Engineering Acoustics

Engineering Acoustics
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Описание книги

A comprehensive evaluation of the basic theory for acoustics, noise and vibration control together with fundamentals of how this theoretical material can be applied to real world problems in the control of noise and vibration in aircraft, appliances, buildings, industry, and vehicles. The basic theory is presented in elementary form and only of sufficient complication necessary to solve real practical problems. Unnecessary advanced theoretical approaches are not included. In addition to the fundamental material discussed, chapters are included on human hearing and response to noise and vibration, acoustics and vibration transducers, instrumentation, noise and vibration measurements, and practical discussions concerning: community noise and vibration, interior and exterior noise of aircraft, road and rail vehicles, machinery noise and vibration sources, noise and vibration in rapid transit rail vehicles, automobiles, trucks, off road vehicles, and ships. In addition, extensive up to date useful references are included at the end of each chapter for further reading. The book concludes with a glossary on acoustics, noise and vibration

Оглавление

Malcolm J. Crocker. Engineering Acoustics

Table of Contents

List of Tables

List of Illustrations

Guide

Pages

Wiley Series in Acoustics, Noise and Vibration

Engineering Acoustics. Noise and Vibration Control

Series Preface

Preface

Acknowledgements

1 Introduction. 1.1 Introduction

1.2 Types of Noise and Vibration Signals

1.2.1 Stationary Signals

1.2.2 Nonstationary Signals

1.3 Frequency Analysis

1.3.1 Fourier Series

Example 1.1

Solution

1.3.2 Nonperiodic Functions and the Fourier Spectrum

1.3.3 Random Noise

Example 1.2

Solution

Example 1.3

Solution

1.3.4 Mean Square Values

Example 1.4

Solution

1.3.5 Energy and Power Spectral Densities

1.4 Frequency Analysis Using Filters

Example 1.5

Solution

1.5 Fast Fourier Transform Analysis

References

2 Vibration of Simple and Continuous Systems. 2.1 Introduction

2.2 Simple Harmonic Motion

2.2.1 Period, Frequency, and Phase

2.2.2 Velocity and Acceleration

Example 2.1

Solution

2.3 Vibrating Systems. 2.3.1 Mass–Spring System

Example 2.2

Solution

Example 2.3

Solution

Example 2.4

Solution

2.4 Multi‐Degree of Freedom Systems

2.4.1 Free Vibration – Undamped

Example 2.5

Solution

2.4.2 Forced Vibration – Undamped

Example 2.6

Solution

Example

Solution

2.4.3 Effect of Damping

Example

Example 2.9

Solution

2.5 Continuous Systems

2.5.1 Vibration of Beams

Example 2.10

Solution

Example 2.11

Solution

2.5.2 Vibration of Thin Plates

Example 2.12

Solution

Example 2.13

References

3 Sound Generation and Propagation. 3.1 Introduction

3.2 Wave Motion

3.3 Plane Sound Waves

Example 3.1

Solution

3.3.1 Sound Pressure

3.3.2 Particle Velocity

3.3.3 Impedance and Sound Intensity

3.3.4 Energy Density

3.3.5 Sound Power

3.4 Decibels and Levels

3.4.1 Sound Pressure Level

3.4.2 Sound Power Level

3.4.3 Sound Intensity Level

3.4.4 Combination of Decibels

Example 3.2

Solution

Example 3.3

Solution

Example 3.4

Example 3.5

Example 3.6

Solution

3.5 Three‐dimensional Wave Equation

3.6 Sources of Sound

3.6.1 Sound Intensity

3.7 Sound Power of Sources. 3.7.1 Sound Power of Idealized Sound Sources

Example 3.7

Solution

Example 3.8

Solution

Example 3.9

Solution

3.8 Sound Sources Above a Rigid Hard Surface

Example 3.10

Solution

3.9 Directivity

3.9.1 Directivity Factor (Q(θ, ϕ))

3.9.2 Directivity Index

Example 3.11

Solution

3.10 Line Sources

3.11 Reflection, Refraction, Scattering, and Diffraction

Example 3.12

Solution

3.12 Ray Acoustics

3.13 Energy Acoustics

3.14 Near Field, Far Field, Direct Field, and Reverberant Field

3.14.1 Reverberation

3.14.2 Sound Absorption

3.14.3 Reverberation Time

Example 3.13

Solution

Example 3.14

Solution

3.15 Room Equation

3.15.1 Critical Distance

3.15.2 Noise Reduction

3.16 Sound Radiation From Idealized Structures

Example 3.15

Solution

3.17 Standing Waves

Example 3.16

Solution

3.18 Waveguides

3.19 Other Approaches. 3.19.1 Acoustical Lumped Elements

3.19.2 Numerical Approaches: Finite Elements and Boundary Elements

3.19.3 Acoustic Modeling Using Equivalent Circuits

References

4 Human Hearing, Speech and Psychoacoustics. 4.1 Introduction

4.2 Construction of Ear and Its Working

4.2.1 Construction of the Ear

4.2.2 Working of the Ear Mechanism

4.2.3 Theories of Hearing

4.3 Subjective Response

4.3.1 Hearing Envelope

4.3.2 Loudness Measurement

Example 4.1

Example 4.2

Solution

Example 4.3

Solution

4.3.3 Masking

4.3.4 Pitch

Example 4.4

Solution

4.3.5 Weighted Sound Pressure Levels

Example 4.5

Solution

4.3.6 Critical Bands

4.3.7 Frequency (Bark)

Example 4.6

Solution

4.3.8 Zwicker Loudness

4.3.9 Loudness Adaptation

4.3.10 Empirical Loudness Meter

4.4 Hearing Loss and Diseases (Disorders)

4.4.1 Conduction Hearing Loss

4.4.2 Sensory‐Neural Hearing Loss

4.4.3 Presbycusis

4.5 Speech Production

References

5 Effects of Noise, Vibration, and Shock on People. 5.1 Introduction

5.2 Sleep Disturbance

5.3 Annoyance

5.4 Cardiovascular Effects

5.5 Cognitive Impairment

5.6 Infrasound, Low‐Frequency Noise, and Ultrasound

5.7 Intense Noise and Hearing Loss

5.7.1 Theories for Noise‐Induced Hearing Loss

5.7.2 Impulsive and Impact Noise

5.8 Occupational Noise Regulations

5.8.1 Daily Noise Dose and Time‐Weighted Average Calculation

Example 5.1

Solution

Example 5.2

Solution

Example 5.3

Solution

Example 5.4

Solution

5.9 Hearing Protection. 5.9.1 Hearing Protectors

Example 5.5

Solution

5.9.2 Hearing Conservation Programs

Example 5.6

Solution

5.10 Effects of Vibration on People

5.11 Metrics to Evaluate Effects of Vibration and Shock on People. 5.11.1 Acceleration Frequency Weightings

5.11.2 Whole‐Body Vibration Dose Value

Example 5.7

Solution

Example 5.8

Solution

Example 5.9

Solution

5.11.3 Evaluation of Hand‐Transmitted Vibration

Example 5.10

Solution

Example 5.11

Solution

References

6 Description, Criteria, and Procedures Used to Determine Human Response to Noise and Vibration. 6.1 Introduction

6.2 Loudness and Annoyance

6.3 Loudness and Loudness Level

6.4 Noisiness and Perceived Noise Level. 6.4.1 Noisiness

Example 6.1

Solution

6.4.2 Effective Perceived Noise Level

6.5 Articulation Index and Speech Intelligibility Index

6.6 Speech Interference Level

Example 6.2

Solution

6.7 Indoor Noise Criteria

6.7.1 NC Curves

Example 6.3

Solution

6.7.2 NR Curves

6.7.3 RC Curves

6.7.4 Balanced NC Curves

Example 6.4

6.8 Equivalent Continuous SPL

Example 6.4

Solution

6.9 Sound Exposure Level

Example 6.6

Solution

Example 6.7

Solution

6.10 Day–Night Equivalent SPL

Example 6.8

Solution

Example 6.9

Solution

6.11 Percentile SPLs

6.12 Evaluation of Aircraft Noise

6.12.1 Composite Noise Rating

6.12.2 Noise Exposure Forecast

6.12.3 Noise and Number Index

6.12.4 Equivalent A‐Weighted SPL Leq, Day–Night Level Ldn, and Day–Evening–Night Level Lden

6.13 Evaluation of Traffic Noise

6.13.1 Traffic Noise Index

6.13.2 Noise Pollution Level

6.13.3 Equivalent SPL

6.14 Evaluation of Community Noise

Example 6.10

Solution

6.15 Human Response

6.15.1 Sleep Interference

6.15.2 Annoyance

Example 6.11

Solution

Example 6.12

Solution

6.16 Noise Criteria and Noise Regulations

6.16.1 Noise Criteria

6.17 Human Vibration Criteria

6.17.1 Human Comfort in Buildings

Example 6.13

Solution

6.17.2 Effect of Vibration on Buildings

References

7 Noise and Vibration Transducers, Signal Processing, Analysis, and Measurements. 7.1 Introduction

7.2 Typical Measurement Systems

7.3 Transducers

7.3.1 Transducer Characteristics

7.3.2 Sensitivity

Example 7.1

Solution

Example 7.2

Solution

Example 7.3

Solution

7.3.3 Dynamic Range

7.3.4 Frequency Response

7.4 Noise Measurements

7.4.1 Types of Microphones for Noise Measurements

7.4.2 Directivity

7.4.3 Transducer Calibration

7.5 Vibration Measurements

7.5.1 Principle of Seismic Mass Transducers

Example 7.4

Solution

Example 7.5

Solution

Example 7.6

Solution

7.5.2 Piezoelectric Accelerometers

Example 7.7

Solution

7.5.3 Measurement Difficulties

Example 7.8

Solution

7.5.4 Calibration, Metrology, and Traceability of Shock and Vibration Transducers

7.6 Signal Analysis, Data Processing, and Specialized Noise and Vibration Measurements. 7.6.1 Signal Analysis and Data Processing

7.6.2 Sound Level Meters (SLMs) and Dosimeters

7.6.3 Sound Power and Sound Intensity

7.6.4 Modal Analysis

7.6.5 Condition Monitoring

7.6.6 Advanced Noise and Vibration Analysis and Measurement Techniques

References

8 Sound Intensity, Measurements and Determination of Sound Power, Noise Source Identification, and Transmission Loss. 8.1 Introduction

8.2 Historical Developments in the Measurement of Sound Pressure and Sound Intensity

8.3 Theoretical Background

8.4 Characteristics of Sound Fields

8.4.1 Active and Reactive Intensity

8.4.2 Plane Progressive Waves

8.4.3 Standing Waves

8.4.4 Vibrating Piston in a Tube

8.5 Active and Reactive Sound Fields

8.5.1 The Monopole Source

Example 8.1

Solution

Example 8.2

Solution

8.5.2 The Dipole Source

8.5.3 General Case

8.6 Measurement of Sound Intensity

8.6.1 The p–p Method

Example 8.3

Solution

Example 8.4

Solution

Example 8.5

Solution

8.6.2 The p–u Method

Example 8.6

Solution

8.6.3 The Surface Intensity Method

8.7 Applications

8.7.1 Sound Power Determination

Example 8.7

Solution

Example 8.8

Solution

8.7.2 Noise Source Identification

8.7.3 Noise Source Identification on a Diesel Engine Using Sound Intensity

8.7.4 Measurements of the Transmission Loss of Structures Using Sound Intensity

8.8 Comparison Between Sound Power Measurements Using Sound Intensity and Sound Pressure Methods

8.8.1 Sound Intensity Method

8.8.2 Sound Pressure Method

8.9 Standards for Sound Intensity Measurements

References

9 Principles of Noise and Vibration Control. 9.1 Introduction

9.2 Systematic Approach to Noise Problems

9.2.1 Noise and Vibration Source Identification

9.2.2 Noise Reduction Techniques

9.3 Use of Vibration Isolators

9.3.1 Theory of Vibration Isolation

Example 9.1

Solution

Example 9.2

Solution

9.3.2 Machine Vibration

9.3.3 Use of Inertia Blocks

9.3.4 Other Considerations

Example 9.3

Solution

9.4 Use of Damping Materials

9.4.1 Unconstrained Damping Layer

9.4.2 Constrained Damping Layer

9.5 Use of Sound Absorption

9.5.1 Sound Absorption Coefficient

9.5.2 Noise Reduction Coefficient

Example 9.4

Solution

9.5.3 Absorption by Porous Fibrous Materials

9.5.4 Panel or Membrane Absorbers

9.5.5 Helmholtz Resonator Absorbers

Example 9.5

Solution

9.5.6 Perforated Panel Absorbers

Example 9.6

Solution

9.5.7 Slit Absorbers

Example 9.7

Solution

9.5.8 Suspended Absorbers

9.5.9 Acoustical Spray‐on Materials

9.5.10 Acoustical Plaster

9.5.11 Measurement of Sound Absorption Coefficients

9.5.12 Optimization of the Reverberation Time

Example 9.8

Solution

9.5.13 Reduction of the Sound Pressure Level in Reverberant Fields

Example 9.9

Solution

9.6 Acoustical Enclosures

9.6.1 Reverberant Sound Field Model for Enclosures

9.6.2 Machine Enclosure in Free Field

9.6.3 Simple Enclosure Design Assuming Diffuse Reverberant Sound Fields

Example 9.10

Solution

Example 9.11

Solution

Example 9.12

Solution

9.6.4 Close‐Fitting Enclosures

9.6.5 Partial Enclosures

9.6.6 Other Considerations

9.7 Use of Barriers

9.7.1 Transmission Loss of Barriers

9.7.2 Use of Barriers Indoors

Example 9.13

Solution

9.7.3 Reflections from the Ground

9.7.4 Use of Barriers Outdoors

Example 9.14

Solution

Example 9.15

Solution

9.8 Active Noise and Vibration Control

References

10 Mufflers and Silencers – Absorbent and Reactive Types. 10.1 Introduction

10.2 Muffler Classification

10.3 Definitions of Muffler Performance

10.4 Reactive Mufflers

10.5 Historical Development of Reactive Muffler Theories

10.6 Classical Reactive Muffler Theory. 10.6.1 Transmission Line Theory

10.6.2 TL of Resonators

Example 10.1

Solution

Example 10.2

Solution

Example 10.3

Solution

Example 10.4

Solution

Example 10.5

Solution

Example 10.6

Solution

10.6.3 NACA 1192 Study on Reactive Muffler TL

10.6.4 Transfer Matrix Theory

10.7 Exhaust System Modeling

10.7.1 Transmission Loss

10.7.2 Insertion Loss

10.7.3 Sound Pressure Radiated from Tailpipe

10.8 Tail Pipe Radiation Impedance, Source Impedance and Source Strength. 10.8.1 Tail Pipe Radiation

10.8.2 Internal Combustion Engine Impedance and Source Strength

10.9 Numerical Modeling of Muffler Acoustical Performance. 10.9.1 Finite Element Analysis

10.9.2 Boundary Element Analysis

10.9.3 TL of Concentric Tube Resonators

10.10 Reactive Muffler IL

10.11 Measurements of Source Impedance

10.12 Dissipative Mufflers and Lined Ducts

10.13 Historical Development of Dissipative Mufflers and Lined Duct Theories

10.14 Parallel‐Baffle Mufflers

10.14.1 Embleton's Method [8]

10.14.2 Ver's Method [11, 12, 136]

10.14.3 Ingard's Method [149]

10.14.4 Bies and Hansen Method [14]

10.14.5 Mechel's Design Curves [152]

10.14.6 Ramakrishnan and Watson Curves [151]

Example 10.7

Solution

10.14.7 Finite Element Approach for Attenuation of Parallel‐Baffle Mufflers

References

11. Noise and Vibration Control of Machines. 11.1 Introduction

11.2 Machine Element Noise and Vibration Sources and Control. 11.2.1 Gears

Example 11.1

Solution

Example 11.2

Solution

11.2.2 Bearings

Example 11.3

Solution

11.2.3 Fans and Blowers

Example 11.4

Solution

Example 11.5

Solution

11.2.4 Metal Cutting

11.2.5 Woodworking

11.3 Built‐up Machines. 11.3.1 Internal Combustion Engines

11.3.2 Electric Motors and Electrical Equipment

Example 11.6

Solution

Example 11.7

Solution

11.3.3 Compressors

Example 11.8

Solution

Example 11.9

Solution

11.3.4 Pumps

Example 11.10

Solution

11.4 Noise Due to Fluid Flow. 11.4.1 Valve‐Induced Noise

Example 11.11

Solution

Example 11.12

Solution

11.4.2 Hydraulic System Noise

11.4.3 Furnace and Burner Noise

11.5 Noise Control of Industrial Production Machinery

11.5.1 Machine Tool Noise, Vibration, and Chatter

11.5.2 Sound Power Level for Industrial Machinery

References

12 Noise and Vibration Control in Buildings. 12.1 Introduction

12.2 Sound Transmission Theory for Single Panels

12.2.1 Mass‐Law Transmission Loss

Example 12.1

Solution

12.2.2 Random Incidence Transmission Loss

Example 12.2

Solution

Example 12.3

Solution

12.2.3 The Coincidence Effect

Example 12.4

Solution

12.3 Sound Transmission for Double and Multiple Panels

12.3.1 Sound Transmission Through Infinite Double Panels

12.3.2 London's Theory

Example

Solution

12.3.3 Empirical Approach

Example

Solution

Example 12.7

Solution

12.4 Sound and Vibration Transmission and Structural Response Using Statistical Energy Analysis (SEA) 12.4.1 Introduction

12.4.2 SEA Fundamentals and Assumptions

Example 12.8

Solution

Example 12.9

Solution

Example 12.10

Solution

Example 12.11

Solution

Example 12.12

Solution

12.4.3 Power Flow Between Coupled Systems

12.4.4 Modal Behavior of Panel

12.4.5 Use of SEA to Predict Sound Transmission Through Panels or Partitions

12.4.6 Design of Enclosures Using SEA

12.4.7 Optimization of Enclosure Attenuation

12.4.8 SEA Computer Codes

12.5 Transmission Through Composite Walls

Example 12.13

Solution

Example 12.14

Solution

Example 12.15

Solution

12.6 Effects of Leaks and Flanking Transmission

Example 12.16

Solution

12.7 Sound Transmission Measurement Techniques

12.7.1 Laboratory Methods of Measuring Transmission Loss

Example

Solution

12.7.2 Measurements of Transmission Loss in the Field

Example 12.18

Solution

12.8 Single‐Number Ratings for Partitions

Example 12.19

Solution

Example 12.20

Solution

12.9 Impact Sound Transmission

12.9.1 Laboratory and Field Measurements of Impact Transmission

Example 12.21

Solution

12.9.1.1 Rating of Impact Sound Transmission

Example 12.22

Solution

12.9.2 Measured Airborne and Impact Sound Transmission (Insulation) Data

12.9.2.1 Gypsum Board Walls

12.9.2.2 Masonry Walls

12.9.2.3 Airborne and Impact Insulation of Floors

12.9.2.4 Doors and Windows

12.9.3 Sound Insulation Requirements

Example 12.23

Solution

Example 12.24

Solution

12.9.4 Control of Vibration of Buildings Caused by Strong Wind

12.9.4.1 Wind Excitation of Buildings

12.9.4.2 Structural Vibration Response of Buildings and Towers

12.9.4.3 Methods of Building Structure Vibration Reduction and Control

12.9.4.4 Human Response to Vibration and Acceptability Criteria

References

13 Design of Air‐conditioning Systems for Noise and Vibration Control. 13.1 Introduction

13.2 Interior Noise Level Design Criteria

13.3 General Features of a Ventilation System

13.3.1 HVAC Systems in Residential Homes

13.3.2 HVAC Systems in Large Buildings

13.3.3 Correct and Incorrect Installation of HVAC Systems

13.3.4 Sources of Noise and Causes of Complaints in HVAC Systems

13.4 Fan Noise

13.4.1 Types of Fans Used in HVAC Systems

13.4.2 Blade passing Frequency (BPF)

Example 13.1

Solution

13.4.3 Fan Efficiency

13.4.4 Sound Power and Frequency Content of Fans

Example 13.2

Solution

13.4.5 Sound Power Levels of Fans and Predictions

13.4.6 Prediction of Fan Sound Power Level

Example 13.3

Solution

13.4.7 Importance of Proper Installation of Centrifugal Fans

13.4.8 Terminal Units (CAV, VAV, and Fan‐Powered VAV Boxes)

13.5 Space Planning

13.6 Mechanical Room Noise and Vibration Control

13.6.1 Use of Floating Floors

Example 13.4

Solution

13.6.2 Vibration Control of Equipment

13.6.3 Selection of Vibration Isolators

Example 13.5

Solution

Example 13.6

Solution

13.6.4 Vibration Isolation of Ducts, Pipes, and Wiring

Example 13.7

Solution

Example 13.8

Solution

13.7 Sound Attenuation in Ventilation Systems

13.7.1 Use of Fiberglass in Plenum Chambers, Mufflers, and HVAC Ducts

13.7.2 Attenuation of Plenum Chambers

Example 13.9

Solution

13.7.3 Duct Attenuation

Example 13.10

Solution

Example 13.11

Solution

Example 13.12

Solution

13.7.4 Sound Attenuators (Silencers)

13.7.5 Branches and Power Splits

13.7.6 Attenuation Due to End Reflection

Example 13.13

Solution

13.7.7 Attenuation by Mitre Bends

13.8 Sound Generation in Mechanical Systems

13.8.1 Elbow Noise

13.8.2 Take‐off Noise

13.8.3 Grille Noise

13.8.4 Diffuser Noise

13.8.5 Damper Noise

13.9 Radiated Noise

13.9.1 Duct‐Radiated Noise

13.9.2 Sound Breakout and Breakin From Ducts

Example 13.14

Solution

13.9.3 Mixing Box Radiated Noise

13.9.4 Radiation From Fan Plenum Walls

13.9.5 Overall Sound Pressure Level Prediction

Example 13.15

Solution

References

14 Surface Transportation Noise and Vibration Sources and Control. 14.1 Introduction

14.2 Automobile and Truck Noise Sources and Control

14.2.1 Power Plant Noise and Its Control

Example 14.1

Solution

Example 14.2

Solution

14.2.2 Intake and Exhaust Noise and Muffler Design

14.2.3 Tire/Road Noise Sources and Control

14.2.4 Aerodynamic Noise Sources on Vehicles

14.2.5 Gearbox Noise and Vibration

14.2.6 Brake Noise Prediction and Control

14.3 Interior Road Vehicle Cabin Noise. 14.3.1 Automobiles and Trucks

14.3.2 Off‐Road Vehicles

14.4 Railroad and Rapid Transit Vehicle Noise and Vibration Sources. 14.4.1 Wheel–Rail Interaction Noise

14.4.2 Interior Rail Vehicle Cabin Noise

Example 14.3

Solution

14.5 Noise And Vibration Control in Ships

Example 14.4

Solution

References

15 Aircraft and Airport Transportation Noise Sources and Control. 15.1 Introduction

15.2 Jet Engine Noise Sources and Control

15.3 Propeller and Rotor Noise Sources and Control

15.4 Helicopter and Rotor Noise

15.5 Aircraft Cabin Noise and Vibration and Its Control. 15.5.1 Passive Noise and Vibration Control

15.5.2 Active Noise and Vibration Control

15.6 Airport Noise Control

15.6.1 Noise Control at the Source

15.6.2 Airport‐specific Noise Control Measures

Example 15.1

Solution

Example 15.2

Solution

References

16 Community Noise and Vibration Sources. 16.1 Introduction

16.2 Assessment of Community Noise Annoyance

Example 16.1

Solution

16.3 Community Noise and Vibration Sources and Control. 16.3.1 Traffic Noise Sources

16.3.2 Rail System Noise Sources

16.3.3 Ground‐Borne Vibration Transmission from Road and Rail Systems

16.3.4 Aircraft and Airport Noise Prediction and Control

16.3.5 Off‐road Vehicle and Construction Equipment Exterior Noise Prediction and Control

16.3.6 Industrial and Commercial Noise in the Community

16.3.7 Construction and Building Site Noise

16.4 Environmental Impact Assessment

16.5 Environmental Noise and Vibration Attenuation. 16.5.1 Attenuation Provided by Barriers, Earth Berms, Buildings, and Vegetation

16.5.2 Base Isolation of Buildings for Control of Ground‐Borne Vibration

16.5.3 Noise Control Using Porous Road Surfaces

16.6 City Planning for Noise and Vibration Reduction and Soundscape Concepts. 16.6.1 Community Noise Ordinances

Example 16.2

Solution

16.6.2 Recommendations for Urban Projects

16.6.3 Strategic Noise Maps

16.6.4 Soundscapes

References

Glossary. References

Glossary

Index

a

b

c

d

e

f

g

h

i

j

k

l

m

n

o

p

q

r

s

t

u

v

w

y

z

WILEY END USER LICENSE AGREEMENT

Отрывок из книги

Malcolm J. Crocker

.....

and substituting w(x,t) from Eq. (2.52) into Eq. (2.51) we get

(2.53)

.....

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