.0.5.0.5 3.0 Time, ses Figure 5.7. Using equation [], figure shows the expeted deays for room before after the absorption is added. 5.8. What has happened is that after an initial time period, the long RT of room has dominated the deay proess. 5.9. Figure shows that areful attention should be given to measured multi deays as they not only indiate uneven absorption, but also show the effets of oupling. The range over whih the RT is evaluated should also be given onsideration. 5.0. Figure shows the deay after adding 75m of absorption to room. The RT(-5,-35) is now.6ses the double deay is evident. fter Needed bsorption 0-0 -0-30 -0-50 -60-0.5 0.0 0.5.0.5.0.5 3.0 Time, ses Figure
5.. The safest way to determine the liely effets of oupling, is to find the oupling fators by measuring the level differenes using equations [3] []. 5.. Lets now assume that we have a bar a restaurant oupled together via an open stairase. 5.3. Complaints have been reeived from people in the restaurant that the noise from ustomers in the bar is too high. 5.. The following are the parameters of the two rooms: Room : Bar, Room : Restaurant Volume (V) : 300m 3 Total surfae area () : 69m Power level (P) : 9dB (bar ustomers) Volume (V) : 50m 3 Total surfae area () : 38m Power level (P) 85dB (restaurant ustomers) Coupling area () : 6m 5.5. ound pressure level differenes between the rooms were measured: L L db, L L 6dB 5.6. Using the above data applying the formulae, we alulate that the urrent signal-to-noise ratio in the restaurant is 5dB (i.e. speeh level in restaurant less noise from bar). 5.7. To inrease the /N ratio in the restaurant, we must inrease the absorption in the bar. We an alulate the mathematial maximum ahievable /N ratio by setting the absorption oeffiient of the bar surfaes to unity. The result is: Maximum ahievable /N(restauran 3.3dB 5.8. Of ourse, this is purely theoretial sine diffuse onditions would no longer our, we would most liely be onsidering the effets of diret level propagation.