Contents
1 Complex Numbers for Harmonic Functions . . . . . . . . . . . . . . . . . . |
1 |
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1.1 |
Review of Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . |
2 |
1.2 |
Complex Numbers in Polar Form . . . . . . . . . . . . . . . . . . . . . . |
3 |
1.3 |
Four Equivalent Forms to Represent Harmonic Waves . . . . . . . |
4 |
1.4 |
Mathematical Identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
6 |
1.5 |
Derivation of Four Equivalent Forms . . . . . . . . . . . . . . . . . . . |
8 |
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1.5.1 Obtain Form 2 from Form 1 . . . . . . . . . . . . . . . . . . . |
8 |
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1.5.2 Obtain Form 3 from Form 2 . . . . . . . . . . . . . . . . . . . |
9 |
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1.5.3 Obtain Form 4 from Form 3 . . . . . . . . . . . . . . . . . . . |
9 |
1.6Visualization and Numerical Validation of Form 1
and Form 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
10 |
1.7Space-Time Harmonic Functions Expressed in Four
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Equivalent Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
16 |
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1.8 |
Homework Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
18 |
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1.9 |
References of Trigonometric Identities . . . . . . . . . . . . . . . . . . |
21 |
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1.9.1 |
Trigonometric Identities of a Single Angle . . . . . . . . . |
21 |
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1.9.2 |
Trigonometric Identities of Two Angles . . . . . . . . . . . |
23 |
1.10A MATLAB Code for Visualization of Form 1
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and Form 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
23 |
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2 |
Derivation of Acoustic Wave Equation . . . . . . . . . . . . . . . . . . . . . . |
27 |
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2.1 |
Euler’s Force Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
29 |
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2.2 |
Equation of Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
31 |
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2.3 |
Equation of State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
34 |
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2.3.1 |
Energy Increase due to Work Done . . . . . . . . . . . . . . |
35 |
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2.3.2 |
Pressure due to Colliding of Gases . . . . . . . . . . . . . . |
37 |
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2.3.3 |
Derivation of Equation of State . . . . . . . . . . . . . . . . . |
38 |
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2.4 |
Derivation of Acoustic Wave Equation . . . . . . . . . . . . . . . . . . |
39 |
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2.5 |
Formulas for the Speed of Sound . . . . . . . . . . . . . . . . . . . . . . |
42 |
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2.5.1 |
Formula Using Pressure . . . . . . . . . . . . . . . . . . . . . . |
42 |
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2.5.2 |
Formula Using Bulk Modulus . . . . . . . . . . . . . . . . . . |
43 |
ix
x |
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Contents |
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2.5.3 |
Formula Using Temperature . . . . . . . . . . . . . . . . . . |
. 45 |
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2.5.4 Formula Using Colliding Speed . . . . . . . . . . . . . . . . |
. 46 |
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2.6 |
Homework Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
. 47 |
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3 Solutions of Acoustic Wave Equation . . . . . . . . . . . . . . . . . . . . . . |
. 49 |
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3.1 |
Review of Partial Differential Equations . . . . . . . . . . . . . . . . |
. 51 |
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3.1.1 Complex Solutions of a Partial Differential |
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Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
. 51 |
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3.1.2 Trigonometric Solutions of a Partial Differential |
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Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
. 53 |
3.2 |
Four Basic Complex Solutions . . . . . . . . . . . . . . . . . . . . . . . |
. 54 |
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3.3 |
Four Basic Traveling Waves . . . . . . . . . . . . . . . . . . . . . . . . |
. 59 |
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3.4 |
Four Basic Standing Waves . . . . . . . . . . . . . . . . . . . . . . . . . |
. 60 |
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3.5 |
Conversion Between Traveling and Standing Waves . . . . . . . |
. 63 |
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3.6 |
Wavenumber, Angular Frequency, and Wave Speed . . . . . . . |
. 68 |
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3.7 |
Visualization of Acoustic Waves . . . . . . . . . . . . . . . . . . . . . |
. 71 |
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3.7.1 |
Plotting Traveling Wave . . . . . . . . . . . . . . . . . . . . . |
. 71 |
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3.7.2 |
Plotting Standing Wave . . . . . . . . . . . . . . . . . . . . . |
. 72 |
3.8 |
Homework Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
. 75 |
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4 Acoustic Intensity and Specific Acoustic Impedance . . . . . . . . . . . |
. 81 |
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4.1 |
Pressure-Velocity Relationship . . . . . . . . . . . . . . . . . . . . . . . |
. 82 |
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4.1.1 Pressure-Velocity Relationships for BTW . . . . . . . . |
. 83 |
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4.1.2 Pressure-Velocity Relationships for BSW . . . . . . . . . |
. 86 |
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4.1.3 Pressure-Velocity Relationships in Complex |
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Function Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
. 88 |
4.2 |
RMS Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
. 89 |
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4.2.1 RMS Pressure of BTW . . . . . . . . . . . . . . . . . . . . . . |
. 90 |
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4.2.2 RMS Pressure of BSW . . . . . . . . . . . . . . . . . . . . . . |
. 91 |
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4.3 |
Acoustic Intensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
. 94 |
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4.3.1 Acoustic Intensity of BTW . . . . . . . . . . . . . . . . . . . |
. 94 |
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4.3.2 Acoustic Intensity of BSW . . . . . . . . . . . . . . . . . . . |
. 95 |
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4.4 |
Specific Acoustic Impedance Expressed as Real Numbers . . . |
. 96 |
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4.4.1 Specific Acoustic Impedance of BTW . . . . . . . . . . . |
. 97 |
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4.4.2 Specific Acoustic Impedance of BSW . . . . . . . . . . . |
. 98 |
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4.5Specific Acoustic Impedance Expressed as Complex
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Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
103 |
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4.5.1 Issues with Real Impedance . . . . . . . . . . . . . . . . . . . |
103 |
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4.5.2 Definition of Complex Impedance . . . . . . . . . . . . . . . |
103 |
4.6 |
Computer Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
105 |
4.7 |
Homework Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
111 |
4.8 |
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
114 |
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4.8.1 Derivatives of Trigonometric and Complex |
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Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . |
114 |
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4.8.2 Trigonometric Integrals . . . . . . . . . . . . . . . . . . . . . . . |
115 |
Contents |
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xi |
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5 |
Solutions of Spherical Wave Equation . . . . . . . . . . . . . . . . . . . . . . . |
117 |
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5.1 |
Spherical Coordinate System . . . . . . . . . . . . . . . . . . . . . . . . . |
118 |
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5.2 |
Wave Equation in Spherical Coordinate System . . . . . . . . . . . |
119 |
5.3Pressure Solutions of Wave Equation in Spherical
Coordinate System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 5.4 Flow Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 5.4.1 Flow Velocity in Real Format . . . . . . . . . . . . . . . . . . 124 5.4.2 Flow Velocity in Complex Format . . . . . . . . . . . . . . . 126
5.5 RMS Pressure and Acoustic Intensity . . . . . . . . . . . . . . . . . . . 128 5.6 Specific Acoustic Impedance . . . . . . . . . . . . . . . . . . . . . . . . . 132 5.7 Homework Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
6 Acoustic Waves from Spherical Sources . . . . . . . . . . . . . . . . . . . . . |
137 |
6.1Review of Pressure and Velocity Formulas
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for Spherical Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
139 |
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6.2 |
Acoustic Waves from a Pulsating Sphere . . . . . . . . . . . . . . . . |
139 |
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6.3 |
Acoustic Waves from a Small Pulsating Sphere . . . . . . . . . . . . |
141 |
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6.3.1 |
Near-Field Solutions of a Small Spherical |
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Source (kr 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
142 |
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6.3.2 |
Far-Field Solutions of a Small Spherical |
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Source (kr 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
143 |
6.4 |
Acoustic Waves from a Point Source . . . . . . . . . . . . . . . . . . . |
146 |
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6.4.1 |
Point Sources Formulated with Source Strength . . . . . |
146 |
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6.4.2 |
Flow Rate as Source Strength . . . . . . . . . . . . . . . . . . |
147 |
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6.4.3 |
Point Source in an Infinite Baffle . . . . . . . . . . . . . . . . |
149 |
6.5 |
Acoustic Intensity and Sound Power . . . . . . . . . . . . . . . . . . . . |
150 |
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6.6 |
Computer Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
157 |
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6.7 |
Project . |
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160 |
6.8 |
Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
161 |
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6.9 |
Homework Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
162 |
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7 Resonant Cavities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
167 |
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7.1 |
1D |
Standing Waves Between Two Walls . . . . . . . . . . . . . . . . |
168 |
7.2 |
Natural Frequencies and Mode Shapes in a Pipe . . . . . . . . . . . |
170 |
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7.3 |
2D |
Boundary Conditions Between Four Walls . . . . . . . . . . . . |
177 |
7.3.12D Standing Wave Solutions of the Wave
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Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
177 |
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7.3.2 2D Nature Frequencies Between Four Walls . . . . . . . |
179 |
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7.3.3 |
2D |
Mode Shapes Between Four Walls . . . . . . . . . . . . |
181 |
7.4 |
3D Boundary Conditions of Rectangular Cavities . . . . . . . . . . |
184 |
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7.4.1 |
3D |
Standing Wave Solutions of the Wave |
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Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
184 |
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7.4.2 |
3D |
Natural Frequencies and Mode Shapes . . . . . . . . . |
184 |
7.5 |
Homework Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
192 |
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xii |
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Contents |
8 Acoustic Waveguides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
. 197 |
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8.1 |
2D Traveling Wave Solutions . . . . . . . . . . . . . . . . . . . . . . . |
. 199 |
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8.1.1 |
Definition of Wavenumber Vectors . . . . . . . . . . . . . |
. 199 |
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8.1.2 |
Wavenumber Vectors in 2D Traveling Wave |
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Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 8.2 Wavenumber Vectors in Resonant Cavities . . . . . . . . . . . . . . . 206 8.3 Traveling Waves in Resonant Cavities . . . . . . . . . . . . . . . . . . 210 8.4 Wavenumber Vectors in Acoustic Waveguides . . . . . . . . . . . . 214 8.5 Traveling Waves in Acoustic Waveguides . . . . . . . . . . . . . . . 220 8.6 Homework Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
9 Sound Pressure Levels and Octave Bands . . . . . . . . . . . . . . . . . . . . |
227 |
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9.1 |
Decibel Scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
228 |
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9.1.1 Review of Logarithm Rules . . . . . . . . . . . . . . . . . . . . |
228 |
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9.1.2 Levels and Decibel Scale . . . . . . . . . . . . . . . . . . . . . |
229 |
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9.1.3 |
Decibel Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . |
230 |
9.2 |
Sound Pressure Levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
231 |
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9.2.1 |
Power-Like Quantities . . . . . . . . . . . . . . . . . . . . . . . |
231 |
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9.2.2 Sound Power Levels and Decibel Scale . . . . . . . . . . . |
232 |
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9.2.3 Sound Pressure Levels and Decibel Scale . . . . . . . . . . |
233 |
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9.2.4 Sound Pressure Levels Calculated |
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in Time Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
233 |
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9.2.5 Sound Pressure Level Calculated in Frequency |
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Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
235 |
9.3 |
Octave Bands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
237 |
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9.3.1 Center Frequencies and Upper and Lower |
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Bounds of Octave Bands . . . . . . . . . . . . . . . . . . . . . . |
237 |
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9.3.2 Lower and Upper Bounds of Octave Band |
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and 1/3 Octave Band . . . . . . . . . . . . . . . . . . . . . . . . |
238 |
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9.3.3 Preferred Speech Interference Level (PSIL) . . . . . . . . |
241 |
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9.4 |
Weighted Sound Pressure Level . . . . . . . . . . . . . . . . . . . . . . . |
242 |
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9.4.1 |
Logarithm of Weighting . . . . . . . . . . . . . . . . . . . . . . |
243 |
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9.4.2 |
A-Weighted Decibels (dBA) . . . . . . . . . . . . . . . . . . . |
244 |
9.5 |
Homework Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
246 |
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10 Room Acoustics and Acoustical Partitions . . . . . . . . . . . . . . . . . . . . 251 10.1 Sound Power, Acoustic Intensity, and Energy Density . . . . . . . 252 10.1.1 Definition of Sound Power . . . . . . . . . . . . . . . . . . . . 252 10.1.2 Definition of Acoustic Intensity . . . . . . . . . . . . . . . . . 253 10.1.3 Definition of Energy Density . . . . . . . . . . . . . . . . . . . 254
10.2Absorption Coefficients, Room Constant,
and Reverberation Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
10.2.1 Absorption Coefficient of Surface . . . . . . . . . . . . . . . 254
10.2.2 Room Constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
10.2.3 Reverberation Time . . . . . . . . . . . . . . . . . . . . . . . . . 258
Contents |
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xiii |
10.3 |
Room Acoustics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
261 |
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10.3.1 Energy Density due to an Acoustic Source . . . . . . . . . |
261 |
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10.3.2 Sound Pressure Level due to an Acoustic Source . . . . |
263 |
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10.4 |
Transmission Loss due to Acoustical Partitions . . . . . . . . . . . . |
264 |
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10.4.1 |
Transmission Coefficient . . . . . . . . . . . . . . . . . . . . . . |
264 |
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10.4.2 |
Transmission Loss (TL) . . . . . . . . . . . . . . . . . . . . . . |
265 |
10.5 |
Noise Reduction due to Acoustical Partitions . . . . . . . . . . . . . |
266 |
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10.5.1 Energy Density due to a Partition Wall . . . . . . . . . . . |
266 |
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10.5.2 Sound Pressure Level due to a Partition Wall . . . . . . . |
268 |
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10.5.3 |
Noise Reduction (NR) . . . . . . . . . . . . . . . . . . . . . . . |
269 |
10.6 |
Homework Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
274 |
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11 Power Transmission in Pipelines . . . . . . . . . . . . . . . . . . . . . . . . . . . |
277 |
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11.1 |
Complex Amplitude of Pressure and Acoustic Impedance . . . . |
278 |
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11.1.1 Definition of Complex Amplitude of Pressure . . . . . . |
278 |
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11.1.2 Definition of Acoustic Impedance . . . . . . . . . . . . . . . |
279 |
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11.1.3 |
Transfer Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . |
281 |
11.2 |
Complex Acoustic Impedance . . . . . . . . . . . . . . . . . . . . . . . . |
283 |
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11.3 |
Balancing Pressure and Conservation of Mass . . . . . . . . . . . . . |
285 |
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11.4 |
Transformation of Pressures . . . . . . . . . . . . . . . . . . . . . . . . . . |
286 |
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11.5 |
Transformation of Acoustic Impedance . . . . . . . . . . . . . . . . . . |
288 |
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11.6 |
Power Reflection and Transmission . . . . . . . . . . . . . . . . . . . . |
297 |
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11.6.1 Definition of Power of Acoustic Waves . . . . . . . . . . . |
297 |
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11.6.2 Power Reflection and Transmission Coefficients |
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of One-to-One Pipes . . . . . . . . . . . . . . . . . . . . . . . . . |
297 |
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11.6.3 Simplified Cases of Power Reflection |
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and Transmission in One-to-One Pipes . . . . . . . . . . . |
298 |
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11.6.4 Special Case of Power Reflection and Transmission: |
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One-to-One Pipes . . . . . . . . . . . . . . . . . . . . . . . . . . . |
298 |
11.7 |
Numerical Method for Molding of Pipelines . . . . . . . . . . . . . . |
303 |
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11.8 |
Computer Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
305 |
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11.9 |
Project . |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
312 |
11.10 |
Homework Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
313 |
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12 Filters and Resonators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
317 |
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12.1 |
Pressure in a One-to-Two Pipe . . . . . . . . . . . . . . . . . . . . . . . . |
318 |
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12.1.1 |
Equivalent Acoustic Impedance |
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of a One-to-Two Pipe . . . . . . . . . . . . . . . . . . . . . . . . |
319 |
12.2 |
Power Transmission of a One-to-Two Pipe . . . . . . . . . . . . . . . |
323 |
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12.2.1 Power Reflection and Transmission |
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of a One-to-Two Pipe . . . . . . . . . . . . . . . . . . . . . . . . |
323 |
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12.2.2 Special Case of Power Reflection |
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and Transmission: A One-to-Two Pipe |
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with No Returning Waves . . . . . . . . . . . . . . . . . . . . . |
324 |
12.3 |
Low-Pass Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
329 |
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xiv |
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Contents |
12.4 |
High-Pass Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
. 335 |
12.5 |
Band-Stop Resonator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
. 343 |
12.6Numerical Method for Modeling of Pipelines
with Side Branches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350 12.7 Project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350 12.8 Homework Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351
Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355
Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359
Appendix 1: Discrete Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . 359
Discrete Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359
Appendix 2: Power Spectral Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368
Power Spectral Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368
Accumulated Sound Pressure Square . . . . . . . . . . . . . . . . . . . . . . . . . 369
Sound Pressure Level in Each Band . . . . . . . . . . . . . . . . . . . . . . . . . . 369
Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373
Chapter 1
Complex Numbers for Harmonic Functions
Sound waves are typically described as simple harmonic motions. Simple harmonic motion can be extended to describe any motion (of the sound wave) by adding a series of simple harmonic motions such as a Fourier series. Simple harmonic motions can be formulated using either trigonometric functions or complex exponential functions. Even though we are more familiar with trigonometric functions such as a cosine or a sine function, the complex exponential functions are commonly used due to their compact and elegant form for derivation and integration in vibration and acoustic analysis.
Since trigonometric functions and complex exponential functions can be used interchangeably in the analysis of vibrations and acoustics, the ability to convert a harmonic function between trigonometric functions and complex exponential functions is essential for the analysis of either vibrations or acoustics. For this reason, conversions between trigonometric functions and complex exponential functions are introduced at the beginning of this course to provide a mathematical tool for analyzing acoustics throughout this course.
This chapter will introduce the four equivalent forms for simple harmonic motions. After you are familiar with the four equivalent forms, you will be able to express any simple harmonic function using either a trigonometric function or a complex exponential function using the four equivalent forms. Furthermore, you will be able to convert any simple harmonic function between the four equivalent forms.
This chapter can be treated as an independent chapter from the rest of the chapters in this book. The materials in this chapter can be used for any vibration-related course to provide students the mathematical skills to handle harmonic functions in vibration analysis.
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 |
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H. Lin et al., Lecture Notes on Acoustics and Noise Control, https://doi.org/10.1007/978-3-030-88213-6_1