EE1.el3 (EEE1023): Electronics III. Acoustics lecture 18 Room acoustics. Dr Philip Jackson.

Transcription

1 EE1.el3 EEE1023): Electronics III Acoustics lecture 18 Room acoustics Dr Philip Jackson

2 Room acoustics Objectives: relate sound energy density to rate of absorption compare direct and reflected sound levels measure and calculate reverberation time explain differences between free and diffuse sound fields define an expression for a room s critical distance Topics: Direct and reflected sound echoes Sound energy stored in a room A room s characteristic measurements Worked examples R.1

3 Preparation for room acoustics What is the absorption coefficient α? find a definition give two examples of materials with different values of α How can we characterise the amount of reflection or absorption in a room? find out what is the meaning of reverberation time how can it be measured? R.2

4 Reflections in a room Direct sound original waveform from source Early reflections reinforce direct sound inform on room size & shape Reverberation late reflections) strengthens room impression sound source acoustical filter, hm) received sound pressure R.3

5 Echoes All surfaces both absorb and reflect sound. The absorption coefficient α gives the ratio of absorbed to incident sound: W ab = αw in W re = 1 α)w in Echoes need delay >50 ms to be heard distinctly Parallel reflective surfaces produce flutter echoes: large separation fast repetitions medium separation buzzing small separation ringing Combination of multiple echoes produces reverberation: sound reflected by all surfaces from all directions with equal probability impulse response exhibits an exponential decay R.

6 Diffuse sound stored in a room Absorption of sound in a room Rooms provide means of storing and absorbing sound. By analysing the effect of sound in a small volume, δv, on a small patch of wall, δs, it can be shown that the intensity of incident sound depends on the energy density, E, in the room: δv r δs I in = Ec where E = p2 rms ρ 0 c 2 1) With surface area S and average absorption α, the rate of energy absorption by the room determines its absorbed sound power: Sα Ec W ab = 2) R.5

7 Calculating sound absorption Frequency Hz) Material concrete glass wood carpet fibreglass acoustic tile audience Table 1. Typical sound absorption coefficients α For a room m L) 3 m W) 2.5 m H) with wooden walls, acoustic tiled ceiling and carpet on the floor, we can calculate the total absorption at 1 khz: A = 2 + 3) ) ) ) = = sabin R.6

8 Growth of sound in a room From eq. 2, we obtain a differential equation to describe the growth of sound energy within a closed volume V : V Sαc V de dt = W src de dt + E = W src Sαc Sα Ec 3) If the sound source is started at t = 0, solution yields Et) = W src 1 e Sαc V t) ) Sαc As t, the energy density reaches equilibrium: E = W src Sαc 5) R.7

9 Absorption rate at equilibrium Energy in a reverberant room acts like a first-order system: t) = W src S αc ON OFF t At equilibrium, the rate of energy absorption matches the input power: W src = W ab = Sα E c 6) R.8

10 Rate of reverberation decay If the source is switched off, eq. 3 gives an expression for the rate of decay of the reverberation: de Sα Ec = 7) dt V The initial conditions then provide the full equation: Et) = E 0 exp Sαc ) V t 8) Also, from eq. 1, we relate energy density to sound level: SPLt) = 10 log 10 p 2 rms p 2 ref ) = 10 log 10 Et)ρ0 c 2 p 2 ref = 10 log 10 Et) + log 10 ρ 0 c 2 log 10 p 2 ref ) ) = 10 log 10 Et) + C = 10 log 10 E 0 Sαc V t log 10 e ) + C 9) R.9

11 Reverberation time The reverberation time, T 60, is the time for the SPL in a room to drop by 60 db, i.e., when t = T 60 in eq. 9: 10 Sαc V T 60 log 10 e = 60 T 60 = KV Sα where K = 2/c log 10 e) 0.16 sm 1 gives Sabine s Eq. 10) Note: T 60 can depend on frequency but is otherwise independent of original sound source R.10

12 Measurement of reverberation time Reverberation time is commonly measured with pink noise from a loudspeaker: With limited dynamic range, T 30 is measured to avoid the noise floor and doubled to give T 60. As reverberation time varies with frequency, band-limited noise can be used. R.11

13 Soundfields in an acoustic environment Free field, no reflections, sound comes direct from source Near field close), more direct sound than reflections Diffuse field reverberant field), more reflected sound from all directions than direct sound Critical distance a.k.a. room radius), d c, equal levels of direct and reflected sound R.12

14 Level of reverberant sound As in eq. 3, we have the ODE for reverberant energy: V Sαc de rev dt V de rev = 1 α) W src Sα E revc dt 1 α) + E rev = W src Sαc E rev t) = 1 α) W src Sαc 1 e Sαc V t) Erev = 1 α) W src Sαc From eq. 1, we can obtain the reverberant SPL: SPL rev = 10 log 10 E rev ρ 0 c 2 p 2 ref 1 α) Wsrc ρ = 10 log 0 c 10 Sα p 2 ref = 10 log 10 Wsrc R A I ref ) ) ) R.13

15 Critical distance calculation The critical distance d c is reached when SPL rev = SPL dir. For direct sound from a point source in free field, we have SPL dir = SIL dir = 10 log 10 Idir I ref ) = 10 log 10 Wsrc πd 2 c I ref which matches reverberant sound level at critical distance SPL rev = 10 log 10 Wsrc R A I ref where R A = Sα/1 α) is the room absorption constant. ) ) Source power W src and reference intensity I ref cancel out, to yield the final result: = 1 R A πd 2 c d c = 1 RA π 11) R.1

16 Worked example: factory noise A machine with a sound power of W src = 1 W is situated in a room that has dimensions 7 m L) 5 m W) 3 m H). 1. Treating it as a point source, what is the intensity at a distance of 2 m? 2. The machine is left running continuously and reflections contribute to build up a reverberant field. If the absorption coefficient is α = 0.1, what will be the asymptotic energy density? 3. What is the corresponding pressure level of the reverberant sound?. What changes would occur if the ceiling was treated with a material having α = 0.6? R.15

17 Room acoustics Reflection and absorption of sound in a room ODE of sound stored in a room Measurement of reverberation time Calculation of critical distance Reference L. E. Kinsler, A. R. Frey, A. B. Coppens and J. V. Sanders, Fundamentals of Acoustics, th ed., Wiley, Chapter 12, [shelf 53 KIN] R.16

18 Preparation for musical acoustics For one instrument from each class, identify what creates a sound source, and a resonator that enhances it? voice woodwind brass percussion string R.17

19 Appendix: Sound stored in a room Absorption of sound in a room Sound from a compact source impinges on a patch δs and in a reverberant field: δw = δs cos θ W src πr 2 δe = δs cos θ E δv πr 2 12) δs r where sound energy density E is related to the RMS pressure of the diffuse field: E = p2 rms ρ 0 c2, 13) δs r θ δs cosθ δv assuming the constituents act like plane waves eq. 7a on N.6). R.18

20 Derivation of the reverberant energy density Considering a hemispherical shell of thickness δr and radius r centred on δs, we first obtain the energy in a ring: δv = 2πr sin θ δr rδθ The total energy in time interval δt = δr/c is: E = = δse = = de = π/2 0 δs E δr 2 δs E δr cos θ π/2 0 π/2 0 δs cos θ E πr 2 dv 2πr sin θ δr r πr 2 cos θ sin θ dθ sin 2θ dθ dθ δs δs δθ r θ θ r δv δr = δs E δr [ 12 cos 2θ ] π/2 0 = δs E δr 1) R.19

21 Surface intensity from a diffuse field From the total sound energy on a small patch of the wall δs E δr E = the rate of incident energy per unit area gives the intensity I in = E δt δs = Ec 15) The result is 1/ of that for a normally-incident plane wave: I = Ec With total sound absorption of surfaces A = α δs = Sα with area S and average absorption α, the rate of energy absorption defines the absorbed sound power: W ab = Sα Ec 16) R.20