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6 CHAPTER 1. FUNDAMENTAL CONCEPTS OF ACOUSTICSthe wave equation by differentiating two times with respect to the time andrepeating the same with respect to the position x :∂ 2 u= −ω 2 u(t,x)∂t 2∂ 2 u= − ω2∂x 2 c2u(t,x) (1.4)and filling these expressions in the wave equation.If damping is present due to absorption of the medium, the solution hasthe following form:U = U 0 exp(−αu) (1.5)Note that the wave can be considered undamped if propagation of soundoccurs in an (unconfined) air volume. Indeed, the damping of sound at 1000Hz is only 5 decibel per km. One can show that the damping is proportionalto the square of the frequency according to :α = ω2 τc(1.6)With τ the relaxation time (around 0.2 ns for monatomic gases). Consequentlyfor high frequencies damping in air may not be neglected. Concerningthe periodic character, we know that the exponential function withimaginary exponent has a periodicity equal to 2π :1. Time periodicity: u(t + T,x) = u(t,x) with period T. From whichfollows that ωT = 2π and further ω = 2π = 2πf with f = 1 theT Tfrequency.2. Space periodicity: u(t,x + λ) with wavelength λ, from which followsthat λk = 2π. Replacing k = ω c yields : ωλ c = 2π and finally ω = 2πλ c3. Combining 1. and 2. gives us the relation between the spatial andfrequency domain wave propagation parameters :with c the speed of sound.1.2.2 The speed of soundλf = c (1.7)The speed of sound is the velocity at which a perturbation, a wave front,propagatesinthegivenmedium. Itdependsonthepropertiesofthismedium,
1.2. PLANE SOUND WAVES 7Material Speed of SoundGlass 5400Steel 5000Aluminium 5200Nickel 4800Wood 4000Copper 3500Plumbum 1300Platinum 2800Silver 2600Water 1460Seawater 1500Mercury 1407Hydrogen 1260Table 1.1: Speed of sound for common materialsespecially density and elasticity.In gases the following equation is applicable [19] :√ √γP O Kc = = = √ γrT (1.8)ρ 0 ρ 0withK = γP 0 the compression– or bulk modulus for gases andwhere γ representsthe heat capacity ratio and P 0 the atmospheric air pressureT the absolute temperature inKr the specific gas constant√BIn fluids : c = with ρB the compression– or bulk modulus for fluids (in Pa)√EIn solids c = with ρE the elastic– or Young’s modulusThe speed of sound for some common materials, gases and fluids are tabulatedin Table 1.1.
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Chapter 4Sound AbsorptionAll(constr
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4.2. REALIZATION OF ACOUSTIC ABSORP
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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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5.3. 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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6.3. 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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144 APPENDIX A. MATERIAL PROPERTIES
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152 BIBLIOGRAPHY[12] Bruel and Kjae
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