Introduction to Acoustics

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642 Part E Music, Speech, Electroacoustics Part E 15.4 and Rossing ([15.5], Sect. 21.15). Inscriptions on the set of 65 tuned chime bells in the tomb of Zeng Hou Yi (433 BC), show that the Chinese had already established a 12-note scale, closely related to our present, but much later, western scale. The ceremonial use of bells and gongs is widespread in religious cultures all over the world and in western countries has the traditional use of summoning worshippers to church and accompanying the dead to their graves. In the 18th century classical symphony orchestra of Haydn and Mozart’s time, the timpani helped emphasise the beat and pitch of the music being played, particularly in loud sections, with the occasional use of cymbals and triangle to emphasise exotic and often Turkish influences. The percussion section of today’s symphony orchestra may well be required to play up to 100 different instruments for a single piece, as in Notations I–IV by Boulez [15.203]. This typical modern score includes timpani, gongs, bells, metals and glass chimes, claves, wooden blocks, cowbells, tom-toms, marimbas, glockenspiels, xylophones, vibraphones, sizzle and suspended cymbals, tablas, timbales, metal blocks, log drums, boobams, bell plates in C and B flat, side drums, Chinese cymbals, triangles, maracas, a bell tree, etc. Modern composers can include almost anything that makes a noise – everything from typewriters to vacuum cleaners. The percussion section of the orchestra is required to play them all – often simultaneously. We will necessarily have to be selective in the instruments that we choose to consider and will concentrate largely on the more familiar instruments of the modern classical symphony orchestra. We will also constrict our attention to instruments that are struck and will ignore instruments like whistles, rattles, scrapers, whips and other similar instruments that percussion players are also often required to play. 01 11 21 02 31 12 1 1.59 2.14 2.30 2.65 2.92 41 22 03 51 32 61 3.16 3.50 3.60 3.65 4.06 4.15 Fig. 15.114 The first 12 modes of a circular membrane illustrating the mode nomenclature, nodal lines and frequencies relative to the fundamental 01 mode 15.4.1 Membranes Circular Membrane A uniform membrane with areal density σ, stretched with uniform tension T over a rigid circular supporting frame, supports acoustically important transverse displacements z perpendicular to the surface described by the wave equation � ∂2z 1 ∂z T + ∂r2 r ∂r + 1 r 2 ∂2z ∂φ2 � = σ ∂2z , (15.147) ∂t2 which has solutions of the form � � z (r,φ,t) = Jm(kmnr) A cos mφ +B sin mφ e iωt . (15.148) Jm(kmnr) are Bessel functions of order m, where n denotes the number of radial nodes and m the number of nodal diameters. The eigenfrequencies ωmn = kmn √ T/σ are determined by the requirement that Jm(kmna) = 0 on the boundary at r = a. The frequency of the fundamental (01) mode is (2.405/2πa) √ T/σ, where J0(k01a) = 0. The relative frequencies of the first 12 modes and associated nodal lines and circles are showninFig.15.114. For ideal circular symmetry, the independent azimuthal cosine and sine solutions result in degenerate modes having the same resonant frequencies. The degeneracy of such modes can be lifted by a nonuniform tension, variations in thickness when calfskin is used, and a lack of ideal circularity of the supporting rim. Any resulting splitting of the frequencies of such modes can result in beats between the sound of given partials, which the player can eliminate by selectively adjusting the tension around the perimeter of the membrane or by hitting the membrane at a nodal position of one of the contributing modes. Unlike transverse waves on a stretched string, the modes of a circular membrane are inharmonic. As a consequence, the waveforms formed by the combination of such partials excited when the drumhead is struck are non-repetitive. Audio illustrates the frequencies of the first 12 modes played in sequence. illustrates their sound when played together as a damped chord, which already produces the realistic sound of a typical drum, having a reasonably well-defined sense of pitch, despite the inharmonicity of the modes. Although percussion instruments may often lack a particularly well-defined sense of pitch, one can nev-
ertheless describe the overall pitch as being high or low. For example, a side drum has a higher overall pitch than a bass drum and a triangle higher than a large gong. From an acoustical point of view, we will be particularly interested in the lower-frequency quasi-harmonic modes. However, one must never forget the importance of the higher-frequency inharmonic modes in defining the initial transient, which is very important in characterising the sound of an instrument. Nonlinear effects arise in drums in much the same wayasinstrings(Sect.15.2.2). The increase in tension, proportional to the square of the vibrational amplitude, leads to an increase in modal frequencies. In addition, nonlinearity can result in mode conversion (Sect. 15.2.2) and the transfer of energy from initially excited lower-frequency modes with large amplitudes to higher partials. Although the pitch of a drum is raised when strongly hit, this may to some extent be compensated by the psychoacoustic effect of a low-frequency note sounding flatter as its intensity increased. Changes in perceived pitch of a drum with time can often be emphasised by the use of digital processing, to increase the frequency of the recorded playback without changing the overall envelope in time (audio ). Air Loading and Radiation The above description of the vibrational states of a membrane neglects the induced motion of the air on either Musical Acoustics 15.4 Percussion Instruments 643 Table 15.7 Ideal and measured frequencies of the modal frequencies of a timpani drum head normalised to the acoustically important (11) mode before and after mounting on the kettle, and the internal air resonances of the kettle. The arrows indicate the sense of the most significant perturbations of drum head frequencies and the asterisks indicate the resulting quasi-harmonic set of acoustically important modes (adapted from Fletcher and Rossing) Mode Ideal membrane Drumhead in air Coupled internal air resonances side of the drum skin. At low frequencies, this adds amass≈ 8 3 ρa3 to the membrane (Fletcher and Rossing [15.5], Sect. 18.1.2), approximating to a cylinder of air with the same thickness as the radius a of the drum head. The added mass lowers the vibrational frequencies relative to those of an ideal membrane vibrating in vacuum. The effect is largest at low frequencies, when the wavelength in air is larger than or comparable with the size of the drumhead. For higher-frequency modes, with a number of wavelengths λ across the width of the drumhead, the induced air motion only extends a distance ≈ λ2π(≪ a) from the membrane. Air loading therefore leaves the higher-frequency modes relatively unperturbed. Drums can have a single drum skin stretched over a hollow body, such as the kettle drum of the timpani, or two drum heads on either side of a supporting cylinder or hollowed out block of wood, like the side drum and southern Indian mrdanga (Fletcher and Rossing [15.5], Sect. 18.5). By stretching the drum head over a hollow body, the sound radiated from the back surface is eliminated, just like mounting a loudspeaker cone in an enclosure. At low frequencies, the drum then acts as a monopole source with isotropic and significantly enhanced radiation efficiency. This is illustrated by the much reduced 60 dB decay time of the (11) dipole mode of a stretched timpani skin, when the drum skin was attached to the kettle – Drumhead on kettle 01 0.63 82 Hz 0.53 (0,1,0) 385 Hz 127 Hz 0.85 ↑ (0,1,1) 11 1.00 160 1.0 (1,1,0) 337 Hz 150 1.00 ↓ *** (1,1,1) 566 Hz 21 1.34 229 1.48 (2,1,0) 537 Hz 227 1.51 ↓ *** (2,1,1) 747 Hz 02 1.44 241 1.55 (0,1,0) (0,2,0) 252 1.68 ↑ 31 1.66 297 1.92 (3,1,0) (3,1,1) 298 1.99 *** 12 1.83 323 2.08 (1,2,0) (1,2,1) 314 2.09 ↓ 41 1.98 366 2.36 366 2.44 *** 22 2.20 402 2.59 401 2.67 03 2.26 407 2.63 (0,1,0) 418 2.79 ↑ 51 2.29 431 2.78 434 2.89 32 2.55 479 3.09 448 2.99 *** 61 2.61 484 3.12 462 3.08 Part E 15.4
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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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Page 596 and 597: 592 Part E Music, Speech, Electroac
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Page 690 and 691: This topic was addressed by Ladefog
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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.
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
Page 1054 and 1055:
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
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
Page 1160 and 1161:
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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