Introduction to Acoustics

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504 Part D Hearing and Signal Processing Part D 14.1 14.8 The Cepstrum ...................................... 517 14.9 Noise.................................................. 518 14.9.1 Thermal Noise........................... 518 14.9.2 Gaussian Noise ......................... 519 14.9.3 Band-Limited Noise .................. 519 14.9.4 Generating Noise ...................... 519 14.9.5 Equal-Amplitude Random-Phase Noise ....................................... 520 14.9.6 Noise Color ............................... 520 14.10 Sampled data...................................... 14.10.1 Quantization and Quantization 520 Noise ....................................... 520 14.10.2 Binary Representation ............... 520 14.10.3 Sampling Operation................... 521 14.10.4 Digital-to-Analog Conversion ..... 521 14.1 Definitions Signal processing begins with signals. The simplest signal is a sine wave with a single spectral component, i. e., with a single frequency, as shown in Fig. 14.1.Itissometimes called a pure tone. A sine wave function of time t with amplitude C, angular frequency ω, andstarting phase ϕ,isgivenby x(t) = C sin(ωt + ϕ) . (14.1) The amplitude has the same units as the waveform x,the angular frequency has units of radians per second, and the phase has units of radians. Because there are 2π radians in one cycle ω = 2π f, (14.2) and (14.1) can be written as or as x(t) = C sin(2π ft + ϕ) (14.3) x(t) = C sin(2πt/T + ϕ) , (14.4) where f is the frequency in cycles per second (or Hertz) and T is the period in units of seconds per cycle, T = 1/ f . A complex wave is the sum of two or more sine waves, each with its own amplitude, frequency, and phase. For example, x(t) = C1 sin(ω1t + ϕ1) + C2 sin(ω2t + ϕ2) (14.5) 14.10.5 The Sampled Signal ................... 522 14.10.6 Interpolation ............................ 522 14.11 Discrete Fourier Transform ................... 522 14.11.1 Interpolation for the Spectrum ... 523 14.12 The z-Transform.................................. 524 14.12.1 Transfer Function ...................... 525 14.13 Maximum Length Sequences ................ 526 14.13.1 The MLS as a Signal.................... 527 14.13.2 Application of the MLS ............... 527 14.13.3 Long Sequences ........................ 527 14.14 Information Theory.............................. 528 14.14.1 Shannon Entropy ...................... 529 14.14.2 Mutual Information ................... 530 References .................................................. 530 x(t) C 0 –C T T T Time Fig. 14.1 A sine wave with amplitude C and period T. A little more than three and a half cycles are shown. The starting phase is ϕ = 0 is a complex wave with two spectral components having frequencies f1 and f2. The period of a complex wave is the reciprocal of the greatest common divisor of f1 and f2. For instance, if f1 = 400 Hz and f2 = 600 Hz, then the period is 1/(200 Hz) or 5 ms. The fundamental frequency is the reciprocal of the period. A general waveform can be written as a sum of N components, N� x(t) = Cn sin(ωnt + ϕn) , (14.6) n=1 and the fundamental frequency is the greatest common divisor of the set of frequencies { fn}. An alternative description of the general waveform can be derived by using the trigonometric identity sin(θ1 + θ2) = sin θ1 cos θ2 + sin θ2 cos θ1 (14.7)
so that x(t) = N� An cos(ωnt) + Bn sin(ωnt) , (14.8) n=1 where An = Cn sin ϕn, andBn = Cn cos ϕn, arethecosine and sine partial amplitudes respectively. Thus the two parameters Cn and ϕn are replaced by two other parameters An and Bn. Because of the trigonometric identity sin 2 θ + cos 2 θ = 1 , (14.9) the amplitude Cn can be written in terms of the partial amplitudes, C 2 n = A2n + B2 n , (14.10) as can the component phase ϕn = Arg(An, Bn) . (14.11) 14.2 Fourier Series The Fourier series applies to a function x(t) thatis periodic. Periodicity means that we can add any integral multiple m of T to the running time variable t and the function will have the same value as at time t, i. e. x(t + mT) = x(t) , for all integral m . (14.13) Because m can be either positive or negative and as large as we like, it is clear that x is periodic into the infinite future and past. Then Fourier’s theorem says that x can be represented as a Fourier series like ∞� x(t) = A0 + [An cos(ωnt) + Bn sin(ωnt)] . (14.14) n=1 All the cosines and sines have angular frequencies ωn that are harmonics, i. e., they are integral multiples of a fundamental angular frequency ωo, ωn = nωo = 2πn/T , (14.15) where n is an integer. The fundamental frequency f0 is given by f0 = ωo/(2π). The fundamental frequency is the lowest frequency that a sine or cosine wave can have and still fit exactly into one period of the function x(t) because f0 = 1/T. In order to make a function x(t) with period T, the only sines and cosines that are allowed to enter the sum are those that fit exactly into the same period T. These are those sines and cosines with frequencies that are integral multiples of the fundamental. Acoustic Signal Processing 14.2 Fourier Series 505 The Arg function is essentially an inverse tangent, but because the principal value of the arctangent function only runs from −π/2toπ/2, an adjustment needs to be made when Bn is negative. In the end, and Arg(An, Bn) = arctan(An/Bn) (for Bn ≥ 0) (14.12) Arg(An, Bn) = arctan(An/Bn) + π (for Bn < 0) . The remaining sections of this chapter provide a brief treatment of real signals x(t) – first as continuous functions of time and then as sampled data. Readers who are less familiar with the continuous approach may wish to refer to the more extensive treatment in [14.1]. The factors An and Bn in (14.14) are the Fourier coefficients. They can be calculated by projecting the function x(t) onto sine and cosine functions of the harmonic frequencies ωn. Projecting means to integrate the product of x(t) and a sine or cosine function over a duration of time equal to a period of x(t). Sines and cosines with different harmonic frequencies are orthogonal over a period. Consequently, projecting x(t) onto, for example cos(3ωot), gives exactly the Fourier coefficient A3. It does not matter which time interval is used for integration, as long as it is exactly one period in duration. Itiscommontousetheinterval−T/2toT/2. The orthogonality and normality of the sine and cosine functions are described by the following equations: 2 T �T/2 −T/2 for all m and n; and 2 T 2 T �T/2 −T/2 �T/2 −T/2 dt sin(ωnt) cos(ωmt) = 0 , (14.16) dt cos(ωnt) cos(ωmt) = δn,m (14.17) dt sin(ωnt) sin(ωmt) = δn,m , (14.18) Part D 14.2
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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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Page 460 and 461: 452 Part D Hearing and Signal Proce
Page 466 and 467: Psychoacoust 13. Psychoacoustics Ps
Page 468 and 469: Absolute threshold (dB SPL) 100 90
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Page 480 and 481: 13.4 Temporal Processing in the Aud
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Page 488 and 489: elaborate place models have been pr
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Page 502 and 503: References 13.1 ISO 389-7: Acoustic
Page 504 and 505: 13.74 D. Ronken: Monaural detection
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Page 508 and 509: 13.211 J. Vliegen, A.J. Oxenham: Se
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556 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
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Vibrations in a punctured artery De
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the veins and diffuse into the inte
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21.5.3 Agitated Saline and Patent F
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transported by those cells and/or r
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1 0.8 0.6 0.4 0.2 0 Relative power
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High intensity focused ultrasound h
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a) b) c) Fig. 21.74a-c B-mode imagi
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provided simple metrics for the lik
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thickness of the carotid arteries,
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Structural 22. Structural Acoustics
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Structural Acoustics and Vibrations
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Magnitude (arb. units) 1.4 1.2 1.0
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This does not include the case wher
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the mass matrix is taken to be cons
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Magnitude (dB) -10 -30 -50 0 1 2 3
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where D(ω) = ω 4 − 2iω 3�
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form y(x, t) = � Φn(x)qn(t) . (2
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first to solve the wave equation (2
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5 3 1 -1 -3 -5 0 1 2 3 4 5 6 7 8 9
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conditions at each end) are necessa
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from which we can derive through (2
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In this case, the eigenfrequencies
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of radiation modes and their link w
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Finally, the displacement is ˜ξ(x
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notes the value of this eigenmode a
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Pm = ˙Q H [Rs + Ra] ˙Q , (22.262)
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where and Ra = 2ζaω0 M Z(s) = zL
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Radiated Power. The mean acoustic p
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(22.318) can be written ξ(x, y, t)
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Denoting the eigenfrequencies and e
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Magnitude (arb. units) 3 2 1 0 -1 -
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for any real symmetric tensor Xij.
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F(N) 1 0.5 0 -0.5 -1 -1 -0.8 -0.6 -
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whose solution is θ1 = A cos ωτ.
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4.5 3.5 2.5 1.5 A 0.5 0 0.5 1.0 1.5
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Equations (22.423) are usually writ
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3. As the amplitude reaches the top
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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.
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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
Page 1052 and 1053:
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
Page 1060 and 1061:
Sound power level (dB re 1 pW) 72 7
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it is relatively straightforward si
Page 1064 and 1065:
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
Page 1134 and 1135:
Brian C. J. Moore Chapter D.13 Univ
Page 1136 and 1137:
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
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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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