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IT IS EARNEStLY DESIRED THAT THE BOOK BE HANDLED WITH CARE AND BE NOT MARKED, UNDERLINED OR DISFIGURED IN ANY OTHER WAY. OTHER WISE IT WILL HAVE TO BE REPLACED OR PAID FOR BY the BORROWED IN THE INTEREST OF THE library.

ELEMENTS OF

Acoustical Engineering

ELEMENTS OF

Acoustical Engineering

By

HARRY F. OLSON, E.E., Ph.D.

Acoustical Research Director, RCA Laboratories Princeton, New Jersey

S ECO SO EDITION — FOURTH PRINTING

D. VAN NOSTRAND COMPANY, Inc.

TORONTO NEW YORK IjONDON

NEW YORK

D. Van Nostrand Company, Inc., 250 Fourth Avenue, New York 3

TORONTO

D. Van Nostrand Company (Canada), Ltd., 228 Bloor Street, Toronto

LONDON

Macmillan & Company, Ltd., St. Martin’s Street, London, W.C. 2

Copyright, 1940, 1947, by D. VAN NOSTRAND COMPANY, Inc.

This book^ or any parts thereof^ may not be reproduced in any form without written per- mission Jrom the author and the publisher

First Published, April 1940

Reprinted, Jan. 1942; Oct. 1943

Second Edition, September 1947 Reprinted, January, 1948, March, 1949 May, 1952

PRINTED IN THE UNITED STATES OF AMERICA

PREFACE TO THE SECOND EDITION

The first edition of this book, published in 1940, was the subject matter of thirty lectures prepared for presentation at Columbia University. It was an exposition of the fundamental principles used in modern acoustics and a description of existing acoustical instruments and systems.

Many and varied advances have been made in acoustical engineering since the issuance of the first edition of this book. Developments and improvements in radio receivers, phonographs, records, sound motion picture, intercommunicating, and sound systems, sound pickup technics, hearing aids, acoustical treatment and a multitude of other devices and systems have kept pace with the ever increasing public interest. In these applications acoustics plays a major role in public acceptance. World War II stimulated activity in both subaqueous and air acoustics and, as a result, many new principles and systems were evolved. In view of the progress in acoustics since the first edition, it appeared advisable to pre- pare a new edition of this book.

In preparing new material and in revising existing material in the second edition, the same principles were followed as in the first edition. Particular efforts have been directed towards the development of analogies between electrical, mechanical and acoustical systems because engineers have found that the reduction of a vibrating system to the analogous electrical net- work is a valuable tool in the analysis of vibrating systems.

Each chapter has been brought up to date and amplified. Two new chapters on Underwater Sound and Supersonics and Ultrasonics have been added. The new edition contains 539 pages as compared to 344 pages in the first edition. The first edition contained 197 illustrations. The new edition contains 342 illustrations of which 50 are revised and 145 are new. As in the first edition, most of the illustrations contain several parts so that a complete theme is depicted in a single illustration.

The author wishes to express his appreciation to Miss Veronica Moran for her work in typing the manuscript and to his wife Lorene E. Olson for assistance in compiling and correcting the manuscript.

June 1947

Harry F. Olson

CONTENTS

Chapter

Pace

1. SOUND WAVES

1.1 Introduction

1.2 Sound Waves ...

1.3 Acoustical Wave Equation

A. Equation of Continuity . .

B. Equation of Motion ....

C. Compressibility of a Gas

D. Condensation

E. D’Alembertian Wave Equation ...

1.4 Plane Sound Waves .

A. Particle Velocity in a Plane Sound Wave

B. Pre.s.sure in a Plane Sound Wave

C. Particle Amplitude in a Plane Sound Wave

1.5 Spherical Sound Waves

A. Pressure in a Spherical Sound Wave ...

B. Particle Velocity in a Spherical Sound Wave

C. Phase Angle between the Pressure and Particle Velocity in a

Spherical Sound Wave.

D. Ratio of the Absolute Magnituiles of the Particle Velocity and

the Pressure in a Spherical Sound Wave

1.6 Stationary Sound Waves

1.7 Sound Energy Density. . .

1.8 Sound Intensity

1.9 Decibels (Bels)

1.10 Doppler Effect

1.11 Refraction and Diffraction ...

1.12 Acoustical Rectprocii v Theorem

1.13 Acoustical Principle of SiMii ARMY

1.14 Longitudinal Waves in a Rod

1.15 Torsional Waves in a Rod . . .

1

2

4

5

6 7

10

11

12

13

14

16

17

18 19 21 23 23 25

II. ACOUSTICAL RADIATING SYSTEMS

2.1 Introduction 26

2.2 Simple Point Source 26

A. Point Source Radiating into an Infinite Medium. Solid Angle

of 4w Stcradians 26

vii

viii

CONTENTS

Chapter Page

B. Point Source Radiating into a Semi-Infinite Medium. Solid Angle

of 2t Steradians 27

C. Point Source Radiating into a Solid Angle of ir Steradians 27

D. Point Source Radiating into a Solid Angle of v/l Steradians 27

E. Application of the Simple Source 27

2.3 Double Source (Doublet Source) 28

2.4 Series of Point Sources 31

2.5 Straight Line Source 32

2.6 Tapered Straight Line Source 33

2.7 Nonuniform Straight Line Source. . . . 34

2.8 Curved Line Source (Arc of a Circle) 34

2.9 Circular Ring Source 37

2.10 Plane Circular Surface Source. . . 38

2.11 Nonuniform Plane Circular Surface Source 39

2.12 Plane Square Surface Source 39

2.13 Plane Rectangular Surface Source 40

2.14 Horn Source 40

A. Exponential Horns 41

B. Conical Horns 42

C. Parabolic Horns 43

2.15 Curved Surface Source 44

2.16 Cone Surface Source 47

III. MECHANICAL VIBRATING SYSTEMS

3.1 Introduction 50

3.2 Strings 50

3.3 Transverse Vibration of Bars 52

A. Bar Clamped at One End 52

B. BarF rce at Both Ends 54

C. Bar Clamped at Both Ends 54

D. Bar Supported at Both Ends 54

E. Bar Clamped at One End and Supported at the Other 55

F. Bar Supported at One End and Free at the Other 55

G. Tapered Cantilever Bars. . . 55

3.4 Stretched Membranes 56

A. Circular Membrane 56

B. Square Membrane 58

C. Rectangular Membrane 58

3.5 Circular Plates 58

A. Circular Clamped Plate 59

B. Circular Free Plate 61

C. Circular Plate Supported at the Center 62

D. Circular Plate Supported at the Outside 62

3.6 Longitudinal Vibration of Bars 62

CONTENTS

IX

Chapter Page

3.7 Torsional Vibration of Bars 64

3.8 Open AND Closed Pipes 65

IV. DYNAMICAL ANALOGIES

4.1 Introduction 67

4.2 Definitions 69

4.3 Elements 74

4.4 Resistance 75

A. Electrical Resistance . . 75

B. Mechanical Rectilineal Resistance 75

C. Mechanical Rotational Resistance 75

D. Acoustical Resistance 76

4.5 Inductance, Mass, Moment of Inertia, Inertance 76

A. Inductance . 76

B. Mass . . 77

C. Moment of Inertia. .... 77

D. Inertance 77

4.6 Electrical Capacitance, Rectilineal Compliance, Rotational

Compliance, Acoustical Capacitance 78

A. Electrical Capacitance. . 78

B. Rectilineal Compliance . . 79

C. Rotational Compliance . . 79

D. Acoustical Capacitance 79

4.7 Representation of Electrical, Mechanical Rectilineal, Mechani- cal Rotational and Acoustical Elements 80

V. ACOUSTICAL ELEMENTS

5.1 Introduction 86

5.2 Acoustical Resistance 86

5.3 Acoustical Impedance of a Tube of Small Diameter 87

5.4 Acoustical Impedance OF A Narrow Slit. . . . .... 87

5.5 Acoustical Resistance of Silk CIoth .... . . 88

5.6 Inertance . . . . 89

5.7 Acou.stical Capacitance. 89

5.8 Mechanical and Acoustical Impedance Load upon a Vibrating

Piston 90

5.9 Mechanical and Acoustical Impedance Load Upon a Pulsating

Sphere . . ..... 91

5.10 Mechanical and Acoustical Impedance Load Upon an Oscillating

Sphere ... ... .... ... 93

5.11 Acoustical Impedance of a Circular Orifice in a Wall of Infinitesi- mal Thickness . ... 94

5.12 Acoustical Impedance of an Open Pipe with Large Flanges 94

5.13 Horns 94

X

CONTENTS

Chapter Page

5.14 Fundamental Horn Equation 95

5.15 Infinite Cylindrical Horn (Infinite Pipe) 96

5.16 Infinite Parabolic Horn .... 96

5.17 Infinite Conical Horn 97

5.18 Infinite Exponential Horn 98

5.19 Infinite Hyperbolic Horn .... 99

5.20 Throat Acoustical Impedance Characteristic of Infinite Para- bolic, Conical, Exponential, Hyperbolic and Cylindrical Horns . . 100

5.21 Finite Cylindrical Horn. . 100

5.22 Finite Conical Horn . 102

5.23 Finite Exponential Horn . 104

5.24 Throat Acoustical Impedance Characteristics of Finite Exponen- tial Horns 106

5.25 Exponential Connectors . . 109

5.26 A Horn Consisting of Manifold Exponential St.ciions Ill

5.27 Closed Pipe with a Flange .... 112

5.28 Sound Transmission in Tubes . . 113

5.29 Transmission from One Pipe to Another Pipe of Different Cross-

sEcnoNAL Area. . ... 114

5.30 Transmission Throigh Three Pipes 117

5.31 Transmission FROM One Medium TO Another Medium. 119

5.32 Transmlssion Through Three Media. .... 119

5.33 Tubes Lined with .Absorbing Material 120

5.34 Response of a Vibrating System of One Decree of Freedom 120

^VI. DIRECT RADIATOR LOUD SPEAKERS

6.1 Introduction 123

f 6.2 Single-Coil, Single-Cone Loud Speaker 124

6.3 Multiple, Single-Cone, SiNCLE-CoiL Loud Speaker 136

♦ 6.4 Single-Coil, Double-Cone Loud Speaker 139

6.5 Double-Coil, Single-Cone Loud Speaker 140

6.6 Double-Coil, Double-Cone Loud Speaker. . . 141

• 6.7 Mechanical Networks for Controlling the High Frequency Re-

sponse of a Loud Speaker . ... 141

A. Conventional Single Coil Ijoud Speaker. . . ... 143

B. Loud Speaker with a Compliance Shunting the Cone Mechanical

Impedance 143

C. Loud Speaker with a Compliance Shunting; a Compliance and

Mass in Parallel, Connected in Series with the Cone Mechanical Impedance 144

D. Loud Speaker with a “ T ” Type Filter Connecting the Voice Coil

Mass and the Cone Mechanical Impedance 144

f 6.8 Loud Speaker Baffles 145

A. Irregular Baffle 145

• B. Large Baffle, Different Resonant Frequencies 146

CONTENTS xi

Chapter Page

C. Low Resonant Frequency, Different Baffle Sizes 146

• D. Different Resonant Frequencies and Different Baffle Sizes 148

• 6.9 Cabinet Loud Speakers 148

A. Low Resonant Frequency, Different Cabinet Sizes 150

B. Different Resonant Frequencies and Different Cabinet Sizes 151

C. Effect of the Depth of the Cabinet 151

^ 6.10 Back Enclosed Cabinet Loud Speaker .. 151

t 6.11 Acoustical Phase Inverter Loud Speaker 154

, 6.12 Acoustical Labyrinth Loud Speaker 155

6.13 Combination Horn and Direct Radiator Loud Speaker 156

^ 6.14 Feedback Applied io a Loud Speaker. ... 158

, 6.15 Transient Response 159

^6.16 Distortion ... 163

A. Nonlinear Suspension System ... 164

B. Distortion Characteristics of a Nonlinear Suspension System 166

C. Response Frequency Characteristics of a Direct Radiator Loud

Speaker with a Nonlinear Suspension System . . . 168

D. Distortion Due to Inhomogcneity of the Air Gap Flux ... . 169

E. Frequency — Modulation Distortion . 171

F. Air Nonlinear Distortion. . .. 172

6.17 Diaphragms, Suspensions, and Voice Coils 173

6.18 High Frequency Sound Disiributor 176

6.19 Field Structi RES ..... 177

VIL HORN LOUD SPEAKERS

7.1 Introduction ..... 184

7.2 Efficiency ... 184

A. The Relation between the Voice Coil Mass, the Load Mechanical

Resistance and the Initial Efficiency 185

B. The Effect of the Mass of the Vibrating System upon the Efficiency 188

C. The Effect of the Air Chamber upon the Efficiency 190

D. The Effect of the Generator Electrical Impedance and the Mechan-

ical Impedance at the Throat of the Horn upon the Efficiency. . 192

E. The Effect of the Voice Coil Temperature upon the Efficiency. . . 193

F. The Effect of the Sound Radiation from the Unloaded Side of the

Diaphragm upon the Efficiency 194

7.3 Distortion 196

A. Distortion Due to Air Overload in the Horn 196

B. Distortion Due to Variation in Volume of the Air Chamber 198

C. Distortion Due to the Diaphragm Suspension System . . . 200

D. Distortion Due to a Nonuniform Magnetic Field in the .Air Gap. . 200

E. Subharmonic Distortion 201

F. Power Handling Capacity and the Voice Coil Temperature 201

G. Power Handling Capacity and the .Amplitude of the Diaphragm. . 202

XU

CONTENTS

Chapter Page

7.4 Horn Loud Speaker Systems 203

A. Single Horn, Single Channel System 203

B. Multiple Horn, Multiple Channel System 206

C. Compound Horn Loud Speaker 208

D. Multiple Horn, Single Channel System 209

E. Folded Horns 210

F. Horn Loud Speaker Mechanisms 211

G. Diaphragms and Voice Coils 212

H. Field Structures 212

I. Horn Walls. Vibration and Absorption 213

/VIIL MICROPHONES

8.1 Introduction 214

8.2 Pressure Microphones 214

A. Carbon Microphones 214

1. Single Button Carbon Microphone 214

2. Double Button Carbon Microphone 218

B. Condenser Microphone 220

C. Piezoelectric (Crystal) Microphones 224

1. Direct Actuated Crystal Microphone 224

2. Diaphragm Actuated Crystal Microphone 226

D. Moving Conductor Microphones 226

1. Moving Coil Microphone (Dynamic Microphone) . 226

2. Inductor Microphone (Straight Line Conductor) . . 229

3. Ribbon Microphone 229

E. Electronic Microphone 233

8.3 Velocity Microphones, First Order Gradient Microphones 237

A. Pressure Gradient Microphone 237

B. Velocity Microphone 241

8.4 Unidirectional Microphones 253

A. Combination Unidirectional Microphones. , . 253

1. The Response of the Unidirectional Microphone as a Func- tion of the Distance and the Frequency 255

2. Efficiency of Energy Response to Random Sounds of the

Unidirectional Microphone as a Function of the Relative Sensitivities of the Bidirectional and Nondirectional Micro- phones 256

3. Efficiency of Energy Response to Random Sounds of a

Unidirectional Microphone as a Function of the Phase Angle Between the Two Units 257

4. Distortion of the Directional Pattern in the Unidirectional

Microphone 260

B. Single Element Unidirectional Microphones 260

1. Phase Shifting Unidirectional Microphone 260

2. Polydirectional Microphone 261

CONTENTS

xiii

Chapter Page

3. Uniphase Dynamic Microphone 266

4. Dipole Microphone 268

5. Differential Microphone. Lip Microphone 269

8.5 Higher Order Gradient Microphones 269

A. Second Order Gradient Microphones 270

B. Gradient Microphones of Any Order 270

C. Noise Discrimination of Gradient Microphones 271

D. Higher Order Unidirectional Gradient Microphones 275

8.6 Wave Type Microphones 276

A. Parabolic Reflector 277

B. Line Microphones 278

1. Line Microphone: Useful Directivity on the Line Axis.

Simple Line 278

2. Line Microphone: Useful Directivity on the Line Axis.

Line with Progressive Delay 279

3. Line Microphone: Useful Directivity on the Line Axis.

Two Lines and a Pressure Gradient Element 281

4. Ultradirectional Microphone 283

8.7 Directional Efficiency of a Directional Sound Collecting System. 285

8.8 Throat Microphone 286

8.9 Lapel and Boom Microphones 287

8.10 Wind Excitation and Screening of Microphones 288

8.11 Nonlinear Distortion IN Microphones 289

8.12 Transient Response of Microphones 289

8.13 Noise in a Sound Pickup System 290

A. Ambient Noise in the Studio 291

B. Noise Due to Thermal Agitation of the Air Molecules 291

C. Noise Due to Thermal Agitation of the Atoms in the Vibrating

System 292

D. Noise Due to Thermal Agitation of the Electrons in the Conductor 292

E. Noise Due to Barkhausen EflFect in the Transformer. . 293

F. Noise in the Vacuum Tube 293

G. Noise Due to Thermal Agitation of the Electrons in the Plate

Resistor 293

H. Example of Noise in a Sound Pickup System 293

IX. MISCELLANEOUS TRANSDUCERS

9.1 Introduction 295

9.2 Telephone Receivers 295

A. Bipolar Telephone Receiver 295

B. Crystal Telephone Receiver 299

C. Dynamic Telephone Receiver 300

D. Inductor Telephone Receiver 301

XIV

CONTENTS

Chapter Page

9.3 Phonographs 303

' A. Recording Systems 304

1. Lateral Recorder 304

2. Vertical Recorder 304

3. Recording Characteristics 305

, B. Mechanical Phonograph 307

C. Phonograph Pickups 308

1. Crystal Pickup 308

2. Magnetic Pickup 309

3. Dynamic Pickup 311

4. Frequency Modulation Pickup 313

5. Electronic Pickup. . . 314

6. Variable Resistance Pickup 315

^ D. Distortion in Record Reproduction. . . 315

^ E. Record Noise 317

9.4 Vibration Pickup 318

9.5 Sound Powered Phones 320

9.6 Electrical Megaphone . . . . 322

* 9.7 Magnetic Wire Sound Reprodi cinc S^sTEM 323

9.8 Motion Picture Film Sound Reproducing System 325

A. Recording System 325

1. Variable Area. . 325

2. V^^riable Density 327

B. Reproducing System 328

9.9 Hearing Aids. . ... 329

9.10 Electrical Musical Instruments. . 331

9.11 Sirens 333

9.12 Compressed Air Loud Speaker. . 333

9.13 Seismic Detectors 334

9.14 Stethoscopes 334

9.15 Ear Defenders 338

X. MEASUREMENTS

10.1 Introduction 340

y l(k2 Caubfation of Microphones 340

A. Response Frequency Characteristic . 340

1. Pressure Response 340

a. Pistonphone 341

b. Thermophone 342

c. Electrostatic Actuator 343

2. Field Response . .... 344

a. Rayleigh Disk 344

b. Reciprocity 345

3. Artificial Voice 349

CONTENTS

XV

Chapter

4. Artificial Throat 349

B. Directional Characteristic 350

C. Nonlinear Distortion Characteristic 351

D. Phase Distortion Characteristic 352

E. Electrical Impedance Frequency Characteristic 352

F. Transient Response Characteristic 353

^ 10.3 Testing of Loud Speakers 353

A. Response Frequency Characteristic 353

1. Pressure Response 353

2. Apparatus for Measuring the Sound Pressure Frequency

Relationship of a Sound Source 354

3* Calibration of the Sound Measuring Equipment 357

4. F'rec Field Sound Room 359

5. Outdcxir Response 363

6. Small and Partially Deadened Room 364

7. Arrangement of Loud Speakers for Test 364

8. Living Room Measurements 365

9. Theater Measurements 365

10. Automobile Measurements 365

B. Directional Characteristic 366

C. Nonlinear Distortion Characteristic 366

D. Efficiency Frequency Characteristic 370

1 . Direct Determination of Radiated Power 370

2. Indirect Determination of Radiated Power 373

E. Phase Distortion Characteristic 374

F. Electrical Impedance Frequency Characteristic 374

G. Transient Response Characteristic 375

H. Subjective Measurements 376

^ 10.4 Testing OF Telephone Receivers 376

A. Subjective Measurements 376

B. Objective Measurements 377

1. Artificial Ear 377

2. Artificial Mastoid 378

10.5 Testing OF Phonographs 378

A. Measurement of the Response of a Phonograph Record by the

Optical Method 378

B. Testing of Phonograph Pickups 380

C. Testing of'Mechanical Phonographs 380

D. Measurement of Mechanical Noise Produced by a Phonograph

Pickup 380

10.6 Measurement OF Wows 381

10.7 Measurement of Acoustical Impedance 381

10.8 Mechanical Impedance Bridge. 384

♦ 10.9 Measurement OF Porosity 387

• 10,10 Measurement of D.C, Acoustical Resistance (Flow Resistance). . . 388

XVI

CONTENTS

^ 10.11 Measurement or Reverberation Time 390

^ 10.12 Measurement of Absorption Coefficient 391

10.13 Measurement OF Noise 392

, 10.14 Measurement OF Transmission Coefficient 394

I 10.15 Audiometry 394

^ 10.16 Articulation Measurements 395

^ 10.17 Testing of Hearing Aids 395

XI. ARCHITECTURAL ACOUSTICS AND THE COLLECTION AND DIS- PERSION OF SOUND

11.1 Introduction 397

11.2 Dispersion OF Sound 398

. A. Sound Absorption and Reverberation 398

• B. Mechanism of Sound Absorption by Acoustical Materials 401

/ C. Functional Sound Absorbers 405

^ D. Articulation and Reverberation Time 407

E. Sound Motion Picture Reproducing System 408

F. Sound Re-enforcing Sy'stem 411

' G. Theater Acoustics 415

- H. Reverberation Time of a Theater for the Reproduction of Sound. . 416

1. Power Requirements for Reproducing Systems 417

' J. Noise at Different Locations 419

* K. Public Address Systems 420

^ L. Orchestra and Stage Shell 424

M. General Announce and Paging Systems 425

N. Intercommunicating Systems 426

O. Radio Receiver Operating in a Living Room 427

P. Radio Receiver Operating in an Automobile 428

Q. Absorption of Sound in Pas.sing Through Air. 429

. R. Sound Transmission Through Partitions 430

yil,3 Collection of Sound 432

A. Sound Collecting System 432

' B. Broadcasting Studios 436

• C. Scoring and Recording Studios 440

• D. Vocal Studios 441

' E. Reverberation Time of Broadcasting, Recording and Scoring

Studios 442

F. Sound Stages for Motion Pictures and Television 442

- G. Synthetic Reverberation 445

• H. Volume Limiters, Compressors and Expanders 446

11.4 Complete Sound Reproducing Systems 447

A. Telephone 447

B. Binaural Sound Reproducing System 448

C. Monaural Sound Reproducing System 449

D. Auditory Perspective Reproducing System 450

CONTENTS xvii

E. Sound Motion Picture Reproducing System 450

F. Radio Sound Reproducing System 452

♦ G. Phonograph Reproducing System 454

• H. Magnetic Sound Reproducing System 455

XII. SPEECH, MUSIC AND HEARING

12.1 Introduction 457

12.2 Hearing Mechanism, . . 457

12.3 Voice Mechanism 459

12.4 Artificial Voice Mechanisms. . 463

A. Artificial Larynx .... 463

B. Voder ... 465

C. Vocoder . . . 466

12.5 Visible Speech ... 466

12.6 Response Frequency Characteristics of Ears . . 467

12.7 Loudness ... .... 468

12.8 Pitch ... ... ... 468

12.9 Masking ... 469

12.10 Nonlineari rv of the Ear ... 470

12.11 Effect of Phase Rei-afions .Among 1 HE Harmonics .. 472

12.12 Modulation (Vibraio) . 472

12.13 Minimum Pfrcep'iible Differences .. ... 473

12.14 'Fimbre (ToNEQuALI^^) .. 474

12.15 Durmion 475

12.16 Growth and Decay 475

12.17 Auditory Localiza I ION ... 475

12.18 Hearing .Aci iiy in ihk United States Population 476

12.19 The Freoi^ency and Voi.i me Ranges of Speech and Music 477

12.20 The Effect of Frequence Discrimination Upon the Articulation

OF Reproduced Speech 478

12.21 The F.ifect of Frequency Discrimination Upon the Quality of

Reproduced Music .... ... . 479

12.22 Absolute Amplit* des and Spectra of Speech, Musical Instruments

AND Orchestras 480

12.23 Noise in Reprodi ci.ng Systems 483

12.24 Room Noise and the Reproduction of Sound 485

12.25 Combination Tones and Nonlinear Transducers 487

12.26 Kffect of Nonlinear Distortion Upon the Quality of Reproduced

Speech and Music 488

12.27 Frequency Ranges of Sound Reproducing Systems 491

12.28 Frequency Range Preference for Reproduced Speech and Music. . 493

12.29 Frequency Range Preference for Live Speech and Music 495

12.30 Musical Scale ... 497

12.31 Fundamental Frequency Ranges of Voices and Musical Instruments 498

XVlll

CONTENTS

XIII. UNDERWATER SOUND

13.1 Introduction 501

13.2 Sound Waves in Water 501

13.3 Direct Radiator Dynamic Subaqueous Loud Speaker 503

13.4 Subaqueous Condenser Microphone 505

13.5 High Frequency Direct Radiator Dynamic Subaqueous Loud

Speaker and Microphone 506

13.6 Magnetic Subaqueous Loud Speaker 507

13.7 Magnetrostriction Subaqueous Loud Speaker 509

13.8 Magnetrostriction Subaqueous Microphone ... 511

13.9 Quartz Crystal Subaqueous Loud Speaker 512

13.10 Quartz Crystal Subaqueous Microphone. . 515

13.11 Quartz Crystal Sandwich Loud Speaker AND Microphone 516

13.12 Rochelle Salt Cry.stal Subaqueous Loud Speaker and Microphone 517

13.13 Echo Depth Sounding Sonar 518

13.14 Echo Direction Ranging Sonar 519

13.15 Communication Sonar 520

XIV. ULTRASONICS AND SUPERSONICS

14.1 Introduction 522

14.2 Supersonic and Ultrasonic Generators 522

14.3 Cavitation Due to Supersonics 523

14.4 Dispersion Due TO Supersonics 523

\4S Emulsification Due to Superso.sics . . 524

14.6 Coagulation Due to Supersonics 524

14.7 Chemical Effects of Supersonics and Ultra.sonics 525

14.8 Biological Effects of Supersonics 525

14.9 Ther.mal Effects OF Supersonics 526

14.10 Supersonics as a Detergent 526

14.11 Television System Modulation by Ultrasonics 526

14.12 Testing of Materials by Means of Ultrasonics 527

ELEMENTS OF ACOUSTICAL ENGINEERING

CHAPTER I

SOUND WAVES

1.1. Introduction. — A knowledge of the elements of acoustics is be- coming increasingly important to any profession depending in any manner upon acoustics. Modern civilization is becoming more critical of sound reproduction. The radio receiver, phonograph, sound motion picture or sound reinforcing system of a few years ago is not acceptable today. Auditoriums and studios must exhibit proper acoustical qualities. Reduc- tion of noise in all types of machinery and appliances is demanded by the consumers. Acoustics, one of the oldest divisions of physics, appeared to be a decadent science a few years ago. Today it is an important and neces- sary branch of Applied Science and its application to every phase of modern civilization is in its infancy.

The widespread interest in the phonograph, radio broadcasting, tele- vision, sound motion pictures, sound reinforcing, architectural acoustics and noise problems has stimulated research and developments in these fields. Acoustics were involved in almost every type of communication system used in World War II. Accelerated by the war, tremendous ad- vances were made in underwater sound. The industrial applications of ultrasonics and su person ics is beginning to unfold a new field in the appli- cation of sound.

During the early stages progress in the development of acoustical devices was made by the trial-and-error method. Later, by the extension and ap- plication of scientific knowledge, results have been obtained that could not have been accomplished by other means. A major portion of the prob- lems in acoustics is concerned with vibrating systems. Comparisons be- tween these problems and those of electricity considered from a dynamical viewpoint have led to impedance methods in acoustics. By a judicious application of dynamical theory and experimental research, the science of acoustics has developed into a wide field of interesting phenomena with countless useful applications.

In this book, the author has attempted to outline the essentials of acous-

1

2

SOUND WAVES

tics from the standpoint of the engineer or applied scientist. The book has been written and illustrated so that the derivations may be taken for granted. The concepts of mechanical and acoustical impedance have been introduced and applied so that anyone who is familiar with electrical circuits will be able to analyze the action of vibrating systems.

1,2. Sound Waves. — Sound is an alteration in pressure, particle dis- placement or particle velocity propagated in an elastic material or the superposition of such propagated alterations.

Sound is also the sensation produced through the ear by the alterations described above.

Sound is produced when air is set into vibration by any means whatso- ever, but sound is usually produced by some vibrating object which is in contact with the air. If a string, such as one used in a banjo or similar in- strument, is stretched between two solid supports and plucked, sound is produced which dies down in a fairly short time. When the string is plucked it tends to spring back into its rest position, but due to its weight (mass) and speed (velocity) it goes beyond its normal position of rest. Then, in returning it again goes beyond its normal position of rest. The excursions become smaller and smaller and finally the string comes to rest. As the string moves forward it pushes air before it and compresses it, while air rushes in to fill the space left behind the moving string. In this way air is set in motion. Since air is an elastic medium, the disturbed portion transmits its motion to the surrounding air so that the disturbance is propa- gated in all directions from the source of disturbance.

If the string is connected in some way to a diaphragm such as a stretched drumhead of a banjo, the motion is transmitted to the drum. The drum, having a large area exposed to the air, sets a greater volume of air in motion and a much louder sound is produced.

If a light piston several inches in diameter, surrounded by a suitable baffle board several feet across, is set in rapid oscillating motion (vibration) by some external means, sound is produced (Fig. 1.1). The air in front of the piston is compressed when it is driven forward, and the surrounding air expands to fill up the space left by the retreating piston when it is drawn back. Thus we have a series of compressions and rarefactions (expan- sions) of the air as the piston is driven back and forth. Due to the elas- ticity of air these areas of compression and rarefaction do not remain sta- tionary but move outward in all directions. If a pressure gage were set up at a fixed point and the variation in pressure noted, it would be found that the pressure varies in regular intervals and in equal amounts above and below the average atmospheric pressure. Of course, the actual varia-

SOUND WAVES

3

tions could not be seen because of the high rate at which they occur. Now, suppose that the instantaneous pressure, along a line in the direction of sound propagation, is measured and plotted with the ordinates representing the pressure; the result would be a wavy line as shown in Fig. 1.1. The points above the straight line represent positive pressures (compressions, condensations); the points below represent negative pressures (expansions, rarefactions) with respect to the normal atmospheric pressure represented by the straight line.

Fig. 1.1. Production of sound waves by a vibrating piston.

From the above examples a few of the properties of sound waves and vibrations in general may be defined

Periodic Quantity. — A periodic quantity is an oscillating quantity the values of which recur for equal increments of the independent variable.

Cycle, — One complete set of recurrent values of a periodic quantity comprises a cycle; or, in other words, any one set of variations starting at one condition and returning once to the same condition is a cycle.

Period. — The period is the time required for one cycle of a periodic quantity.

Frequency. — -The number of cycles occurring per unit of time, or which would occur per unit of time if all subsequent cycles were identical with the cycle under consideration is the frequency. The unit is the cycle per second.

Fundamental Frequency. — A fundamental frequency is the lowest com- ponent frequency of a periodic wave or quantity.

Harmonic. — A harmonic is a component of a periodic wave or quantity having a frequency which is an integral multiple of the fundamental fre- quency. For example, a component, the frequency of which is twice the fundamental frequency, is called the second harmonic.

4

SOUND WAVES

Subharmonic. — A subharmonic is a component of a complex wave hav- ing a frequency which is an integral submultiple of the basic frequency.

IVavelength, — The wavelength of a periodic wave in an isotropic medium is the perpendicular distance between two wave fronts in which the dis- placements have a phase difference of one complete cycle.

Octave. — An octave is the interval between two frequencies having a ratio of two to one.

Transducer. — A transducer is a device by means of which energy may flow from one or more transmission systems to one or more other trans- mission systems. The energy transmitted by these systems may be of any form (for example, it may be electrical, mechanical or acoustical) and it may be the same form or different froms in the various input and out- put systems.

The example of Fig. 1.1 has shown graphically some of the properties of wave motion. It is the purpose of the next section to derive the fun- damental wave equation. It is not necessary that the reader digest all the assumptions and processes involved in order to obtain valuable infor- mation concerning the properties of a sound wave.

1.3. Acoustical Wave Equation. — The general case of sound propaga- tion involves three dimensions. The general relation for sound propaga- tion of small amplitudes in three dimensions will be derived and then these relations will be applied to special problems.

A. Equation oj Continuity. — The fundamental equation of hydrokinetics is the equation of continuity. This equation is merely a mathematical statement of an otherwise obvious fact that matter is neither created nor destroyed in the interior of the medium. That is, the amount of matter which enters the boundaries of a small volume equals the increase of matter inside. Consider the influx and efflux through each pair of faces of the cube of dimensions A^ and Az, the difference between the latter and the former for the whole cube is

where Xy y^z = coordinates of a particle in the medium,

UyVyW ^ component velocities of a particle in the medium, and p' = density of the medium.

The rate of growth of mass — A^ Ay Az in the cube must be equal to

ACOUSTICAL WAVE EQUATION

5

the expression 1.1. This may be written as

dt dx dy dz

where t = time.

This is the equation of continuity which signifies the conservation of mat- ter and the three dimensionality of space.

B. Equation of Motion, — Referring again to the space Lx Ly Az the

acceleration of momentum parallel to a; is p' A^ Az ^ • The mean pres- sures on the faces perpendicular to x are

^ f) f )

where />o' = pressure in the medium.

The difference is a force — A.v Ly Az in the direction of increasing x.

Equating this to the acceleration of momentum, the result is the equation of motion,

^ dt dx ^ dt dy ^ dt dz

The equation of motion may be written

^ + - Grad po' = 0 1.4

at p

C. Compressibility of a Gas. — The next property of a gas which is used to derive the wave equation depends upon the thermodynamic properties of gases. The expansions and contractions in a sound wave are too rapid for the temperature of the gas to remain constant. The changes in pres- sure and density are so rapid that practically no heat energy has time to flow away from the compressed part of the gas before this part is no longer compressed. When the gas temperature changes, but its heat energy does not, the compression is termed adiabatic.

In the case of an adiabatic process,

6

SOUND WAVES

where pQ « static pressure. The static pressure is the pressure that would exist in the medium with no sound waves pres- ent. The unit is the dyne per square centimeter, p = static or original density,

= total pressure (static + excess), p' = instantaneous density (static + change), and 7 = ratio of specific heat at constant pressure to that at con- stant volume and has a value of 1 .4 for air.

D. Condensation. — A new term will now be introduced. Condensation is defined as the ratio of the increment of density change to the original density.

P

Combining equations 1 .5 and 1 .6

= (-7= (1 = 1 +7J 1.7

/>0 \P /

or pd = />o + />o7'^ 1*8

The excess pressure, or instantaneous sound pressure />, is — />o»

p = pQys 1.9

The instantaneous sound pressure at a point is the total instantaneous pressure at that point minus the static pressure. The unit is the dyne per square centimeter. This is often called excess pressure.

The effective sound preSvSure at a point is the root-mean-square value of the instantaneous sound pressure over a complete cycle, at that point. The unit is the dyne per square centimeter. The term “ effective sound pressure is frequently shortened to “ sound pressure.”

The maximum sound pressure for any given cycle is the maximum absolute value of the instantaneous sound pressure during that cycle. The unit is the dyne per square centimeter. In the case of a sinusoidal sound wave this maximum sound pressure is also called the pressure am- plitude.

The peak sound pressure for any specified time interval is the maxi- mum absolute value of the instantaneous sound pressure in that interval. The unit is the dyne per square centimeter.

A dyne per square centimeter is the unit of sound pressure.

ACOUSTICAL WAVE EQUATION

7

E. jy Alemhertian Wave Equation, — The three equations 1.2, 1.4 and 1.5 characterize disturbances of any amplitude. The first two are non- linear save for small amplitudes. In general, acoustic waves are of in- finitesimal amplitudes, the alternating pressure is small compared with the atmospheric pressure and the wavelength is so long that «, y, w and s change very little with y and z. Substituting equation 1.6 in 1.2 and neglecting high order terms,

ds du dv dw dt~^ dx dy^ dz

1.10

The type of motion to be considered is irrotational, that is Curl Fuvw = 0. That is a necessary and sufficient condition for the existence of a scalar velocity potential <p which is defined as

dcf) d(l>

« = — I V = - - f

dx dy

f^uvw = Grad

w —

dz

1.11

or

Substitute equations 1.11 in 1.3 and multiply by dxy dy and dz

^</<#> = - \ dpo' ot p

1.12

or integrating

Cdp{\

^ J "7”

Since the density changes very little, the mean density, p, may be used. The fdp^ is the excess pressure; then

dt p

1.13

where p = excess pressure.

From equations 1.9, 1.10, 1.11 and 1.13

d^4> _ ypo/^ 1 ^ _i_ = 0

d/2 p d/ dzV

1.14

d^<b

or this may be written

8

SOUND WAVES

which is the standard D’Alcmbertian wave equation for <t>. The velocity of propagation is

2£? = ,*

p

1.15

For the velocity of sound in various mediums see Table 1.1.

Table l.I. young’s modulus in dynes per square centimeter, poisson’s ration, DENSITY p, IN GRAMS PER CUBIC CENTIMETER, VELOCITY OF SOUND r, IN METERS PER SECOND, AND THE SPECIFIC ACOUSTICAL RESISTANCE pf, IN GRAMS PER SECOND PER SQUARE CENTIMETER

Metals

Substance		a		)	c	pc
Aluminum	1	3	X	101»	.33	2	7	52(K)	140	X	10^
Antimony	1	8	X	10»	.33	h	6	34(X)	220	X	10*
Beryllium	12	7	X	10^1	.33	1	8	84(K)	150	X	10*
Bismuth	3	19 >	c 10»»	35	9	7	I8(M)	170	X	10*
Cadmium	5	3	X	10“	30	8	r>	25(X)	215	X	10*
Cobalt	19	0	X	10“	30	S	7	47(KJ	410	X	10*
Sir: :::::	11	0	X	10“	35	8	9	3500	310	X	10*
8	0	X	10»‘	35	19	3	2'XX)	390	X	10*
Iridium	5	2	X	10“	.33	22	4	1 500	340	X	10*
Iron Cast	9	0	X	10“	29	7	8	34v'X)	270	X	10*
Iron Wrought	20	0	X	10“	.28	7	9	5100	400	X	10*
L«ad	1	7	X	10“	43	11	3	12(X)	130	X	10*
Magnesium	4	0	X	10“	.33	1	7	48(X)	82	X	10*
Mercury					. .	13	5	14(K)	190	X	10*
Nickel.'	21	0	X	10“	.31	8	8	4900	430	X	10*
Palladium	12	0	X	10“	.39	12	0	32(X)	380	X	10*
Platinum	17	0	X	10“	33	21	4	28(K)	WX)	X	10*
Rhodium	30	0	X	10“	.34	12	4	4900	610	X	10*
Silver	7	8	X	10“	37	10	5	27»XJ	280	X	10*
Tantalum	19	0	X	10“	31	U)	6	34(X)	560	X	10*
Tin	4	5	X	10“	.33	7	3	25(X)	180	X	10*
Tungsten	35	0	X	10“	.17	19	0	43(X)	830	X	10*
Zinc	8	2	X	10“	17	7	1	34(X)	240	X	10*

Alloys

Alnico	17	0	X	10“	32	7 0	4900	340	X	10*
Beryllium Copper ....	12	5	X	10“	33	8.2	39(X)	320	X	10*
Brass	9	5	X	10“	33	8.4	34(X)	290	X	10*
Bronze Phosphor	12	0	X	10“	35	8 8	37(K)	330	X	10*
Duraluminum	7	0	X	UP*	.33	2 8	50(K)	140	X	10*
German Silver	11	6	X	10“	.37	8.1	3800	310	X	10*
Monel	18	0	X	10“	.32	8 8	45(X)	400	X	10*
Steel C.08	19	0	X	10“	.27	7.7	5000	390	X	10*
Steel C.38	20	0	X	10"	29	7 7	SUX)	390	X	10*

ACOUSTICAL WAVE EQUATION

Ceramics, Rocks

9

Substance	a	<r	P	c	pc
Brick	2 5 X 10*1		1.8	3700	67 X 10*
Clay Rock	2.5 X 10^^		2.2	3400	75 X 10*
Concrete	2 5 X 10“		2.6	3100	81 X 10*
Glass, Hard	8 7 X 10“		2.4	6000	144 X 10*
Glass, Soft	6 0 X 10^'		2.4	5000	120 X 10*
Ciranite	9 8 X 10^1		2.7	6000	162 X 10*
1 solan tite	5 0 X 10“		2.4	4600	no X 10*
Limestone	2 9 X 1011		2.6	3300	86 X 10*
Marble	3 8 X 10“		2.6	3800	99 X 10*
Porcelain	4 2 X 1011		2.4	4200	102 X 10*
Quartz, Fused	5 2 X 1011		2.7	4400	118 X 10*
Quartz, 11 Optic	10 3 X 1011		2.7	6200	168 X 10*
Quartz, 1 Optic	7 95 X 1011		2.7	5400	146X10*
^ate	5 8 X 1011		2 9	4500	131 X 10*
Ice	94 X 1011		.92	3200	29 X 10*

Woods (w’ith the grain)

Ash	1 3 X 10»	.64	4500	29 X 10*
Beech	1 0 X 10“	.65	3900	25 X 10*
Cork	CU62 X 10^*	.25	500	1.2 X 10*
Elm	1 0 X 10“	.54	4300	23 X 10*
Fir	1.1 X 10“	.51	4700	24 X 10*
Maliogany	1 1 X 10“	.67	4000	27 X 10*
Maple	1 3 X 10“	.68	4300	29 X 10*
Oak, White	1.2 X 10“	.72	4100	29 X 10*
Pine, White	6 X 10“	.45	3600	16 X 10*
Poplar	1 0 X 10“	.46	4600	21 X 10*
Sycamore	1 0 X 10“	.54	4300	23 X 10*
Walnut	1.2 X	.56	4600	26 X 10*

Across the grain, i to J of the above values fore.

Plastics

Cellulose Acetate,				1
Sheet . . .	1.4 X lO'o		1.3	' 1000	13 X 10*
Cellulose Acetate,
Molded . . .	2 1 X 10^0		1.3	1300	17 X 10*
Cellulose Acetate,
Butyrate Cellulose Acetate, Py-	17.0 X 10'°		1.2	3700	44X 10*
roxylin	21 0 X 10'°		1.5	1 3700	55 X 10*
Ethyl Cellulose . ...	2 1 X lO'o		1.1	1 1400	15 X 10*
Ivorv Methyl Metha-Crylate	9 0 X 10'"		1.8	2200	40 X 10*
3.5 X 10“		1.2
Resin, Cast		1700	20 X 10*
Methyl Metha-Crylate	2.8 X 10“		1.2
Resin, Molded . . .		1500	18 X 10*
Paper, Parchment . . .	4 8 X 10“		1.0	2200	22 X 10*
Paraffin, 16** C.	1 5 X 10“	...	.9	1300	12 X 10*

10

SOUND WAVES

Plastics (continued)

Substance		<r	P	c	pc
Phenol- Form aldehyde Wood F'iller	8.4 X 10'®		1.35	2500	34 X 10^
Phenol-Formaldehyde Paper Base .	7.0 X 10“		1.3	2300	30 X 10*
Phenol-Formaldehyde Fabric Base	8 4 X 101°		1.35	2500	34 X 10^
Phenol-Formaldehyde Mineral Filler. . . .	10 5 X 101°		1.8	2400	43 X 10*
Rubber, Hard ... .	2.3 X 101°		1.1	14(X)	15 X 10*
Rubber, Soft .	5 X 10“		.95	70	67 X 10*
Sheepskin	2 0 X 10»		.9	470	4 2 X 10*
Shellac Compound . . .	3 8 X 101°		1.7	1500	26 X 10*
Styrene Resin	3 1 X 101°		1.1	1 1700	19 X 10*

Liquids

Alcohol, Methyl	.81	1240	10 0 X 10*
Benzine	90	1170	10.5 X 10*
Chloroform Ether	1.5 .74	983 1020	14.7 X 10* 7.6 X 10*
Gasoline Turpentine Water, 13° C W’ater, Salt	.68 .87 1.0 1.03	1390 1330 1441 1504	9.4 X 10* 11.6 X 10* 14.4 X 10* 15 5 X 10*

Gases

Air, 0° C.	.00129	331	42 7
Air, 20^ C.. . .	.00120	344	41 4
Carbon Monoxide . . .	.00125	337	42 0
Carbon Dioxide. . .	.00198	258	51.2
Chlorine	.00317		65.0
Hydrogen	00(XJ9	1270	11.4
Methane	.00072	432
Nitrogen	.00125	336	42 0
Oxygen	.00143	317	45 5
Steam	.00058	405	23.5

1.4. Plane Sound Waves. — Assume that a progressive wave proceeds along the axis of x. Then <ff is a function of x and / only and the wave equation 1.14 reduces to

^ = 2 ^ a/2 ^ 3x2

1.16

A solution of this equation for a simple harmonic wave traveling in the positive X direction is

0 = y/ cos i(c/ — x)

1.17

PLANE SOUND WAVES

11

where A = amplitude of 0, k = 27r/X,

X = wavelength, in centimeters,

c — fK — velocity of sound, in centimeters per second, and / = frequency, in cycles per second.

A. Particle Velocity in a Plane Sound fPave. — The particle velocity, «, employing equations 1.11 and 1.17 is

d(i>

u ^ — = kA sin k{ct — x) 1.18

dx

The particle velocity in a sound wave is the instantaneous velocity of a given infinitesimal part of the medium, with reference to the medium as a whole, due to the passage of the sound wave.

B. Pressure in a Plane Sound IVave. — h'rom equations 1.9, 1.13 and 1.15 the following relation may be obtained

1.19

The condensation in a plane wave from equations 1.19 and 1.17 is given by

j = — sin k{ct — .v) 1.20

From equations 1.9 and 1.15 the following relation may be obtained

p = rpj 1.21

Then, from equations 1.20 and 1.21 the pressure in a plane wave is

p == kcpA sin k{ct — .v) 1.22

Note: the particle velocity, equation 1.18, and the pressure, equation 1.22, are in phase in a plane wave.

C. Particle Amplitude in a Plane Sound W ave, — The particle amplitude of a sound wave is the maximum distance that the vibrating particles of the medium are displaced from the position of equilibrium.

From equation 1.18 the particle velocity is

{ = « = kA sin k{ct — ;c) 1.23

where { = amplitude of the particle from its equilibrium position, in centimeters.

12

SOUND WAVES

The particle amplitude, in centimeters, is

£ = — cos k(ct — x)

From equations 1 .20 and 1 .24 the condensation is

s =

dx

1.24

1.25

1.6. Spherical Sound Waves. — Many acoustical problems are concerned with spherical diverging waves. In spherical coordinates x = r sin ^ cos y = r sin 0 sin yp and z = r cos 6 where r is the distance from the center, $ is the angle between r and the oz axis and ^ is the angle between the projection of r on the xy plane and ox. Then becomes

dr^ r dr r- sin 6 dd

dd sin2 Q

1.26

For spherical symmetry about the origin

The general wave equation then becomes, d- d'^

— {r<l>) = f-— {r<t>) 1.28

dr dr^

The wave equation for symmetrical spherical waves can be derived in another way. Consider the flux across the inner and outer surfaces of the spherical shell having radii of r — Ar/2 and r + Ar/2, the difference is

1.29

The velocity is

dr _ d^ dt Or

1.30

where <l> = velocity potential.

The expression 1.29 employing equation 1.30 becomes

1.31

SPHERICAL SOUND WAVES

13

The rate of growth of mass in the shell is

dp^

47rr^ — Ar

a/

U2

The difference in flux must be equal to the rate of growth of mass^ expres- sions 1.31 and 1.32,

Using equations 1.6, 1.9 and 1.13, equation 1.33 may be written,

1.34

a/- dr \ dr/

Equation 1.34 may be written

d-ir4>) , d^(.r4>)

dr-

— c-

dr^

= 0

1.35

which is the same as equation 1.28. The solution of equation 1.35 for diverging waves is

r

From equations 1.19 and 1.36 the condensation is given by

__ L ^ _ ltd.

dt cr

1.36

1.37

A. Pressure in a Spherical Sound Wave. — The pressure from equation 1.21 is

1.38

p = c^ps

The pressure then from equations 1.37 and 1.38 is

' p ^ — r)

^ r

Retaining the real part of equation 1.39 the pressure is

kcA , . , .

p = p sin k{ct — r)

r

1.39

1.40

B. Particle Velocity in a Spherical Sound Wave. — The particle velocity

14

SOUND WAVES

from equations 1.11 and 1.36 is

« = - 1.41

Retaining the real part of equation 1.41 the particle velocity is

Ah f” 1 ”1

u — 1 ~cos k(ct — r) — sin k{ct — r) j 1.42

C. Phase Angle between the Pressure and the Pai tide V elocity in a Spheri- cal Sound IVave. — The particle velocity given by equation 1.42 may be written

u = sin [k{ct — r) — d] 1.43

r \ r-

where tan 0=1 /hr.

Comparing equation 1.43 with equation 1.40 for the pressure it will be seen that the phase angle between the pressure and velocity in a spherical wave is given by

6 = tan“* 7“ 1.44

hr

For very large values of kr^ that is, plane waves, the pressure and par- ticle velocity are in phase. The phase angle as a function of kr is depicted in Fig. 1.2.

D. Ratio of the Absolute Magnitudes of the Particle V elocity and the Pres- sure in a Spherical Sound IVave, — From equations 1.40 and 1.43 the ratio of the absolute value of the particle velocity to the absolute value of the pressure is given by

Ratio

Vl 4- k^A

pckr

1.45

The ratio in equation 1.45, as a function of ir, is depicted in Fig. 1.3.

1.6. Stationary Sound Waves. — Stationary waves are the wave system resulting from the interference of waves of the same frequencies and are characterized by the existence of nodes or partial nodes.

Consider two plane waves of equal amplitude traveling in opposite direc- tions; the velocity potential may be expressed as

<l> = A [cos k{ct — x) + cos k{ct + x)]

1.46

STATIONARY SOUND WAVES

15

The pressure in this wave system from equations 1.19 and 1.21 is d<t}

p = ~ a7 “ — *) + sin k(ct + x)] 1.47

p = 2kcpA [sin kct cos ht\ 1.48

Fig. 1.2. Phase angle between the pressure and particle velocity in a spherical sound wave in

27r

terms of kr^ where k — \ = wavelength and r = distance from the source.

The particle velocity in this wave system from equations 1.11 and 1.46 is

d<b

— = k/^ [sin k{ct — x) dx

— sin k{ct + i^f)]

1.49

—Ik A [cos kct sin

1.50

IkA j^sin ^ct — ^ cos

(-1)]

1.51

Equations 1.48 and 1.51 show that the maxima of the particle velocity and pressure are separated by a quarter wavelength. The maxima of p and u differ by 90° in time phase.

A stationary wave system is produced by the reflection of a plane wave

16

SOUND WAVES

by an infinite wall normal to the direction of propagation. This is the simplest type of standing wave system. Complex stationary wave sys- tems are produced when a sound source operates in a room due to the re- flections from the walls, ceiling and floor.

Fio. 1.3. Ratio of the absolute magnitude of the particle velocity to the pressure in a spheri- cal sound wave in terms of kr^ where k = A = wavelength and r = distance from the source.

1.7. Sound Energy Density. — Sound energy density is the sound energy per unit volume. The unit is the erg per cubic centimeter.

The sound energy density in a plane wave is

where ^ *= sound pressure, in dynes per square centimeter, p = density, in grams per cubic centimeter, and c = velocity of sound, in centimeters per second.

The positive radiation pressure in dynes per square centimeter exerted

DECIBELS (BELS)

17

by sound waves upon an infinite wall is

p = (y + l)E 1.53

where E = energy density of the incident wave train in ergs per cubic centimeter, and

7 = ratio of specific heats, 1.4 for air.

Instruments for measuring the radiation pressure have been built, con- sisting of a light piston mounted in a large wall with means for measuring the force on the piston. Since the radiation pressure is very small these instruments must be quite delicate.

1.8. Sound Intensity. — I'he sound intensity of a sound field in a speci- fied direction at a point is the sound energy transmitted per unit of time in the specified direction through a unit area normal to this direction at the point. The unit is the erg per second per square centimeter. It may also be expressed in watts per square centimeter.

The intensity, in ergs per second per square centimeter, of a plane wave is

iP"

I — = pti — pciP 1.54

pc

where p = pressure, in dynes per square centimeter,

u = particle velocity, in centimeters per second, c = velocity of propagation, in centimeters per second, and p = density of the medium, in grams per cubic centimeter.

The product pc is termed the specific acoustical resistance of the medium. The specific acoustical resistance of various mediums is shown in Table 1.1.

1.9. Decibels (Bels). — In acoustics the ranges of intensities, pressures, etc., are so large that it is convenient to use a scale of smaller numbers termed decibels. The abbreviation db is used for the term decibel. The bel is the fundamental division of a logarithmic scale for expressing the ratio of two amounts of power, the number of bels denoting such a ratio being the logarithm to the base ten of this ratio. The decibel is one tenth of a bel. For example, with Pi and P2 designating two amounts of power and n the number of decibels denoting their ratio:

« = 10 logio > decibels “2

1.55

When the conditions are such that ratios of currents or ratios of voltages (or the analogous quantities such as pressures, volume currents, forces and particle velocities) are the square roots of the corresponding power ratios, the number of decibels by which the corresponding powers differ is ex-

18

SOUND WAVES

pressed by the following formulas:

w = 20 logio ^ > decibels /2

w = 20 logio — > decibels ei

1.56

1.57

where /1//2 and ei/e^ are the given current and voltage ratios, respectively.

For relation between decibels and power and current or voltage ratios, see Table 1.2.

Table 1.2. the relation between decibels and power and current or voltage ratios

Power Ratio	Decibels
1	0
2	3.0
3	4 8
4	6.0
5	7.0
6	7.8
7	8.5
8	9 0
9	9.5
10	10
100	20
1000	30
10,000	40
100,000
1,000,000

Current or V’oltage Ratio	Decibels
1	0.
2
3
4	12 0
5	14 0
6	15.6
7	16 9
8	18 1
9	19.1
10	20
100	40
KXX)	60
10,000	80
100,000	100
1,000,000	120

1.10. Doppler Effect.^ — The change in pitch of a sound due to the rela- tive motion of the source and observer is termed the Doppler Effect. When the source and observer are approaching each other the pitch ob- served by the listener is higher than the actual frequency of the sound source. If the source and observer are receding from each other the pitch is lower.

* Perrine, J. O., Amer.Jour. Phys.^YoX, 12, No. 1, p. 23, 1944. This paper de- scribes sixteen versions of the Doppler and Doppler Echo Effects. In addition to systems given in the text above are systems involving moving and fixed reflectors.

REFRACTION AND DIFFRACTION

19

The frequency at the observation point is

/o =

1.58

where v == velocity of sound in the medium,

Vq = velocity of the observer, v, == velocity of the source, and /, = frequency of the source.

All the velocities must be in the same units.

No account is taken of the effect of wind velocity or motion of the me- dium in equation 1.58. In order to bring in the effect of the wind, the velocity v in the medium must be replaced hy v + w where w is the wind velocity in the direction in which the sound is traveling. Making this substitution in 1.58 the result is

yo =

r + tt? — t;o r + w — y.

/.

1.59

Equation 1 .59 shows that the wind does not produce any change in pitch unless there is some relative motion of the sound source and the observer.

WARM ACR - H»CH VELOCITY

WARM AIR - HIGH VELOCITY

COOL AIR - LOW VELOCITY ' /

ftURTACC or TMC EARTH

SURFACE OF THE EARTH

Fig. 1.4. The refraction of a sound wave in air.

1.11. Refraction and Diffraction. — The change in direction of propa- gation of sound, produced by a change in the nature of the medium which affects the velocity, is termed refraction. Sound is refracted when the density varies over the wave front (see ecjuation 1.15). A sound wave may be bent either downward or upward depending upon the relative temperatures (densities) of the air,* Fig. 1.4. The distance over which sound may be heard is greater when the wave is bent downward than when it is bent upward. The first condition usually obtains during the early morning hours while the latter condition prevails during the day.

» For other phenomena of atmospheric acoustics such as the effects of wind and temperature upon the propagation of sound waves and the ranging and signaling in air, see Stewwt and Lindsay, Acoustics, D. Van Nos- trand company, New York, N. Y., 1930.

20

SOUND WAVES

DiifFraction is the change in direction of propagation of sound due to the passage of sound around an obstacle. It is well known that sound vnO, travel around an obstacle. The larger the ratio of the wavelength to the dimensions of the obstacle the greater the diffraction. The dif- fraction around the head is important in both speaking and listening.

Fio. 1.5. The diffraction of a sound wave by a cylinder, cube and sphere. (After Muller,

Black and Dunn.)

The diffraction of sound by microphones and loud speakers is important in the performance of these instruments. The diffraction ® of sound by a sphere, a cube and a cylinder as a function of the dimensions is shown in Fig. 1.5. These data may be used to predict the diffraction of sound by objects of these general shapes. As, for example, the sphere may be used to predict the diffraction of sound by the human head.

Another example of diffraction of sound is illustrated by the zone plate shown in Fig. 1 .6. The path lengths of the sound from the source to the focus vary by an integral wavelength. As a consequence, all the pencils

•Muller, Black and Dunn, Jour, Acous. Soc, Amer.^ Vol. 10, No. 1, p. 6, 1938.

ACOUSTICAL RECIPROCITY THEOREM

21

of sound are in phase at the focus with the result that the sound pressure is considerably greater at this point than any other position behind the zone plate.

rRONT VIEW CROSS’ SECTIONAL VIEW

Fio. 1.6. Z*nc plate. The source and the focus ^ are equidistant from the zone plate.

1.12. Acoustical Reciprocity Theorem.^* — The acoustical reciproc- ity theorem, as developed by Helmholtz, states: If in a space filled with air which is partly bounded by finitely extended bodies and is partly un- bounded, sound waves may be excited at a point the resulting velocity potential at a second point B is the same in magnitude and phase as it would have been at A had B been the source of sound. It is the purpose of this section to derive the acoustical reciprocity theorem.

Consider two independent sets of pressures />', p" and particle veloci- ties vf and y". Multiply equation 1.4 by the p and v of the other set.

o" V - u' — + - o" grad po' - - o' grad po" =0 1.60

at at p p

If p and V vary as a harmonic of the time, equation 1.60 becomes

- u" grad po' — ~ ti' grad />o" =0 1.61

P P

There is the following relation:

V grad p = div vp -- p div v 1.62

From equations 1.9 and 1.10

— ^ + div t; = 0 1.63

7/)o ot

^ Rayleigh, “ Theory of Sound,” Macmillan and Company, London, 1926.

‘ Ballentine, S., Proc., I.R.E., Vol. 17, No. 6, p. 929, 1929.

•Olson, H. F., RAC Review, Vol. 6, No. 1, p. 36, 1941.

’Olson, “Dynamical Analogies,” D. Van Nostrand Company, New York, N. Y.. 1943.

22

SOUND WAVES

From equations 1.61, 1.62 and 1.63,

div (»"/-»'/>")= 0 1.64

The relation of equation 1.64 is for a point. Integration of equation 1.64 over a region of space gives

//

(v’y - v'p”)ds = 0

1.65

If. in an acoustical system comprising a medium of uniform density and propagating irrotational vibrations of small amplitude, a pressure p* produces a particle velocity v' and a pressure />" produces a particle velocity ti", then

J f ivy - v'y')js = 0

1.66

where the surface integral is taken over the boundaries of the volume.

In the simple case in which there are only two pressures, as illustrated in the free field acoustical system of Fig. 1.7, equation 1.66 becomes

pV' = p'V 1.67

riCLO LUMPCO CONSTANTS

Fig. 1.7. Reciprocity in field and lumped constant acoustical systems.

where />', /)" and v" are the pressures and particle velocities depicted in the free field acoustical system of Fig. 1 .7.

The above theorem is applicable to all acoustical problems. However, it can be restricted to lumped constants* as follows: In an acoustical sys- tem composed of inertance, acoustical capacitance and acoustical resist- ance, let a set of pressures /)/, />2^ pz • • • pny all harmonic of the same frequency acting in n points in the system, produce a volume current dis- tribution Xi\ Xz . . . Xy,\ and let a second set of pressures pi\ p2\ pz* . . . Pn", of the same frequency as the first, produce a second volume current distribution X\ \ X^'y Xz' . . . Xn'* Then

^p/^/' = 1.68

i « i i - 1

» Olson, “Dynamical Analogies,” D. Van Nostrand Company, New York, N. Y., 1943.

LONGITUDINAL WAVES IN A ROD

23

This theorem is valid provided the acoustical system is invariable, contains no internal source of energy or unilateral device, linearity in the relations between pressures and volume currents and complete re- versibility in the elements, and provided the applied pressures /)i, />2, pz • • • pn are all of the same frequency.

In the simple case in which there are only two pressures, as illustrated in the acoustical system of lumped constants in Fig. 1.7, equation 1.68 becomes

p'X^' = 1.69

where p\ />" and JiT" are the pressures and volume currents depicted in the acoustical system of lumped constants in Fig. 1.7.

1.13. Acoustical Principle of Similarity.® — The principle of similarity in acoustics states: For any acoustical system involving diffraction phe- nomena it is possible to construct a new system on a different scale, which will exhibit similar performance, providing the wavelength of the sound is altered in the same ratio as the linear dimensions of the new system.

The principle of similarity is useful in predicting the performance of similar acoustical systems from a single model. A small model can be built and tested at very high frequencies to predict the performance of similar large systems at lower frequencies. For example: in the diffraction of sound by objects, if the ratio of the linear dimensions of the two objects is : 1, the corresponding configuration of the frequency characteristics will be displaced 1 ; A' in frequency. This is illustrated in Fig. 1.5. Other examples, are the directional characteristics of various sound sources Figs. 2.3 to 2.18 inclusive, the air load upon a diaphragm. Fig. 5.2, etc.

1.14. Longitudinal Waves in a Rod. — The preceding considerations have been concerned with sound waves in gases and fluids. In the case of solids, longitudinal waves in rods are of practical interest in many applications. It is the purpose of this section to derive the equations for longitudinal sound waves in a rod of homogeneous material and constant cross section.

The longitudinal axis of the bar will be assumed to coincide with the a: axis. Consider an element of the bar Sxy determined by two planes per- pendicular to X and initially at distances x and x + 8x from x = 0, Assume that the planes are displaced by distances f and f The distance

between the planes is now

dx + 8^ 8x + ’^Bx 1.70

ox

•Olson, H. F., RCJ Review, Vol. 6, No. 1, p. 36, 1941.

24

SOUND WAVES

The increase in distance between the planes is — 6a: •

dx

The increase in length of the bar per unit length at this point is •

dx

Young’s modulus is defined as the ratio of the longitudinal stress to the corresponding extension. At the first face of the element Young’s modulus

^ a?

where ^ = Young’s modulus, in dynes per square centimeter,

F = force, in dynes,

6* = cross-sectional area of the rod, in square centimeters, and

extension.

The force acting on the element across the first face is

F = ^Sp

The force acting across the second face of the element is

The resultant force on the element is

dx^

The acceleration of momentum of the element is

where p = density, in grams per cubic centimeter.

TORSIONAL WAVES IN A ROD

25

Equating the resultant force on the element to the acceleration of mo- mentum, the result is

p dx^

1.77

This is the wave equation for Equation 1.77 is analogous to equation 1.16 for plane waves in a gas and the solution of the differential equation is similar. The velocity of propagation, in centimeters per second, of longitudinal waves in a rod is

c =

1.78

where ^ = Young’s modulus, in dynes per square centimeter (see Table 1.1), and

p = density, in grams per cubic centimeter (see Table 1.1).

The velocity of sound, Young’s modulus and the density for various solids are given in Table 1.1.

1.16. Torsional Waves in a Rod. — A rod may be twisted about an axis of the rod in such a manner that each transverse section remains in its own plane. If the section is not circular there will be motion parallel to the axis of the bar. For a circular cross section and a homogeneous bar the equations of motion are analogous to those of longitudinal waves in the rod. The velocity of propagation, in centimeters per second, of tor- sional waves in a rod, is

where ^ = Young’s modulus, in dynes per square centimeter (see Table

1.1),

p = density, in grams per cubic centimeter (see Table 1.1), and

G = Poisson’s ratio (see Table 1.1).

CHAPTER II

ACOUSTICAL RADUTING SYSTEMS

2.1 Introduction. — There are almost an infinite number of different types of sound sources. The most common of these are the human voice, musical instruments, machinery noises and loud speakers. The most important factors which characterize a sound source are the directional pattern, the radiation efficiency and the output as a function of the fre- quency. In the case of some sound sources as, for example, musical instru- ments, it is almost impossible to analyze the action. However, in the case of most sound reproducers the action may be predicted with amazing accuracy. It is the purpose of this chapter to consider some of the simple sound sources that are applicable to the problems of sound reproduction.

2.2. Simple Point Source. — A point source is a small source which alternately injects fluid into a medium and withdraws it.

A. Point Source Radiating into an Infinite Medium, Solid Angle of 47r Steradians, — Consider a point source having a maximum rate of fluid emission of \rA cubic centimeters per second. The momentary rate at a time t is ^-kA cos oj/. The maximum rate of fluid emission may be written

-AirA = 2.1

where = area of the surface of the source, in square centimeters, and |o = maximum velocity, in centimeters per second over the sur- face S,

The velocity potential of a point source from equation 1.36 is

2.2

r

The particle velocity at a distance r from equation 1.42 is

_ MV

r

cos k {ct — r) — sin k{ct — r)

The pressure at a distance r from equation 1.40 is

pkcA .

P =

sin k{ct — r)

2.3

2.4

26

SIMPLE POINT SOURCE

27

The intensity or average power, in ergs per second, transmitted through a unit area at a distance r, in centimeters, is the product of p and u and is given by

P =

2^-2

2.5

The total average power in ergs per second emitted by the source is

Pt = liTrpckrA"^ 2.6

where p = density of the medium, in grams per cubic centimeter, c = velocity of sound, in centimeters per second, k = 27r/X,

X = wavelength, in centimeters, and A is defined by equation 2.L

B. Point Source Radiating into a Semuinjinite Medium. Solid Angle of It Steradians. — The above example considered a point source operating in an infinite medium. The next problem of interest is that of a point source operating in a semi-infinite medium, for example, a point source near an infinite wall.

In this case we can employ the principle of images as shown in Fig. 2.1. The pressure, assuming the same distance from the source, is two times that of the infinite medium. The particle velocity is also two times that of the infinite medium. The average power transmitted through a unit area is four times that of the infinite medium. The average power out- put of the source, however, is two times that of a simple source operating in an infinite medium.

C. Point Source Radiating into a Solid Angle of tt Steradians. — Em- ploying the method of images Fig. 2.1 the pressure is four times, the par- ticle velocity is four times and the average power transmitted through a unit area is sixteen times that of an infinite medium for the same distance. The average power output of the source is four times that of a simple source operating in an infinite medium.

D. Point Source Radiating into a Solid Angle of t/2 Steradians. — Em- ploying the method of images. Fig. 2.1, the pressure is eight times, the particle velocity eight times and the average power transmitted through a unit area is sixty-four times that of the same source operating in an infinite medium at the same distance. The average power output is eight times that of the same simple source operating in an infinite medium.

E. Application of the Simple Source. — The above data may be applied to acoustic radiators in which the dimensions are small compared to the

28

ACOUSTICAL RADIATING SYSTEMS

wavelength and located close to the boundaries indicated above. For example, A would correspond to a loud speaker, which acts as a simple source, suspended in space at a large distance from any walls or boundaries. B would correspond to a loud speaker placed on the floor in the center of

SOLID ANGLE PRESSURE POWER ENERGY OF SOUND AT A OUTPUT DENSITY

EMISSION DISTANCE r DISTANCE r

SOURCE

4TT

P W I

SOURCE

ZTT

2p 2W 41

SOURCE

IMAGES

4p 4W 16)

6p 8W 641

Fig. 2.1. The pressure, total power output and energy density delivered by a point source operating in solid angles of 47r, 27r, tt and ir/2 stcradians.

the room. C would correspond to a loud speaker placed on the floor along a wall, and D would correspond to a loud speaker placed in the corner of the room. Of course, as pointed out above, these examples hold only when the dimensions of the radiator and the distance from the wall are small compared to the wavelength.

2.3. Double Source (Doublet Source). — A double source consists of two point sources equal in strength dz47r//', but opposite in phase sepa-

^ Lamb, “ Dynamical Theory of Sound,” E. Arnold, London, 1931. *

® Davis, ” Modern Acoustics,” The Macmillan Co., New York, N. Y., 1934.

* Wood, ” A Textbook of Sound,” Bell and Sons, London, 1930.

^ Crandall, ” Theory of Vibrating Systems and Sound,” D. Van Nostrand Com- pany, New York, N. V., 1926.

DOUBLE SOURCE (DOUBLET SOURCE)

29

rated by a vanishingly small distance Sr. The strength of the doublet is \vA%r, Let A' hr ^ A, In these considerations A' corresponds to A of equation 2.1, that is Air A' = *S’|o.

At a distance r in a direction inclined at an angle a to the axis of the doublet the velocity potential is

<t> =

' ■ JHct-

^jkict-r)

cos a

The pressure from equation 2.7 is d<j)

,pCkA (\ \

Jkict—T)

cos Ot

Retaining the real parts of equation 2.8

p — r_ ^ ^ — r) 1 cos a

r Lr J

At a very large distance

k-A

p oc cos a

^ r

At a very small distance

kA

p oc — - cos a

2.7

2.8

2.9

2.10

2.11

d(/>

The particle velocity has two components, the radial ■— and the trans- 1 d</>

verse ~ ^ ‘ * "The radial component of the particle velocity from equation 2.7 is.

Retaining the real parts of equation 2.12

cos a

2.12

- A

[0-7)

At a very large distance

2/t

cos k(^ct — r) r sin ^(f/

Tr

Ak^

U OC cos Of

r

— r)J cos a 2.13

2.14

30

ACOUSTICAL RADIATING SYSTEMS

At a very small distance

A

a « — cos a. r*

2.15

The transverse component of the particle velocity is

“ rda \ r* /

\k{ci—T)

sin a

Retaining the real parts of equation 2.16

[1 k . “1 .

— cos k{ct — r) ; sin k{ct — ;*) J sin ol

At a very large distance

ucc — sin a r-

At a very small distance

A .

«oc sin a

yZ

2.16

2.17

2.18

2.19

PRcssuae

Fig. 2.2. The pressure and particle velocity at a constant distance from a doublet source. The magnitude of the pressure is indicated by the circle. The particle velocity has two com- ponents: a radial and a transverse component. The direction and mag- nitude of these two components are indicated by vectors.

Fig. 2.2 shows the velocity compo- nents and the pressure for various points around a doublet source. A common example of a doublet source is a direct radiator loud speaker mounted in a small baffle. (Dimensions of the baffle are small compared to the wavelength.) If the response of such a loud speaker is measured with a pressure microphone for various angles at a constant distance, the result will be a cosine characteristic. If the response is measured with a ve- locity microphone keeping the axis pointed toward the loud speaker, the result will be a cosine directional char- acteristic. If the same is repeated keep- ing the axis of the velocity microphone normal to the line joining the microphone

SERIES OF POINT SOURCES

31

and the loud speaker, the result will be a sine directional characteristic. The total power, in ergs, emitted by a doublet source is

P =

2.20

where p = pressure, in dynes per square centimeter, p = density, in grams per cubic centimeter, c = velocity of sound, in centimeters per second, and dS = area, in square centimeters, over which the pressure is p.

Taking the value of p from equation 2.9 (for r very large), the total average pov/er in ergs per second emitted by a doublet source is

Pt

pck^A^

2r^

cos^ a sin a

2.21

Pt ~ ^Tpck^/^^

2.22

where p = density, in grams per cubic centimeter, k = 27r/X,

X = wavelength, in centimeters, c = velocity of sound, in centimeters per second, and // is defined in the first paragraph of this section.

The power output from a simple source (equation 2.6) is proportional to the square of the frequency, while the power output from a doublet source (equation 2.22) is proportional to the fourth power of the frequency. For this reason the power output of a direct radiator loud speaker falls off rapidly with frequency when the dimensions of the baffle are small com- pared to the wavelength (see Sec. 6.8).

2.4. Series of Point Sources. — The directional characteristic^'®’^ of a source made up of any number of equal point sources, vibrating in phase, located on a straight line and separated by equal distances is given by

2.23

I* Wolff, I., and Malter, L., Jour. Acous. Soc. Amer.y Vol. 2, No. 2, p. 201, 1930. « Stenzel, H., Eiek. Nach. Tech., Vol. 4, No. 6, p. 239, 1927.

» Stenzel, H., Elek. Nach. Tech., Vol. 6, No. 5, p. 165, 1929.

32

ACOUSTICAL RADIATING SYSTEMS

where = ratio of the pressure for an angle a to the pressure for an angle a = 0. The direction a = 0 is normal to the line, n = number of sources,

d = distances between the sources, in centimeters, and X = wavelength, in centimeters.

The directional characteristics of a two point source are shown in Fig. 2.3. It will be noted that the secondary lobes are equal to the main lobe.

OlSrANCE=^X 0ISTANCE=5X distance=x

Fig. 2.3. Directional characteristics of two separated equal small sources vibrating in phase as a function of the distance between the sources and the wavelength. The polar graph depicts the pressure, at a fixed distance, as a function of the angle. The pressure for the angle 0® is arbitrarily chosen as unity. The direction corresponding to the angle 0® is perpendicular to the line joining the two sources. I'he directional characteristics in three dimensions are surfaces of revolution about the line joining the two sources as an axis.

2.6. Straight Line Source. — A straight line source may be made up of a large number of points of equal strength and phase on a line sepa- rated by equal and very small distances. If the number of sources n approach infinity and dy the distance between the sources, approaches zero in such a way that

nd = /

the limiting case is the line source. If this is carried out, equation 2.23 becomes

2.24

The directional characteristics of a continuous line source are shown in

TAPERED STRAIGHT LINE SOURCE

33

Fig. 2.4. The directional characteristics are symmetrical about the line as an axis. Referring to Fig. 2.4, it will be seen that there is practically no directivity when the length of the line is small compared to the wave- length. On the other hand, the directional characteristics are sharp when the length of the line is several wavelengths.

LCNCTH -4 LENGTH-lx LENCTH-X LENGTH-lix LENCTH-3X

Fig. 2.4. Directional characteristics of a line source as a function of the length and the wave- length. The polar graph depicts the pressure, at a large fixed distance, as a function of the angle. The pressure for the angle O*’ is arbitrarily chosen as unity. The direction corre- sponding to the angle 0° is perpendicular to the line. The directional characteristics in three dimensions are surfaces of revolution about the line as an axis.

2.6. Tapered Straight Line Source. — The directional characteristic * of a line source, all parts vibrating in phase, in which the strength varies linearly from its value at the center to zero at either end, is given by

Ra-

2.25

where /?„ = ratio of the pre: sure for an angle a to the pressure for an angle « = 0. The direction a = 0 is normal to the line, / = total length of the line in centimeters, and X = wavelength, in centimeters.

The directional characteristics of a tapered line source are shown in Fig. 2.5. Comparing the directional characteristics of Fig. 2.5 with those

* Menges, Karl, Jkus, ZeiL, Vol. 6, No. 2, p. 90, 1941.

34

ACOUSTICAL RADIATING SYSTEMS

of the uniform line of Fig. 2.4, it will be seen that the main lobe is broader and the secondary lobes are reduced in amplitude.

LENCTHS3X

Fio. 2.5. Directional characteristics of a tapered line source as a function of the length and the wavelength. The volume current output along the line vanes linearly from a maximum at the center to zero at the two ends. The polar graph depicts the pressure, at a fixed distance, as a function of the angle. 'I'he pres.sure for the angle 0° is arbitrarily chosen as unity. The direction corresponding to the angle 0® is perpendicular to the line. The directional characteristics in three dimensions arc surfaces of revolution about the line as an axis.

2.7. Nonuniform Straight Line Source. — The directional characteristic of a line, all parts vibrating in phase, in which the strength varies as a function of the distance x along a line is given by

S <1 ‘

2.26

///«■*

i

distance from the center of the line, in centimeters, total length of the line, in centimeters, strength distribution function and the other quantities arc the same as those in equation 2.25.

2.8. Curved Line Source (Arc of a Circle). — A curved line source may be made up of a large number of point sources vibrating in phase on the arc of a circle separated by very small distances. The directional char-

where x = d = /(^) =

CURVED LINE SOURCE (ARC OF A CIRCLE)

35

acteristics of such a line in the plane of the arc are, Ra = ^

2m + 1

k	r^wR , “1
X) cos	cos (a + ke)
k<^ —m	-A J

+j 2- S'” rir

k^-m L ^

COS (a + kd)

]|

2.27

where Ra = ratio of the pressure for an angle a to the pressure for an angle a = 0,

a = angle between the radius drawn through the central point and the line joining the source and the distant observation point,

X = wavelength, in centimeters,

R = radius of the arc, in centimeters,

2m + 1 = number of points,

6 = angle subtended by any two points at the center of the arc, and k = variable.

Another method ® is to break up the arc into a large number of equal chords. The strength is assumed to be uniform over each chord. Also the phase of all of the chords is the same. In this case the result takes the form,

/?a =

1

2m + 1

k =»m

cos

k »» — m

— — COS (a + ^)

ird

^ . nrR , ^

+ j 2^ sin j— — cos (a + ^)[ l!»-m I A J

sin (a + kO)

I

sin I — sin (a + kO)

A

sin -

]

— Sin (a + kB)

X

2.28

where Ra = ratio of the pressure for an angle a to the pressure for an artgle a = 0,

X = wavelength, in centimeters, k == variable,

R = radius of the arc, in centimeters,

2m + 1 = number of chords,

B = angle subtended by any of the chords at the center of circumscribing circle, and d = length of one of the chords, in centimeters.

•Wolff, I. and Malter, L., Jour, Acous, Soc, Amer.^ Vol. 2, No. 2, p. 201, 1930.

36

ACOUSTICAL RADIATING SYSTEMS

The directional characteristics for an arc of 60°, 90° and 120° are shown in Figs. 2.6, 2.7 and 2.8. The interesting feature of the directional char-

PA0lUSs2X RADIUS- 4X RADIUS -8X RADIUS -IdX

Fig. 2.6. Directional characteristics of a 60® arc as a function of the radius and the wave- length. 'I he polar graph depicts the pressure, at a large fixed distance, as a function of the angle in the plane of the arc. The pressure for the angle 0° is arbitrarily chosen as unity.

acteristics of an arc is that the directional characteristics are very broad for wavelengths large compared to the dimensions, and are narrow for

Fio. 2.7. Directional characteristics of a 90® arc as a function of the radius and the wave- length. The polar graph depicts the pressure at a large fixed distance, as a function of the angle in the plane of the arc. The pressure for the angle 0® is arbitrarily chosen as unity.

wavelengths comparable to the dimensions and are broad again for wave- lengths small compared to the dimensions of the arc. The arc must be

CIRCULAR RING SOURCE

37

several wavelengths in length in order to yield a “wedge-shaped” di- rectional characteristic.

2.9. Circular Ring Source. — The directional characteristics of a circular ring source of uniform strength and the same phase at all points on the ring is

Ra = /o

where Ra = ratio of the pressure for an angle a to the pressure for an angle a = 0,

Jo = Bessel function of zero order,

R = radius of the circle, in centimeters, and

a = angle between the axis of the circle and the line joining the point of observation and the center of the circle.

Fig. 2.8. Directional characteristics of a 120° arc as a function of the radius and the wave- length. The polar graph depicts the pressure, at a large fixed distance, as a function of the angle in the plane of the arc. The pressure for the angle 0° is arbitrarily chosen as unity.

The directional characteristics of a circular-ring source as a function of the diameter and the wavelength are shown in Fig. 2.9. The shapes are quite similar to those of a straight line. The characteristic is somewhat sharper than that of a uniform line of length equal to the diameter of the circle, but has almost the same form. The amplitudes of the secondary lobes are greater than those of the uniform line.

Stenzel, H., Elek, Nach, Tech.y Vol. 4, No. 6, p. 1, 1927.

“ Wolff, I. and Malter, L., Jour, Acous, Soc, Amer,y Vol. 2, No. 2, p. 201, 1930.

38

ACOUSTICAL RADIATING SYSTEMS

2.10. Plane Circular Surface Source. — The directional characteris- tics of a circular surface source with all parts of the surface vibrating

F'ig. 2.9. Directional characteristics of a circular line or ring source as a function of the diameter and wavelength. The polar graph depicts the pressure, at a large fixed distance, as a function of the angle. The pressure for the angle 0® is arbitrarily chosen as unity. The direction corresponding to the angle 0® is the axis. The axis is the center line perpen- dicular to the plane of the circle. The directional characteristics in three dimensions are surfaces of revolution about the axis.

with the same strength and phase are

2.30

where Ra = ratio of the pressure for an angle a to the pressure for an angle a = 0,

/i = Bessel function of the first order,

R = radius of the circle, in centimeters,

a = angle between the axis of the circle and the line joining the point of observation and the center of the circle, and X = wavelength, in centimeters.

The directional characteristics of a plane circular surface source as a function of the diameter and wavelength are shown in Fig. 2.10. The characteristic is somewhat broader than that of the uniform line of length

« Stenzel, H., FJek. Nach. Tech., Vol. 4, No. 6, p. 1, 1927.

” WolflF, I. and Maker, L., Jour, Acous, Soc. Amer,, Vol. 2, No. 2, p. 201, 1930.

PLANE SQUARE SURFACE SOURCE

39

equal to the diameter of the circle, but has approximately the same form. The amplitudes of the secondary lobes are smaller than those of the uniform line.

DIAMETER - X

DIAMETER -llx 10 •

diameter -JX

DIAMETER- flX

lO

Fig. 2.10. Directional characteristics of a circular piston source as a function of the diameter and wavelength. 'I'he polar graph depicts the pressure, at a large fixed distance, as a function of the angle. The pressure for the angle 0® is arbitrarily chosen as unity. The direction corresix)nding to the angle 0® is the axis. The axis is the center line perpendicular to the plane of the piston. The directional characteristics in three dimensions are surfaces of revolution about the axis.

2.11. Nonuniform Plane Circular Surface Source.'^ — The integratio of the expression for a plane circular surface source in which the strength varies as a function of the distance from the center cannot be obtained in simple terms. An approximate method may be employed in which the plane cir- cular surface with nonuniform strength is divided into a number of rings with the proper strength assigned to each ring. An alternative method may be employed in which the strength distribution is obtained by super- posing a number of plane circular surface sources of different radii with the proper strength assigned to each surface.

2.12. Plane Square Surface Source. — The directional characteristics of a plane square surface source, with all parts of the surface vibrating with the same intensity and phase, in a normal plane parallel to one side, is the same as that of a uniform line source having a length equal to one side of the square (equation 2.24).

Jones, R. Clark, Jour, Acous, Soc, Amer,^ Vol. 16, No. 3, p. 147, 1945. This is a comprehensive paper on the study of directional patterns of plane surface sources with specified normal velocities. A number of directional patterns and tables are given.

40

ACOUSTICAL RADIATING SYSTEMS

The directional characteristics of a plane square surface source, with all parts of the surface vibrating with the same strength and phase, in a normal plane containing the diagonal is the same as that of the tapered line source having a length equal to the diagonal (equation 2.25).

2.13. Plane Rectangular Surface Source. — The directional characteris- tics of a rectangular surface source with all p3rts of the surface vibrating with the same strength and phase are

Ra^

2.31

where /« = length of the rectangle, lb = width of the rectangle,

a = angle between the normal to the surface source and the pro- jection of the line joining the middle of the surface and the observation point on the plane normal to the surface and parallel to 4, and

= angle between the normal to the surface source and the pro- jection of the line joining the middle of the surface and the observation point on the plane normal to the surface and parallel to 4.

The directional characteristic of a plane rectangular surface source with uniform strength and phase is the same as the product of the characteristic of two line sources at right angles to each other and on each of which the strength and phase are uniform.

2.14. Horn Source. — The directional characteristics of a horn depend upon the shape, mouth opening and the frequency. It is the purpose of this section to examine and consider some of the factors which influence the directional characteristics of a horn.

The phase and particle velocity of the various incremental areas which may be considered to constitute the mouth determines the directional characteristics of the horn. The particular complexion of the velocities and phase of these areas is governed by the flare and dimensions and shape of the mouth. In these considerations the mouth will be of circular cross section and mounted in a large flat baffle. The mouth of the horn plays a major role in determining the directional characteristics in the range where the wavelength is greater than the mouth diameter. The flare is the major factor in determining the directional characteristics in the range where the wavelength is less than the mouth diameter.

HORN SOURCE

41

A. Exponential Horns, — The eflFect of the diameter of the mouth for a constant flare upon the directional characteristics of an exponential horn is depicted in Fig. 2.1 1. At the side of each polar diagram is the diam- eter of a vibrating piston which will yield approximately the same direc- tional characteristic. It will be seen that up to the frequency at which the wavelength becomes comparable to the mouth diameter, the directional characteristics are practically the same as those of a piston of the size of the mouth. Above this frequency the directional characteristics are prac-

1000 ~ 2000^ 4000^ 7000^ 10.000^

Fig. 2.11. The directional characteristics of a group of exponential horns, with a constant flare and throat diameter of inch as a function of the mouth diameter. The number at the right of each polar diagram indicates the diameter of a circular piston which will >ield the same directional characteristic. The polar graph depicts the pressure, at a fixed distance, as a function of the angle. The pressure for the angle 0° is arbitrarily chosen as unity. The direction corresponding to 0° is the axis of the horn. The directional charac- teristics in three dimensions are surfaces of revolution about the horn axis.

tically independent of the mouth size and appear to be governed primarily by the flare.

To further illustrate the relative effects of the mouth and flare, Fig. 2.12 shows the eflFect of different rates of flare, for a constant mouth diameter, upon the directional characteristics of an exponential horn. These results also show that, for the wavelengths larger than the mouth diameter, the

» Olson, H. F., /?CY Review, Vol. 1, No. 4, p. 68, 1937.

Goldman, S., Jour, Acous, Soc, Amer,, Vol. 5, p. 181, 1934, reports the results of an investigation upon the directional characteristics of exponential horns at 15,000 and 25,000 cycles. A comparison can be made with the results shown in h'igs. 2.11 and 2.12 by increasing the dimensions of the horns used by him to con- form with those shown here and decreasing the freauency by the factor of increase in dimensions. Such a comparison shows remarkable agreement between the two sets of data.

42

ACOUSTICAL RADIATING SYSTEMS

directional characteristics are approximately the same as those of a vibrat- ing piston of the same size as the mouth. Above this frequency the directional characteristics are broader than those obtained from a piston the size of the mouth. From another point of view, the diameter of the piston which will yield the same directional characteristic is smaller than

\000^ 2000^ AOOOr^ 7000^ 10000^

Fig. 2.12. The directional characteristics of a group of exponential horns, with a mouth diameter of 12 inches and a throat diameter of ^ inch, as a function of the flare. The number at the right of each polar diagram indicates the diameter of a circular piston which will yield the same directional characteristic. The polar graph dej>icts the pressure, at a fixed distance, as a function of the angle. The pressure for the angle 0° is arbitrarily chosen as unity. The direction corresponding to 0° is the axis of the horn. 'I'he direc- tional characteristics in three dimensions are surfaces of revolution about the horn axis.

the mouth. These results also show that the directional characteristics vary very slowly with frequency at these smaller wavelengths. Referring to Fig. 2.12, it will be seen that for any particular high frequency, 4000, 7000 or 10,000 cycles per second, the directional characteristics become progressively sharper as the rate of flare decreases.

B. Conical Horns. — In the case of the circular conical horn the direc- tional pattern should be the same as that of a circular, spherical surface source. The radius of the spherical surface is the distance along the side of the horn from the apex to the mouth. The directional characteristics of two conical horns are shown in Fig. 2.13. At the lower frequencies the

HORN SOURCE

43

directional pattern is approximately the same as that of a piston of the same size as the mouth. The directional pattern becomes sharper with an in- crease of the frequency. However, at the higher frequencies where the diameter of the mouth is several wavelengths, the pattern becomes broader as would be expected from a spherical surface source. The directional characteristics of a conical horn as depicted in Fig. 2.13 are practically the same as those of a spherical surface source.

Fig. 2.13. The directional characteristics of two conical horns with mouth diameters of 12 inches and throat diameters of inch and lengths of 12 inches and 24 inches. The polar graph depicts the sound pressure, at a fixed distance, as a function of the angle. The sound pressure for the angle 0° is arbitrarily chosen as unity. The direction corresponding to 0® is the axis of the horn. The directional characteristics in three dimensions are surfaces of revolution about the horn axis.

C. Faraholic Horns, — In the parabolic horn the sectional area is pro- portional to the distance from the apex. This horn may be constructed as shown in Fig. 2.14 in which two opposite horn walls are parallel and the other two are inclined at an angle with respect to each other. The direc- tional characteristics of a 90° parabolic horn are shown in Fig. 2.14. The source at the mouth is essentially a curved-line source described in Sec. 2.8. Therefore, the directional characteristics in a plane parallel to the two parallel sides of the horn shovild be essentially the same as that of a 90° arc. Comparing Fig. 2.14 with the 90° arc source of Fig. 2.7 it will be seen that the two directional patterns are quite similar.

From the directional patterns of horn type radiators described in the preceding sections, it is evident that a wide range of directional patterns is possible in simple horns by variations in the shape of the horn and the mouth opening.

The results of Figs. 2.11, 2.12, 2.13 and 2.14 are applicable to other geo- metrically similar horns by changing the wavelength (or the reciprocal of the frequency) in the same ratio as the linear dimensions in accordance with the principle of similarity of Sec. 1.13,

44

ACOUSTICAL RADIATING SYSTEMS

R:4X Rr8X Rsl2X Rsl^X

Fig. 2.14. The directional characteristics of a parabolic horn of the shape and the dimensions shown in the sketches on the left. The patterns were obtained in the plane midway be- tween and parallel to the two parallel sides. The polar graph depicts the sound pressure, at a fixed distance, as a function of the angle. The sound pressure for the angle 0® is arbitrarily chosen as unity. The direction correspjonding to 0° is spaced midway between the two nonparallel sides of the horn. = 12 inches. The ratio of R/\ is also given for comparison with Fig. 2.7.

2.16. Curved Surface Source. — A sphere vibrating radially radiates sound uniformly outward in all directions. A portion of a spherical surface, large compared to the wavelength and vibrating radially, emits uniform sound radiation over a solid angle subtended by the surface at the center of curvature. To obtain uniform sound distribution over a certain solid angle, the radial air motion must have the same phase and amplitude over the spherical surface intercepted by the angle having its center of curvature at the vertex and the dimensions of the surface must be large compared to the wavelength. When these conditions are satisfied for all frequencies, the response characteristic will be independent of the position within the solid angle.

A loud speaker consisting of a large number of small horns with the axis passing through a common point will satisfy, for all practical purposes, the requirement of uniform phase and amplitude over the spheri- cal surface formed by the mouths of the horns. A cellular or multihorn of this type is shown in Fig. 2,\SA, This particular horn system consists of fifteen horns arranged in five vertical rows and three horizontal rows. The mouth opening of each horn is 8 X 8 inches. The horizontal and ver- tical angle between the axis of the individual horn is 17°.

The directional characteristics of a multihorn loud speaker may be

»’Wente, E. C., and Thuras A. L., ^our. A, 7. E. Vol. 53, No. 1, p. 17, 1934.

** Hilliard, J. K., Tech, Bull. Acad. Res. Council March, 1936.

Olson, H. F., RCA Review, Vol. 1, No. 4, p. 68, 1937,

CURVED SURFACE SOURCE

45

predicted theoretically’® from the directional characteristics of an in- dividual horn and the geometrical configuration of the assembly of horns. Assume that the point of observation is located on the OY axis. Fig. 2.155, at a distance several times the length of the horn. The amplitude of the vector contributed by an individual horn for the angle <t> can be deter- mined from its individual directional characteristic. In this illustration, the plane XOZ is chosen as reference plane for the phase of the vector. The phase angle of the vector associated with an individual horn is

2.32

where d = the distance between the center of the mouth of the horn and the reference plane X'0'Z\ in centimeters, and X = wavelength, in centimeters.

Fio. 2.15. A spherical radiating surface consisting of 15 individual exponential horns B, Geometry for predicting the directional characteristics of a cluster of small horns.

The vectors, having amplitudes //,, /f2y /fa, /f4, etc., determined from the directional characteristics and having phase angles 0i, S^y 03, 04, etc., de- termined from equation 2.32, are added vectorially as shown in Fig. 2.155. This method of predicting the directional characteristics assumes that there is no interaction between individual horns which changes the com- plexion of the velocities at the mouth from that which obtains when operating an individual horn. Obviously, this condition is not absolutely satisfied. Apparently, the discrepancy has no practical significance be- cause it has been found that this method of analysis agrees quite well with experimental results.

The directional characteristics of the cellular horn of Fig. 2.15/f are shown in Figs. 2.16 and 2.17. Above 2000 cycles the dimensions of the

46

ACOUSTICAL RADIATING SYSTEMS

total mouth surface are several wavelengths and the directional character- istics are fairly uniform and defined by the total angular spread. Where the dimensions are comparable to the wavelength the directional char-

250^ 500<v 1000^

Fio. 2.16. Directional characteristics of the 15-cell cellular horn shown in Fig. 2.15/f in a plane containing the line B-B* and the axis of the center horn. The polar graph depicts the pressure, at a fixed distance, as a function of the angle. The pressure for the angle 0® is arbitrarily chosen.

250A^ SOOa# 1000a/

Fio. 2.17. Directional characteristics of the 15-cell cellular horn shown in Fig. 2.\SA in a plane containing the line and the axis of the center horn. The polar graph depicts

the pressure, at a fixed distance, as a function of the angle. The pressure for the angle 0® is arbitrarily chosen.

acteristics become very sharp, as shown by the polar curves for 500 and 1000 cycles. Then, as the dimensions of the surface become smaller than the wavelength, 250 cycles, the angular spread broadens, as is illustrated

CONE SURFACE SOURCE

47

by the larger spread for the smaller vertical dimension when compared to the smaller spread for the larger horizontal dimension.

The directional characteristics of a cellular horn show a striking resem- blance to those of an arc of the same angular spread. For example, the angular spread of the horn of Fig. 2.15 in the plane containing the line AA^ and the axis is 87^°. This may be compared to the arc of Fig. 2.7. In this case X/4, X/2, X, 2X, 4X and 8X will correspond to 145, 290, 580, 1160, 2320 and 4640 cycles. The angular spread in the plane containing the line and the axis is 52^°. This may be compared to the 60° arc of Fig. 2.6 with the same relation between the wavelengths and frequencies, as noted above. It will be seen that there is a marked resemblance be- tween corresponding frequencies. Of course, there is some variation to the fact that the frequencies do not correspond exactly. Further, there is some difference in the angular spread. For most spherical sur- faces of this type the directional characteristics in various planes corre- spond very closely to the directional characteristics of the corresponding arc.

sectional view vector diagram

Fio. 2.18. Geometry for obtaining the directional pattern of a cone type radiator.

2.16. Cone Surface Source.^^ — The directional characteristics of a paper or felted paper cone used in the direct radiator type loud speaker may be predicted theoretically from the dimensions and shape of the cone and the velocity of sound propagation in the material. For this type of analysis the cone is divided into a number of ring type radiators as shown in Fig. 2.18. The dimension of the ring along the cone should be a small fraction of the wavelength o( sound in the paper. The output of the cone at any angle is the vector sum of the vectors Aq^ Aiy A^ ... An where the A"^ are the amplitudes of the individual rings.

Carlisle, R. W., lour. Acous. Soc. Amer.y Vol. 15, No. 1, p. 44, 1943.

The analysis in this section assumes that there is no reflected wave at the outer boundary. In order to obtain a uniform response frequency characteristic the reflected wave must be small. If the reflected wave is small, the effect upon the directional pattern may be neglected.

48

ACOUSTICAL RADIATING SYSTEMS

The phase angle of the amplitude of the first ring is

flo = 0 2.33

The phase angle of the amplitude of the second ring is

The phase angle of the amplitude of the third ring is

Di D2^

62 — ^TT f

\p

-)

cos a

2.34

2.35

The phase angle of the amplitude of the nth ring is + do • • • dn Di + D2 •

On = 27r

Xp

cos a

2.36

where diy ^2, . . . = axial distances shown in Fig. 2.18 in centimeters, and Di, D2, . . . = distances along the cone shown in Fig. 2.18 in centi- meters,

Xa = wavelength of sound in air, in centimeters,

Xp = wavelength of the sound in the paper cone, in centi- meters, and

a = angle between the axis of the cone and the line joining the observation point and the center of the first ring.

The relative amplitude of the vector /in is given by

/in —

2.37

where = radius of the «th ring, in centimeters,

Dn = width of the wth ring along the cone, in centimeters,

Xa = wavelength of sound in air, in centimeters, a = angle between the axis of the cone and the line joining the observation point and the center of the cone, and Jo = Bessel function of zero order.

The directional characteristic of the cone is

Ra

K^n /T-n

^ Ak cos Qk ^ j ^ Ak sin Bk

K^Q A'-O

K^n

£

A-0

2.38

CONE SURFACE SOURCE

49

where Ra = ratio of the pressure for an angle a to the pressure for an angle

a = 0.

A consideration of equation 2.38 shows that the directional pattern is a function of the frequency and becomes sharper as the frequency increases. For a particular frequency, cone angle and material the directional patterns are practically similar for the same ratio of cone diameter to wavelength. For a particular frequency and the same cone material the directional pattern becomes broader as the cone angle is made larger. For a particu- lar frequency and cone angle the directional pattern becomes broader as the velocity of propagation in the material decreases (see sec. 6.2).

CHAPTER III

MECHANICAL VIBRATING SYSTEMS

3.1. Introduction. — The preceding chapters have been confined to the considerations of simple systems, point sources, homogeneous mediums and simple harmonic motion. Sources of sound such as strings, bars, membranes and plates are particularly liable to vibrate in more than one mode. In addition, there may be higher frequencies which may or may not be harmonics. The vibrations in solid bodies are usually termed as longitudinal, transverse or torsional. In most cases it is possible to con- fine the motion to one of these types of vibrations. For example, the vibrations of a stretched string are usually considered as transverse. It is also possible to excite longitudinal vibrations which will be higher in fre- quency. If the string is of a fairly large diameter torsional vibrations may be excited. The vibrations of a body are also afifected by the me- dium in which it is immersed. Usually, in the consideration of a particular example it is necessary to make certain assumptions which will simplify the problem. The mathematical analysis of vibrating bodies is extremely complex and it is beyond the scope of this book to give a detailed analysis of the various systems. For complete theoretical considerations, the reader is referred to the treatises which have been written on this subject. It is the purpose of this chapter to describe the most common vibrators in use today, to illustrate the form of the vibrations and to indicate the resonant frequencies.

3.2. Strings. — In all string instruments the transverse and not the longitudinal vibrations are used. In the transverse vibrations all parts of the string vibrate in a plane perpendicular to the line of the string. For the case to be described it is assumed: that the mass per unit length is a constant, that it is perfectly flexible (the stiffness being negligible) and that it is connected to massive nonyielding supports. Fig. 3.1. Since the string is fixed at the end, nodes will occur at these points. The fun- damental frequency of the string is given by

50

3.1

STRINGS

51

where T = tension, in dynes,

m = mass per unit length, in grams,

/ = length of the string, in centimeters.

The shape of the vibration of a string is sinusoidal. In addition to the fundamental, other modes of vibration may occur, the frequencies being 2, 3, 4, 5, etc., times the fundamental. The first few modes of vibration

FIRST OVERTONE SECOND HARMONIC

SECOND OVERTONE THIRD HARMONIC

THIRD OVERTONE FOURTH HARMONIC

FIFTH OVERTONE SIXTH HARMONIC

Fig. 3.1. Modes of vibration of a stretched string. The nodes and loops are indicated by

N and L.

of a string are shown in Fig. 3.1. The points which are at rest are termed nodes and are marked N, The points between the nodes where the ampli- tude is a maximum are termed antinodes or loops and are marked L.

The above example is the simplest form of vibration of a string. A few of the problems which ha^^e been considered by different investigators ^*2, 3.4.6 are as follows: nonuniform strings, loaded strings, stiff strings, nonrigid

^ Rayleigh, “ Theory of Sound,** Macmillan and Company, London, 1926.

* Crandall, “ Theory of Vibrating Systems and Sound,*’ D. Van Nostrand Com- pany, New York, N. Y., 1926.

* Wood, A Text Book of Sound,** Bell and Sons, I^ndon, 1930.

^ Morse, “ Vibration and Sound,** McGraw-Hill Book Company, New York, N. Y., 1936.

* Lamb, “ Dynamical Theory of Sound,** E. Arnold, London, 1931.

52

MECHANICAL VIBRATING SYSTEMS

supports, the effect of damping and the effect of different types of excita- tion. These factors of course alter the form of vibration and the overtones.

3.3. Transverse Vibration of Bars.^*®*^*® — In the preceding section the perfectly flexible string was considered where the restoring force due to stiffness is negligible compared to that due to tension. The bar under no tension is the other limiting case, the restoring force being entirely due to stiffness. For the cases to be considered it is assumed that the bars are straight, the cross section is uniform and symmetrical about a central

A

THIRD OVERTONE

D

THIRD OVERTONE

B

THIRD OVERTONE

E

FUNDAMENTAL

THIRD OVERTONE

c

THIRD OVERTONE

F

THIRD OVERTONE

Fio. 3.2. Modes of transverse vibrations of bars. //. A bar clamped at one end and free at the other. B. A bar clamped at one end and supported at the other. C. A bar supported at one end and free at the other. D. A bar free at both ends. £. A bar

supported at both ends. A bar clamped at both ends.

plane and, as in the case of the string, only the transverse vibrations will be considered.

A. Bar Clamped at One End. — Consider a bar clamped in a rigid sup- port at one end with the other end free (Fig. 3.2/f). The fundamental frequency is given by

. _ .5596

l^ y P

3.2

where / = length of the bar, in centimeters,

p = density, in grams per cubic centimeter, see Table 1.1,

^ = Young’s modulus, in dynes per square centimeter, see Table LI, and

K = radius of gyration.

TRANSVERSE VIBRATION OF BARS

53

For a rectangular cross section the radius of gyration is

Vl2

where a == thickness of the bar, in centimeters, in the direction of vibration. For a circular cross section.

where a = radius of the bar, in centimeters. For a hollow circular cross section,

Va^ + ai^

where a = outside radius of the pipe, in centimeters, and ai = inside radius of the pipe, in centimeters.

The modes of vibration of a bar clamped at one end are shown in Fig. 3,2//. The table below gives the position of the nodes and the frequencies of the overtones.

No. of Tone	No. of Nodes	Distances of Nodes from Free End m Terms of the Length of the Bar	Frequencies as a Ratio of the Fundamental
1	0		/.
2	1	.2165	6.267/,
3	2	.1321, .4999	17.55/1
4	3 j	.0944, .3558, .6439	34.39/1

It will be seen that the overtones are not harmonics. The first overtone of a bar or reed has a higher frequency than the sixth harmonic of a string. The tuning fork is the most common example of a bar clamped at one end, because it can be considered to be two vibrating bars clamped at the lower ends. The overtone or the high frequency sound of a tuning fork is quickly damped out leaving almost a pure sound.

54

MECHANICAL VIBRATING SYSTEMS

B. Bar Free at Both Ends. — Consider a perfectly free bar (Fig. 3.2D). The fundamental frequency is given by

/■ = ^^/f

where / = length of the bar, in centimeters. All the other quantities are the same as in equation 3.2.

The modes of vibration of a perfectly free bar are shown in Fig. 3. 2D. The table which follows gives the position of the nodes and the frequencies of the overtones.

No. of Tone	No. of Nodes	Distances of Nodes from One Fnd in Terms of the Length of the Bur	Frequencies as a Ratio of the h'undamental
1	2	2242, .7758	/.
2	3	1321, 50, 8679	2 75r/i
3	4	0944, 3558, .6442, 9056	5 404/,
4	5 1	.0734, .277, .5, .723, .9266	8 933/i

C. Bar Clamped at Both Ends. — Consider a bar rigidly clamped at both ends (Fig. 3. IF). The same tones are obtained as in the case of the per- fectly free bar.

D. Bar Supported at Both Ends. — Consider a bar supported on knife edges at the two edges at the two ends (Fig. 3.2E.). The fundamental fre- quency is given by

where / = length of the bar, in centimeters. All the other quantities are the same as in equation 3.2.

The overtones are

/* = 4/x /» = 9/i /4 = \6fi etc.

The nodes are equidistant as in case of the string.

TRANSVERSE VIBRATION OF BARS

55

E. Bar Clamped at One End and Supported at the Other. — Consider a bar clamped at one end and supported at the other end (Fig. 3.25). The fundamental frequency is given by

. _ 2.45 to

‘ \ P

3.5

The overtones are

/2= 3.25/1 /3= 6.75/1 /4= 11.5/1,

/5 = 17.7/1

The modes of vibration are shown in Fig. 3.25.

F. Bar Supported at One End and Free at the Other. — Consider a bar supported at one end and free at the other (Fig. 3.2C). The fundamental frequency is zero. The first overtone is given by

. 2.45 to

■ "F-VV

3.6

The overtones arc

/i= 0

/3= 3.25/2 /4= 6.75/2

/s = 11.5/2,

and

h = 17.7/2

The modcvS of vibration are shown in Fig. 3.2C.

G. Tapered Cantilever Bars, — In the preceding, considerations have been concerned with bars of uniform cross section. It is the purpose of this section to give the formulas for the resonant frequencies of tapered cantilever bars.

The resonant frequency of a wedge-shaped bar vibrating normal to the two parallel sides of the wedge, Fig. 3.3/f, is

_ 1.14 g

\l2p

3.7

where b = thickness of the bar in the direction of vibration, in centimeters.

56

MECHANICAL VIBRATING SYSTEMS

Fig. 3.3. Tapered cantilever bars, that is, bars clamped at one end and free at the other. A, A wedge-shaped bar vi- brating in a direction normal to the two parallel sides. B. A wedge-shaped bar vibrating in a direction parallel to the two parallel sides. C. A conical bar.

The resonant frequency of a wedge- shaped bar vibrating parallel to the two parallel sides of the wedge, Fig. 3.3£, is

^ pyjiip

3.8

The resonant frequency of a conical bar. Fig. 3.3C, is

\ 4p

3.9

where a = radius of the cone at the base, in centimeters.

3.4. Stretched Membranes.®*^*®*®*^® — The ideal membrane is assumed to

be flexible and very thin in cross section, and stretched in all directions by a force which is not affected by the motion of the membrane. Complete theoretical analyses have been made of circular, square and rectangular membranes. For cases of practical interest the membrane is assumed to be rigidly clamped and stretched by a massive surround. It is the purpose of this section to consider circular, square and rectangular stretched membranes.

A. Circular Membrane, — The fundamental frequency of a circular stretched membrane is given by

r -382 fr

/oi

where m = mass, in grams per square centimeter of area,

R = radius of the membrane, in centimeters, and T = tension, in dynes per centimeter.

The fundamental vibration is with the circumference as a node and a maximum displacement at the center of the circle (Fig. 3.4//) The fre-

•Lamb, “ Dynamical Theory of Sound,” E. Arnold, London, 1931.

^ Rayleigh, “ Theory of Sound,” Macmillan and Company, London, 1926.

• Morse, “ Vibration and Sound,” McGraw-Hill Book Company, New York, N.y., 1936.

•Wood, ” A Text Book of Sound,” Bell and Sons, London, 1930.

Crandall, “ Theory of Vibrating Systems and Sound,” D. Van Nostrand Company, New York, N. Y., 1926.

STRETCHED MEMBRANES

57

quencies of the next two overtones with nodal circles are

/o2 = 2.3Q/0I /o3 = 3.6(yoi

Fio. 3.4. Modes of vibration of a stretched circular membrane. Shaded segments are dis- placed in opposite phase to unshaded.

and are shown in Figs. 3.45 and 3.4C. The frequencies of the first, second and third overtones with nodal diameters are

/u = 1.59/01

/21 = 2.1^01

/si = 2.6S/01

These nodes are shown in Figs. 3.4D, 3.4E, and 3.4F. Following these simpler forms of vibration are combinations of nodal circles and nodal diameters. The frequency of one nodal circle and one nodal diameter, Fig. 3.4G, is

/is = 2.92/01

58 MECHANICAL VIBRATING SYSTEMS

The frequency of one nodal circle and two nodal diameters, Fig. 3.4//, is

/22 = 3.50/

The frequency of two nodal circles and one nodal diameter. Fig. 3.47, is

/i3 = 4.22/01

The stretched circular membrane is used in the condenser microphone (see Sec. 8.2fi). The fundamental resonance frequency is placed at the upper limit of the frequency range. A resistive load is coupled to the diaphragm for damping the response in the neighborhood of the funda- mental resonance frequency. This resistance is incorporated in the back plate which serves as the stationary electrode.

A stretched circular membrane is also used in all types of drums. In this case the air enclosure as well as the characteristics of the membrane controls the modes of vibration.

B. Square Me^nhrane. — The fundamental frequency of a square stretched membrane is given by

f

.705 It

a ?n

3.11

where m = mass, in grams per square centinjeter of area, a = length of a side, in centimeters, and T = tension, in dynes per centimeter.

C. Rectangular Membrane, — The fundamental frequency of a rectangu- lar stretched membrane with the sides in the ratio of 1 to 2 is given by

^ /r

3.12

where m = mass, in grams per square centimeter,

a = 2b = length of the long side, in centimeters, b = length of the short side in centimeters, and T = tension, in dynes per centimeter.

3.6. Circular Plates.* — The circular plates shown in Fig. 3.5 are assumed to be of uniform cross section and under no tension. It is

“ Rayleigh, “ Theory of Sound,*' Macmillan and Company, London, 1926. Morse, “ Vibration of Sound,” McGraw-Hill Book Company, New York, N. Y., 1936.

'®Wood, ” A Text Book of Sound,** Bell and Sons, London, 1930.

Crandall, ” Theory of Vibrating Systems and Sound,** D. Van Nostrand Com- pany, New York, N. Y., 1926.

^^Lamb, ” Dynamical Theory of Sound,” R. Arnold, London, 1931.

CIRCULAR PLATES

59

the purpose of this section to consider the vibration of circular plates for the various support means of Fig. 3.5.

CLAMPtO CDCC

B

i

SUPPORTED EDGE

SUPPORTED CENTER FREE

Fig. 3.5. Circular plates. A circular plate clamped at the edge. B. A circular plate

supported at the edge. C. A circular plate supported at the center. D. A free circular

plate.

A. Circular Clamped Plate, — Consider a circular clamped plate as shown in Fig. 3.5//. The fundamental frequency is given by

where / = thickness of the plate, in centimeters,

R = radius of the plate up to the clamping boundary, in centimeters, p = density, in grams per cubic centimeters (see Table 1.1),

O' = Poisson’s ratio (see Table 1.1), and

^ = Young’s modulus, in dynes per square centimeter (see Table

1.1).

.467/ / ^

R} \p(l ~ (t2)

60

MECHANICAL VIBRATING SYSTEMS

The fundamental frequency is with the circumference as a node and a maximum displacement at the center (Fig. 3.6^).

The frequency of the next two overtones with nodal circles, Fig. 3.65 and 3.6C, are,

/o2 = 3.91/01 /08 = 8,75/01

f2t*a74foi

Fio. 3.6. Modes of vibration of a clamped circular plate. Shaded segments are displaced in

opposite phase to unshaded.

The frequencies of the first, second and third overtones with nodal diame- ters are

/u *= 2.O9/01 /2I as 3.43/01 /3I = 4.95/01

These nodes are shown in Figs. 3.6D, 3.65 and 3.6F.

Following these simpler forms of vibration are combinations of nodal

CIRCULAR PLATES

61

circles and nodal diameters. The frequency of one nodal circle and one nodal diameter, Fig. 3.6G, is

/i2 = 5.98/01

The frequency of one nodal circle and two nodal diameters. Fig. 3.6^, is

/22 = 8.74/01

The frequency of two nodal circles and one nodal diameter. Fig. 3.67, is

/18 = 11.9/01

The clamped plate is used in electromagnetic telephone receivers in which the steel diaphragm serves as the armature (see Sec. 9,2A). It is used in carbon microphones (see Sec. 8.2/7). It is used in the subaqueous condenser microphone (Sec. 13.4) and the magnetic subaqueous loud speaker (Sec. 13.6). Clamped plate diaphragms have been used in minia- ture condenser microphones. The disadvantage of a plate is the difficulty of mounting a thin plate to give a small mass per unit area for high sensi- tivity and still have sufficient stiffness to yield a high fundamental fre- quency.

In telephone receivers, microphones and loud speakers employing a clamped diaphragm, the effective mass and effective area of the diaphragm, in terms of the velocity at the center, are needed when the system is re- duced to a lumped element representation. The effective mass or effective area for this condition is one third of the total mass or total area of the diaphragm. The air or water load on the diaphragm can be determined by assuming the effective radius of the equivalent piston to be .55 times the radius of the diaphragm (see Sec. 5.8).

B. Circular Free Plate — Consider a circular plate under no tension, uniform in cross section and perfectly free (Fig. 3.5D). For a vibration with nodal circle, as depicted in Fig. 3.45, the frequency is

, >412/ / ^

^ R} \p(l - (T»)

where / = thickness of the plate, in centimeters,

R == radius of the plate, in centimeters, p s= density, in grams per cubic centimeter (see Table 1.1), a *= Poisson’s ratio (see Table 1.1), and

^ — Young’s modulus, in dynes per square centimeter (see Table

1.1).

For a vibration with two nodal diameters, as depicted in Fig. 3.4£, the

62

MECHANICAL VIBRATING SYSTEMS

frequency is

. _ .193/ ^

^ E? \p(l - <r*)

3.15

C. Circular Plate Supported at the Center. — Consider a circular plate under no tension, uniform in cross section, edges perfectly free and supported at the center (Fig. 3. SC). The frequency, for the umbrella mode, is

.172/ / ^

\p(l-

3.16

D. Circular Plate Supported at the Outside. — Consider a plate under no tension, uniform in cross section, edges simply supported at the periphery (Fig. 3.55). The fundamental frequency is

.233/ ^

^ B? \p(l - ff*)

3.17

3.6. Longitudinal Vibration of — Consider an entirely

free rod of homogeneous material and constant cross section (see Sec. 1.14). The simplest mode of longitudinal vibration of a free rod is one in which a loop occurs at each end and a node in the middle, that is, when the length of the rod is one half wavelength. The fundamental frequency of longitudinal vibration of a free rod, F'ig. 3.7, may be obtained from equation 1.78 as follows,

/i =

c

X

= l /I

2/ 2/\ p

3.18

where / = length of the rod, in centimeters,

p = density of the material, in grams per cubic centimeter (see Table 1.1),

^ = Young’s modulus, in dynes per square centimeter (see Table

1.1),

c = velocity of sound, in centimeters per second (see Table 1.1, and equation 1.78), and

\ = wavelength of the sound wave, in centimeters.

Rayleigh, “ Theory of Sound,” Macmillan and Company, London, 1926. Morse, “ Vibration and Sound,” McGraw-Hill Book Company, New York, N. Y., 1936.

Wood, ” A Text Book of Sound,” Bell and Sons, London, 1930.

*®Lamb, ” Dynamical Theory of Sound,” E. Arnold, London, 1931.

LONGITUDINAL VIBRATION OF BARS

63

The overtones of the free rod are harmonics of the fundamental; that is h = 2/i,/3 = 3/i,/4 = 4/i, etc., Fig. 3.7.

The fundamental resonance frequency occurs when the length of the rod is one-half wavelength. This fact provides a means of computing the velocity of sound when the density, Young’s modulus and the frequency are known, or the frequency of sound when the velocity, density and Young’s modulus are known.

Rods in which the longitudinal waves are excited by striking the ends are used as standards of high-frequency sounds, 5000 cycles and above, where a tuning fork is not very satisfactory.

L M L M L

FIRST OVERTONE SECOND HARMONIC

SECOND OVERTONE THIRD HARMONIC

Fio. 3.7. Modes of longitudinal vibrations of a free rod. The nodes and loops are indicated

by iV and L.

Longitudinal waves in a rod may be set up by electromagnetic, electro- static or magnetostriction means. In the first case, if the rod is of mag- netic material and is held near an electromagnet in which an alternating current is flowing a longitudinal force will be set up in the rod. If the frequency of the driving current is continuously variable, the rod will be set into violent vibrations at the fundamental resonant frequency. If the plane end of a rod is placed near a metallic disk, the two plane surfaces may be used to serve as plates of a condenser. An alternating current sent through the condenser will cause an alternating force to be exerted upon the end of the rod. The rod will be sent into violent vibrations when the frequency of the impressed alternating current corresponds to the fundamental frequency or one of the overtones. Magnetization of magnetic materials produces small changes in the dimensions of these materials. A rod of magnetic material placed in a coil of wire will expe- rience a change in length corresponding to the alterations in the actuating current. If the coil is part of the circuit of a vacuum tube oscillator the

64

MECHANICAL VIBRATING SYSTEMS

rod will vibrate and the vacuum tube will oscillate at the fundamental frequency of the rod. Such a system is termed a magnetostriction sonic, ultrasonic or supersonic generator and may be used to produce sound waves in air or any other medium (see Secs. 13.7 and 13.8).

3.7. Torsional Vibration of Bars.^^*“ — A solid bar or tube may be twisted about the axis of the rod in such a manner that each transverse section remains in its own plane (see Sec. 1.15). If the section is not circu- lar there will be motion parallel to the axis of the bar. Consider an entirely free rod of homogeneous material and circular cross section. The simplest

JL N L

(T) ) M M t < * * M M ril)

fundamental first harmonic

*

L N L N L

( )) ) M ♦ * ♦ * M } ) ) ) M * * • * M n ))

FIRST OVERTONE SECOND HARMONIC

JL N L N L N L

( )) M ♦ ♦ M ) ) M » M 1 ) t t . » I ) j)

SECOND OVERTONE THIRD HARMONIC

Fig. 3.8. Modes of torsional vibration of a free rod. The nodes and loops arc indicated by

iV and L.

or fundamental mode of torsional vibration occurs when there is a node in the middle and a loop at each end; that is, when the length of the rod Is one- half wavelength. The fundamental resonant frequency, Fig. 3.8, may be obtained from equation 1.79 as follows,

where / = length of the rod, in centimeters,

p = density, in grams per cubic centimeter (see Table 1.1),

^ = Young’s modulus, in dynes per square centimeter (see Table 1.1),^

cr = Poisson’s ratio (see Table 1.1),

Pierce, G. W., Proc, Am, Acad, Arts and Sci,^ Vol. 63, p. 1, 1928.

^ Wood, “ A Text Book of Sound,” Bell and Sons, London, 1930.

** Rayleigh, ” Theory of Sound,” Macmillan and Company, London, 1926.

OPEN AND CLOSED PIPES

65

c = velocity of propagation of torsional waves, in centimeters per second, see equation 1.79, and X = wavelength of the torsional wave, in centimeters.

The overtones, as in the case of longitudinal vibrations, are harmonics of the fundamental. That is, /j = 2/i, /a = 3/i, /a = 4/i, etc. The nodes and antinodes for the various harmonics are formed as in the case of longi- tudinal vibrations.

Torsional vibrations may be set up in bars by any means which applies tangential forces to the free end. From a comparison of the longitudinal and torsional vibrations in the same bar, Poisson^s ratio may be determined.

rUNOAMENTAL flRST HARMONIC FUNDAMENTAL FIRST HARMONIC

SECOND OVERTONE THIRD HARMONIC FIRST OVERTONE THIRD HARMONIC

Fio. 3.9. Mcnies of vibration of the air column in a pipe open at both ends and in a pipe closed at one end and open at the other end. The velocity nodes and loops are indicated by iV and L.

3.8. Open and Closed Pipes. — The vibrations of a column of gas or fluid in a cylindrical tube are analogous to the longitudinal vibrations in a solid bar. For the open pipe there must be a loop of displacement at the open ends.

The fundamental resonant frequency of a pipe, open at both ends, Fig. 3.9, is

3.20

where / = length of the pipe, in centimeters,

c = velocity of sound, in centimeters per second (see Table 1.1), and X = wavelength, in centimeters.

The overtones of an open pipe are harmonics of the fundamental. That

is, /z = 2/i, /s = 3/i, /a = ^/u etc.

The fundamental resonant frequency of a pipe closed at one end and

66

MECHANICAL VIBRATING SYSTEMS

open at the other end, Fig. 3.9, is

e

4/

3.21

The overtones of the pipe closed at one end are the odd harmonics. That is /2 = 3/i, fz - Sfu etc.

In the above examples the end correction has been omitted. Rayleigh** shows the added length at the open end to be .HR where R is the radius of the pipe. If the pipe is terminated in a large flange the end correction will be that given in Sec. 5.12.

Organ pipes and whistles have been built to cover the range from 16 cycles to 30,000 cycles. The frequency of open and closed pipes may be computed from the above equations. The sound vibrations in the pi{>e are set up by the stream of air which is controlled by the vibration in the pipe. It is an oscillatory system fed by a direct current of air or gas.

“ Rayleigh, “ Theory of Sound/* Macmillan and Company, London, 1926.

CHAPTER IV

DYNAMICAL ANALOGIES

4.1. Introduction. — Analogies are useful when It Is desired to compare an unfamiliar system with one that Is better known. The relations and actions are more easily visualized, the mathematics more readily applied and the analytical solutions more readily obtained in the familiar system. Analogies make it possible to extend the line of reasoning into unexplored fields.

A large part of engineering analysis is concerned with vibrating systems. Although not generally so considered, the electrical circuit is the most common example and the most widely exploited vibrating system. The equations of electrical circuit theory may be based on Maxwell’s dynamical theory in which the currents play the role of velocities. Expressions for the kinetic energy, potential energy and dissipation show that network equations are deducible from general dynamic equations. In other words, an electrical circuit may be considered to be a vibrating system. This immediately suggests analogies between electrical circuits and other dy- namical systems as, for example, mechanical and acoustical vibrating systems.

The equations of motion of mechanical systems were developed a long time before any attention was given to equations for electrical circuits. For this reason, in the early days of electrical circuit theory, it was natural to explain the action in terms of mechanical phenomena. However, at the present time, electrical circuit theory has been developed to a much higher state than the corresponding theory of mechanical systems. The number of engineers and scientists versed in electrical circuit theory is many times the number equally familiar with mechanical systems.

Almost any work involving mechanical or acoustical systems also includes electrical systems and electrical circuit theory. The acoustical engii^eer is interested in sound reproduction or the conversion of electrical or me- chanical energy into acoustical energy, the development of vibrating systems and the control of sound vibrations. This involves acoustical, electroacoustical, mechanoacoustical or electromechanoacoustical systems. The mechanical engineer is interested in the development of various mechanisms or vibrating systems involving masses, springs and friction.

67

68

DYNAMICAL ANALOGIES

Electrical circuit theory is the branch of electromagnetic theory which deals with electrical oscillations in linear electrical networks.^ An elec- trical network is a connected set of separate circuits termed branches or meshes. A circuit may be defined as a physical entity in which varying magnitudes may be specified in terms of time and a single dimension.^ The branches or meshes are composed of elements. Elements are the constituent parts of a circuit. Electrical elements are resistance, induct- ance and capacitance. Vibrations in one dimension occur in mechanical systems made up of mechanical elements, as, for example, various as- semblies of masses, springs and brakes. Confined acoustical systems in which the dimensions are small compared to the wavelength are vibrations in a single dimension.

The number of independent variables required to completely specify the motion of every part of a vibrating system is a measure of the number of degrees of freedom of the system. If only a single variable is needed the system is said to have a single degree of freedom. In an electrical circuit the number of degrees of freedom is equal to the number of inde- pendent closed meshes or circuits.

The use of complex notation has been applied extensively to electri- cal circuits. Of course, this operational method can be applied to any analytically similar system.

Mathematically the elements in an electrical network are the coeffi- cients in the differential equations describing the network. When the electric circuit theory is based upon Maxwell’s dynamics, the network forms a dynamical system in which the currents play the role of velocities. In the same way the coefficients in the differential equations of a mechani- cal or acoustical system may be looked upon as mechanical or acoustical elements. Kirchhoff's electromotive force law plays the same role in set- ting up the electrical equations as D’Alembert’s principle does in setting up the mechanical and acoustical equations. That is to say, every elec- trical, mechanical or acoustical system may be considered as a combination of electrical, mechanical or acoustical elements. Therefore, any mechani- cal or acoustical system may be reduced to an electrical network and the problem may be solved by electrical circuit theory.

In view of the tremendous amount of study which has been directed toward the solution of circuits, particularly electrical circuits, and the

^ The use of the terms “ circuit ” and “ network in the literature is not estab- lished. The term “ circuit ” is often used to designate a network with several branches.

* The term “ single dimension ” implies that the movement or variation occurs along a path. In a field problem there is a variation in two or three dimensions.

DEFINITIONS

69

engineer’s familiarity with electrical circuits, it is logical to apply this knowledge to the solution of vibration problems in other fields by the same theory as that used in the solution of electrical circuits.

It is the purpose of this chapter to develop the analogies between ele- ments* in electrical, mechanical and acoustical systems.

4.2. Definitions. — A few of the terms used in dynamical analogies will be defined in this section. Terms not listed below will be defined in sub- sequent sections.

Abvolt — An abvolt is the unit of electromotive force.

Instantaneous Electromotive Force — The instantaneous electromotive force between two points is the total instantaneous electromotive force. The unit is the abvolt.

Effective Electromotive Force — The effective electromotive force is the root mean square of the instantaneous electromotive force over a complete cycle between two points. The unit is the abvolt.

Maximum Electromotive Force — The maximum electromotive force for any given cycle is the maximum absolute value of the instantaneous electro- motive force during that cycle. The unit is the abvolt.

Peak Electromotive Force — The peak electromotive force for any speci- fied time interval is the maximum absolute value of the instantaneous electromotive force during that interval. The unit is the abvolt.

Dyne — A dyne is the unit of force or mechanomotive force.

Instantaneous Force (Instantaneous Mechanomotive Force) — The instantaneous force at a point is the total instantaneous force. The unit is the dyne.

Effective Force (Effective Mechanomotive Force) — The effective force is the root mean square of the instantaneous force over a complete cycle. The unit is the dyne.

Maximum Force (Maximum Mechanomotive Force) — The maximum force for any given cycle is the maximum absolute value of the instanta- neous force during that cycle. The unit is the dyne.

Peak Force (Peak Mechanomotive Force) — The peak force for any specified interval is the rhaximum absolute value of the instantaneous force during that interval. The unit is the dyne.

Dyne Centimeter — A dyne centimeter is the unit of torque or rotato- motive force.

Instantaneous Torque (Instantaneous Rotatomotive Force) — The in- stantaneous torque at a point is the total instantaneous torque. The unit is the dyne centimeter.

• For further considerations of analogies see Olson, “ Dynamical Analogies,** D. Van Nostrand Company, New York, N. Y., 1943.

70

DYNAMICAL ANALOGIES

Effective Torque (Effective Rotatomotive Force) — The effective torque is the root mean square of the instantaneous torque over a complete cycle. The unit is the dyne centimeter.

Maximum Torque (Maximum Rotatomotive Force) — The maximum torque for any given cycle is the maximum absolute value of the instanta- neous torque during that cycle. The unit is the dyne centimeter.

Peak Torque (Peak Rotatomotive Force) — The peak torque for a speci- fied interval is the maximum absolute value of the instantaneous torque during that interval. The unit is the dyne centimeter.

Dyne per Square Centimeter — A dyne per square centimeter is the unit of sound pressure.

Static Pressure — The static pressure is the pressure that would exist in a medium with no sound waves present. The unit is the dyne per square centimeter.

Instantaneous Sound Pressure (Instantaneous Acoustomotive Force) — The instantaneous sound pressure at a point is the total instantaneous pressure at the point minus the static pressure. The unit is the dyne per square centimeter.

Effective Sound Pressure (Effective Acoustomotive Force) — The effec- tive sound pressure at a point is the root mean square value of the instan- taneous sound pressure over a complete cycle at the point. The unit is the dyne per square centimeter.

Maximum Sound Pressure (Maximum Acoustomotive Force) — The maximum sound pressure for any given cycle is the maximum absolute value of the instantaneous sound pressure during that cycle. The unit is the dyne per square centimeter.

Peak Sound Pressure (Maximum Acoustomotive Force) — The peak sound pressure for any specified time interval is the maximum absolute value of the instantaneous sound pressure in that interval. The unit is the dyne per square centimeter.

Abampere — An abampere is the unit of current.

Instantaneous Current — The instantaneous current at a point is the total instantaneous current at that point. The unit is the abampere.

Effective Current — The effective current at a point is the root mean square value of the instantaneous current over a complete cycle at that point. The unit is the abampere.

Maximum Current — The maximum current for any given cycle is the maximum absolute value of the instantaneous current during that cycle. The unit is the abampere.

Peak Current — The peak current for any specified time interval is the

DEFINITIONS

71

maximum absolute value of the instantaneous current in that interval. The unit is the abampere.

Centimeter per Second — A centimeter per second is the unit of velocity.

Instantaneous Velocity — The instantaneous velocity at a point is the total instantaneous velocity at that point. The unit is the centimeter per second.

Effective Velocity — The effective velocity at a point is the root mean square value of the instantaneous velocity over a complete cycle at that point. The unit is the centimeter per second.

Maximum Velocity — The maximum velocity for any given cycle is the maximum absolute value of the instantaneous velocity during that cycle. The unit is the centimeter per second.

Peak Velocity — The peak velocity for any specified time interval is the maximum absolute value of the instantaneous velocity in that interval. The unit is the centimeter per second.

Radian per Second — A radian per second is the unit of angular velocity.

Instantaneous Angular Velocity — The instantaneous angular velocity at a point is the total instantaneous angular velocity at that point. The unit is the radian per second.

Effective Angular Velocity — The effective angular velocity at a point is the root mean square value of the instantaneous angular velocity over a complete cycle at the point. The unit is the radian per second.

Maximum Angular Velocity — The maximum angular velocity for any given cycle is the maximum absolute value of the instantaneous angular velocity during that cycle. The unit is the radian per second.

Peak Angular Velocity — The peak angular velocity for any specified time interval is the maximum absolute value of the instantaneous angular velocity in that interval. The unit is the radian per second.

Cubic Centimeter per Second — A cubic centimeter per second is the unit of volume current.

Instantaneous Volume Current — The instantaneous volume current at a point is the total instantaneous volume current at that point. The unit is the cubic centimeter per second.

Effective Volume Current — The effective volume current at a point is the root mean square value of the instantaneous volume current over a complete cycle at that point. The unit is the cubic centimeter per second.

Maximum Volume Current — The maximum volume current for any given cycle is the maximum absolute value of the instantaneous volume current during that cycle. The unit is the cubic centimeter per second.

Peak Volume Current — The peak volume current for any specified

72

DYNAMICAL ANALOGIES

time interval is the maximum absolute value of the instantaneous volume current in that interval. The unit is the cubic centimeter per second.

Electrical Impedance — Electrical impedance is the complex quotient of the alternating electromotive force applied to the system by the resulting current. The unit is the abohm.

Electrical Resistance — Electrical resistance is the real part of the elec- trical impedance. This is the part responsible for the dissipation of energy. The unit is the abohm.

Electrical Reactance — Electrical reactance is the imaginary part of the electrical impedance. The unit is the abohm.

Inductance — Inductance in an electrical system is that coefficient which, when multiplied by 27r times the frequency, gives the positive imaginary part of the electrical impedance. The unit is the abhenry.

Electrical Capacitance — Electrical capacitance in an electrical system is that coefficient which, when multiplied by 2ir times the frequency, is the reciprocal of the negative imaginary part of the electrical impedance. The unit is the abfarad.

Mechanical Rectilineal Impedance^ (Mechanical Impedance) — Me- chanical rectilineal impedance is the complex quotient of the alternating force applied to the system by the resulting linear velocity in the direction of the force at its point of application. The unit is the mechanical ohm.

Mechanical Rectilineal Resistance (Mechanical Resistance) — Mechani- cal rectilineal resistance is the real part of the mechanical rectilineal impedance. This is the part responsible for the dissipation of energy. The unit is the mechanical ohm.

Mechanical Rectilineal Reactance (Mechanical Reactance) — Mechani- cal rectilineal reactance is the imaginary part of the mechanical recti- lineal impedance. The unit is the mechanical ohm.

Mass — Mass in a mechanical system is that coefficient which, when multiplied by 2w times the frequency, gives the positive imaginary part of the mechanical rectilineal impedance. The unit is the gram.

Compliance — Compliance in a mechanical system is that coefficient which, when multiplied by 2t times the frequency, is the reciprocal of the negative imaginary part of the mechanical rectilineal impedance. The unit is. the centimeter per dyne.

^ The word “ mechanical ” is ordinarily used as a modifier to designate a mechani- cal system with rectilineal displacements and the word “ rotational ** is ordinarily used as a modifier to designate a mechanical system with rotational displacements. To avoid ambiguity in this book, where both systems are considered concurrently, the words “ mechanical rectilineal ** are used as modifiers to designate a mechanical system with rectilineal displacements and the words “ mechanical rotational ** arc used as modifiers to designate a mechanical system with rotational displacements.

DEFINITIONS

73

Mechanical Rotational Impedance (Rotational Impedance) — Mechani- cal rotational impedance is the complex quotient of the alternating torque applied to the system by the resulting angular velocity in the direction of the torque at its point of application. The unit is the rotational ohm.

Mechanical Rotational Resistance (Rotational Resistance) — Mechani- cal rotational resistance is the real part of the mechanical rotational impedance. This is the part responsible for the dissipation of energy. The unit is the rotational ohm.

Mechanical Rotational Reactance (Rotational Reactance) — Mechani- cal rotational reactance is the imaginary part of the mechanical rotational impedance. The unit is the rotational ohm.

Moment of Inertia — Moment of inertia in a mechanical rotational system is that coefficient which, when multiplied by 27r times the frequency, gives the positive imaginary part of the mechanical rotational impedance. The unit is the gram centimeter to the second power.

Rotational Compliance — Rotational compliance in a mechanical rota- tional system is that coefficient which, when multiplied by Itt times the frequency, is the reciprocal of the negative imaginary part of the mechanical rotational impedance. The unit is the radian per centimeter per dyne.

Acoustical Impedance — Acoustical impedance is the complex quotient of the alternating pressure applied to the system by the resulting volume current. The unit is the acoustical ohm.

Acoustical Resistance — Acoustical resistance is the real part of the acoustical impedance. This is the part responsible for the dissipation of energy. The unit is the acoustical ohm.

Acoustical Reactance — Acoustical reactance is the imaginary part of the acoustical impedance. The unit is the acoustical ohm.

Inertance — Inertance in an acoustical system is that coefficient which, when multiplied by Itt times the frequency, gives the positive imaginary part of the acoustical impedance. The unit is the gram per centimeter to the fourth power.

Acoustical Capacitance — Acoustical capacitance in an acoustical sys- tem is that coefficient which, when multiplied by 27r times the frequency, is the reciprocal negative imaginary part of the acoustical impedance. The unit is the centimeter to the fifth power per dyne.

Element — An element or circuit parameter in an electrical system de- fines a distinct activity in its part of the circuit. In the same way, an element in a mechanical rectilineal, mechanical rotational or acoustical system defines a distinct activity in its part of the system. The elements in an electrical circuit are electrical resistance, inductance and electrical capacitance. The elements in a mechanical rectilineal system are mechani-

74

DYNAMICAL ANALCXJIES

cal rectilineal resistance, mass and compliance. The elements in a me- chanical rotational system are mechanical rotational resistance, moment of inertia, and rotational compliance. The elements in an acoustical sys- tem are acoustical resistance, inertance and acoustical capacitance.

Electrical System — An electrical system is a system adapted for the transmission of electrical currents consisting of one or all of the electrical elements: electrical resistance, inductance and electrical capacitance.

Mechanical Rectilineal System — A mechanical rectilineal system is a system adapted for the transmission of vibrations consisting of one or all of the following mechanical rectilineal elements: mechanical rectilineal resistance, mass and compliance.

Mechanical Rotational System — A mechanical rotational system is a system adapted for the transmission of rotational vibrations consisting of one or all of the following mechanical rotational elements: mechanical rotational resistance, moment of inertia and rotational compliance.

Acoustical System — An acoustical system is a system adapted for the transmission of sound consisting of one or all of the following acoustical elements: acoustical resistance, inertance and acoustical capacitance.

Electrical Abohm — An electrical resistance, reactance or impedance is said to have a magnitude of one abohm when an electromotive force of one abvolt produces a current of one abampere.

Mechanical Ohm — A mechanical rectilineal resistance, reactance or impedance is said to have a magnitude of one mechanical ohm when a force of one dyne produces a velocity of one centimeter per second.

Rotational Ohm — A mechanical rotational resistance, reactance or impedance is said to have a magnitude of one rotational ohm when a torque of one dyne centimeter produces an angular velocity of one radian per second.

Acoustical Ohm — An acoustical resistance, reactance or impedance is said to have a magnitude of one acoustical ohm when a pressure of one dyne per square centimeter produces a volume current of one cubic centi- meter per second.

4.3. Elements. — An element or circuit parameter in an electrical system defines a distinct activity in its part of the circuit. In an electrical system these elements are resistance, inductance and capacitance. They are dis- tinguished from the devices: resistor, inductor and capacitor. A resistor, inductor or capacitor idealized to have only resistance, inductance and capacitance is a circuit element. As indicated in the preceding chapter, the study of mechanical and acoustical systems is facilitated by the intro- duction of elements analogous to the elements of an electric circuit. In

RESISTANCE

75

this procedure, the first step is to develop the elements in these vibrating systems. It is the purpose of this chapter to define and describe electrical, mechanical rectilineal, mechanical rotational and acoustical elements.

4.4. Resistance. — A. Electrical Resistance. — Electrical energy is changed into heat by the passage of an electrical current through a re- sistance. Energy is lost by the system when a charge q of electricity is driven through a resistance by a voltage e. Resistance is the circuit ele- ment which causes dissipation.

Electrical resistance in abohms, is defined as

€

VE = - 4.1

/

where e = voltage across the electrical resistance, in abvolts, and i = current through the electrical resistance, in abamperes.

Equation 4.1 states that the electromotive force across an electrical resistance is proportional to the electrical resistance and the current.

B. Mechanical Rectilineal Resistance. — Mechanical rectilineal energy is changed into heat by a rectilinear motion which is opposed by linear resistance (friction). In a mechanical system dissipation is due to friction. Energy is lost by the system when a mechanical rectilineal resistance is displaced a distance x by a force Jm.

Mechanical rectilineal resistance (termed mechanical resistance) rjif, in mechanical ohms, is defined as

y V . _

rv = — 4.2

ii

where Jm — applied mechanical force, in dynes, and

u = velocity at the point of application of the force, in centimeters per second.

Equation 4.2 states that the driving force applied to a mechanical recti- lineal resistance is proportional to the mechanical rectilineal resistance and the linear velocity.

C. Mechanical Rotational Resistance. — Mechanical rotational energy is changed into heat by a rotational motion which is opposed by a rotational resistance (rotational friction). Energy is lost by the system when a mechanical rotational resistance is displaced by an angle </> by a torque Jr.

Mechanical rotational resistance (termed rotational resistance) rjj, in

76

DYNAMICAL ANALOGIES

rotational ohms, is defined as

4.3

where Jr = applied torque, in dyne centimeters, and

0 s= angular velocity at the point of application about the axis, in radians per second.

Equation 4.3 states that the driving torque applied to a mechanical rotational resistance is proportional to the mechanical rotational resistance and the angular velocity.

D. Acoustical Resistance. — In an acoustical system dissipation may be due to the fluid resistance or radiation resistance. At this point the former typ^ of acoustical resistance will be considered. Acoustical energy is changed into heat by the passage of a fluid through an acoustical resistance. The resistance is due to viscosity. Energy is lost by the system when a volume Y of a fluid or gas is driven through an acoustical resistance by a pressure p.

Acoustical resistance ra, in acoustical ohms, is defined as

4.4

where p == pressure, in dynes per square centimeter, and

U = volume current, in cubic centimeters per second.

Equation 4.4 states that the driving pressure applied to an acoustical resistance is proportional to the acoustical resistance and the volume current.

4.6. Inductance, Mass, Moment of Inertia, Inertance A. Inductance. — Electromagnetic energy is associated with inductance. Electromagnetic energy increases as the current in the inductance increases. It decreases when the current decreases. It remains constant when the current in the inductance is a constant. Inductance is the electrical circuit element which opposes a change in current. Inductance L, in abhenries, is defined as

e = 4.5

dt

where e ■■ electromotive or driving force, in abvolts, and

dijdt = rate of change of current, in abamperes per second.

INDUCTANCE, MASS

77

Equation 4.5 states that the electromotive force across an inductance is proportional to the inductance and the rate of change of current.

B. Mass. — Mechanical rectilineal inertial energy is associated with mass in the mechanical rectilineal system. Mechanical rectilineal energy increases as the linear velocity of a mass increases, that is, during linear acceleration. It decreases when the velocity decreases. It remains constant when the velocity is constant. Mass is the mechanical element which opposes a change of velocity. Mass w, in grams, is defined as

JM ^ m— 4.6

where dufdt = acceleration, in centimeters per second per second, and Jm = driving force, in dynes.

Equation 4.6 states that the driving force applied to the mass is pro- portional to the mass and the rate of change of velocity.

C. Moment of Inertia, — Mechanical rotational inertial energy is associ- ated with moment of inertia in the mechanical rotational system. Me- chanical rotational energy increases as the angular velocity of a moment of inertia increases, that is, during angular acceleration. It decreases when the angular velocity decreases. It remains a constant when the angular velocity is a constant. Moment of inertia is the rotational ele- ment which opposes a change in angular velocity. Moment of inertia /, in gram (centimeter), is given by

Jr =

4.7

where d^/dt = angular acceleration, in radians per second per second, and Jr = torque, in dyne centimeters.

Equation 4.7 states that the driving torque applied to the moment of inertia is proportional to the moment of inertia and the rate of change of angular velocity.

D. Inertance. — Acoustical inertial energy is associated with inertance in the acoustical system. Acoustical energy increases as the volume cur- rent of an inertance increases. It decreases when the volume current decreases. It remains constant when the volume current of the inertance is a constant. Inertance is the acoustical element that opposes a change in volume current. Inertance A/, in grams per (centimeter), is defined as

P

4.8

78

DYNAMICAL ANALOGIES

where M — inertance, in grams per (centimeter),

dUjdt = rate of change of volume current, in cubic centimeters per second per second, and

p = driving pressure, in dynes per square centimeter.

Equation 4.8 states that the driving pressure applied to an inertance is proportional to the inertance and the rate of change of volume current. Inertance® may be expressed as

where m = mass, in grams,

S = cross-sectional area in square centimeters, over which the driving pressure acts to drive the mass.

The inertance of a circular tube is

4.10

where R = radius of the tube, in centimeters,

/ = effective length of the tube, that is, length plus end correction, in centimeters, and

p = density of the medium in the tube, in grams per cubic centi- meter.

4.6. Electrical Capacitance, Rectilineal Compliance, Rotation Com- pliance, Acoustical Capacitance — A. Electrical Capacitance, — Electro- static energy is associated with the separation of positive and negative charges, as in the case of the charges on the two plates of an electrical capacitance. Electrostatic energy increases as the charges of opposite polarity are separated. It is constant and stored when the charges remain unchanged. It decreases as the charges are brought together and the electrostatic energy released. Electrical capacitance is the electrical cir- cuit element which opposes a change in voltage. Electrical capacitance C«, in abfarads, is defined as

de

4.11

Equation 4.11 may be written

e =

i dt =

Ce

4.12

‘ See Sec. 5.6.

COMPLIANCE

79

where q = charge on electrical capacitance, in abcoulombs, and e == electromotive force, in abvolts.

Equation 4.12 states that the charge on an electrical capacitance is pro- portional to the electrical capacitance and the applied electromotive force.

B. Rectilineal Compliance, — Mechanical rectilineal potential energy is associated with the compression of a spring or compliant element. Me- chanical energy increases as the spring is compressed. It decreases as the spring is allowed to expand. It is a constant, and is stored, when the spring remains immovably compressed. Rectilineal compliance is the mechanical element which opposes a change in the applied force. Rec- tilineal compliance Cm (termed compliance) is defined as

Jm =

X

4.13

where x = displacement, in centimeters, and Jm = applied force, in dynes.

Equation 4.13 states that the displacement of a compliance is propor- tional to the compliance and the applied force.

Stiffness is the reciprocal of compliance.

C. Rotational Compliance, — Mechanical rotational potential energy is associated with the twisting of a spring or compliant element. Mechanical energy increases as the spring is twisted. It decreases as the spring is allowed to unwind. It is constant, and is stored when the spring remains immovably twisted. Rotational compliance is the mechanical element which opposes a change in the applied torque. Rotational compliance C«, or moment of compliance, is defined as

4.14

where 4> == angular displacement, in radians, and Jr == applied torque, in dyne centimeters.

Equation 4.14 states that, the rotational displacement of the rotational compliance is proportional to the rotational compliance and the applied torque.

D. Acoustical Capacitance, — Acoustical potential energy is associated with the compression of a fluid or gas. Acoustical energy increases as the gas is compressed. It decreases as the gas is allowed to expand. It is constant, and is stored when the gas remains immovably compressed.

80

DYNAMICAL ANALOGIES

Acoustical capacitance is the acoustic element which opposes a change in the applied pressure. The pressure, in dynes per square centimeter, in terms of the condensation, is from equation 1 .21

p = c^ps

4.15

where c = velocity, in centimeters per second,

p = density, in grams per cubic centimeter, and s = condensation, defined in equation 4.16.

The condensation in a volume V due to a change in volume from V to ^'is

s =

4.16

The change in volume V — F', in cubic volume displacement, in cubic centimeters.

centimeters, is equal to the

V ^ ^ X

4.17

where X « volume displacement, in cubic centimeters. From equations 4.15, 4.16, and 4.17, the pressure is

Acoustical capacitance Ca is defined as

X

where p = sound pressure in dynes per square centimeter, and X = volume displacement, in cubic centimeters.

4.18

4.19

Equation 4.19 states the volume displacement in an acoUvStical capaci- tance is proportional to the pressure and the acoustical capacitance.

From equations 4.18 and 4.19 the acoustical capacitance of a volume is

4.20

where V = volume, in cubic centimeters.

4.7. Representation of Electrical, Mechanical Rectilineal, Mechanical Rotational and Acoustical Elements. — Electrical, mechanical rectilineal, mechanical rotational and acoustical elements have been defined in the

REPRESENTATION OF ELEMENTS

81

preceding sections. Fig. 4.1 illustrates schematically the four elements in each of the four systems.

The electrical elements, electrical resistance, inductance and electrical capacitance are represented by the conventional symbols.

Mechanical rectilineal resistance is represented by sliding friction which causes dissipation. Mechanical rotational resistance is represented by a wheel with a sliding friction brake which causes dissipation. Acoustical resistance is represented by narrow slits which cause dissipation due to viscosity when fluid is forced through the slits. These elements are analogous to electrical resistance in the electrical system.

	1a	Tm	>R
	Mam .
	M	□
Ct	Ca	Cm	Cr
HI-	-J\|
		RCCTILINCAL	ROTATIOMAL
ELECTRICAL	ACOUSTICAL	MECHANICAL

Fig. 4.1. Graphical representation of the three basic elements in electrical, mechanical rectilineal, mechanical rotational and acoustical systems.

rjj = electrical resistance

TR = mechanical rotational resistance m = mass

Ca = acoustical capacitance

TH = acoustical resistance

L — inductance

/ == moment of inertia Cm — compliance

VM = mechanical rectilineal resistance M = inertance

Ce = electrical capacitance Cr = rotational compliance

Inertia in the mechanical rectilineal system is represented by a mass. Moment of inertia in the mechanical rotational system is represented by a flywheel. Inertance in the acoustical system is represented as the fluid contained in a tube in which all the particles move with the same phase when actuated by a force due to pressure. These elements are analogous to inductance in the electrical system.

Compliance in the mechanical rectilineal system is represented as a spring. Rotational compliance in the mechanical rotational system is represented as a spring. Acoustical capacitance in the acoustical system

82

DYNAMICAL ANALOGIES

IS represented as a volume which acts as a stiffness or spring element. These elements are analogous to electrical capacitance in the electrical system.

In the preceding discussion of electrical, mechanical rectilineal, mechani- cal rotational and acoustical systems it was observed that the four systems are analogous. As pointed out in the introduction, using the dynamical concept for flow of electrical currents in electrical circuits the fundamental laws are of the same nature as those which govern the dynamics of a moving body. In general, the three fundamental dimensions are mass, length, and time. These quantities are directly connected to the mechanical rec- tilineal system. Other quantities in the mechanical rectilineal system may be derived in terms of these dimensions. In terms of analogies the dimen- sions in the electrical circuit corresponding to length, mass and time in the mechanical rectilineal system are charge,* self-inductance and time. The corresponding analogous dimensions in the rotational mechanical system are angular displacement, moment of inertia and time. The corresponding analogous dimensions in the acoustical system are volume displacement, inertance and time. The above-mentioned fundamental dimensions in each of the four systems are shown in tabular form in Table 4.1. Other quantities in each of the four systems may be expressed in terms of the dimensions of Table 4.1. A few of the most important quantities have been tabulated in Table 4.2. Tables 4.1 and 4.2 depict analogous quanti- ties in each of the four systems. Further, they show that the four systems are dynamically analogous.

The dimensions given in Table 4.1 should not be confused with the classi- cal dimensions of electrical, mechanical and acoustical systems given in Table 4.3. Table 4.3 uses mass A/, length L and time T, In the case of electrical units dielectric and permeability constants are assumed to be dimensionless.

For further considerations of dynamical analogies, as, for example, electrical, mechanical rectilineal, mechanical rotational and acoustical systems of one, two and three degrees of freedom, corrective networks, wave filters, transients, driving systems, generating systems, theorems and applications, the reader is referred to Olson, “ Dynamical Analogies,” D. Van Nostrand Company, New York, N. Y., 1943.

Table 4.1

REPRESENTATION OF ELEMENTS

83

Symbol

■V*

S c

E.2

3 “

i

%

1

t

1

.-H

"c3

u

CA

§

c

o

s

.2

*Z3

«A

§

ii

c/3

1

H

c:?

8

<

c

8

c

rt

u

C

V

o

CQ

'o.

w

Q

u

2

"o

>

Time

<J

<

Quantity

Volume

Current

Pressure

Acoustical

Resistance

Acoustical

Capacitance

Energy

Power

'rt

c

2

E

in

-0-

c

o

8 G £.2

C

7

M

X,

s

*9

i.

.2

r9

1

rt

O

Vi

c

u

g

O

E "o (A ^

o

•'O-

G

"cl

<J

'S

n

jG

a

s

Quantity

foment of Inerti

4>

U

C«

s-

Q

k.

J2

*3

bb

C

<

V

.E

ri

Ui

u

5

‘S

JG

u

s

>>

'C

c

««

*3 ’u

-2 o

3 "o

g>>

<

Torque

Rotational

Resistance

Rotational

Compliance

Energy

Power

CO

"rt

V

1

B

>«

<

H

V

C

E 2

G

7

><

r

1

7

B

c

C/3

,

Cl

u

9J

c

u

E

o

u

C3c:

E "o in

u

O

•X

?

CJ

u

'c

Vi

>.

C

u

J2

’H,

tf)

u

'c

rt

JC

X

v

^ u

Vi c

fj rj

U

g

u

s

&

Mass

Q

S

c

Time

w

s

c

ct

Linear

Velocii

Force

1.2

2 Ss

V K

.2

”01

E

o

U

Energy

Power

Symbol

■S*

Dimen-

sion

T

3-

s.

iq

'

■i

u

•5

w

u

•S

Synr

bol

m

M

{?

a

Quantity

Self-Inductance

Electrical Charg

Time

u

a

Quantity

Current

Electromotive

Force

Electrical

Resistance

Electrical

Capacitance

Energy

Power

84

DYNAMICAL ANALOGIES

Table 4.3

Electrical j	Mechanical Rectilineal
Quantity	Unit	Sym- bol	Dimension	Quantity	Unit	Sym- bol	Dimension
Electromo- tive Force	Volts X W	e		Force	Dynes	Jm	MLT-^
Charge or Quantity				Linear Dis- placement	Centimeters	X	L
Current	Amperes X 10“^	fl		Linear Velocity	Centimeters per Second	X or V	LT-^
Electrical Imped- ance	Ohms X 10*	ze	LT-^	Mechanical Impedance	Mechanical Ohms	zm	A/r-i
Electrical Resist- ance	Ohms X 10*	rs	LT-^	Mechanical Resistance	Mechanical Ohms	ru
Electrical Reactance	Ohms X 10*	xs	LT-^	Mechanical Reactance	Mechanical Ohms	1 XM
Inductance	Henries X 10*	L	L	Mass	(irams	m	M
Electrical Capaci- tance	Farads X 10~*	Ce	l-xjn	Compliance	Centimeters per Dyne	Cv
Power	Ergs per Second	Pb	MDT-*	Power	Ergs per Second		MDT-*

REPRESENTATIONS OF ELEMENT

85

TABLE 4.3 — Continued

Mechanical Rotational	Acoustical
Quantity	Unit	Sym- bol	Dimension	Quantity	Unit	Sym- bol	Dimension
Torque	Dyne Centimeter	/«		Pressure	Dynes per Square Centimeter	P
Angular Displace- ment	Radians	0	■		Cubic Cen- timeters	X	D
Angular Velocity	Radians per Second	^otB	7-1	Volume Current	Cubic Cen- timeters per Second	X or U
Rotational Imped- ance	Rotational Ohms	zr	MDT-^	mm mu	Acoustical Ohms	za
Rotational Resist- ance	Rotational Ohms	fR	MDT-^	Acoustical Resistance	Acoustical Ohms	rA
Rotational Reactance	Rotational Ohms	XR		Acoustical Reactance	Acoustical Ohms	XA
Moment of Inertia	(Gram) (Cent- imeter)*	I	ML^	Inertance	Grams per (Centime- ter)^	M j	MLr^
Rotational Compli- ance	Radians per Dyne per Centimeter	Cr		Acoustical Capaci- tance	(Centime- ter)^ per Dyne	Ca
Power	Ergs per Second	Ph	MDT-^	Power	Ergs per Second	Pa

CHAPTER V

ACOUSTICAL ELEMENTS

6.1. Introduction. — The preceding chapter is devoted to analogies be- tween electrical, mechanical and acoustical systems. The purpose of draw- ing these analogies is to facilitate the solution of problems in mechanical and acoustical vibrating systems by converting these problems into the corresponding electrical analogies and solving the resultant electrical circuits by conventional electrical circuit theory. An electrical circuit is composed of electrical elements. In the same way the acoustical system is composed of acoustical elements. The type of element, that is, acousti- cal resistance, inertance or acoustical capacitance, will depend upon the characteristic manner in which the medium behaves for different sources of sound and in the different ways of confining the medium. It is the purpose of this chapter to consider acoustical elements and combination of elements.

6.2. Acoustical Resistance. Acoustical resistance may be obtained by forcing air through a small hole. The resistance is due to viscosity which may be considered as friction between adjacent layers of air. In the ordinary transmission of sound in a large tube the motion of all the particles in a plane normal to the axis is the same, therefore the frictional losses are small. When sound travels in a small tube the particle velocity varies from zero at the boundary to a maximum at the center. The same is true when a steady stream of air is forced through a small hole or tube, the velocity of adjacent layers varies from zero at the boundary to a maximum at the center. The smaller the hole the higher will be the resistance be- cause of the greater effect of the sides.

A small tube also has inertance. Therefore, the reactive component increases with frequency. The inertive reactance increases as the size of the hole decreases as does the acoustical resistance, but at a slower rate. Therefore, the inertive reactance may be made negligible compared to the acoustical resistance if the hole is made sufficiently small.

Acoustical resistance employing viscosity may be made in various forms as, for example, a large number of small holes or a large number of slits. The acoustical impedance of fine holes and slits will be considered in the next two sections.

86

IMPEDANCE OF A TUBE

87

6.3. Acoustical Impedance of a Tube of Small Diameter. — The trans- mission of sound waves or direct currents of air in a small tube is influenced by acoustical resistance due to viscosity, "^e diameter is assumed to be sm^l compared to the length so that the end correction may be neglected. The length is assumed to be small compared to the wavelength.

The acoustical impedance, in acoustical ohms, of a small diameter tube is given by

where R = radius of the tube, in centimeters,

H = viscosity coefficient, 1.86 X 10“^ for air, (0 = 27r/,/ = frequency, in cycles per second, / = length of the tube, in centimeters, and p = density, in grams per cubic centimeter.

The eflFect of viscosity is to introduce acoustical resistance in the form of dissipation as well as to add to the acoustical reactance.

The acoustical resistance of a single hole is ordinarily much too high. The desired acoustical resistance may be obtained by using a sufficient number of holes.

6.4. Acoustical Impedance of a Narrow Slit. — A narrow slit acts in a manner quite similar to the narrow tube. The length is assumed to be small compared to the wavelength. The thickness is assumed to be small compared to the length.

The acoustical impedance, in acoustical ohms, of a narrow slit is given by

1 IfiW , . SpWiO

5.2

where p = viscosity coefficient, 1.86 X 10“^ for air, p = density, in grams per cubic centimeter,

d = thickness of the slit normal to the direction of flow, in centi- meters,

^ Crandall, “ Vibrating Systems and Sound,” D. Van Nostrand Company, New York, N. Y., 1926.

* Lamb, ” Dynamical Theory of Sound,” E. Arnold, London, 1931.

* Rayleigh, “ Theory of Sound,” Macmillan and Company, London, 1926.

* Crandall, “ Vibrating Systems and Sound,” D. Van Nostrand Company, New York, N. Y., 1926.

•Lamb, ” Dynamical Theory of Sound,” E. Arnold, London, 1931.

* Rayleigh, ” Theory of Sound,” Macmillan and Company, London, 1926.

88

ACOUSTICAL ELEMENTS

/ = width of the slit normal to the direction of flow, in centimeters, w = length of the slit in the direction of the flow, in centimeters, w = 2Trfy and

/ = frequency, in cycles per second.

In equation 5.2 the acoustical resistance varies inversely as the cube of d and the inertance inversely as d. Therefore, practically any ratio of inertance to acoustical resistance may be obtained. The magnitude may be obtained by a suitable choice of w and /. A slit type of acoustical re- sistance may be formed by using a pile of