Introduction to Acoustics

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the mass matrix is taken to be constant with time. In this case, (22.46) becomes with ⎧ M ¨X + KX = 0 (22.47) ⎪⎨ ⎪⎩ Mkj = M jk = ∂2 Ek ∂ ˙qk∂ ˙q j Kkj = K jk = ∂2 V ∂qk∂q j 22.2.2 Eigenmodes and Eigenfrequencies . (22.48) Natural modes (or eigenmodes) are the sinusoidal (or harmonic) solutions of (22.47) with frequency ω in the absence of a driving term. As a consequence, the natural modes are solutions of the equation � − ω 2 � M + K X = 0 (22.49) with roots (or eigenvalue, or eigenfrequency) ωn solutions of the characteristic equation � det − ω 2 � M + K = 0 . (22.50) The eigenvector Φn associated to each eigenfrequency ωn is given by � − ω 2 � nM + K Φn = 0 . (22.51) Φn is defined with an arbitrary multiplicative constant. The natural modes are defined by the set of eigenvalues ωn and associated eigenvectors Φn. Spectral analysis theory shows that the Φn form an M and K orthogonal basis set, so that t ΦmMΦn = 0 and t ΦmKΦn = 0 for m �= n (22.52) The orthogonality properties of the eigenmodes mean that the inertial (stiffness) forces developed in a given mode do not affect the motion of the other modes. The modes are mechanically independent. As a consequence, it is possible to expand any motion onto the eigenmodes. Given a force distribution F, the motion of the system is governed by the equation M ¨X + KX = F . (22.53) Structural Acoustics and Vibrations 22.2 Discrete Systems 909 The modal projection is written in the form: X = � Φnqn(t) . (22.54) n The functions qn(t)in(22.54) are the modal participation factors. Inserting (22.54) into(22.53) and, after taking the scalar product of both sides of the equation with an eigenfunction Φm,wefind 〈Φn, MΦn〉 ¨qn +〈Φn, KΦn〉qn =〈Φn, F〉 (22.55) where the notation 〈A, B〉 denotes the scalar product t A.B between vectors the A and B. Equation (22.55) shows that the generalized displacements are uncoupled. Each qn is governed by a single-DOF oscillator differential equation. The quantity mn =〈Φn, MΦn〉 (22.56) is the modal mass associated with the mode n. These coefficients are defined with an arbitrary multiplicative constant. Similarly, κn =〈Φn, KΦn〉 (22.57) is the modal stiffness, related to the modal mass through the relationship κn = mnω 2 n . (22.58) Finally, the quantity fn =〈Φn, F〉 (22.59) is the projection of the nonconservative forces onto mode n. Each independent oscillator equation can then be rewritten as ¨qn + ω 2 n qn = fn . (22.60) mn Remark. Since the eigenvectors are defined with arbitrary multiplicative constants Cn,(22.56) shows that the modal mass is proportional to C2 n . In addition, (22.58) shows that fn is also proportional to Cn. Therefore, through (22.60), qn is proportional to C−1 n and from (22.54), X is independent of Cn. Normal Modes. The normal modes Ψn correspond to the case where the arbitrary constant is such that the modal masses become unity. In this case, we can write Ψn = Φn √mn . (22.61) Part G 22.2
910 Part G Structural Acoustics and Noise Part G 22.2 Consequently, (22.57) becomes 〈Ψn, KΨn〉=ω 2 n . (22.62) 22.2.3 Admittances In general, a given structure vibrates as a result of the action of localized and distributed forces and moments. For both numerical and experimental reasons, one often has to work on a discrete representation of the structure. In practice, the geometry of the structure is represented by a mesh consisting of a number N of areas with dimensions smaller than the wavelengths under consideration. This results in having to consider the structure as an N-DOF system. At each point of the mesh, the motion is defined by three translation components plus three rotation components. Similarly, the action of the external medium on the structure can be reduced to three force components plus three moment components, here denoted by Fl. The translational and rotational velocity components Vk at each point are thus related to the forces and moments through a 6 × 6 admittance matrix, such that V = YF (22.63) For each force component at a given point j on the structure, denoted Fj|l, a motion at another point i, denoted by Vi|k, can be generated. As a consequence one can define the transfer admittance Yij|kl = Vi|k Fj|l with 1 ≤ i, j ≤ N and 1 ≤ k, l ≤ 6 . In summary, the transfer admittance matrix is defined by 6N ×6N coefficients such as Yij|kl. Notation. In what follows, the indices (k, l) are omitted, for the sake of simplicity. The transfer admittances are written as Yij, which reduces to Yii (or, even to Yi) in the case of a driving-point admittance. Attention here is mostly focused on translation, though the formalism remains valid for rotation. The previous results obtained on modal decomposition are now used to investigate the frequency dependence of the admittances. Here, X(xi) ≡ Xi is one component of displacement at point xi and F(x j) ≡ Fj is one force component at point x j. The structure is subjected to a forced motion at frequency ω.Wehave � K − ω 2 M � Xi = Fj , (22.64) where Xi = � Φn(xi)qn(t)and � ω 2 n − ω 2� qn = fn mn tΦn(x j)Fj(ω) = . (22.65) mn For a structure discretized on N points, the displacement at point xi is given by Xi(ω) = N� Φn(xi) tΦn(x j) � ω2 n − ω2� Fj(ω) . (22.66) n=1 mn The complete set of functions Xi(ω) givenin(22.66) is the operating deflexion shape (ODS) at frequency ω.The transfer admittance between points xi and x j is written Yij(ω) = iω N� Φn(xi) tΦn(x j) � ω2 n − ω2� . (22.67) n=1 mn The driving-point admittance at point xi becomes Yi(ω) = iω N� mn n=1 [Φn(xi)] 2 � ω 2 n − ω 2� (22.68) Remark. To include damping, (22.67) can be written Yij(ω) = iω N� mn n=1 Φn(xi) tΦn(x j) � ω2 n + 2iζnωnω − ω2� , (22.69) where ζn is a nondimensional damping factor. The physical origin of the damping terms will be presented in more detail in Sect. 22.6.1. Frequency Analysis and Approximations For weak damping, and low modal density, the modulus of Yij(ω) passes through maxima at frequencies close to the eigenfrequencies ω = ωn (Fig. 22.4). • For ω ≈ ωn, the main term in Yij(ω) is equal to iω Φn(xi) tΦn(x j) � ω2 n − ω2� . (22.70) mn • For ω ≫ ωn, the admittance becomes Yij ≈ iω � Φl(xi) tΦl(x j) −mlω2 ≈ 1 iMω l>n with 1 � Φl(xi) = M l>n tΦl(x j) ml (22.71) It can be seen that the modes with a rank larger than a given mode n play the role of a mass M.
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Springer Handbook of Acoustics
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Springer Handbook of Acoustics Thom
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Foreword The present handbook cover
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List of Authors Iskander Akhatov No
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Philippe Roux Université Joseph Fo
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XIV Contents 3.7 Attenuation of Sou
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XVI Contents 10 Concert Hall Acoust
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XVIII Contents 17.8 Physical Modeli
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XX Contents 24.5 Free-Field Microph
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XXII List of Abbreviations K KDP po
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Introduction 1. Introduction to Aco
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of noise have been the subject of c
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in order to understand how objectiv
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A Brief 2. A Brief History of Acous
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of 350 m/s for the speed of sound [
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number and relative strength of its
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To his contemporaries, Koenig was p
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Probably the most important use of
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piezoelectric ceramic compositions
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surveys together [2.46]. Masking of
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ther of computer music, since he de
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Basic 3. Linear Basic Linear Acoust
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M average molecular weight n unit v
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a) b) S v.n ∆t ∆S V |v| ∆t n
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temperature, q =−κ∇T , (3.16)
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The equivalent density M/V is conse
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Basic Linear Acoustics 3.3 Equation
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Because any vector field may be dec
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The latter and (3.84), in a manner
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assumed to have no ambient motion,
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Because the friction associated wit
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3.5 Waves of Constant Frequency One
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3.5.4 Time Averages of Products Whe
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as well as symmetry considerations,
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The auxiliary internal variables th
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Then the transient at a distant pos
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ω ′ = 0, and this is consistent
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1.0 10 -1 10 -2 10 -3 10 -4 10 -5 A
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This yields the interpretation that
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direction of propagation when a pla
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so the time-averaged incident energ
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Although the simple result of (3.31
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ut, also in keeping with the linear
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equation, the two differential equa
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Rodrigues relation, one has �1
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for the derivative.) In the asympto
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The differential scattering cross s
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elow) is F(t) = (2c) 1/2 �t −
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0.5 0 -0.5 -1.0 -1.5 0 Y0(η) (π/2
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is the Wronskian for the Bessel and
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The function qS is termed the sourc
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The appropriate identification for
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where jℓ is the spherical Bessel
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this subtlety taken into account, o
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U(x) Basic Linear Acoustics 3.15 Wa
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is ordinarily valid. With these ass
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along rays. These can be regarded a
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A(x0) x0 A(x) Fig. 3.53 Sketch of a
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possible, and one where neither the
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which is independent of the z-coord
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[3.108], and Carslaw [3.109]. In th
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where C(X)andS(X) are the Fresnel i
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Fig. 3.60 Characteristic diffractio
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of elastic solids, Trans. Camb. Phi
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3.101 J.B. Keller: Geometrical acou
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114 Part A Propagation of Sound Par
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Underwater 5. Underwater Acoustics
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5.1 Ocean Acoustic Environment The
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Long-Range Propagation Paths Figure
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tal direction, there is no loss ass
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equivalent circuit, as shown in Fig
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Attenuation á (dB/km) 1000 100 10
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5.2.5 Ambient Noise There are essen
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ubble natural acoustic resonance ω
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a bubbly medium (and for the simple
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As discussed, because detection inv
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(1/A)∇ 2 A ≪ K 2 )sothat(5.34)
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water, where the deep sound channel
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egion in the form p(r, z) = ψ(r, z
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5.4.6 Propagation and Transmission
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tegration interval. The source puls
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5.6 SONAR Array Processing Signal p
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former output is: �∞ b(θ,t) =
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Fig. 5.37 Angle-versus-time represe
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5.7 Active SONAR Processing Depth M
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a factor when there is a reverberan
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depth measurements that correspond
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along the cross-shelf track taken b
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time as a result of multiple paths,
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technique is very sensitive, and ex
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Fig. 5.57a,b Typical echogram obtai
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Frequency (Hz) 150 100 50 0 0 a) b)
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out. These data allow for study of
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5.59 G. Raleigh, J.M. Cioffi: Spati
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Physical 6. Physical Acou Acoustics
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where I is the intensity of the sou
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Wave velocity The wall exerts a dow
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destructively interfered with one a
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or: � � p2 � rms SPL = 10 log
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6.1.4 Wave Propagation in Solids Si
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where c is again the wave speed and
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measured. Some resonances are cause
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as 50 µmto5µm through one oscilla
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the number of false positives and m
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with the small mass), and frequency
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Laser Lens 1 Circular aperture Fig.
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Ground ring Insulator Sample Resist
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Energy ratio 1.0 0.8 0.6 0.4 Calcul
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terms. In this equation γ is the r
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directly. The nonlinearity paramete
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Thermoacoust 7. Thermoacoustics The
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Table 7.1 The acoustic-electric ana
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walls of the pores. (Positive G ind
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a) b) c) C reso d) δtherm e) p 0 U
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a) b) UA,l c) d) E 0 Q A Q A TA TA
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tor. This eliminates the need to le
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to this consumption of acoustic pow
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The traditional Stirling refrigerat
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Washington 1986) pp. 550-554, Softw
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258 Part B Physical and Nonlinear A
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Acoustics 9. Acoustics in Halls for
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parameters, so we can assist in bui
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weeks or even years is less reliabl
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9.3.1 Reverberation Time Reverberan
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selected). A distance different fro
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ing sound, as will naturally be exp
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of the sound fields. Consequently,
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e derived from interrupted noise de
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Acoustics in Halls for Speech and M
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espectively, can be calculated from
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ated by the architects. However, on
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of C and G. However, as all the ind
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F b d F n I S Fb Fn S d F n/F b d/d
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HS S èn ã d0 P0 rn Position of ey
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Time T (s) 2.5 2 1.5 1 5 10 15 Acou
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floor can be tilted to reduce volum
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ÄLcurv (dB) 10 8 6 4 2 0 -2 -4 -6
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Seating capacity 2662 1 915 + 324 2
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4 3 2 1 Acoustics in Halls for Spee
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cially in auditoria with T values l
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esonance (AR)andmultichannel reverb
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Concert 10. Concert Hall Acoustics
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Concert Hall Acoustics Based on Sub
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0 0 Concert Hall Acoustics Based on
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mately by Concert Hall Acoustics Ba
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10.1.5 Specialization of Cerebral H
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The other (n − 1) bits indicated
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As for the conflicting requirements
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have mainly been concerned with tem
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a) c) S S -4.3 -2.8 -2.0 -3.5 -3dB
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a model of the auditory-brain syste
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Building 11. Building Acou Acoustic
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Pressure Maximum Minimum 0 D Distan
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Table 11.1 Absorption coefficients
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Absorption coefficient α 1 0 Frequ
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is of key importance in rooms where
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Table 11.4 Transmission loss and ST
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TL of wall - (TL of door, window or
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Table 11.7 Generalized noise reduct
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Masking sound pressure level (dB) 5
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As for the room criterion (RC) meth
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11.5 Noise Control Methods for Buil
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Neoprene pads (30 durometer) with a
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Concrete Caulk around perimeter Vib
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Trim board to conceal gap (fasten o
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Gypsum board partition (as schedule
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Double-layer ribbed or waffle neopr
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11.6 Acoustical Privacy in Building
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Table 11.9 AI,SIIandPIforopenplanof
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Electronic sound masking in plenum
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E 1179 Standard Specification for S
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430 Part D Hearing and Signal Proce
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Psychoacoust 13. Psychoacoustics Ps
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Absolute threshold (dB SPL) 100 90
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the signal. By using this off-cente
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is as a crude indicator of the exci
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Relative response (dB) 100 90 80 70
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In a variation of this procedure, t
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Level of matching noise (dB) 100 90
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13.4 Temporal Processing in the Aud
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Most models include an initial stag
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The components were either uniforml
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(Frequency DL)/ERBN 0.2 0.1 0.05 0.
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elaborate place models have been pr
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13.6 Timbre Perception 13.6.1 Time-
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the duplex theory of sound localiza
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harmonic (by shifting the frequency
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sive. While some studies have been
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sentences (see Fig. 13.20). They va
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components is usually only perceive
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References 13.1 ISO 389-7: Acoustic
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13.74 D. Ronken: Monaural detection
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asymmetric function, J. Acoust. Soc
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13.211 J. Vliegen, A.J. Oxenham: Se
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534 Part E Music, Speech, Electroac
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The 16. The Human Human Voice in Sp
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uation, the effect of gravity is in
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piratory muscles (the internal inte
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Mean Ps (cm H2O) 50 40 30 20 10 0 D
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Glottal flow Closed Opening Open Cl
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Transglottal airflow (l/s) 0.4 0.2
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Mean spectrum level (dB) -30 -40 -5
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20 10 0 -10 -20 -30 -40 -50 0 i y u
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Mean level (dB) 0 -10 -20 -30 -40 1
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This topic was addressed by Ladefog
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Fig. 16.29 Acoustic consequences of
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Bass i e u Alto Tenor o ε œ æ Th
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100 80 60 40 20 0 -20 100 80 60 40
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tours for [� ]and[Ù ] sampled at
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Rapp [16.100] asked three native sp
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Utterance command T0 T3 Accent comm
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The # symbol indicates the possibil
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Second formant frequency (Hz) 2500
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tional rather than absolute (à la
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16.7 P. Ladefoged, M.H. Draper, D.
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esonance imaging: Vowels, J. Acoust
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16.139 A. Eriksson: Aspects of Swed
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Computer 17. Computer Mu Music This
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Quantum step Amplitude Quantization
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Some might notice that linear inter
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pers, textbooks, patents, etc. Furt
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0:00.0 0.5 0 Computer Music 17.3 Ad
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(0,0) (0,1) (1,1) (0,2) (1,2) (0,3)
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synthesizing vocoder. The main diff
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x(n) + z -1 z -1 Scattering junctio
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230 Hz FOF 1100 Hz FOF 1700 Hz FOF
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0 y + - Computer Music 17.8 Physica
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-1 (1-ß )×length Velocity + Veloc
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0 -30 (dB) -60 0 1.5 3 4.5 (kHz) Fi
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17.10 Composition The history of co
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and variance of the features can be
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17.20 J.L. Kelly Jr., C.C. Lochbaum
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Audio 18. Audio and Electroacoustic
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Alexander Graham Bell filed his pat
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on magnetic stripes. A common arran
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60 40 20 0 SPL-sound pressure level
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Interaural intensity difference (dB
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with equalization, without incurrin
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Fig. 18.8 Sine wave with crossover
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18.3.8 Dynamic Range Dynamic range
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Directly actuated type Diaphragm ty
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effective pickup pattern of the arr
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attempts to apply complementary com
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Most of the issues relating to spea
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nificant design considerations. At
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Some appreciation of the performanc
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pre-digital reverberators were elec
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and ‘s’ in English text - may b
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content of rest of the signal spect
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cross the head, in opposite directi
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18.2 J. Sterne: The Audible Past (D
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18.71 T. Sporer: Creating, assessin
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786 Part F Biological and Medical A
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Medical 21. Medical Acous Acoustics
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21.1 Introduction to Medical Acoust
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Auscultation location Right & left
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portance. The Strouhal number has b
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Fig. 21.3 Phono-cardiograph transdu
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Amplitude 8 6 4 2 0 -2 -4 -6 Displa
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λ1 = c1 / fus θ1 λ2 = c2 / fus
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lood flow can be resolved if the si
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vided in the image thickness direct
Page 856 and 857: not as predicted, because the wave
Page 858 and 859: Fig. 21.18a,b Quadrature Doppler de
Page 860 and 861: a) b) c) 100 0 V mV 10 0 a = 13 a =
Page 862 and 863: Ultrasound contact gel Position enc
Page 864 and 865: a) 10 cm b) c) 10 cm Fig. 21.32a-c
Page 866 and 867: a) Pin B Pin A Ultrasound line b) M
Page 868 and 869: Organ Patient Ultrasound transducer
Page 870 and 871: systems cannot tell the difference
Page 872 and 873: 40 µm can be resolved. For a 10 kH
Page 874 and 875: a) b) c) d) Medical Acoustics 21.4
Page 876 and 877: Fig. 21.54 Doppler spectral wavefor
Page 878 and 879: 10 20 30 10 20 30 Pre-exercise B-mo
Page 880 and 881: Vibrations in a punctured artery De
Page 882 and 883: the veins and diffuse into the inte
Page 884 and 885: 21.5.3 Agitated Saline and Patent F
Page 886 and 887: transported by those cells and/or r
Page 888 and 889: 1 0.8 0.6 0.4 0.2 0 Relative power
Page 890 and 891: High intensity focused ultrasound h
Page 892 and 893: a) b) c) Fig. 21.74a-c B-mode imagi
Page 894 and 895: provided simple metrics for the lik
Page 896 and 897: thickness of the carotid arteries,
Page 898 and 899: Structural 22. Structural Acoustics
Page 900 and 901: Structural Acoustics and Vibrations
Page 902 and 903: Magnitude (arb. units) 1.4 1.2 1.0
Page 904 and 905: This does not include the case wher
Page 908 and 909: Magnitude (dB) -10 -30 -50 0 1 2 3
Page 910 and 911: where D(ω) = ω 4 − 2iω 3�
Page 912 and 913: form y(x, t) = � Φn(x)qn(t) . (2
Page 914 and 915: first to solve the wave equation (2
Page 916 and 917: 5 3 1 -1 -3 -5 0 1 2 3 4 5 6 7 8 9
Page 918 and 919: conditions at each end) are necessa
Page 920 and 921: from which we can derive through (2
Page 922 and 923: In this case, the eigenfrequencies
Page 924 and 925: of radiation modes and their link w
Page 926 and 927: Finally, the displacement is ˜ξ(x
Page 928 and 929: notes the value of this eigenmode a
Page 930 and 931: Pm = ˙Q H [Rs + Ra] ˙Q , (22.262)
Page 932 and 933: where and Ra = 2ζaω0 M Z(s) = zL
Page 934 and 935: Radiated Power. The mean acoustic p
Page 936 and 937: (22.318) can be written ξ(x, y, t)
Page 938 and 939: Denoting the eigenfrequencies and e
Page 940 and 941: Magnitude (arb. units) 3 2 1 0 -1 -
Page 942 and 943: for any real symmetric tensor Xij.
Page 944 and 945: F(N) 1 0.5 0 -0.5 -1 -1 -0.8 -0.6 -
Page 946 and 947: whose solution is θ1 = A cos ωτ.
Page 948 and 949: 4.5 3.5 2.5 1.5 A 0.5 0 0.5 1.0 1.5
Page 950 and 951: Equations (22.423) are usually writ
Page 952 and 953: 3. As the amplitude reaches the top
Page 954 and 955: Shallow Spherical Shells and Plates
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22.20 L. Cremer, M. Heckl: Structur
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Noise Noise is discussed in terms o
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where the overbars represent time a
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Typical outdoor setting A-weighted
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90 dB(A)” is widely used, and imp
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Microphone, amplifier and A/D conve
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When each segment of the measuremen
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found from � LW = Lp + 10 log A +
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surement method is the ultimate use
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An intensity analyzer is more compl
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23.2.5 Criteria for Noise Emissions
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Table 23.5 Table of limit values fr
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a) Air flows freely to rotor Weak t
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Static efficiency normalized to its
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ful applications. Good results are
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Table 23.8 Crossover speeds for var
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Reduction of airplane engine noise
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A variety of materials are used to
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∆LF (dB) 0 -10 -20 α -30 0.05 0.
Page 994 and 995:
Absorption coefficient 1.0 0.8 0.6
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high enough that there is little so
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23.4.4 Criteria for Noise Immission
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Table 23.10 (cont.) Some features o
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Noise 23.4 Noise and the Receiver 1
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view of administrative procedures r
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provides recommendations for a foll
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sound power levels of noise sources
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23.68 ISO: ISO 9296:1988 Acoustics
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in impedance tubes - Part 2: Transf
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23.194 Commission of the European C
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1022 Part H Engineering Acoustics P
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Sound 25. Intens Sound Intensity So
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Under such conditions the intensity
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a) Pressure and particle velocity 0
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τ is a dummy time variable. The ca
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Error in intensity (dB) 5 0 -5 -10
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8 7 6 5 4 3 2 1 0 -1 -2 -3 Error du
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crophones are calibrated with a pis
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Sound power level (dB re 1 pW) 72 7
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it is relatively straightforward si
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intensity normal to the source. Thi
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25.9 W. Maysenhölder: The reactive
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25.77 ISO: ISO 11205 Acoustics - No
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Optical 27. Optical Methods Metho f
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course also present in ordinary hol
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Optical Methods for Acoustics and V
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50 100 150 200 250 300 350 400 450
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a) b) Optical Methods for Acoustics
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nm 1000 0 -1000 150 y (mm) 100 50 O
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Laser Optical Methods for Acoustics
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References 27.1 E.F.F. Chladni: Die
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27.75 E.-L.Johansson, L. Benckert,
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About the Authors Iskander Akhatov
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Neville H. Fletcher Chapter F.19 Au
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Brian C. J. Moore Chapter D.13 Univ
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Detailed Contents List of Abbreviat
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3.8.3 Acoustic Power ..............
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4.8.3 Typical Speed of Sound Profil
Page 1142 and 1143:
7.3 Engines .......................
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10 Concert Hall Acoustics Based on
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13.4 Temporal Processing in the Aud
Page 1148 and 1149:
15.2.5 String-Bridge-Body Coupling
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19.6 Birds ........................
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22.4.5 Combinations of Elementary S
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24.8 Overall View on Microphone Cal
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Subject Index 3 dB bandwidth 463 A
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British Medical Ultrasound Society
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electric circuit analogues - acoust
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hyperspeech 703 hypospeech 703 I IA
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- participation factors 909 - stiff
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- musical instruments 541 - musical
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- nonlinear vibrations 957 - plate
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subjective preference theory 353 su
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Introduction to Acoustics

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