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

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5.5 Mass Concentrated Bars 97Figure 5.4. Plot of 0.2 × kL (Curve a) and tan kL (Curve b) vs. kL. The intersections of the two sets of curves as shownabove provide the first five solutions kn L(kL = 0 being trivial) of the transcendental equation (5.20) for the fundamentaland overtones of a vibrating bar with 20% concentrated loading at location x = L. The fundamental occurs at k1L = 1.571,the first overtone at k2L = 3.792, the second overtone at k3L = 4.712; and neither of the latter two are harmonics of thefundamental.
98 5. Vibrating BarsFigure 5.5. Location of node in a free mass-loaded bar and fundamental mode of vibration.But overtones corresponding to the higher frequencies are not harmonics of thefundamental. In the example of Figure 5.4, in which the plots of (m/m b )kL and tankL vs. kL yield intersections which constitute the solutions to Equation (5.20), theratio of the first overtone to the fundamental is 3.792/1.571 = 2.414, not the valueof 2.0 that would have specified the overtone to be a harmonic of the fundamental.Solutions exist where the occurrence of non-harmonics overtones could be useful:for example, in a properly mass-loaded loudspeaker, a pure monofrequency inputwould not result in harmonics that could arise from the driving signal, if at all.The nodes of the vibration in the bar exist at locations where cos kx = 0. Thefundamental mode where kL = 3.792 engenders a node at 3.792x/L = π/2, orx = 0.414L, not x = 0.5L that would normally occur in a second harmonic. Incontrast of the free–free bar, the node in a free mass-loaded bar is no longerat the center—it shifts toward the loading mass as shown in Figure 5.5. In thisparticular case, the bar could be supported at this nodal position without affectingthe fundamental mode of vibration.5.6 General Boundary Conditions for a FreelyVibrating BarConsider a freely vibrating bar with arbitrary loadings at each end. We shall establishthe normal modes of vibration in terms of the mechanical impedances at bothends of the bar. Designating the mechanical impedance of the support at x = 0byZ mi0 we express the force acting at this support due to the bar asf 0 =−Z mi0 u(0, t)Here the minus sign is introduced to indicate that a positive (compressive) forcegenerates an acceleration of the support in the negative x-direction. But a positivecompressive force at the end x = L causes an acceleration of the other support tothe right, in the positive x-direction; the force acting on this support isf L =+Z miL u(L, t)where Z miL represents the mechanical impedance of the support at x = L. Thepreceding two equations can be restated in terms of particle displacements by
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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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Page 73 and 74: 3.18 The Monopole Source 63The thre
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Page 77 and 78: References 67medium. The time avera
Page 79 and 80: Problems for Chapter 3 69(a) the wa
Page 81 and 82: 4Vibrating Strings4.1 IntroductionI
Page 83 and 84: 4.4 General Solution of the Wave Eq
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Page 87 and 88: 4.7 Standing Waves 77Figure 4.3. Di
Page 89 and 90: 4.8 The Effect of Initial Condition
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Page 93 and 94: 4.11 Strings of Finite Lengths: For
Page 95 and 96: References 854.12 Real Strings: Fre
Page 97 and 98: Problems for Chapter 4 878. A devic
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Page 111 and 112: 102 5. Vibrating BarsFigure 5.7. An
Page 113 and 114: 104 5. Vibrating Barsthe light beam
Page 115 and 116: 106 5. Vibrating BarsFigure 5.8. Tr
Page 117 and 118: 108 5. Vibrating BarsFigure 5.10. T
Page 119 and 120: 110 5. Vibrating Barsof 0.005 m dia
Page 121 and 122: 112 6. Membrane and PlatesFigure 6.
Page 123 and 124: 114 6. Membrane and PlatesBecause t
Page 125 and 126: 116 6. Membrane and PlatesThe left-
Page 127 and 128: 118 6. Membrane and PlatesTable 6.1
Page 129 and 130: 120 6. Membrane and PlatesIn real a
Page 131 and 132: 122 6. Membrane and PlatesTable 6.2
Page 133 and 134: 124 6. Membrane and PlatesAt low fr
Page 135 and 136: 126 6. Membrane and PlatesThe elast
Page 137 and 138: 128 6. Membrane and PlatesFor the f
Page 139 and 140: 130 6. Membrane and Plates(b) Deter
Page 141 and 142: 132 7. Pipes, Waveguides, and Reson
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148 7. Pipes, Waveguides, and Reson
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150 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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582 19. Sound Reproductionuse of MP
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584 19. Sound ReproductionDickason,
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586 20. Vibration and Vibration Con
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618 21. Nonlinear Acousticsby∇ 2
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620 21. Nonlinear Acousticsmay be i
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622 21. Nonlinear Acousticswhereδ
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624 21. Nonlinear AcousticsThe shoc
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626 21. Nonlinear AcousticsBut the
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Appendix APhysical Properties of Ma
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Appendix A. Physical Properties of
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Appendix BBessel FunctionsB.1 The B
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B.8 Tables of Bessel Functions, Zer
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Appendix CUsing Laplace Transforms
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C.3 Solving Differential Equations
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C.4 Equations with Multiple-Order R
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SolutionC.5 Equations with Complex
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C.5 Equations with Complex Roots 64
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SolutionC.5 Equations with Complex
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650 IndexAuditoriums, design of, 26
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652 IndexElectrostatic speakers, 58
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654 IndexKey notation, 518Kinetic e
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656 IndexPascal (unit), 48Percent i
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658 IndexSound intensity growth in
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660 IndexWater-hammer arresters, 37
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