THE SCIENCE AND APPLICATIONS OF ACOUSTICS - H. H. Arnold ...

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634 Appendix B. Bessel FunctionsB.4 Approximations for Small Argument x < 1J 0 → 1 − x 24J 1 → x 2 − x 3Y 0 → 2 π ln xY 1 →− 2 πB.5 Approximations for Large Arguments x > 2πH (1)nH (2)nB.6 Recursion Relations1x16√2J n →(xπ x cos − nπ 2 − π )4√2Y n →(xπ x sin − nπ 2 − π )4√2→π x ei(x− nπ 2 − π 4 )√2→π x e−i(x− nπ 2 − π 4 )The relations listed below hold true for C being any of the Bessel functions of thefirst, second, or third kind or for linear combinations of these functions.C n+1 + C n−1 = 2nx C nddxddxdC 0dx =−C 1dC ndx = 1 2 (C n−1 − C n+1 )(x n )C n = x n C n−1( ) 1=− 1 x n x C n n+1
B.8 Tables of Bessel Functions, Zeros, and Inflection Points 635B.7 Modified Bessel FunctionsI n (x) = i −n J n (ix)I 0 (x) = J 0 (ix) = 1 + x 22 + x 42 2 2 · 4 + x 62 2 2 · 4 2 · 6 +···∫2 xI 0 (x)dx = xI 1 (x)∫I 1 (x)dx = I 0 (x)I 0 (x) − I 2 (x) = 2 x I 1(x)B.8 Tables of Bessel Functions, Zeros, and Inflection PointsB.1. Bessel Functions or Order 0, 1, and 2x J 0 (x) J 1 (x) J 2 (x) I 0 (x) I 1 (x) I 2 (x)0.0 1.0000 0.0000 0.0000 1.0000 0.0000 0.00000.2 0.9900 0.0995 0.0050 0.0100 0.1005 0.00500.4 0.9604 0.1960 0.0197 0.0404 0.2040 0.02030.6 0.9120 0.2867 0.0437 1.0921 0.3137 0.04640.8 0.8463 0.3688 0.0758 1.1665 0.4329 0.08431.0 0.7652 0.4401 0.1149 1.2661 0.5652 0.13581.2 0.6711 0.4983 0.1593 1.3937 0.7147 0.20261.4 0.5669 0.5419 0.2074 1.5534 0.8861 0.28761.6 0.4554 0.5699 0.2570 1.7500 1.0848 0.39401.8 0.3400 0.5815 0.3061 1.9895 1.3172 0.52602.0 0.2239 0.5767 0.3528 2.2796 1.5906 0.68902.2 0.1104 0.5560 0.3951 2.6292 1.9141 0.88912.4 +0.0025 0.5202 0.4310 3.0492 2.2981 1.11112.6 −0.0968 0.4708 0.4590 3.5532 2.7554 1.43382.8 −0.1850 0.4097 0.4777 4.1575 3.3011 1.79943.0 −0.2601 0.3391 0.4861 4.8808 3.9534 2.24523.2 −0.3202 0.2613 0.4835 5.7472 4.7343 2.78843.4 −0.3643 0.1792 0.4697 6.7848 5.6701 3.44953.6 −0.3918 0.0955 0.4448 8.0278 6.7926 4.25383.8 −0.4026 +0.0128 0.4093 9.5169 8.1405 5.23234.0 −0.3971 −0.0660 0.3641 11.302 9.7594 6.42244.2 −0.3766 −0.1386 0.3105 13.443 11.705 7.86834.4 −0.3423 −0.2028 0.2501 16.010 14.046 9.62594.6 −0.2961 −0.2566 0.1846 19.097 16.863 11.7614.8 −0.2404 −0.2985 0.1161 22.794 20.253 14.3555.0 −0.1776 −0.3276 +0.0466 27.240 24.335 17.5055.2 −0.1103 −0.3432 −0.0217 32.584 29.254 21.3325.4 −0.0412 −0.3453 −0.0867 39.010 35.181 25.9805.6 +0.0270 −0.3343 −0.1464 46.738 42.327 31.6215.8 0.0917 −0.3110 −0.1989 56.039 50.945 38.472(Continued)
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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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574 19. Sound Reproductionon, machi
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576 19. Sound ReproductionIt is fai
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578 19. Sound ReproductionINNER SUS
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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
Page 623 and 624: 618 21. Nonlinear Acousticsby∇ 2
Page 625 and 626: 620 21. Nonlinear Acousticsmay be i
Page 627 and 628: 622 21. Nonlinear Acousticswhereδ
Page 629 and 630: 624 21. Nonlinear AcousticsThe shoc
Page 631 and 632: 626 21. Nonlinear AcousticsBut the
Page 633 and 634: Appendix APhysical Properties of Ma
Page 635 and 636: Appendix A. Physical Properties of
Page 637: Appendix BBessel FunctionsB.1 The B
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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