Room Acoustics and Sound Reinforcement Systems Tadeusz Fidecki Contents Introduction ................................................................................................................................. 3 1. Wave theory approach ................................................................................................... 3 1.1 Eigenfrequencies and eigenmodes .......................................................................... 3 1.2 Modal density ............................................................................................................ 8 1.3 Loss factor and reverberation time ........................................................................... 8 1.4 Conclusion ................................................................................................................ 11 2. Statistical method ........................................................................................................... 12 2.1 The room response to the sound radiation ............................................................... 12 2.2 Reverberation in enclosures ..................................................................................... 13 2.3 Growth of energy in enclosure .................................................................................. 13 2.4 Decay of energy in enclosure ................................................................................... 14 2.5 Reverberation time ................................................................................................... 15 2.6 Direct and diffuse sound field in enclosure ............................................................... 19 3. Geometric Acoustics and Diffuse Sound Fields ............................................................ 21 3.1 Mirror Sound Sources and Ray Tracing ................................................................... 21 3.2 Flutter Echoes ........................................................................................................... 23 3.3 Impulse Responses of Rectangular Rooms ............................................................. 24 3.4 Diffuse Sound Fields ................................................................................................. 25 3.5 Ray tracing and mirroring ......................................................................................... 26 3.6 Sources and receivers .............................................................................................. 26 3.7 Planes ....................................................................................................................... 27 3.8 Multiple reflecting planes .......................................................................................... 27 3.9 Geometrical impulse response ................................................................................. 28 3.10 Impulse response of real rooms ............................................................................... 30 3.11 Room frequency response ........................................................................................ 31 4. Absorption, reflection and diffusion ................................................................................ 32 4.1 Absorption mechanisms ........................................................................................... 32 4.2 Fundamental quantities and their measurement ...................................................... 32 4.3 Measurement of absorption coefficient for normal sound incidence ........................ 34 4.4 Measurement for random incidence ......................................................................... 34 4.5 Porous absorbers ..................................................................................................... 36 4.6 Porous materials in front of a rigid wall and effects of air gap .................................. 37 4.7 Sound absorption by resonators ............................................................................... 38 4.8 Basic principles and properties of individual Helmhotz resonators .......................... 38 4.9 Perforated panels ..................................................................................................... 39 4.10 Microperforated sound absorbers ............................................................................. 41 4.11 Plate and membrane absorbers ............................................................................... 42 5. Diffusion ......................................................................................................................... 42 5.1 The need of diffusion ................................................................................................ 42 5.2 Basic principles of sound diffusers ........................................................................... 43 6. Subjective room acoustics ............................................................................................. 46 6.1 Subjective and objective evaluation of sound in rooms ............................................ 46 6.2 Metrics for perceived sound and relation to physical measuresl .............................. 47 6.3 Subjective preference for room acoustical quantities ............................................... 52 7. Measuring techniques for the evaluation of room acoustics parameters ...................... 54 7.1 Conventional interrupted noise methods of measurement ....................................... 56 7.2 New measurement method based on impulse response ......................................... 57 7.3 Auditorium measures derived from impulse responses ........................................... 58 7.4 Measurement of the pulse response ........................................................................ 69 8. Applied room acoustics .................................................................................................. 77 8.1 Design procedures in room acoustics planning ........................................................ 77 8.2 The basic recommended values of objective parameters ........................................ 77 8.3 Concert Hall Acoustical Design ................................................................................ 83 8.4 Concert Hall Architectural Design ............................................................................. 85 8.5 Hall Design Procedure .............................................................................................. 86 8.6 Case Studies ............................................................................................................ 86 8.7 Acoustics for Pop and Rock Music ........................................................................... 94 Bibliography ................................................................................................................................ 99 Introduction Sound that propagates inside enclosures is reflected by the enclosure boundaries. The reflected waves are modified by the reflection properties of the walls and other surfaces. The size and shape of the surfaces as well as the wave frequency affect the creation and properties of the reflected wave. The room volumes we live, work, speak and listen to music vary and the acoustic properties of that interiors strongly change over the audio frequency range. One of the important acoustical features of the rooms is the large range of the wavelength of sound in the audio frequency, from about 17 m at 20 Hz to several cm at 20 kHz. The listening in small rooms is different from the listening in large rooms such as auditoria or concert halls. The physical analyses and listening experience indicate that the room properties depend on the frequency, particularly at low frequencies. This requires a combined approach using separate acoustics methods for the small and large rooms and for the low and high audio frequencies: wave theory, statistical and geometrical acoustic methods. 1. Wave theory approach 1.1 Eigenfrequencies and eigenmodes *) The free waves in enclosures are standing waves that are created by the wave reflections from the enclosure walls. Figure 1.1 shows a rectangular enclosure with the dimensions Lx, Ly, Lz. Fig. 1.1. Dimensions of rectangular room *) "Eigenvalue" comes from the German "Eigenwert" which means proper or characteristic value. "Eigenfunction" is from "Eigenfunktion" meaning "proper or characteristic function". The sound pressure is a function of three coordinates. The wave equation for the sound pressure of harmonic waves: In rectangular coordinate system, the eigenfunction can be separated into three functions that depend on one coordinate only. Substituting into the wave equation results in the separation into three spatial functions: where kx, ky, kz are the wave numbers for wave components that propagate in one direction only. Equation can be split into three equations, each depending on one variable only [1]. For the variable x: Applying the boundary condition ux = 0 at x = Lx, solution of the second-order differential equation results in: kxLx = m m = 0,1,2,3,… and The wave number km is for a plane standing wave that propagates in the direction of x- axes and is therefore called axial wave. The general name of this wave is a mode and the frequencies fm,n,l are called characteristic frequencies or eigenfrequencies. Whenever the distance between the walls is a multiple of half wavelengths, a standing wave can exist. The calculation may be repeated for the variables y and z. The expressions