**After reading this article you will learn about Sound:- 1.** **Introduction to Sound 2. Propagation of Sound 3. Complex Waves 4. Octaves 5. Energy 6. Reflection 7. Decibel Scale.**

**Contents:**

- Introduction to Sound
- Propagation of Sound
- Complex Sound Waves
- Octaves
- Energy in Sound Waves
- Reflection of Sound Waves and Impedance
- Decibel Scale

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** 1. Introduction to Sound:**

The field of noise pollution and its control is an interdisciplinary subject and, consequently not an easy area to master. The subject of noise pollution borrows generously from many fields of science and engineering such as mathematics, dynamics, theory of vibrations, fluid mechanics, thermodynamics, acoustics, electronics, psychophysics, materials science and, of course, acoustical engineering.

Another difficulty in mastering the subject of noise- control engineering is that the apparently insidious nature of the way sound is generated and propagated often defies the intuitional noise-control solutions.

Thus an understanding of the fundamentals of sound generation and propagation is essential to the engineer (or other person) responsible for initiating and carrying out noise-control programmes. In the following sections, some fundamentals of sound and vibration (relevant to the subject of noise pollution and its control) will be covered, mostly in a qualitative manner.

From the physiological point of view, sound is the result of pressure variations in the air which act on the surface of the eardrum. The ear converts these variations in pressures into electrical signals which are then interpreted by the brain as sound. The pressure variations associated with sound in air, however, are quite small.

The fundamentals of sound in relation to noise pollution and its control are the generation of sound by a source and the propagation of this sound from the source to the receiver (i.e., a human being, for the purpose of noise control). We note here that all noise control techniques are based on the alteration of the sound pressure as it propagates in the atmosphere from the source to the receiver.

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** 2. Propagation of Sound:**

In order to reach the ear, sound must travel (or propagate) some distance through the air, from the source here it originated. Since air has mass as well as elasticity, propagation of sound through the air can be thought of in terms of momentum and elastic restoring forces.

It is very helpful to imagine the air medium sliced into very thin layers of particles (i.e., air molecules). These particles are in random motion such that the average of all the particle velocities is zero. This is the state of the medium when there is no imposed disturbance.

Consider now a rigid panel (attached to some machine) that is moving back and forth from its static equilibrium position, as shown in Fig. 1. As the panel moves to the right, the molecules of air against the panel must also move to the right.

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Since these particles have acquired a net velocity (and, therefore, momentum), they are capable of applying a force (or pressure) on neighbouring particles to the right. Having done this, the neighbouring particles are also set in motion which, in turn, transfer their momentum to their neighbours, and so on.

Fig. 1 clearly shows a disturbance, or a wave, travelling to the right. This may be called a** “positive pressure” **wave or a condensative wave.

While this wave is progressing, the vibrating panel has attained its maximum displacement to the right and has reversed its direction so that it is now travelling to the left. As before, the particles of air next to the panel remain in contact with the panel and faithfully move to the left.

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Again, momentum is transferred to the neighbouring particles by the imposed motion to the left. This process is repeated with subsequent neighbouring particles; each particle moving, in turn, to the left as the disturbance passes through.

This may be called the “negative pressure” wave or the rarefactive wave. We note here that the rarefactive wave travels in the same direction as the condensative wave, away from the source of disturbance (the vibrating panel) and to the right, as shown in Fig. 1.

There are two different types of motion involved here, viz., the particle motion and the wave motion. In the first case, each of the particles moves back and forth about its equilibrium position in a simple harmonic motion.

The wave motion, on the other hand, is really a manifestation of particle motion in the sense that it is a measure of the speed with which momentum transfer occurs through the particles. This is the speed at which the disturbance (i.e., the momentum transfer from particle to particle) propagates through the medium.

Although there is no net mass transfer, the sound wave does have energy by virtue of the pressure variations in the mediums. The rate at which the momentum transfer occurs is known as the speed of sound. The speed of sound v is a function of the density and elasticity of the medium and, therefore, it is strongly dependent on temperature in the case of a gaseous medium.

In air, the speed of sound v is given by the relation

where T = ambient temperature (in degrees Celsius).

If the wave motion were to be stopped at some instant in time and the pressure fluctuations were plotted as a function of distance, the resulting wave would look something like that shown in the lower part of Fig. 1. The distance between successive similar points of the waveform (such as the distance between successive pressure maxima) is called the wavelength of the wave, denoted by the symbol λ.

Now if the wave motion were to be viewed through a narrow slit, then the number of successive similar points in the wave which passed by the slit in one second would be called the frequency of the wave, denoted by the symbol f. There is a simple relation between the frequency of wavelength λ and the speed of the wave v:

f = v/λ cycles/sec. … (2)

It is an accepted practice now to use Hertz (abbreviated to “Hz”), in honour of the well-known scientist H.R. Hertz (1857-1897), as the unit of frequency (1 Hz = 1 cycle per second). The range of audible frequencies extends approximately from 20 to 20,000 Hz.

The manner in which the sound waves propagate depends on the relationship between the size of the source and the wavelength. If the source is large in relation to the wavelength, the result is a plane wave, i.e., at any instant, the sound pressure throughout a plane perpendicular to the direction of wave propagation, is constant.

On the other hand, if the size of the source is small compared to the wavelength, the wave will spread out in all directions, thereby forming a spherical wave.

In practice, plane waves usually occur near a large source (in terms of its physical size). As the distance from this source increases, the wave tends to assume a spherical condition, regardless of the size of the source. Plane waves most commonly occur in pipes or ducts, where the wavelength is large compared to the diameter of the pipe or cross-section of the duct.

** 3. Complex Sound Waves: **

The simplest form of sound wave that can be generated is composed to only one frequency, and is sinusoidal with time.

This is called a pure tone. Common examples of devices which emit pure tones are tuning forks and organ pipes It is rare in industry for a process or a machine to emit only a single pure tone’ but there are a number of cases (e.g., furnaces, material mixers, electric motors and generators, transformers, etc.) where a single pure tone dominates.

Acoustic or mechanical resonances may also result in a pure-tone sound. A pure tone can be generated with a rigid vibrating panel by forcing it to move back and forth about an equilibrium position with a displacement y given by the relation

y = A sin (2 π ft.) (3)

where A = maximum displacement from equilibrium position (known as the amplitude), f = frequency and t = time. The sinusoidal motion given by Eq. (3) is known as simple harmonic motion.

Using the ideas of Fourier analysis, the displacement y (t) in the case of a more complex, but still periodic, motion may be written as a sum of smusoidal or simple harmonic motion as follows:

y (t) = A_{1} sin (2π ft.) + A_{2} sin (4π f t) + A_{3} sin (6 πft.) + … + A_{n} sin (2π n ft.)………….. (4)

In Eq. (4), A_{1} A_{2}, A_{3}…, A_{n} are the amplitudes and f, 2f, 3f,…, nf are the frequencies of these sinusoidal waves. The first term A_{1} sin (2 π ft.) in Eq. (4) is known as the fundamental wave, while other terms are known as higher harmonics. For example, the term A_{n} sin (2πn ft.) is known as the n^{th} harmonic We note from Eq. (4) that the frequencies of higher harmonics are integral multiples of the fundamental frequency f.

In general the sharper or more discontinuous the motion producing the sound the greater would be the number of significant harmonics in the sound wave and the greater would be the frequency range over which they extend.

For example diesel engines are notable for an extremely rapid rise in pressure in the combustion chamber during the combustion process, and the resulting sound from such engines contains hundreds of measurable harmonics extending over the entire audio-frequency range.

On the other hand, if a sound wave were to be constructed by closely grouping sinusoidal waves of different frequency, but not harmonically related (i.e., higher frequencies not being integral multiples of the lowest frequency) and, further, if each sine wave were to have an amplitude that varied individually with time, the resulting sound is known as random sound.

Such sound would not have any discernible periodic behaviour and the pressure amplitude has to be determined on a time average basis. If the time average amplitude of the random sound is approximately constant over a broad range of frequencies, the resulting sound is known as white noise (because of the analogy to white light which contains all the visible colours of the spectrum).

White noise occurs often in nature, e.g., the sound of falling water and resulting leaves, and is also a by-product of some machinery and machining processes (e.g., the noise of escaping steam or air, and the noise from weaving mills). Random sounds, in general, accompany all machinery and machining processes in combination with single and complex sounds.

** 4. Octaves: **

Frequency is an important parameter in the description of sound waves. Frequencies are laid out in octaves in the same manner as a piano keyboard. The term **“octave”** from the field of music. It means the interval between any two sounds having a frequency ratio 2:1.

The frequency of 256 Hz is approximately that of middle C on the piano, and an octave higher is about 512 Hz. The top note on the piano keyboard corresponds to about 4,096 Hz.

The series of frequencies most commonly used for the measurement of noise frequencies is as follows: 37.5 – 75 Hz; 75 – 150 Hz; 150 – 300 Hz; 300 – 600 Hz; 600 Hz – 1.2 kHz (kHz = kilo Hertz =1,000 Hz); 1.2-2.4 kHz; 2.4 -4.8 kHz; and 4.8 – 9.6 kHz. Each of these intervals is a frequency octave. The most important frequency ranges are shown in Table 1.

The instrument used to obtain measurements involving frequency is known as “sound analyser” (or “frequency analyser”). In order to measure any noise adequately, both the sound level meter (used to measure the sound pressure level) and the sound analyser must be used. These two instruments, when used together, will give the intensity of any noise in each frequency range.

Most noises are not pure tones. They are usually composed of several or many frequencies. Each frequency, moreover, may have a different intensity. The overall noise level measured with a sound level meter cannot tell whether or not a given noise is a high frequency or a low-frequency noise.

A sound analyser, on the other hand, can do this by measuring the intensity in each frequency range. These intensities of each frequency range add up logarithmically to the overall intensity level.

** 5. Energy in Sound Waves: **

Since sound is propagated by momentum transfer, there must be energy involved in the process. This is dramatically evident in the destruction caused by shock waves from explosives or sonic booms of supersonic aircraft.

**The process is the same on a lesser scale:**

The energy from the vibrating surface or other source is transmitted in the propagating wave at the speed of sound.

At any point in the path of the Wave, the acoustic energy consists of the sum of the potential and kinetic energies associated with the motion of particles at that point. The potential energy arises from the elastic compression of the medium and kinetic energy from the velocities of particles.

The average sound energy flowing through a given area perpendicular to the direction of wave propagation is equivalent to the average work done on that area. Since work is defined as force times distance, or pressure times area times distance (because pressure = force per unit area), and since power is defined as the time rate at which work is performed, sound power W may be defined as

W = p̅ a y̅/t

= p̅ v y̅ watt, … (5)

with v̅ = y̅/t …(6)

where p̅= average pressure, v̅ = average particle velocity y̅ = displacement, a = area and t = time. The bars in Eq. (5) and (6) indicate the time-averaged values of respective quantities.

The intensity of sound I, an often used measure of sound, is defined as the amount of sound power passing through a unit area. From Eq. (5), we find that

I = W/a = p̅ v̅ a/a

= p̅ v̅ watt/m^{2}. …. (7)

The form of sound intensity given by Eq. (7), though obtained from basic principles, is not convenient in practice because particle velocity is not easily measured. A form for sound intensity better suited for engineering calculations will be presented later.

** 6. Reflection of Sound Waves and Impedance: **

When a sound wave strikes a plane surface, part of the sound energy is reflected back into the space. This a very common experience and terms like echo and reverberation are used to describe this phenomenon.

In discussing the phenomenon of reflection of sound waves, it is helpful to introduce the concept of impedance. As the word implies, impedance is a measure of resistance to movement from an applied force or pressure. The mechanical impedance Z_{m} is defined as

Z_{m} = force/velocity. … (8)

In the case of wave propagation, the corresponding quantity is the specific acoustic impedance Z_{s} (at a point) which is defined for a single propagating wave as follows:

It can be shown that for a progressive plane wave or for a spherical wave at sufficient distance from the source, the specific acoustic impedance, i.e., the resistance of the medium (solid, liquid or gas) to the propagating sound wave is given by

Z_{s}=pv, …(10)

where p = density of the medium and v = speed of sound in that medium. The product (pv) is often called the characteristic impedance of the medium. The values of characteristic impedance for air and some other media are listed in Table 2. In wave propagation, the product wave propagation, the product (pv) plays the same part as the mass in mechanical phenomena.

**Reflection and Transmission Coefficients: **

It is a well-known fact that the phenomenon of reflection is a result of “impedance mismatching.” Let us consider two acoustic media with characteristic impedances (p_{1}V_{1}) and (p_{2}v_{2}), as shown in Fig. 2. A plane wave from the left (the incident wave) is propagating perpendicular to the interface.

If (p_{1}V_{1}) is different from (p_{2}v_{2}), then all of the energy from the incident wave cannot be transmitted across the plane interface; any energy remaining must go into a reflected wave, as shown in Fig. 2.

By considering the conservation of energy and momentum at the interface, it can be shown that the acoustic reflection coefficient a_{R} and the acoustic transmission coefficient a_{T}, are given by the relations,

Moreover, if A_{I} is the amplitude of the incident wave, then the amplitude of the reflected wave A_{R} and the amplitude of the transmitted wave A_{T }are given by the relations

A_{R} = α_{R}A_{I}, A_{T}. = α_{T}. A_{I}. …(13)

It is obvious from Eq. (11 -13) that if p_{1}v_{1} = p_{2}v_{2}, A_{R} = 0and A_{T}= 1. That is, when there is no change in the characteristic impedance at the interface, the propagation of sound wave continues undisturbed.

When p_{2}v_{2} is larger than p_{1}V_{2}, A_{R} has the same sign as A_{I} (see Eq. (11) and (13)), i.e., the reflected wave is in phase with the incident wave. But when p_{2} v_{2} is smaller than p_{1}v_{2}, the signs of A_{R} and A_{I} are opposite so that they are 180 degrees out of phase. In both of these cases, the reflected wave occurs at the interface.

In practical situations, it is quite rare that a simple situation such as that described above occurs. When both incident and reflected acoustic waves are present in a medium, a term known as acoustic impedance is used. The acoustic impedance Z_{A} is defined by the relation

where total acoustic pressure=sum of the pressure of the incident and reflected wave at a point and, similarly, total particle velocity = sum of the particle velocity in the incident and reflected wave at a point.

The difference between Eq. (9) and Eq. (14) is that the specific acoustic impedance Z_{s} given by Eq. (9) is defined for one propagating wave in one direction, while the acoustic impedance Z_{A} given by Eq. (14) accounts for waves in opposite directions. In general, the waves travelling in opposite directions will be out of phase with each other and the associated particle velocities will also be out of phase.

In most noise-control applications of a sound wave incident on an acoustic material, the precise magnitude and phase angle of the reflected wave are not of interest; but the fraction of the acoustic wave that is absorbed (or dissipated) by the acoustic material is of interest.

The acoustic absorption coefficient a is a measure of the fraction of acoustic energy absorbed (and subsequently dissipated as heat) by an acoustic material.

It is related to the characteristic impedance (p_{1}v_{1}) of the medium and the acoustic impedance Z_{A }of the acoustic material by the relation where the vertical bars in Eq. (15) and (16) denote the magnitude (or modulus) of the quantity concerned since the phase angle is not of concern in this form.

The acoustic absorption coefficient a is a function of frequency. It is measured experimentally and reported as a property of acoustic materials by the manufacturer.

** 7. Decibel Scale:**

The decibel is a mathematical scale similar in use to a logarithmic scale. It is used to describe the energy level or intensity of a physical quantity. In addition to its application in acoustics, the decibel is also used for electrical power and, in certain cases, mechanical energy presentation.

Fundamentally, the decibel is ten times the logarithm (on base ten) of the ratio of a power or energy quantity with respect to a reference base of the same physical quantity. The basic equation for expressing this concept is

L_{E }= 10 log_{10}(E/E_{ref})dB, …(17)

where Lg = energy or power level (in decibels, abbreviated to dB), E = energy or power quantity of interest (in watts), and E_{ref} = reference energy or power quantity of interest (in watts).

It is evident from Eq. (17) that the original units of energy or power quantity of interest are cancelled in the ratio. Energy or power, moreover, can be described in various ways such as in terms of velocity, acceleration, pressure, voltage, etc.

**In acoustics, the decibel scale is used for acoustic power, acoustic intensity and acoustic pressure, as expressed by the following equations: **

L_{w }= 10 log_{10}(W/W_{ref})dB, …(18)

L_{I} = 10 log_{10} (I/I_{ref}) dB, …(19)

L_{p} = 10 log_{10} (P/P_{ref})^{2} dB, …(20)

where L_{w} = sound power level (in dB), L_{I} = sound intensity level (in dB) and L_{p} = sound pressure level (in dB). Moreover, W = observed power, I = observed intensity and P = observed pressure. The subscript “ref” attached to these symbols denotes the reference values of corresponding physical quantities.

**These reference values are taken to be: **

W_{ref} = 10-^{12} watts, ….(21)

P_{ref} = 0.00002 dynes/cm^{2}

= 0.00002 microbar. … (22)