Sound Visualization and Manipulation E-Book

Yang-Hann Kim

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 609

Veröffentlichungsjahr: 2013

Beschreibung

Unique in addressing two different problems - sound visualization and manipulation - in a unified way Advances in signal processing technology are enabling ever more accurate visualization of existing sound fields and precisely defined sound field production. The idea of explaining both the problem of sound visualization and the problem of the manipulation of sound within one book supports this inter-related area of study. With rapid development of array technologies, it is possible to do much in terms of visualization and manipulation, among other technologies involved with the spatial distribution of sound. This book aims to explore various basic functions for the visualization and manipulation and demonstrate to the reader how these properties determine the quality of visualization and manipulation. The first half of the book introduces some basic and general concepts and theories and the second part of the book explains a number of techniques in sound visualization and manipulation. It offers a unified presentation to two very different topics - sound field visualization techniques based on microphone arrays, and techniques for generation of controlled sound fields using loudspeaker arrays. The authors emphasize the similarities between these two physical problems and between the mathematical methods used for solving them. With extensive examples throughout the book, chapters include: Acoustic Wave Equation and its Basic Physical Measures, Acoustic Wave Equation and its Basic Physical Measures, Basic Theory of Sound Visualization, Acoustic Holography, Beamforming, Basic Theory of Sound Manipulation, Sound Focusing, and Sound Field Reproduction. * The first book to combine both the visualization and manipulation of sound technologies in one comprehensive volume * Presents the basic concepts using simple one dimensional cases and then extends the concept to three dimensional cases, enabling easier understanding of the fundamental concepts through the use of minimum mathematics * Provides a solid understanding of associated physics as well as mathematical concepts for understanding the technologies, addressing diffraction problems in an integrated format by using Kirchhoff-Helmholtz integral equation * Uses extensive examples demonstrating the benefits and drawbacks of various applications, including beamforming and acoustic holography A valuable resource forpost/graduate students, acoustic engineers, audio and noise control system developers

Leseprobe

Table of Contents

Title Page

Copyright

About the Author

Preface

Acknowledgments

Part I: Essence of Acoustics

Chapter 1: Acoustic Wave Equation and Its Basic Physical Measures

1.1 Introduction

1.2 One-Dimensional Acoustic Wave Equation

1.3 Three-Dimensional Wave Equation

1.4 Acoustic Intensity and Energy

1.5 The Units of Sound

1.6 Analysis Methods of Linear Acoustic Wave Equation

1.7 Solutions of the Wave Equation

1.8 Chapter Summary

References

Chapter 2: Radiation, Scattering, and Diffraction

2.1 Introduction/Study Objectives

2.2 Radiation of a Breathing Sphere and a Trembling Sphere

2.3 Radiation from a Baffled Piston

2.4 Radiation from a Finite Vibrating Plate

2.5 Diffraction and Scattering

2.6 Chapter Summary

2.7 Essentials of Radiation, Scattering, and Diffraction

References

Part II: Sound Visualization

Chapter 3: Acoustic Holography

3.1 Introduction

3.2 The Methodology of Acoustic Source Identification

3.3 Acoustic Holography: Measurement, Prediction, and Analysis

3.4 Summary

References

Chapter 4: Beamforming

4.1 Introduction

4.2 Problem Statement

4.3 Model-Based Beamforming

4.4 Signal-Based Beamforming

4.5 Correlation-Based Scan Vector Design

4.6 Subspace-Based Approaches

4.7 Wideband Processing Technique

4.8 Post-Processing Techniques

References

Part III: Sound Manipulation

Chapter 5: Sound Focusing

5.1 Introduction

5.2 Descriptions of the Problem of Sound Focusing

5.3 Summing Operator ()

5.4 Product Theorem (×)

5.5 Differential Operator and Super-Directivity ()

5.6 Optimization with Energy Ratios (÷)

References

Chapter 6: Sound Field Reproduction

6.1 Introduction

6.2 Problem Statement

6.3 Reproduction of One-Dimensional Sound Field

6.4 Reproduction of a 3D Sound Field

6.5 Field-Matching Approach

6.6 Mode-Matching Approach

6.7 Surface Integral Equations

6.8 Single-layer Formula

References

Appendix A: Useful Formulas

A.1 Fourier Transform

A.2 Dirac Delta Function

A.3 Derivative of Matrices

A.4 Inverse Problem

Appendix B: Description of Sound Field

B.1 Three-Dimensional Acoustic Wave Equation

B.2 Wavenumber Domain Representation of the Rayleigh Integral

B.3 Separation of Variables in Spherical Coordinates

References

Index

This edition first published 2013

© 2013 John Wiley & Sons Singapore Pte. Ltd.

Registered office

John Wiley & Sons Singapore Pte. Ltd., 1 Fusionopolis Walk, #07-01 Solaris South Tower, Singapore 138628

For details of our global editorial offices, for customer services and for information about how to apply for permission to reuse the copyright material in this book please see our website at www.wiley.com.

All Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as expressly permitted by law, without either the prior written permission of the Publisher, or authorization through payment of the appropriate photocopy fee to the Copyright Clearance Center. Requests for permission should be addressed to the Publisher, John Wiley & Sons Singapore Pte. Ltd., 1 Fusionopolis Walk, #07-01 Solaris South Tower, Singapore 138628, tel: [phone], fax: [phone], email: [email protected].

Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books.

Designations used by companies to distinguish their products are often claimed as trademarks. All brand names and product names used in this book are trade names, service marks, trademarks or registered trademarks of their respective owners. The Publisher is not associated with any product or vendor mentioned in this book. This publication is designed to provide accurate and authoritative information in regard to the subject matter covered. It is sold on the understanding that the Publisher is not engaged in rendering professional services. If professional advice or other expert assistance is required, the services of a competent professional should be sought.

Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. It is sold on the understanding that the publisher is not engaged in rendering professional services and neither the publisher nor the author shall be liable for damages arising herefrom. If professional advice or other expert assistance is required, the services of a competent professional should be sought.

Library of Congress Cataloging-in-Publication Data

Kim, Yang-Hann.

Sound visualization and manipulation / Yang-Hann Kim, Jung-Woo Choi.

pages cm

Includes bibliographical references and index.

ISBN 978-1-118-36847-3 (cloth)

1. Sound-waves--Mathematical models. 2. Helmholtz equation. I. Choi, Jung-Woo. II. Title.

QC243.K46 2013

534.01′5153533--dc23

About the Author

Yang-Hann Kim

The research area of Yang-Hann Kim is mainly acoustics, noise/vibration. Experimental approaches and associated digital signal processing are used the most. Research projects include sound field visualization, noise source identification using array microphones, detection and estimation of moving noise source, structural acoustics, duct acoustics, silencer design, diagnostics of machines, and active noise/vibration control. Recently, he has been recognized as a pioneer in the field of sound visualization and manipulation. The latter is to make any sound field or shape in the selected region/regions. Therefore, it can be used for having very focused sound field, private sound zone/zones, or 3D listening field.

Dr Kim joined the Department of Mechanical Engineering as an Associate Professor in 1989. Previously he worked for five years at the Korea Institute of Technology as an Assistant and Associate Professor of the Department of Mechatronics. From 1979 to 1984, he was a research assistant at the Acoustics and Vibration Laboratory of Massachusetts Institute of Technology while pursuing Ph.D. degree in the field of acoustics and vibration, and obtained Ph.D. in February 1985 at M.I.T., Mechanical Engineering (O.E. Program).

He has been on the editorial board of Mechanical Systems and Signal Processing (MSSP), editorial advisor of the Journal of Sound and Vibration (JSV) and Journal of Noise Control Engineering. He also served KSNVE as an editor for three years (1995–97). His research has been recognized in the professional societies and institutes in many respects, including the best paper award by KSNVE (1998), the best research award by ASK (1997), second place award in the sound visualization competition by the Acoustical Society of America (1997), the best international cooperation award from KAIST, and KSNVE, and the best teaching award from KAIST, department of M.E (2010). He is elected as co-chairman of inter-noise 2015, San Francisco, also a director of I-INCE. He is a Fellow of the Acoustical Society of America.

Dr Kim has published more than 100 papers, mostly in the field of sound visualization and manipulation, in the well-known journals, including the Journal of the Acoustical Society of America, Journal of Sound and Vibration, and Journal of Acoustics and Vibration, the Transaction of ASME. He is an author of well-known acoustics text, Sound Propagation: An Impedance Based Approach, published by John Wiley & Sons, Inc. He also wrote the chapter “Acoustic holography” in the Handbook of Acoustics, published by Springer Verlag. He has delivered two plenary lectures in ICA (2004), and Inter-Noise (2012) and one keynote lecture in ICSV (2009). All of these lectures were on acoustic holography, sound visualization, and manipulation.

Jung-Woo Choi

Jung-Woo Choi's primary research area includes active sound control and array signal processing for loudspeaker/microphone arrays. His research interests also include sound field reproduction, sound focusing, and their application for audio systems. From 1999, he has been working on sound/noise control over elected regions based on the concept of acoustic contrast, which has been widely adopted for the implementation of personal sound zones. Recently, his research has extended to interactive 3D sound/multi-channel audio systems that can be manipulated in real-time by exploiting the beauty of direct integral formulas.

Dr Choi received his B.Sc., M.Sc., and Ph.D. degrees in Mechanical Engineering from the Korea Institute of Science and Technology (KAIST), Korea, in 1999, 2001, and 2005, respectively. He was a Postdoctoral Research Associate with the Center for Noise and Vibration Control (NOVIC), KAIST, Korea, 2005–06. From 2006 to 2007, he was a Visiting Postdoctoral Researcher at the Institute of Sound and Vibration Research (ISVR), University of Southampton, UK. From 2007 to 2011, he was with Samsung Electronics at the Samsung Advanced Institute of Technology (SAIT) in Korea, working on array-based audio systems as a Research & Development staff member and a Senior Engineer. In 2011, he joined the Department of Mechanical Engineering, KAIST, Korea, and has since been a Research Professor there. He is the author of more than 50 papers/conference articles and 15 patent applications, including five registered patents on loudspeaker array systems.

Preface

If only we could see sound propagation in space with our eyes, and if only the sound could be created in any desired shape! Such a fantastic concept is being realized. New approaches to acoustics and noise engineering have allowed innovative changes in these fields.

So far, extensive efforts have been made using various methods to explain how a medium changes as sound propagates in space or how the shape of the sound propagation changes depending on its frequency and wavelength. There are two main approaches being employed to resolve these questions: theoretical and experimental. The theoretical approach is adopted to develop an understanding of the phenomena of sound propagation in acoustic waves and, through this understanding, to attempt to find a solution. The characteristics of acoustic wave equations in certain cases are interpreted by numerically solving the so-called linear acoustic wave equations. Popular numerical techniques are the finite element method and boundary element method; both have achieved incredible developments owing to continuous evolutions in their background theories and improvements in the arithmetic capacity of computers. The experimental approach has also seen rapid improvements. Developments in semiconductor technologies have reduced the microphone size to eliminate unnecessary scattering induced by them, and the reduction in cost allows for tens and hundreds of microphones to be used at the same time. We can now sample, record, and analyze signals from hundreds of microphones in almost real time. These developments allow us to “visualize” sound using our eyes in the real world, which is something that human beings have long dreamt of. The first half of this book explains various methods to visualize sound.

From a mathematical point of view, sound visualization can be regarded as an exploration of methods to transform measured data into information that is visible to the human eyes. Most information transformation is determined during selection of the desired basis function because information transformation can produce different results depending on its mapping functions, as shown in the figure below. Thus, we need to deal with problems such as selecting a basis function and expressing a sound field as a visible image using the selected basis function. This book explains the planar, cylindrical, and spherical basis functions used in acoustic holography and the functions employed for beamforming methods. Their advantages and disadvantages as well as practical applicability are addressed. The advantage of the acoustic holography method includes visualization of information with great physical significance, such as acoustic pressure, velocity, intensity, and energy. On the other hand, the beamforming method can provide a variety of visualization information depending on the type of basis function used for the beamformer.

The concept that visualization results vary significantly depending on the basis functions can be reasonably expanded to an idea of sound manipulation where arbitrary or desired forms of sound can be created in space. Desired sounds can be produced by selecting basis functions so that the sounds generated from the sound sources arranged in the space are of these types of basis functions, as shown in the figure. Well-known methods include wave field synthesis (WFS) and Ambisonics. WFS is a representative method based on the so-called Kirchhoff–Helmholtz integral

equation, whereas Ambisonics is a technique that expresses sound fields using spherical harmonics and embodies the desired shapes of sound in space using such expressions. From a unifying point of view, the manipulation of a sound field is an issue in obtaining the desired output using the available sound sources; accordingly, we can select the best basis function depending on the definition of the desired function. Based on this idea, the sound focusing problem of concentrating the sound in desired areas or dividing an area into acoustically bright and dark zones and maximizing the ratio of sound energies between the two areas can also be explained.

Both the sound visualization method and sound manipulation method demand considerable theoretical knowledge of mathematics and acoustics as well as knowledge of signal processing to understand their principles and realize their practical applications. To aid potential readers who want to understand the basic concepts or those who will practically apply the methods, the simplest one-dimensional theories are introduced in this book, and their mathematical and theoretical explanations are presented in every chapter. Chapter 1 is intended to aid understanding of the basic physical quantities in acoustics. Part I consists of two chapters. Chapter 1 explains three physical quantities in acoustics using one-dimensional examples: interrelationships among acoustic pressure, particle velocity, and acoustic density. This approach is justified in that the principle of superposition holds for a linear system; hence, most of the concepts explained in one dimension can be extended to multidimensional cases.

Part II introduces the sound visualization methods and explains how their basic principles can be varied depending on certain basis functions. Accordingly, basis functions and approaches for the acoustic holography and the beamforming method are introduced. An appropriate basis function should be used depending on what we want to visualize; depending on this basis function, the information of visualized sound fields can be varied.

In Part III, we deal with sound manipulation techniques. Sound manipulation is carried out using two main methods; both are discussed with respect to how they are embodied in one-dimensional situations. Sound manipulation involves a sound focusing technique that concentrates the sound in specific areas in space and a sound field reproduction method that generates a wave front in the desired forms. For realizing these two methods, unique inputs to generate sound fields in the desired forms need to be determined. Therefore, the sound focusing and reproduction problems are defined as inverse problems corresponding to the beamforming and acoustic holography methods, respectively. Thus, the sections on acoustic holography and sound field reproduction are organized to complement each other. The sections on beamforming and sound focusing address similar issues but explain them from different points of view. The chapter on beamforming focuses on a signal processing technique to extract the parameter determining the locations of sound sources, whereas the chapter on sound focusing explains resolution variations depending on the geometric configuration of arrays and beam pattern variations depending on the basic aperture functions. Thus, Parts II and III address different and similar issues from complementary points of view; readers interested in visualization are strongly recommended to read the manipulation part. It would be efficient for readers of this book to use Part I as a reference when they need to know more about acoustics while reading Parts II and III.

In conclusion, this book introduces and explains methods for sound visualization and manipulation. The book is organized such that readers can gain a profound understanding of basic concepts and theoretical approaches from the one-dimensional case. The methods of visualization and manipulation are explained as a unifying approach for creating certain assumed or desired shapes in space based on the measured or available information using basis functions.

Yang-Hann KimJung-Woo Choi

Acknowledgments

It was around 1990 that the first author had an idea about the basis function described in this book. He visited his old friend, Prof. J. K. Hammond of the Institute of Sound and Vibration Research (ISVR), University of Southampton, who was giving a lecture on nonlinear signal processing for a group of people in the industry at the time. The first page of the handout made for that class included a primitive version of the picture that is published in this book's preface. In fact, this picture originated and evolved from the image in Science with a Smile (Robert L. Weber, Institute of Physical Publishing, Bristol and Philadelphia, 1992, pp. 111–12). The moment he looked at this picture, it occurred to him that a part of this picture could be used to explain the processing of every signal. Signal processing essentially involves finding desired information using available data. Thus, this picture symbolically shows that the quality of information obtained eventually depends on how well the processing method represents substantial, physical, or mathematical situations. Ultimately, the result is fully dependent on the processing method one has chosen, that is, a basis function. If so, how do we select a basis function? Although it is a very basic question, it is self-evident that if we can suggest the best method to be selected, it would serve as a very innovative and useful method in this discipline.

In fact, since the first author was at the time working on issues such as mechanical noise diagnosis and fault detection using signal processing, he looked at the picture that Prof. Hammond had used in a symbolic manner and gained an idea to view various problems in a unifying manner. He came to realize that both sound visualization techniques—acoustic holography and beamforming—eventually produced different results with regard to visualization owing to differences in the basic functions that were used. If so, in-depth knowledge of whether a basis function used has a mathematical function to perform something well would allow one to have a good understanding of the result of the sound visualization, that is, the picture. Thus, to clearly interpret the visualized information gained through the acoustic holography method and to accurately analyze the desired information, it is necessary to analyze how well the basis function expresses the desired visual information. Similarly, the following questions can be approached from an understanding of the basis function that was used: what is the specific information that can be gained from the spatial distribution of the beamforming power obtained from the beamforming method? Does the maximum value of the beaming power correctly describe the locations of sound or noise sources? What properties of the sources does the spatial distribution of the beaming power represent? In this regard, the visualization described in this book was greatly inspired by the discussions with Prof. Hammond.

The sound manipulation study described in the second half of this book started, it is recalled, around 1999 when the first author thought he had found some improvements in the study of visualization or was getting bored with studying sound visualization. The second author was studying how to focus sounds in an arbitrary space as part of the work for a master's degree, and based on this, he started a full-scale study on sound manipulation. The first result aimed to practically implement a system that allows one to hear a desired sound without disturbing others by focusing the sound in a specific space. Fortunately, the experiment was successful, and in 2000, the authors succeeded in focusing sound in a specific space by using six loudspeakers. Later on, this study was expanded to a study of a home/mobile speaker array system by the second author in the industry and to another study of monitor speaker array development by the first author at KAIST. The monitor speaker array system led to the implementation of a personal audio system that focuses sound using nine speakers. Among those who participated in the theoretical development and the experiment are Chan-hee Lee, currently working at Hyundai Heavy Industries Co., Ltd.; Dr Ji-ho Chang, currently at DTU after completing his doctoral degree; and Jin-young Park, currently pursuing his doctoral degree. At that time, the result attracted so much attention that it was broadcast on national TV. Thanks to this, the first author was granted an unexpected research fund, and his team could build a set of experimental equipment consisting of 32 speakers supported by KAIST's HRHR project. Using this experimental equipment, the research team implemented a method for focusing sounds on specific spots in various ways, and the effects were found to be better than expected. One day, a question was raised about what results would be produced if the focusing point was moved to an arbitrary location. Min-ho Song developed an interface using an iPhone, in which as a finger moved to a location, the location at which sound was focused was also changed, making the sound audible in real time. In fact, from a theoretical viewpoint, they knew that the sound focusing solution had nothing to do with 3D sound. However, the listener could feel the effect of the location of the sound source moving through the sound focusing solution only. As a matter of fact, studies have reported on a focused source using the time-reversed driving function in wave field synthesis, but these did not have sufficient theoretical basis, and no perfect integral equation form was available for the array in the form of surrounding the listener. The theoretical explanation of the experimental results was completed by the second author in 2011, and it was proved that a general solution can be drawn by combining Porter-Bojarski integral with a multipole virtual source. The first work to create sounds using this solution aimed to relocate a mosquito's sound to a desired space, and it was a great success. This substantial success became a motivation for the book's third part. The doctoral students, Jeong-min Lee and Dong-su Kang, made substantial contributions to developing a speaker system that implemented a sound ball. In addition, the authors would like to acknowledge Dr Min-Ho Song, who performed great research while completing his doctoral degree at the Graduate School of Cultural Technology and who contributed to developing one particular interface.

In fact, the graduates' wonderful studies were greatly helpful to the authors in writing the sound visualization part. In particular, studies by Dr Jae-Woong Choi, who has made great achievements in the areas of spherical beamforming and music, and Dr Young-Key Kim, who founded a company and has been disseminating sound visualization technology, were very helpful in writing the beamforming chapter. Dr Hyu-sang Kwon developed moving frame acoustic holography (MFAH), and he is expected to realize great achievements as an expert in this area. Furthermore, Dr Soon-hong Park of the Korea Aerospace Research Institute has made a great contribution to the method by applying MFAH to moving sound sources. The authors also want to acknowledge Dr Sea-Moon Kim of the Korea Institute of Ocean Science and Technology who successfully lead the acoustic holography experiment on the King Seong-deok Bell; Dr Kyung-Uk Nam of Hyundai Motor Company who greatly contributed to the partial field acoustic holography; and Dr Chun-Su Park who developed the time domain acoustic holography technique using the spatio-temporal complex envelope.

Credit for Chapter 4 of this book also belongs to Ku-Hwan Kim, who programmed most of the codes for beamforming simulations. The authors would like to express their appreciation to all the laboratory members—Jung-Min Lee, Dong-Soo Kang, Dae-Hoon Seo, Ki-Won Kim, Myung-Ryun Lee, Seong-Woo Jung—for their enthusiasm in correcting errors and giving advice to improve the content of this book.

The second author also wants to thank his former advisors at KAIST—Yoon-Sik Park, Chong-Won Lee, Jeong-Guon Lee, and Young-Jin Park—for teaching him the fundamentals of sound and vibration. Special thanks must be given to Prof. P. A. Nelson and S. J. Elliott and Dr F. M. Fazi of ISVR for many hours of fruitful discussions with him regarding the sound field reproduction and sound focusing projects. The experiences with his former colleagues at Samsung Electronics—Youngtae Kim, Jungho Kim, Sang-Chul Ko, and Seoung-Hun Kim—were greatly helpful in summarizing the techniques discussed in Chapters 4–6.

Finally, the authors would like to express their special thanks to James Murphy and Clarissa Lim of John Wiley & Sons for their consistent help and cooperation with regard to editing this book. Without their encouragement, this book would not have been possible.

Yang-Hann KimJung-Woo Choi

Part I

Essence of Acoustics

Sound is an important part of our lives. Even in the womb, human beings are capable of detecting sounds. We create and enjoy sounds, and we can identify information conveyed by sound. We live with sound and are familiar with the fundamental concepts associated with it. Fundamental concepts of sound visualization and manipulation can also be explained on the basis of the mechanisms by which a sound wave is generated, propagated, and decayed by various internal and external disturbances.

Acoustics is a vast field of study that explains the propagation of waves in different media, and it cannot be completely covered in merely the first two chapters of this book. In this book, however, we limit the scope by focusing on the general idea of acoustics in terms of its essential physical measures. This part of the book discusses essential measures that refer to the primary measures or physical variables used in acoustics, such as acoustic pressure, velocity, intensity, and energy, which can be used to describe sound propagation. Various impedances, radiations, scatterings, surfaces, and so on, are also considered as important measures that affect wave propagation in space. In order to uniquely and conveniently explain the physics of acoustics, this part of the book relies heavily on the concept of impedance as a window to study sound propagation in time and space.

Chapter 1 introduces the essential physical parameters used in acoustics measurements. The significance of physical parameters other than impedance, such as sound pressure, speed, energy, power, and intensity, are explained (Figure I.1). It is emphasized that these parameters form the fundamental concepts required for understanding the propagation of sound waves. The aforementioned parameters are explained by using a one-dimensional approach. The Euler equation is used to describe the relation between the sound pressure and the particle velocity in a given medium. The state equation, on the other hand, is used to evaluate the relation between the acoustic density and the fluctuating pressure, which is the acoustic pressure that causes the sound propagation. The third equation used is the law of conservation of mass for the compressible fluid, which defines how the fluctuating density and the fluid particle velocity are associated with each other. Therefore, the three essential variables: sound pressure, particle velocity, and fluctuating density, follow these three equations. This enables the derivation of the acoustic wave equation that governs all the parameters associated with acoustic wave propagation. Chapter 1 discusses the two different approaches that can be used to solve this acoustic wave equation. The first one is based on the eigenfunction analysis, in which the solution is determined as the superposition of eigenmodes. Another approach uses Green's function, which describes how a sound field is constructed when the field has a monopole source at an arbitrary position in space. This approach leads to the Kirchhoff–Helmholtz equation.

Figure I.1 Pictorial relation between three variables that govern acoustic wave propagation ( and express the mean pressure and static density, respectively; and denote acoustic pressure and fluctuating density, respectively; denotes the speed of propagation, and is the velocity of the fluctuating medium)

Chapter 2 takes a rather ambitious route to describe how sound wave reacts under impedance mismatch in space and time using the concept of radiation, scattering, and diffraction. It is believed that the scattering and the diffraction of sound can be explained by acoustic radiation. For instance, a scattered sound field is a result of radiation scattering (Figure I.2), whereas diffraction is a result of the radiation from an object that has spatial impedance mismatch.

Figure I.2 Reflection and transmission phenomena using the principle of superposition

An understanding of the first two chapters is expected to help in analyzing and explaining the results obtained by the sound visualization described in Chapters 3 and 4, and the manipulation described in Chapters 5 and 6.

Chapter 1

Acoustic Wave Equation and Its Basic Physical Measures1

1.1 Introduction

The waves along a string propagate along its length, but the string itself moves perpendicular to the propagation direction. It therefore forms a transverse wave. If the particle of a medium moves in the direction of propagation, we refer to it as a longitudinal wave. The waves in air, water, or any compressible medium are longitudinal waves, which are often referred to as acoustic waves. This chapter explores the underlying physics and sensible physical measures related to acoustic waves, including pressure, velocity, intensity, and energy. Impedance plays a central role with regard to its effect on these measures.

In the area of sound visualization, our objectives are to determine a rational means to convert essential acoustic variables such as pressure, velocity, and density, or other physically sensible acoustic measures such as intensity or energy, into visible representations. One very straightforward way to accomplish these objectives is to express acoustic pressure by using a color code. Notably, there are many ways to visualize a sound field, depending on a mapping or general basis function, which relates acoustic variables to visual expressions. Therefore, this chapter starts with a discussion on the visualization of a one-dimensional acoustic wave.

1.2 One-Dimensional Acoustic Wave Equation

The simplest case is illustrated in Figure 1.1. The end of a pipe or duct which is filled with a homogeneous compressible fluid (air, water, etc.) is excited with a radian frequency (, : frequency in Hz). If the pipe is semi-infinitely long, then the pressure in the pipe can be mathematically written as

1.1

where is the pressure magnitude and is an initial phase. Here, represents the spatial frequency (, : wavelength in m) of the pressure field, which is often called wavenumber.

Figure 1.1 Relation between forces and motion of an infinitesimal fluid element in a pipe (expressing momentum balance: the left-hand side shows the forces and the right exhibits the change of momentum)

If the pipe is of finite length L, then the possible acoustic pressure in the pipe can be written as

1.2

Equations (1.1) and (1.2) are different simply because of the boundary conditions: the former has no boundary condition prescribed at , but the latter has a rigid-wall condition (velocity is zero).

To understand what is happening in the pipe, we have to understand how pressures and velocities of the fluid particles behave and are associated with each other. This motivates us to look at an infinitesimal element of the volume of the fluid in the pipe; specifically, we will investigate the relation between force and motion.

As illustrated in Figure 1.1, the forces acting on the fluid between and and its motion will follow the conservation of momentum principle. That is,

1.3

We can mathematically express this equality as

1.4

where it has already been assumed that the viscous force, which likely exists in the fluid, is small enough (relative to the force induced by pressure) to be neglected.

The rate of change of velocity can be expressed by

1.5

where is a function of position () and time () and velocity is the time rate change of the displacement. Therefore, we can rewrite Equation (1.5) as

1.6

If the cross-section between and is maintained constant and becomes small , then Equation (1.4) can be expressed as2

1.7

where

1.8

1.9

1.10

Note that the pressure is composed of the static pressure () and the acoustic pressure (), which is induced by the small fluctuation of fluid particles. The density also has two components: the static density () and the small fluctuating density ().

Equation (1.10) is the total derivative, and is often called the material derivative. The first term expresses the rate of change with respect to time, and the second term can be obtained by examining the change with respect to space as we move with the velocity .3 As can be anticipated, the second term is generally smaller than the first.

If the static pressure () and density () do not vary significantly in space and time, then Equation (1.7) becomes

1.11

where is acoustic pressure and is directly related to acoustic wave propagation. As already implied in Equation (1.8), acoustic pressure is considerably smaller than static pressure.4 Equation (1.11) essentially means that a small pressure change across a small distance () causes the fluid of mass/unit volume to move with the acceleration of . This equation is generally referred to as a linearized Euler equation. Equation (1.7), on the other hand, is an Euler equation.

Equations (1.7) and (1.11) describe three physical parameters, pressure, fluid density, and fluid particle velocity. In other words, they express the relations between these three basic variables. In order to completely characterize the relations, two more equations are needed.

The relation between density and fluid particle velocity can be obtained by using the conservation of mass. Figure 1.2 shows how much fluid enters the cross-section at and how much exits through the surface at . If we apply the principle of conservation of mass law to the fluid volume between and , the following equality can be written.

Figure 1.2 Conservation of mass in an infinitesimal element of fluid (increasing mass of the infinitesimal volume results from a net decrease of the mass through the surfaces of the volume)

Expressing this equality mathematically leads to

1.12

as illustrated in Figure 1.2. As assumed before, if the area of the cross-section () remains constant, then Equation (1.12) can be rewritten as

1.13

We can linearize this equation by substituting Equation (1.9) into Equation (1.13). Equation (1.13) then becomes

1.14

Equations (1.11) and (1.14) express the relation between sound pressure and fluid particle velocity, as well as the relation with fluctuating density and fluid particle velocity, respectively. One more equation is therefore needed to completely describe the relations of the three acoustic variables: acoustic pressure, fluctuating density, and fluid particle velocity. The other equation must describe how acoustic pressure is related to fluctuating density. Recall that a pressure change will induce a change in density as well as other thermodynamic variables, such as entropy. This leads us to postulate that acoustic pressure is a function of density and entropy, that is,

1.15

where denotes entropy. We can then write the change of pressure, or fluctuating pressure, or , by modifying Equation (1.15) as follows:

1.16

Table 1.1 The dependency of the speed of sound on temperature

Table 1.2 The dependency of the speed of sound on relative humidity and on frequency

This equation simply states that a pressure change causes a density change () and entropy variation (). It is noticeable that the fluid obeys the law of isentropic processes when it oscillates within the range of the audible frequency: 20 Hz to 20 kHz.5 The second term on the right-hand side of Equation (1.16) is therefore negligible. This implies that the small change of sound pressure with regard to the infinitesimal change of density can be assumed to have certain proportionality. (An alternative way to deduce the same relation can be found in Appendix B, Section B.1.3.) Note that the second relation of Equation (1.16) is mostly found experimentally. This reduces Equation (1.16) to the form

1.17

where is the bulk modulus that expresses the pressure required for a unit volume change and is the speed of sound. We may obtain Equation (1.17) by introducing a gas dynamics model. This equation is an equation of state. Tables 1.1 and 1.2 summarizes the speed of sound in accordance with the state of gas [2]. An alternative method of deducing Equations (1.16) and (1.17) can be found in Appendix B, Section B.1.3.

Note that Equation (1.17) expresses how the access pressure or acoustic pressure communicates with the fluctuating density. Equations (1.11) and (1.14) completely express the laws that govern the waves in which we are interested. Therefore, we can summarize the relations as

1.11

1.14

1.17

Figure 1.3 demonstrates how these equations and physical variables are related. If we eliminate and from Equations (1.11), (1.14), and (1.17), then we obtain

1.18

Figure 1.3 Pictorial relation between three variables that govern acoustic wave propagation ( and express the mean pressure and static density, respectively; and denote acoustic pressure and fluctuating density, respectively; denotes the speed of propagation, and is the velocity of the fluctuating medium)

This is a linearized acoustic wave equation.6

Equation (1.18) is essentially a general one-dimensional acoustic wave, that is, the waves in compressible fluid. A similar relation can be found from the propagation of a string wave. The only difference between the waves along a string and acoustic waves lies in whether the directions of wave propagation and velocity fluctuation of medium are collinear or perpendicular. Note that the propagation direction of the waves along a string is perpendicular to that of the motion of the string. Conversely, the acoustic wave propagates in the direction of the fluid particle's velocity.7 The former is called a transverse wave, while the latter is considered a longitudinal wave.8 This means that the acoustic wave, at least a one-dimensional case, is analogous to that of a wave on a string. It therefore appears that a unified concept or theory is very possible.

We can also optimistically consider that any three-dimensional wave can be decomposed into a one-dimensional wave, at least conceptually. The principle of superposition holds for the waves that are governed by the linearized acoustic wave equation. Therefore, any combination of the solutions with respect to each component of the coordinate – for example, rectangular coordinate, cylindrical, and spherical coordinates – also satisfies the governing equation. Simply extending this concept leads us to use the Fourier series or transform. This means that we attempt to look at what we are interested in, in terms of the orthogonal series. Many sound visualization and manipulation techniques, such as acoustic holography in Chapter 3, Section 3.3.3 and mode-matching approach in Chapter 6, Section 6.6, utilize orthogonal series to predict or reproduce the sound field.

The second conclusion involves the relations between three acoustic variables, which are illustrated in Figure 1.3. Euler equation (Equation (1.11)), states that the spatial pressure change makes the fluid particle move. Equation (1.14) shows that a higher compression rate in time makes a steeper negative velocity gradient in space. Equation (1.17) indicates that speed of propagation depends on the characteristics of the medium. If we have a smaller density change for a unit pressure change, then we will obtain faster propagation speed (see Tables 1.1 and 1.2). Lastly, the wave equation defines the relation between the spatial and temporal change of sound. From Equations (1.1) and (1.18), we have

1.19

From Equation (1.19), we can observe that the frequency and wavenumber are bound by the wave equation. This relation, called dispersion relation of one-dimensional acoustic waves, indicates that the possible spatial fluctuation of wave is limited when the frequency of sound is determined.

1.2.1 Impedance

If we use Equation (1.11) to examine the ratio between the pressure acting on the waves and the corresponding particle velocities (their magnitudes), then it leads to the concept of acoustic impedance.

1.20

We normally regard the acoustic impedance as a complex variable, which allows us to express any possible phase difference between the pressure and particle velocity.

For example, examine the sound propagation along the semi-infinite duct. The pressure field of Equation (1.1) is the real part of the complex exponential

1.21

Note that represents the complex amplitude including both the magnitude and phase of the pressure field (). The real-valued pressure can then be expressed as

the same token, the particle velocity can be written as

1.22

Since only access pressure and velocity (, ) are involved with this form of wave propagation, we can derive the acoustic impedance from the ratio of Equation (1.21) to (1.22). That is,

1.23

The right hand side of Equation (1.23) is composed of the medium density and the speed of sound in the medium. Because it represents the characteristics of the medium, we call this type of impedance as the characteristic impedance.

1.3 Three-Dimensional Wave Equation

We now extend Equations (1.11), (1.14), and (1.18) to a three-dimensional case. First, Euler equation can be written as

1.24

where we use coordinate for convenience. Each equation simply states that the change of pressure with respect to the direction, that is, the pressure gradient, makes the fluid particle move. , , and denote the velocity with respect to the coordinate system.

We may use a vector notation to express Equation (1.24), which will yield a more compact form. This gives

1.25

This is the linearized Euler equation, where

1.26

Similarly, Equation (1.14) can also be extended to the three-dimensional form, that is,

1.27

The right-hand term of Equation (1.27) represents the net mass flow into the unit volume in space. Appendix B, Section B.1 provides the detailed derivation and theoretical background.

If we eliminate and using Equations (1.25), (1.27), and (1.17), then

1.28

is obtained, which is a three-dimensional form of a wave equation.9

To summarize, three physical variables (acoustic pressure, fluid particle velocity, and fluctuating density) govern acoustic waves in a compressible fluid. These variables are not independent. The relations between them are expressed by laws or equations, that is, the linearized Euler equation, conservation of mass, and the state equation of gas. Lastly, we envisage that three-dimensional acoustic waves can also be understood using theory developed to explain one-dimensional acoustic waves.

1.4 Acoustic Intensity and Energy

We acknowledged that acoustic pressure (), density (), and fluid particle velocity ( for one dimension, or for the three-dimensional case) are major physical variables that determine everything related to acoustic wave propagation. We also recall that the impedance measures how the waves behave in response to the pressure/velocity perturbation. Power and energy are also major descriptors of wave propagation along a string. Understanding that the waves in a compressible fluid also obey what we have learned in the case of one-dimensional string waves, we explore what is physically meant by the energy and intensity of acoustic waves.

For simplicity, we consider a one-dimensional case (Figure 1.4). We denote acoustic pressure () as , and fluctuating density () as .

Figure 1.4 Volume change and energy for a one-dimensional element ( is potential energy density and is for convenience)

As illustrated in Figure 1.4, there will be a volume change of because of the pressure difference along the element. The length of the element will be shortened by due to the small pressure change . The energy stored in the unit volume (potential or elastic energy), can then be written as

1.29

where has to obey the conservation of mass. We therefore have

1.30

Rearranging this, we obtain

1.31

Note that the last term on the right-hand side is much smaller than the others. Equation (1.31) therefore reduces to

1.32

This equation simply states that the change of density is proportional to the decrease of volume, , or the length reduction ().

Substituting Equation (1.32) into (1.29) then gives

1.33

Using the state equation (Equation 1.17), and changing to , then gives

1.34

where denotes the acoustic potential energy.

The kinetic energy per unit volume can be written as

1.35

If we assume that the dissipated energy in the fluid is much less than the potential energy or kinetic energy, then the total energy has to be written:

1.36

Note that the potential and kinetic energy are identical if the wave of interest is a plane wave in an infinite domain;10 in other words, the wave propagates without any reflection. This can be readily understood by recalling the impedance of a plane wave, which is .

The next question then is how acoustic energy changes with time. We can see that the energy per unit volume has to be balanced by the net power flow through the surfaces that enclose the volume of interest, as illustrated in Figure 1.5. This observation can be written conceptually as

1.37

Figure 1.5 Relation between energy and one-dimensional intensity (energy in the volume and the intensity through the surface at and must be balanced)

This can be translated into a mathematical expression as follows:

1.38

Equation (1.38) can then be reduced to

1.39

where , which we call “acoustic intensity” or “sound intensity.” Acoustic intensity is the acoustic power per unit area.

If we simply extend Equation (1.39) to a three-dimensional case, then

1.40

The second term on the left-hand side of Equation (1.40) expresses net outflow power through the fluid surfaces (Figure 1.5). Two major points must be noted in relation to the expression of the intensity. First, intensity is a vector which has direction.11 Second, intensity is a product of two different physical quantities. Recall that the impedance is also composed of two physical quantities, pressure and velocity, but in a division form; it can therefore be regarded as having an input and output relation. Note, however, that intensity expresses how much power is transported in which direction.

When we have two physical variables, the phase difference between them has significant meaning; for example, the phase relationship between force (pressure) and velocity of impedance: the phase between the force and velocity expresses how well the force generates the velocity (response). In this regard, the intensity can be classified as two different categories: active intensity and reactive intensity. The phase of the former is in-phase and that of the latter has 90° () phase difference.

To understand the meaning of the intensities in physical terms, we look again at the simplest case: the intensity of waves propagating in a one-dimensional duct. Figure 1.6 depicts the waves in an infinite-length duct and Figure 1.7 shows the waves for a finite-length duct.12 These waves are typical examples of sound visualization. As illustrated in these figures, the visualization can be attempted in various ways, and accordingly all the possible approaches to understanding the propagation of sound waves in space and time can be attained.

Figure 1.6 The acoustic pressure and intensity in an infinite duct. Note that the pressure and velocity are in phase with each other. Also, the active intensity (or average intensity with respect to time) is constant

Figure 1.7 The acoustic pressure and intensity in a duct of finite length of . Note that the phase difference between the pressure and velocity is 90° ()

When the waves propagate in an infinite duct, where no reflection is possible, the pressure and velocity have the same phase and the frequency of the intensity is double that of the frequency of the pressure and velocity. It can also be observed that the average intensity with respect to time is constant, as can be seen in Figure 1.6. The instantaneous intensity oscillates with different phase, however, with regard to the position along the duct.

The general behavior of sound intensity is closely related to the energy transfer at the driving point ( in Figure 1.6). This implies that we can use the impedance at the driving point (driving point impedance) as a representative measure to indicate how well pressure responds to external excitation (velocity). For the case of Figure 1.6, the driving point impedance is given by

which is matched precisely with the characteristic impedance of the medium (). Therefore, the generated waves only propagate in the right-going direction. The fact that the waves all propagate to the right implies that only active intensity is allowed. Accordingly, the excitation effectively supplies energy to the system, which means that the pressure and velocity are in phase. Conversely, if we have the same excitation at one end, the duct has a finite length of , and a rigid boundary condition exists at the other end , then the phase difference between the pressure and velocity will be 90° () as depicted in Figure 1.7. Therefore, it is not possible to effectively put energy into the system.

Instantaneous acoustic intensity in this case is completely different from what we have for the infinite duct case. The mean intensity is zero, independent of the position of the duct. Also, we see that intensity is always zero at the nodal point of the duct , but it oscillates between these points where the energy vibrates and does not propagate anywhere. This is a typical characteristic of a finite duct.

We can envisage that a more general case will exhibit behavior which is between those two extreme cases depicted in Figures 1.6 and 1.7. In general, intensity may be both active and reactive.

We have studied fundamental physics and its implications by examining two typical but extreme cases. We now need to explore more specific characteristics of the sound intensity, such as how to calculate and measure the intensity.

The mathematical definition of intensity can be written as

1.41

The one-dimensional expression is simply

1.42

The velocity can be obtained from the Euler equation (Equation (1.11)):

1.43

To obtain the derivative with respect to space, we may use two microphones. This means that we approximate the derivative as

1.44

This equation allows us to measure the rate of change of pressure in space. The pressure () at the position of the measurement can be approximately obtained as:

1.45

where the pressure fluctuates in time and is therefore a dynamic quantity. This means that the microphones that are used to implement Equations (1.44) and (1.45) have to be dynamically identical. In particular, the phase difference has to be treated carefully; this requires accurate calibration for magnitude and phase.

1.4.1 Complex-Valued Pressure and Intensity

We now look at intensity measurement and calculation by considering a plane wave with a radian frequency . The expression of pressure as a complex variable can then be written as

1.46

where denotes the pressure magnitude which has a real value and represents the possible phase change in space. This is a more general expression than what we had before, and is devised specially to look at the physics associated with the phase.

To obtain the velocity using the linearized Euler equation a pressure gradient is needed, that is,

1.47

Equations (1.47) and (1.43) then allow us to obtain the following expression for the complex-valued particle velocity:

1.48

The first term in Equation (1.48) has the same phase as that of the pressure of Equation (1.46); however, the second has a 90° phase difference. These two different velocities obviously result in very physically distinct intensities.

The intensity generated by the real part of pressure (Equation (1.46)) and the corresponding velocity (Equation (1.48)), which is in phase with the real part of the pressure, can be obtained as

1.49

This is normally referred to as the “active component of sound intensity.” The time average of this intensity is often referred to as a mean intensity, or an active intensity, and can be written:

1.50

This intensity can effectively supply power to space, because the velocity and pressure have the same phase.

On the other hand, the multiplication of the real part of the pressure and the imaginary part of the velocity that has 90° phase difference (with respect to the real part of the pressure, Equation (1.46)) will generate the following intensity:

1.51

We refer to this intensity as the “reactive component of sound intensity.” The time average of this intensity is 0 and, therefore, there is no net energy transport; it only oscillates.

We now look at the direction of intensity. From Equations (1.49) and (1.51), we see that the active intensity is proportional to the change of phase with space and the reactive intensity is linearly related to the rate of change of pressure squared with space. From these observations, we can conclude that the direction of the active intensity is perpendicular to the wavefront where the phase is constant. In addition, the direction of reactive intensity has to be perpendicular to the surface over which the mean square pressure is constant.13

In order to clearly distinguish the intensities at a specific time (Equations (1.49) and (1.51)) from the mean intensity (Equation (1.50)), referred to as the instantaneous active intensity and instantaneous reactive intensity, respectively, are employed. Normally, when we say active intensity, we are referring to a time average of the instantaneous active intensity, that is, Equation (1.50). For the reactive intensity case, we describe its amplitude

1.52

as reactive.

Note that the instantaneous intensity expressed by Equation (1.42) is composed of two components: the instantaneous active intensity (Equation (1.49)) and instantaneous reactive intensity (Equation (1.51)). We can therefore write them as:

1.53

Using a complex function, Equation (1.53) can be expressed in simpler form, that is,

1.54

where . This is often referred to as complex intensity.

The real part of complex intensity is active intensity (Equation (1.50)) and its imaginary part is the amplitude of reactive intensity (Equation (1.52)). (Appendix B, Section B.1.5 provides details associated with the derivation of intensities.)

To summarize, acoustic energy and intensity represent how acoustic waves propagate in space and time. Acoustic energy is composed of two components: acoustic kinetic energy and acoustic potential energy. The former is due to the motion of the fluid and the latter is induced by compression and expansion of the medium. The rate of propagation of these energies with regard to space and time is known as the intensity. In other words, intensity is energy per unit area in time and space. Intensity is the product of pressure and velocity; it therefore has both magnitude and direction. Because it is a product of two physical quantities, the phase relationship is an important characteristic. Note that impedance is also composed of pressure and velocity but is a division of two physical quantities.

It has been stressed that the phase relation can be simply described by introducing complex notations. In the following sections and chapters, the complex notation is used as a default descriptor, that is, anddenote the complex pressure and velocityand.

1.5 The Units of Sound

We have studied how physical quantities, such as pressure, velocity, intensity, energy, and impedance, are related to the propagation characteristics of acoustic waves. We have also seen that a one-dimensional acoustic system such as a duct can express the associated fundamentals.

In considering these physical quantities, it is necessary to know how to communicate them. The units of the physical quantities are the basis of such means. In fact, units are a currency that forms the basis of exchanging these physical quantities. Note also that sound exists before the governing equation is formulated; the measures and units therefore form the essential tenets of acoustics.

The units relevant to sound can be classified into two groups: absolute units and subjective units. The former express the acoustic expression of physical dimensions while the latter take into account the subjective recognition of sound.

Let us first start with the absolute units. The force per unit area, which is pressure, has the units:

Velocity, which is defined as the time rate change of displacement, has units:

Intensity, which is the power per unit area, has units:

Finally, the unit of energy is defined:

Although these absolute units are the necessary parameters that have to be visualized in terms of space and time, it should be noted that subjective units have to be visualized in space and time as well. To understand subjective units, we need to understand how we hear, which means that we need to study our hearing system. Figure 1.8a depicts the human hearing system. Sound arrives at the ear and vibrates the ear drum (Figure 1.8a, external ear, Figure 1.8b). This vibration is amplified by three ossicles before arriving at the cochlea (middle ear, Figure 1.8b). The cochlea has a membrane referred to as the basilar membrane in its middle region, which is filled with fluid (Figure 1.8c). Hair cells are planted on the membrane (Figure 1.8d) and the motion of these cells generates signals that are transported to the brain. The brain processes the signals, feeding them to our cognition system. We recognize the sound and the information that it carries.14 An interesting characteristic of hair cells is that they behave like a spatially distributed frequency band-pass filter, as illustrated in Figure 1.8e,f.15

Figure 1.8 The structure of the ear and its frequency band characteristics. (a) The structure of a human ear (Adapted from [3] with the permission of L. Chittka, this figure was redrawn from the original source: D. Purves et al., Neuroscience, 3rd edition, 2004, pp. 288 (Figure 12.3), Sinauer Associates, Inc., Massachusetts, USA. ©2004 Sinauer Associates, Inc.) (b) external, middle, and inner ear (c) Basilar Membrane and Organ of Corti (d) the cross section of the cochlea shows the sensory cells (located in the organ of Corti) surrounded by the cochlear fluids (e) space-frequency map: moving along the cochlea, difference locations are preferentially excited by different input acoustic frequencies. (f) Tonotopic organization. (Figure 1.8(b–f) Drawings by Stéphan Blatrix, from “Promenade around the cochlea” EDU website http://www.cochlea.org by Rémy Pujol et al. INSERM and University Montpellier.)

Figure 1.9 Octave, 1/3 octave, and 1/n octave

It is well known that humans do not hear the frequency of sound in absolute scale, but rather relatively.16 Due to this, we normally use relative units for the frequencies. The octave band is a typical relative scale (Figure 1.9). The band between the reference frequency () and the frequency that is twice that () is referred to as an “octave”. The 1/3 octave band, which is popular in noise control engineering, has a frequency of 21/3; the reference frequency has the frequency of upper band . The center frequency () of each band is at the geometrical center of the band.

According to this definition, the center frequency can be any frequency. However, for convenience, we use the standard center frequencies (Table 1.3). The frequency bandwidth of the octave band is about 70% and that of the 1/3 octave band is about 23% of the corresponding band (see Figure 1.9). Therefore, the higher the center frequency, the wider the frequency band.

Table 1.3 The center frequency of octave and 1/3 octave

As we can see in the standard frequency band, humans can hear sound at a broad range of frequencies. The audible frequency range is between 20 Hz and 20 kHz.

There are two important reasons why we scale or measure the frequency of our interest by an octave or 1/3 of an octave: (i) we hear frequency in a relative scale and (ii) we cannot express our audible frequency range by using a linear frequency scale.

For the amplitude of the sound pressure, we use the sound pressure level (SPL or ). It is defined as

1.55

and is measured in units of decibels (dB); is the reference pressure, is the average pressure, and log10 is a log function that has a base of 10. is . The reference pressure is the smallest sound that a human being can hear. The first reason why we use a log scale and do not use a linear scale is that we can hear sound in a range from small amplitude to extremely large amplitude (Figure 1.10). From Figure 1.10, we can see that the human can hear from about 0 dB to somewhere in the range of 130–140 dB [4].

Figure 1.10 Equal-loudness contour: each line shows the SPL with respect to the frequency which corresponds to a loudness (phon) of 1 kHz pure sound.

Adapted from ISO 226: 2003 revision “Equal loudness contour”

The second reason is that we recognize the level of sound relatively and not in an absolute scale, similar to frequency recognition. Equation (1.55) essentially expresses how large the sound is compared to the reference SPL, and is therefore a good representation of the human hearing system. Table 1.4 collects some typical samples of sound levels that we can encounter, providing some practical references of the SPL [5].

Table 1.4 Daily life noise level in SPL

SPL (dB ref. 20 μPa)

Example

120

When a jet takes off (60 m)

110

Construction site

100

Loud shouting (1.5 m)

90

Truck noise (15 m)

80

Busy street

70

Moving car interior

60

Normal conversation (1 m)

50

Office

40

Living room

30

Bedroom (night)

20

Broadcasting studio

10

Leaves oscillating by a breeze

0

Minimum audible sound pressure level

Reproduced from Thomas D. Rossing, The science of sound, 2nd Edition, Addison-Wesley, pp 86, 1990

In order to calculate SPL, we write the mean-squared value of the real-valued pressure as

1.56

where T denotes the measurement time. Equation (1.56) can be expressed as the sum of every frequency component, equivalent to

1.57

where is the complex amplitude of the th frequency component ().

We then use the well-known relation

1.58

where * denotes the complex conjugate. If we rearrange Equation (1.57) using Equation (1.58), then we obtain

1.59

Equation (1.59) states that the mean square average of sound pressure is the sum of the mean square average of each frequency's sound pressure. This is only valid if and only if Equation (1.59) has a maximum when . Note that when