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12 CHAPTER 1. FUNDAMENTAL CONCEPTS OF ACOUSTICSTransformation into a spherical coordinate system one can obtain (see[19]) the following form of the wave equation :∂ 2 rp∂r 2 + = 1 c 2 ∂ 2 rp∂t 2 (1.20)with r = √ x 2 +y 2 +z 2 .Taking rp as one variable we get a spherical wave equation of the sameform as the equation for plane waves. A general solution of this equation, astraveling waves, is thus :rp(r,t) = p 1 (t−r/c)+p 2 (t+r/c) or p(r,t) = 1 r p 1(t−r/c)+ 1 r p 2(t+r/c)(1.21)withthefirsttermrepresenting adivergent spherical wave andthesecondonea convergent spherical wave. Both waves exhibit the same propagation speedc and their amplitudes decreases with increasing distance r, radially fromthe point source. Convergent waves have so to say no acoustical application,while the divergent waves have . Indeed, this latter is present as soon as thedistance from the sound source becomes larger than the physical dimensionsof that source. If the point source produces an harmonic wave one may writefor the divergent wave :p(r,t) = A rexp(iωt−ikr) (1.22)1.4 Cylindrical sound wavesFor a cylindrical line sound source one can show that the solution can bewritten as (see [19]) :p(r,t) = A √ rexp(iωt−ikr) (1.23)An important conclusion concerning the three simple types of sources is thefollowing :For plane sound waves the sound pressures does not decrease with thedistance.For spherical sound waves the sound pressure decreases linearly withincreasing distance.
1.5. SOUND LEVELS 13For cylindrical sound waves the sound pressure decreases inversely proportionalto the square root of the distance.1.5 Sound levels1.5.1 The effective sound pressureSuppose a given source of sound produces sound, i.e. the quickly fluctuatingair pressure makes our eardrum to vibrate which causes through our earand nervous system a sensation of sound. Total pressure can be writtenas : p = p atm + p sound . One could think that (subjectively) if we havethe impression of a constant sound level, intensity or loudness it implies(objectively) a constant sound pressure in time. Nothing appears to be lesstrue. The sound perceived with a constant loudness may be both a pure sinetone and a stochastic sound generated by a source with constant parameter:p(t) is extremely complicated, and yet the human ear have the impression ofa constant loudness. Our purpose is to describe mathematically this soundloudness. Afirst conclusion : theinstantaneous valueandthe algebraicmeanvalue does not matter (this latter is eventually zero). The human hearingsystem is quite insensitive for sharp positive and negative peaks, which maybe cut off. In contrast, it seems to be sensitive to the energy of sound waves.This led to the consideration of the effective– or Root-Mean-Square (RMS)value of the sound pressure, over a certain time interval, as an measure ofintensity :p eff =√1t 2 −t 1∫ t2t 1p 2 (t)dt (1.24)Foranharmonicwave(withDCvalueequal tozero) thisis: p eff = p max / √ 2.As can be seen in the definition the effective value p eff depends of the timeinterval as well as the type of sound. E.g. :1. We listen to a noise which seems to be constant in time. To describethe intensity one can thus use p 2 averaging over a relative small orrelative large time interval – this changes not much to the result of themeasurement. In this case the time interval does not play a major role.2. When a plane flies over one hear the sound surging followed by extinctionduring a specific time interval. If we want to use p eff in thiscase to describe the instantaneous numerical value giving a reasonableimpression of the perceived sound intensity, we would have to average
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Chapter 5Sound InsulationWhen consi
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5.1. MEASURING SOUND INSULATION 83F
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5.1. MEASURING SOUND INSULATION 85F
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5.2. AIRBORNE SOUND INSULATION OF A
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5.3. THE ACOUSTICAL BARRIER 95For t
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5.3. THE ACOUSTICAL BARRIER 97Figur
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Chapter 6Noise control6.1 Origin of
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6.1. ORIGIN OF NOISE 101Figure 6.2:
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6.2. REDUCING NOISE AT THE LEVEL OF
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6.3. TACKLING NOISE TRANSMISSION 11
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6.4. RADIATION NOISE 115Figure 6.17
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6.4. RADIATION NOISE 117Figure 6.20
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Part IIINoise directives119
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124CHAPTER 7. DIRECTIVE 2000/14/EG:
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128 CHAPTER 8. NOISE ON THE WORK FL
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140 CHAPTER 9. COMMUNITY NOISE2. No
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142 CHAPTER 9. COMMUNITY NOISE
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144 APPENDIX A. MATERIAL PROPERTIES
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152 BIBLIOGRAPHY[12] Bruel and Kjae
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