FUNDAMENTALS OF ACOUSTICS AND NOISE CONTROL Finn Jacobsen, Torben Poulsen, Jens Holger Rindel, Anders Christian Gade and Mogens Ohlrich Department of Electrical Engineering, Technical University of Denmark December 2009 Note no 31200 ii CONTENTS Page 1 An elementary introduction to acoustics .................................................................... 1 Finn Jacobsen 1.1 Introduction .................................................................................................................... 1 1.2 Fundamental acoustic concepts ...................................................................................... 1 1.2.1 Plane sound waves ............................................................................................. 4 1.2.2 Spherical sound waves ..................................................................................... 13 1.3 Acoustic measurements ................................................................................................ 15 1.3.1 Frequency analysis ........................................................................................... 15 1.3.2 Levels and decibels ......................................................................................... 18 1.3.3 Noise measurement techniques and instrumentation ....................................... 21 1.4 The concept of impedance ............................................................................................ 27 1.5 Sound energy, sound intensity, sound power and sound absorption ............................ 31 1.5.1 The energy in a sound field .............................................................................. 32 1.5.2 Sound absorption .............................................................................................. 35 1.6 Radiation of sound ....................................................................................................... 37 1.6.1 Point sources .................................................................................................... 37 1.6.2 Sound radiation from a circular piston in an infinite baffle ............................. 42 1.7 References .................................................................................................................... 49 1.8 Bibliography ................................................................................................................. 50 1.9 Appendix: Complex notation ....................................................................................... 51 2 Ear, Hearing and Speech ........................................................................................... 55 Torben Poulsen 2.1 Introduction .................................................................................................................. 55 2.2 The ear .......................................................................................................................... 55 2.2.1 The outer ear ..................................................................................................... 56 2.2.2 The middle ear .................................................................................................. 56 iii 2.2.3 The inner ear ..................................................................................................... 57 2.2.4 The frequency analyzer at the Basilar membrane ............................................ 59 2.3 Human hearing ............................................................................................................. 61 2.3.1 The hearing threshold ....................................................................................... 61 2.3.2 Audiogram ........................................................................................................ 62 2.3.3 Loudness level .................................................................................................. 63 2.4 Masking ........................................................................................................................ 64 2.4.1 Complete masking ............................................................................................ 65 2.4.2 Partial masking ................................................................................................. 67 2.4.3 Forward masking .............................................................................................. 67 2.4.4 Backward masking ........................................................................................... 67 2.5 Loudness ....................................................................................................................... 68 2.5.1 The loudness curve ........................................................................................... 68 2.5.2 Temporal integration ........................................................................................ 69 2.5.3 Measurement of loudness ................................................................................. 69 2.6 The auditory filters ....................................................................................................... 71 2.6.1 Critical bands .................................................................................................... 71 2.6.2 Equivalent rectangular bands ........................................................................... 72 2.7 Speech .......................................................................................................................... 73 2.7.1 Speech production ............................................................................................ 73 2.7.2 Speech spectrum, speech level ......................................................................... 74 2.7.3 Speech intelligibility ........................................................................................ 75 2.8 References .................................................................................................................... 78 3 An introduction to room acoustics ............................................................................ 81 Jens Holger Rindel 3.1 Sound waves in rooms .................................................................................................. 81 3.1.1 Standing waves in a rectangular room ............................................................. 81 3.1.2 Transfer function in a room .............................................................................. 83 3.1.3 Density of natural frequencies .......................................................................... 83 3.2 Statistical room acoustics ............................................................................................. 85 3.2.1 The diffuse sound field ..................................................................................... 85 iv 3.2.2 Incident sound power on a surface ................................................................... 86 3.2.3 Equivalent absorption area ............................................................................... 87 3.2.4 Energy balance in a room ................................................................................. 87 3.2.5 Reverberation time. Sabine’s formula .............................................................. 88 3.2.6 Stationary sound field in a room. Reverberation distance ............................... 89 3.3 Geometrical room acoustics ......................................................................................... 92 3.3.1 Sound rays and a general reverberation formula .............................................. 92 3.3.2 Sound absorption in the air ............................................................................... 93 3.3.3 Sound reflections and image sources ............................................................... 94 3.3.4 Reflection density in a room ............................................................................ 95 3.4 Room acoustical design ................................................................................................ 96 3.4.1 Choice of room dimensions .............................................................................. 96 3.4.2 Reflection control ............................................................................................. 96 3.4.3 Calculation of reverberation time ..................................................................... 97 3.4.4 Reverberation time in non-diffuse rooms ......................................................... 98 3.4.5 Optimum reverberation time and acoustic regulation of rooms ....................... 99 3.4.6 Measurement of reverberation time ............................................................... 100 3.5 References .................................................................................................................. 101 4 Sound absorbers and their application in room design ........................................ 113 Anders Christian Gade 4.1 Introduction ................................................................................................................ 103 4.2 The room method for measurement of sound absorption .......................................... 103 4.3 Different types of sound absorbers ............................................................................. 104 4.3.1 Porous absorbers ............................................................................................ 105 4.3.2 Membrane absorbers ...................................................................................... 106 4.3.3 Resonator absorbers ....................................................................................... 108 4.4 Application of sound absorbers in room acoustic design ........................................... 109 4.5 References .................................................................................................................. 112 5 An introduction to sound insulation ....................................................................... 113 Jens Holger Rindel 5.1 The sound transmission loss ....................................................................................... 113 5.1.1 Definition ....................................................................................................... 113 v 5.1.2 Sound insulation between two rooms ............................................................. 113 5.1.3 Measurement of sound insulation .................................................................. 114 5.1.4 Multi-element partitions and apertures .......................................................... 114 5.2 Single leaf constructions ............................................................................................ 117 5.2.1 Sound transmission through a solid material ................................................. 117 5.2.2 The mass law .................................................................................................. 119 5.2.3 Sound insulation at random incidence ........................................................... 120 5.2.4 The critical frequency ..................................................................................... 121 5.2.5 A general model of sound insulation of single constructions ........................ 122 5.3 Double leaf constructions ........................................................................................... 123 5.3.1 Sound transmission through a double construction ........................................ 123 5.3.2 The mass-air-mass resonance frequency ........................................................ 124 5.3.3 A general model of sound insulation of double constructions ....................... 124 5.4 Flanking transmission ................................................................................................ 126 5.5 Enclosures .................................................................................................................. 127 5.6 Impact sound insulation ............................................................................................. 127 5.7 Single-number rating of sound insulation .................................................................. 128 5.7.1 The weighted sound reduction index ............................................................. 128 5.7.2 The weighted impact sound pressure level .................................................... 129 5.8 Requirements for sound insulation ............................................................................. 131 5.9 References .................................................................................................................. 131 6 Mechanical vibration and structureborne sound .................................................. 133 Mogens Ohlrich 6.1 Introduction ................................................................................................................ 133 6.1.1 Sources of vibration ....................................................................................... 134 6.1.2 Measurement quantities .................................................................................. 134 6.1.3 Linear mechanical systems ............................................................................ 135 6.2 Simple mechanical resonators .................................................................................... 136 6.2.1 Equation of motion for simple resonator ........................................................ 137 6.2.2 Forced harmonic response of simple resonator .............................................. 138 6.2.3 Frequency response functions ........................................................................ 145 vi 6.2.4 Forced vibration caused by motion excitation ............................................... 147 6.3 Vibration and waves in continuous systems .............................................................. 148 6.3.1 Longitudinal waves ........................................................................................ 149 6.3.2 Shear waves .................................................................................................... 150 6.3.3 Bending waves ............................................................................................... 151 6.3.4 Input mobilities of infinite systems ................................................................ 152 6.4 Vibration isolation and power transmission ............................................................... 153 6.4.1 Estimation of spring stiffness and natural frequency ..................................... 154 6.4.2 Transmission of power in rigidly coupled systems ........................................ 155 6.4.3 Vibration isolated source ................................................................................ 156 6.4.4 Design considerations for resilient elements .................................................. 159 6.5 References .................................................................................................................. 162 List of symbols ....................................................................................................................... 163 Index… ................................................................................................................................... 167 vii viii 1 AN ELEMENTARY INTRODUCTION TO ACOUSTICS Finn Jacobsen 1.1 INTRODUCTION Acoustics is the science of sound, that is, wave motion in gases, liquids and solids, and the effects of such wave motion. Thus the scope of acoustics ranges from fundamental physical acoustics to, say, bioacoustics, psychoacoustics and music, and includes technical fields such as transducer technology, sound recording and reproduction, design of theatres and concert halls, and noise control. The purpose of this chapter is to give an introduction to fundamental acoustic con- cepts, to the physical principles of acoustic wave motion, and to acoustic measurements. 1.2 FUNDAMENTAL ACOUSTIC CONCEPTS One of the characteristics of fluids, that is, gases and liquids, is the lack of constraints to deformation. Fluids are unable to transmit shearing forces, and therefore they react against a change of shape only because of inertia. On the other hand a fluid reacts against a change in the volume with a change of the pressure. Sound waves are compressional oscillatory distur- bances that propagate in a fluid. The waves involve molecules of the fluid moving back and forth in the direction of propagation (with no net flow), accompanied by changes in the pres- sure, density and temperature; see figure 1.2.1. The sound pressure, that is, the difference be- tween the instantaneous value of the total pressure and the static pressure, is the quantity we hear. It is also much easier to measure the sound pressure than the other quantities. Note that sound waves are longitudinal waves, unlike bending waves on a beam or waves on a stretched string, which are transversal waves in which the particles move back and forth in a direction perpendicular to the direction of propagation. Figure 1.2.1 Fluid particles in the sound field generated by a pulsating sphere. (From ref. [1].) 1 In most cases the oscillatory changes undergone by the fluid are extremely small. One can get an idea about the orders of magnitude of these changes by considering the variations in air corresponding to a sound pressure level1 of 120 dB, which is a very high sound pressure level, close to the threshold of pain. At this level the fractional pressure variations are about 2104, the fractional changes of the density are about 1.4104, the oscillatory changes of the temperature are less than 0.02 °C, and the particle velocity2 is about 50 mm/s, which at 1000 Hz corresponds to a displacement of less than 8μm. In fact at 1000 Hz the particle dis- placement at the threshold of hearing is less than the diameter of a hydrogen atom!3 Sound waves exhibit a number of phenomena that are characteristics of waves; see figure 1.2.2. Waves propagating in different directions interfere; waves will be reflected by a rigid surface and more or less absorbed by a soft one; they will be scattered by small obsta- cles; because of diffraction there will only partly be shadow behind a screen; and if the me- dium is inhomogeneous for instance because of temperature gradients the waves will be re- fracted, which means that they change direction as they propagate. The speed with which sound waves propagate in fluids is independent of the frequency, but other waves of interest in acoustics, bending waves on plates and beams, for example, are dispersive, which means that the speed of such waves depends on the frequency content of the waveform. Figure 1.2.2 Various wave phenomena. A mathematical description of the wave motion in a fluid can be obtained by combin- ing equations that express the facts that i) mass is conserved, ii) the local longitudinal force caused by a difference in the local pressure is balanced by the inertia of the medium, and iii) sound is very nearly an adiabatic phenomenon, that is, there is no flow of heat. The observa- tion that most acoustic phenomena involve perturbations that are several orders of magnitude smaller than the equilibrium values of the medium makes it possible to simplify the mathe- matical description by neglecting higher-order terms. The result is the linearised wave equa- tion. This is a second-order partial differential equation that, expressed in terms of the sound pressure p, takes the form 1 See section 1.3.2 for a definition of the sound pressure level. 2 The concept of fluid particles refers to a macroscopic average, not to individual molecules; therefore the particle velocity can be much less than the velocity of the molecules. 3 At these conditions the fractional pressure variations amount to about 2.51010. By comparison, a change in altitude of one metre gives rise to a fractional change in the static pressure that is about 400000 times larger, about 10-4. Moreover, inside an aircraft at cruising height the static pressure is typically only 80% of the static pressure at sea level. In short, the acoustic pressure fluctuations are extremely small compared with com- monly occurring static pressure variations. 2