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21.3 Progressive Waves in Fluids 625Analytical studies have been made of cylindrical explosions and line sourceenergy releases (Lin, 1954; Plooster, 1970, 1971). Lin was working on the problemof shock generation by meteors or missiles, whereas Plooster was studying thedevelopment and decay of cylindrical shocks with more realistic initial energydistributions and atmospheres. A strong shock estimate of the decay of cylindricalwaves from an instantaneous line source in air (γ = 1.4) at standard pressure isgiven byp = 0.216Er 2Thus, a cylindrical decay proportional to r –2 is characteristic of cylindrical strongshocks.Nonlinearity in SolidsThe propagation of ultrasonic waves of finite amplitudes in a crystal of cubicsymmetry can be analyzed with a nonlinear differential equation similar to thatused for fluids.The propagation of a finite-amplitude ultrasonic wave in any direction in anisotropic solid is given by∂ 2 ξρ 0∂t = κ ∂ 2 ξ2 2∂a + (3κ 2 2 + k 3 ) ∂2 ξ ∂ξ(21.24)∂a 2 ∂awhere ξ is the particle displacement and a is the distance measured in the directionof propagation. κ 2 and κ 3 represent elastic constants that depend on the directionof propagation in the solid.The solution of the nonlinear equation (21.24) is made by assuming that the waveis initially sinusoidal at a = 0, with ξ = A 1 sin (ka – ωt). On this assumption wecan obtain a perturbation solution in the formξ = A 1 sin(ka − ωt) + β A2 2 k2 acos 2(ka − ωt) +···. (21.25)4where β is the nonlinear parameter. The negative ratio of the coefficient of thenonlinear term in Equation (21.24)2β =−(3 + κ 3k 2)is the quantity to be determined from measurement. From 2β we can establish κ 3since κ 2 = ρc0 2 is known.Equation (21.25) indicates that an initially sinusoidal ultrasonic wave generatesa second harmonic as it propagates. In order to measure the nonlinear parameter2β, the absolute value of the fundamental displacement amplitude A 1 needs to bemeasured and also that of the second harmonicA 2 = β A2 1 k2 a4
626 21. Nonlinear AcousticsBut the propagation constant k = 2 π/λ = ω/c0 2 is known and sample length a canbe measured. We can then write2β = 8A 2c 2 0A 2 1 ω2 aand determine 2β directly. A 2 is usually plotted as a function of A 2 1, and the slopeis then evaluated to yield 2β.ReferencesBeyer, Robert T. 1974. Nonlinear Acoustics. Washington, DC: Navy Sea Systems Command.(Reissued in 1997 by the Acoustical Society of America, with an appendix containingreferences to new developments.)Beyer, Robert T. (ed.). 1984. Nonlinear Acoustics in Fluids. New York, NY: Van Nostrand.(A collection of fundamental papers on nonlinear acoustics.)Blackstock, David T. 1964. On plane, spherical, and cylindrical sound waves of finiteamplitude in lossless fluids. Journal of the Acoustical Society of America 36:217–219.Blackstock, David T. 1985. Generalized Burgers equation for plane waves. Journal of theAcoustical Society of America 77:2050–2053.Blackstock, David T. 1972. Nonlinear acoustics (theoretical). In: Gray, D. E. (ed.).American Institute of Physics Handbook. New York, NY: McGraw-Hill, pp. 3-183–3-205.Burgers, J. M. 1948. A mathematical model illustrating the theory of turbulence. In: VonMises, R. and von Kàrmàn, T. (eds.). Advances in Applied Mechanics, Vol. I. New York,NY: Academic, pp. 171–191.Crocker, Malcolm F. (ed.). 1998. Part II: Nonlinear Acoustics and Cavitation. Handbookof Acoustics, pp. 175–253.Fay, R. D. 1931. Plane sound waves of finite amplitude. Journal of the Acoustical Societyof America 3:222–241.Fubini, E. 1935. Anomalies in the propagation of acoustic waves of great amplitude. AltaFrequenza 4:530–581 (in Italian).Hamilton, Mark F. and Blackstock, David T. 1988. On the coefficient of nonlinearity ofnonlinear acoustics. Journal of the Acoustical Society of America 83:1555–1561.Hopf, E. 1950. The partial differential equation u t + uu x = μu xx . Communications in Pureand Applied Mathematics 3:201–230.Landau, Lev D. 1945. On shock waves at large distances from the place of their origin.U.S.S.R. Journal of Physics 9:496–503.Lighthill, James. 1972. The propagation of sound through moving fluids. Journal of Soundand Vibration 24:471–492.Lighthill, James. 1978. Waves in Fluids. New York, NY: Cambridge University Press.(Classic text by a former holder of the Lucasian chair at Cambridge University oncehold by Isaac Newton and now occupied by Stephen Hawking.)Lin, S.-C. 1954. Cylindrical shock waves produced by instantaneous energy release.Journal of Applied Physics 25:54–57.Plooster, M. N. 1970. Shock waves from line sources. Numerical Solutions and experimentalmeasurements. Physics of Fluids 13:2665–2675.
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THE SCIENCE ANDAPPLICATIONS OFACOUS
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xPrefaceReferencesBacon, Sir Franci
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xiiContents16. Ultrasonics 44317. C
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2 1. A Capsule History of Acoustics
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1. A Capsule History of Acoustics 5
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12 1. A Capsule History of Acoustic
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14 2. Fundamentals of AcousticsFigu
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16 2. Fundamentals of Acoustics2.2
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18 2. Fundamentals of Acousticsof t
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20 2. Fundamentals of AcousticsQ z+
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22 2. Fundamentals of Acousticsin t
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24 2. Fundamentals of AcousticsWe c
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26 2. Fundamentals of Acoustics(2)
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28 2. Fundamentals of AcousticsEqua
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30 2. Fundamentals of Acoustics11.
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32 3. Sound Wave Propagation and Ch
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3.14 Performance Indices for Enviro
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3.14 Performance Indices for Enviro
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or in terms of complex exponential
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3.17 Sound Intensity 61Here the sur
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3.18 The Monopole Source 63The thre
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3.21 Energy Density 653.20 The Hemi
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References 67medium. The time avera
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Problems for Chapter 3 69(a) the wa
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4Vibrating Strings4.1 IntroductionI
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4.4 General Solution of the Wave Eq
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4.6 Simple Harmonic Solutions of th
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4.7 Standing Waves 77Figure 4.3. Di
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4.8 The Effect of Initial Condition
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4.10 Forced Vibrations in an Infini
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4.11 Strings of Finite Lengths: For
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References 854.12 Real Strings: Fre
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Problems for Chapter 4 878. A devic
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90 5. Vibrating BarsFigure 5.1. A b
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92 5. Vibrating BarsSettingc 2 = E/
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94 5. Vibrating BarsThe allowable f
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96 5. Vibrating Barsthe acceleratio
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98 5. Vibrating BarsFigure 5.5. Loc
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100 5. Vibrating Barsmore difficult
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102 5. Vibrating BarsFigure 5.7. An
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104 5. Vibrating Barsthe light beam
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106 5. Vibrating BarsFigure 5.8. Tr
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108 5. Vibrating BarsFigure 5.10. T
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110 5. Vibrating Barsof 0.005 m dia
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112 6. Membrane and PlatesFigure 6.
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114 6. Membrane and PlatesBecause t
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116 6. Membrane and PlatesThe left-
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118 6. Membrane and PlatesTable 6.1
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120 6. Membrane and PlatesIn real a
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122 6. Membrane and PlatesTable 6.2
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124 6. Membrane and PlatesAt low fr
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126 6. Membrane and PlatesThe elast
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128 6. Membrane and PlatesFor the f
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130 6. Membrane and Plates(b) Deter
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132 7. Pipes, Waveguides, and Reson
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152 8. Acoustic Analogs, Ducts, and
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9Sound-Measuring Instrumentation9.1
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9.3 Microphones 175Figure 9.1. A cu
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SolutionEquation (9.3) is applied t
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9.5 Selection and Positioning of Mi
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9.6 Vector Sound Intensity Probes 1
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9.8 Proper Procedures for Using the
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9.8 Proper Procedures for Using the
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Example Problem 39.10 Dosimeters 18
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9.11 Noise Measurement in Selected
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9.11 Noise Measurement in Selected
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9.12 Real Time Analysis 193analyzer
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9.13 Fast Fourier Transform Analysi
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9.14 Data Windows and Selection of
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9.16 Measurement Error 199whereβ =
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9.18 Measurement of Sound in a Free
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9.20 Sound Power Measurement in a D
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9.20 Sound Power Measurement in a D
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9.23 The Addition Method for Measur
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References 209acoustical output and
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Problems for Chapter 9 2112. Determ
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214 10. Physiology of Hearing and P
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11Acoustics of Enclosed Spaces:Arch
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11.2 Sound Fields 245Figure 11.1. P
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11.4 Sound Intensity Growth in a Li
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11.5 Sound Absorption Coefficients
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11.6 Growth of Sound with Absorbent
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11.7 Decay of Sound 253Following Sa
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11.8 Decay of Sound in Dead Rooms 2
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11.9 Reverberation as Affected by S
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11.13 Sound Levels due to Direct an
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11.15 Concert Halls and Opera House
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Figure 11.9. A view of the Boston S
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11.16 Band Shells and Outdoor Audit
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11.16 Band Shells and Outdoor Audit
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11.17 Subjective Preferences in Sou
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References 277preferred subsequent
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Problems for Chapter 11 279Siebein,
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12Walls, Enclosures, and Barriers12
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12.3 Mass Control Case 283Figure 12
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12.5 Effect of Frequencies on Sound
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12.6 Coincidence Effect and Critica
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12.7 The Double-Panel Partition 289
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an ideal gas) varies in the followi
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12.8 Measuring Transmission Loss 29
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12.10 Combined Sound Transmission C
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12.11 Noise Insulation Ratings 297F
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12.11 Noise Insulation Ratings 299s
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12.12 Noise Reduction of a Wall 301
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12.12 Noise Reduction of a Wall 303
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12.14 Enclosures 305to the extent t
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12.14 Enclosures 307and similarly f
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12.15 Small Enclosures 309walls. Ac
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12.16 Acoustic Barriers 311Figure 1
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12.16 Acoustic Barriers 313In the c
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Equation (12.67) becomes(11D = λ+
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Problems for Chapter 12 3173. A 0.7
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13Criteria and Regulations forNoise
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13.3 The Occupational Safety and He
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13.4 Perception of Noise 323inspect
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13.6 Indoor Noise Criteria 325accou
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13.6 Indoor Noise Criteria 327Room
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13.7 Equivalent Sound Level, Day-Ni
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13.7 Equivalent Sound Level, Day-Ni
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Composite Noise Rating13.9 Rating o
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13.10 Evaluation of Traffic Noise 3
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Highway Construction Noise13.10 Eva
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13.11 Evaluation of Community Noise
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13.12 Guidelines and Regulations in
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References 353Computer Program, HIC
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Problems for Chapter 13 355Problems
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14Machinery Noise Control14.1 Intro
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14.4 Fan or Blower Noise 359Table 1
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14.4 Fan or Blower Noise 361Figure
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14.6 Pumps and Plumbing Systems 367
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14.6 Pumps and Plumbing Systems 369
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14.8 Gears 371fromwheref BRC = N r
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14.8 Gears 373Figure 14.7. Gear noi
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14.8 Gears 375Figure 14.8. Gear tra
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14.8 Gears 377The subscripts 1 and
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14.10 Ball and Roller Bearings 379l
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14.10 Ball and Roller Bearings 381c
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14.11 Other Mechanical Drive Elemen
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Belt Drivesf CL = n 1N 1 2400 × 12
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14.12 Gas-Jet Noise 387said to be c
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14.12 Gas-Jet Noise 389power discha
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Solution14.12 Gas-Jet Noise 391Usin
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14.13 Gas Jet Noise Control 393Solu
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14.13 Gas Jet Noise Control 395Figu
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14.13 Gas Jet Noise Control 397Figu
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14.14 Mufflers and Silencers 399Fig
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(1 + m)A I 1=A 314.14 Mufflers and
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14.14 Mufflers and Silencers 403Let
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References 405Figure 14.21 illustra
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Problems for Chapter 14 407the nois
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15Underwater Acoustics15.1 Sound Pr
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15.3 Speed of Sound in Seawater 411
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15.4 Velocity Profiles in the Sea 4
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15.4 Velocity Profiles in the Sea 4
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15.5 Underwater Transmission Loss 4
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15.6 Parametric Variation of Absorp
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15.8 Underwater Refraction 421Figur
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15.9 Mixed Layer 423When G is const
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15.11 Sonar Transducers and Their P
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Example Problem 115.12 The Sonar Eq
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Active and Passive Equations15.12 T
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15.13 Noise, Echo, and Reverberatio
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15.13 Noise, Echo, and Reverberatio
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15.14 Transient Form of Sonar Equat
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Example Problem 315.16 Shortcomings
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15.17 Theoretical Target Strength o
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Problems for Chapter 15 441Wilson,
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444 16. Ultrasonicscatching small i
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446 16. UltrasonicsPlanck constant
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448 16. UltrasonicsEquation (16.5)
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450 16. UltrasonicsFigure 16.1. Sou
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452 16. Ultrasonicsthe positive hal
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454 16. Ultrasonicson each and ever
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456 16. Ultrasonicscomplete as a li
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458 16. Ultrasonicsinitial of their
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460 16. Ultrasonicsoptic axis, deno
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462 16. Ultrasonicstransducer, it h
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464 16. UltrasonicsTable 16.1. Valu
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466 16. Ultrasonicsthe mechanical r
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468 16. UltrasonicsThe Physics of M
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470 16. UltrasonicsFigure 16.7. The
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472 16. Ultrasonicscurvilinear arra
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474 16. Ultrasonics(a)amplitudeinpu
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476 16. Ultrasonicsslow to allow th
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478 16. Ultrasonics5. In the cases
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480 17. Commercial and Medical Ultr
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510 18. Music and Musical Instrumen
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Figure 18.7. The frequencies of the
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Figure 18.20. The violin octet deve
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570 19. Sound ReproductionAlbert, a
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572 19. Sound ReproductionD. Magnet
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Page 585 and 586: 580 19. Sound Reproductiontuned ape
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Page 589 and 590: 584 19. Sound ReproductionDickason,
Page 591 and 592: 586 20. Vibration and Vibration Con
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Page 627 and 628: 622 21. Nonlinear Acousticswhereδ
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Page 633 and 634: Appendix APhysical Properties of Ma
Page 635 and 636: Appendix A. Physical Properties of
Page 637 and 638: Appendix BBessel FunctionsB.1 The B
Page 639 and 640: B.8 Tables of Bessel Functions, Zer
Page 641 and 642: Appendix CUsing Laplace Transforms
Page 643 and 644: C.3 Solving Differential Equations
Page 645 and 646: C.4 Equations with Multiple-Order R
Page 647 and 648: SolutionC.5 Equations with Complex
Page 649 and 650: C.5 Equations with Complex Roots 64
Page 651 and 652: SolutionC.5 Equations with Complex
Page 653 and 654: 650 IndexAuditoriums, design of, 26
Page 655 and 656: 652 IndexElectrostatic speakers, 58
Page 657 and 658: 654 IndexKey notation, 518Kinetic e
Page 659 and 660: 656 IndexPascal (unit), 48Percent i
Page 661 and 662: 658 IndexSound intensity growth in
Page 663: 660 IndexWater-hammer arresters, 37
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