Introduction to Acoustics

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Pm = ˙Q H [Rs + Ra] ˙Q , (22.262) where ˙Q H is the Hermitian conjugate (conjugate transpose) of ˙Q. Generalization The mean power for a multiple-DOF system is written: Pm(T) = �T � 1 �n (ri +rai) ˙q T 2 i + n� n� � γij ˙qi ˙q j dt 0 i=1 with γij =−miCij . i=1 j=1 j�=i (22.263) This mean power can be written in the same form as (22.262), where the resistance matrix is defined as ⎛ ⎞ r1 +ra1 ... γ1i ... γ1j ... γ1n ⎜ ⎟ ⎜ ... ... ... ... ... ... ... ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ . ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ γi1 ... ri +rai ... γij ... γin ... ... ... ... ... ... ... γ j1 ... γji ... r j +raj ... γjn ... ... ... ... ... ... ... γn1 ... γni ... γnj ... rn +ran (22.264) Remark. The resistance matrix can again be viewed as the sum of a structural resistance matrix Rs and an acoustical resistance matrix Ra. This leads to the expression for the mean acoustic power Pa = ˙Q H Ra ˙Q (22.265) and acoustical efficiency ηm = ˙Q H [Ra] ˙Q ˙Q H [Rs + Ra] ˙Q (22.266) which generalizes (22.32). Note that we have assumed that the structural resistance matrix Rs is diagonal, which is usually a reasonable assumption for lightly damped structures. However, strong structural damping can also be the source of intermodal coupling. In this case, Rs is no longer diagonal, but the general results expressed in (22.262)and(22.266) remain valid. A comparison of efficiencies between the cases of light and heavy fluids, respectively has been conducted by Rumerman [22.30]. Structural Acoustics and Vibrations 22.5 Structural–Acoustic Coupling 933 Radiation Filter Because Ra is real, symmetric and positive definite, we can write this matrix in the form Ra = P t �P , (22.267) where � is a diagonal matrix [22.25]. As a consequence, the acoustic power becomes, removing for simplicity the integration time T, where Pa = b H �b b = P ˙Q . (22.268) This can be written explicitly as Pa = � Ωn|bn| 2 . (22.269) n Equation (22.269) shows that, defining the appropriate basis, the acoustic power can be expressed as a sum of quadratic terms, thus removing the cross products between the qi in the previous subsections. Another interesting consequence of the properties of Ra is that the acoustic power can be decomposed using the Cholesky method. This leads to the expression: Pa = ˙Q H Ra ˙Q = ˙Q H G H G ˙Q = z H z = � |zn| 2 , (22.270) where, by comparison with (22.269), the vector z(ω) can be viewed as the output of a set of radiation filters whose transfer functions G(ω)aregivenby G(ω) = � �(ω)P(ω) (22.271) with input vector ˙Q,sothat z = G ˙Q . (22.272) Impulsively Excited Structure: Total Radiated Energy For an impulsively excited structure, the total radiated energy is given by: �∞ ET = ˙Q H Ra ˙Q �∞ dω = z H (ω)z(ω)dω (22.273) 0 which, by Parseval’s theorem, is equivalent to [22.31] �∞ ET = z t (t)z(t)dt . (22.274) 0 0 n Part G 22.5
934 Part G Structural Acoustics and Noise Part G 22.5 State-Space Analysis The interest of formulating structural acoustic coupling in terms of state-space variables is now emphasized using an example. Denoting by r the internal state of filter G with input X (or ˙Q) and output z, we can use a state-space realization of the form: ⎧ ⎨ ˙r = AGr + BG X , (22.275) ⎩z = CGr + DGX . Combining these equations with the equations of motion of the structure gives � � � �� ˙X A 0 X B = + F (22.276) ˙r BG AG r 0 with the output � � z = DG CG � � X . (22.277) r If, for example, the purpose is to minimize the total energy radiated by a source, then the cost function can be defined as C f = � ∞ 0 zt (t)z(t)dt . (22.278) max{ET} 22.5.3 Oscillator Coupled to a Tube of Finite Length Presentation of the Model The purpose of this section is to present the fundamental concepts for a vibrating structure coupled to an acoustic cavity. As a simple example, we consider a single-DOF oscillator coupled to a 1-D tube of cross-sectional area S and finite length L loaded at its end x = L by an impedance ZL, defined here as the ratio between the pressure and acoustic velocity. The selected structure is a mechanical oscillator of mass M, stiffness K = Mω 2 0 and dashpot R = 2Mζ0ω0 driven by a force T(0, t) at T K R M ZL 0 L x Fig. 22.15 Single-DOF mechanical oscillator coupled to a finite loaded tube position x = 0(Fig.22.15). The amplitude of the motion of the mass is assumed to be small, so that the acoustic velocity v(0, t) is equal to the mechanical velocity ˙ξ(t). We assume lossless wave propagation in the tube itself. The set of equations for the model is written: ⎧ 1 c ⎪⎨ ⎪⎩ 2 ∂2 p ∂t2 = ∂2 p ∂x2 for 0 < x < L ; ρ ∂v ∂t =−∂p ∂x p(L, s) = ZLv(L, s) ; v(0, t) = ˙ξ(t) M � ¨ξ + 2ζ0ω0 ˙ξ + ω2 0ξ� =−Sp(0, t) + T(t) (22.279) Mass Displacement Using the Laplace transforms, we obtain ⎧ p(x, s) = exp ⎪⎨ ⎪⎩ � − sx � � � sxc c F(s) + exp G(s) , v(x, s) = 1 � ρc exp � − sx � c F(s)− exp � � � sx c G(s) , (22.280) where F(s)andG(s) are two functions to be determined. The boundary condition at x = L gives G(s) = F(s) ZL − ρc ZL + ρc exp � − 2Ls� . (22.281) c The continuity of the displacement at position x = 0 yields: v(0, s) = 1 ρc F(s) � 1 − ZL − ρc ZL + ρc exp � − 2Ls �� c = sξ(s) . (22.282) Finally, the equation governing the oscillator motion becomes � 2 s + 2ζ0ω0s + ω 2� 0 ξ(s) with = T(s) M zL = ZL − s Ra M zL + tanh �sL � c 1 + zL tanh �sL c � ξ(s) (22.283) and Ra = ρcS . (22.284) ρc Equation (22.283) shows that, due to the loading of the oscillator by the finite tube, the damping term 2ζ0ω0 becomes: 2ζ0ω0 + 2ζaω0Z(s) , (22.285)
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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
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not as predicted, because the wave
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Fig. 21.18a,b Quadrature Doppler de
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a) b) c) 100 0 V mV 10 0 a = 13 a =
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Ultrasound contact gel Position enc
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a) 10 cm b) c) 10 cm Fig. 21.32a-c
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a) Pin B Pin A Ultrasound line b) M
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Organ Patient Ultrasound transducer
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systems cannot tell the difference
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40 µm can be resolved. For a 10 kH
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a) b) c) d) Medical Acoustics 21.4
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Fig. 21.54 Doppler spectral wavefor
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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 906 and 907: the mass matrix is taken to be cons
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 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
Page 956 and 957: 22.20 L. Cremer, M. Heckl: Structur
Page 958 and 959: Noise Noise is discussed in terms o
Page 960 and 961: where the overbars represent time a
Page 962 and 963: Typical outdoor setting A-weighted
Page 964 and 965: 90 dB(A)” is widely used, and imp
Page 966 and 967: Microphone, amplifier and A/D conve
Page 968 and 969: When each segment of the measuremen
Page 970 and 971: found from � LW = Lp + 10 log A +
Page 972 and 973: surement method is the ultimate use
Page 974 and 975: An intensity analyzer is more compl
Page 976 and 977: 23.2.5 Criteria for Noise Emissions
Page 978 and 979: Table 23.5 Table of limit values fr
Page 980 and 981:
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
Page 988 and 989:
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
Page 996 and 997:
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
Page 1008 and 1009:
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
Page 1052 and 1053:
τ is a dummy time variable. The ca
Page 1054 and 1055:
Error in intensity (dB) 5 0 -5 -10
Page 1056 and 1057:
8 7 6 5 4 3 2 1 0 -1 -2 -3 Error du
Page 1058 and 1059:
crophones are calibrated with a pis
Page 1060 and 1061:
Sound power level (dB re 1 pW) 72 7
Page 1062 and 1063:
it is relatively straightforward si
Page 1064 and 1065:
intensity normal to the source. Thi
Page 1066 and 1067:
25.9 W. Maysenhölder: The reactive
Page 1068 and 1069:
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
Page 1096 and 1097:
Optical Methods for Acoustics and V
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Page 1102 and 1103:
50 100 150 200 250 300 350 400 450
Page 1104 and 1105:
a) b) Optical Methods for Acoustics
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nm 1000 0 -1000 150 y (mm) 100 50 O
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Page 1110 and 1111:
Page 1112 and 1113:
Laser Optical Methods for Acoustics
Page 1114 and 1115:
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
Page 1132 and 1133:
Neville H. Fletcher Chapter F.19 Au
Page 1134 and 1135:
Brian C. J. Moore Chapter D.13 Univ
Page 1136 and 1137:
Detailed Contents List of Abbreviat
Page 1138 and 1139:
3.8.3 Acoustic Power ..............
Page 1140 and 1141:
4.8.3 Typical Speed of Sound Profil
Page 1142 and 1143:
7.3 Engines .......................
Page 1144 and 1145:
10 Concert Hall Acoustics Based on
Page 1146 and 1147:
13.4 Temporal Processing in the Aud
Page 1148 and 1149:
15.2.5 String-Bridge-Body Coupling
Page 1150 and 1151:
19.6 Birds ........................
Page 1152 and 1153:
22.4.5 Combinations of Elementary S
Page 1154 and 1155:
24.8 Overall View on Microphone Cal
Page 1156 and 1157:
Subject Index 3 dB bandwidth 463 A
Page 1158 and 1159:
British Medical Ultrasound Society
Page 1160 and 1161:
electric circuit analogues - acoust
Page 1162 and 1163:
hyperspeech 703 hypospeech 703 I IA
Page 1164 and 1165:
- participation factors 909 - stiff
Page 1166 and 1167:
- musical instruments 541 - musical
Page 1168 and 1169:
- nonlinear vibrations 957 - plate
Page 1170 and 1171:
subjective preference theory 353 su
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Introduction to Acoustics

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