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90 CHAPTER 5. SOUND INSULATIONIt can be shown that for a rectangular wall with height a, length b andthickness h the resonance frequencies are given by [21] :f mn = π4 √ 3 c Lh(( m a )2 +( n )b )2(5.24)where the indices m and n are natural numbers designating different modesand c L the quasi-longitudinal wave velocity given by :√Ec L =(5.25)ρ s (1−ν 2 )The calculation of these frequencies allow us to estimate if the wall is eithermass controlled or stiffness controlled.5.2.3 The coincidence effectTheresonancesofthewalldescribedinpreviousparagraphoccuratrelativelylow frequencies. At higher frequencies higher order bending waves will occurin the wall (f mn for mn large). When the wavelength of these bending wavescoincides with the wavelength of the acoustic waves after projection on thewall a so-called coincidence phenomenon will occur (see Figure 5.4). Theprojection of the acoustic wavelength on the wall is called wave trace (thisis given by λasinθ with λ a the wavelength of the sound and θ the angle ofincidence). In what follows we will derive a formula for the frequencies atwhich coincidence will occur.One can show that the bending waves in a clamped wall can be describedby following equation :−B ∂4 v∂x 4 = m∂2 v∂t 2 (5.26)where B = EI1−ν 2 represents the bending stiffness of the wall (I = d 3 l/12).For an harmonic wave we have v = exp(iωt − ikx). After substitution inEquation 5.26 one gets the velocity of the bending wave :c b = √ 2πf 4 √Bm(5.27)By definition coincidence will happen when λ b = λa and thus whensinθc b = ca (because λf = c for both the acoustic wave and the bending wave).sinθThis yields :f(sinθ) 2 = c22π√ mB(5.28)
5.2. AIRBORNE SOUND INSULATION OF A WALL 91Figure 5.4: Sketch of the coincidence effect.An important frequency for the coincidence phenomenon is the so-calledcritical frequency f crit . It is the lower limit at which coincidence can occur.The critical frequency can be found by using θ = π/2 (indeed the acousticwavelength can be projected on a bigger wavelength by varying theta but noton a smaller one, θ = π/2 gives the smallest possible projection). Therefore,the critical frequency f crit is equal to :f crit = c22π√ mBFor air we can simplify this to the following expression :(5.29)√with c L =Eρ s(1−ν 2 )f crit = 64000d c L(5.30)the so-called quasi-longitudinal wave velocity of the wallmaterial. The critical frequency for somematerials isshown in Table5.2. Fora concrete wall of 10 cm thickness the value of f crit = 138.5 Hz This means
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VRIJE UNIVERSITEITBRUSSELAcousticsB
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iiWhen sound becomes noise, it will
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ContentsI Introduction to acoustics
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CONTENTSvii4.3 Measuring the acoust
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Part IIntroduction to acoustics1
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4 CHAPTER 1. FUNDAMENTAL CONCEPTS O
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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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