Acoustics and Psychoacoustics - Introduction to Sound - Part 1

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Acoustics and Psychoacoustics: Introduction tosound - Part 1

David Howard and Jamie Angus

2/27/2008 12:35 PM EST

Brush up on the nature of sound with this excerpt from the book "Acoustics And Psychoacoustics." Part1 covers pressure waves and sound transmission, and offers some example calculations. Sound issomething most people take for granted. Our environment is full of noises, which we have been exposedto from before birth. What is sound, how does it propagate, and how can it be quantified?

The purpose of this chapter is to introduce the reader to the basic elements ofsound, the way it propagates, and related topics. This will help us to understandboth the nature of sound, and its behaviour in a variety of acoustic contexts and

allow us to understand both the operation of musical instruments and theinteraction of sound with our hearing.

1.1 PRESSURE WAVES AND SOUND TRANSMISSIONAt a physical level sound is simply a mechanical disturbance of the medium,which may be air, or a solid, liquid or other gas. However, such a simplisticdescription is not very useful as it provides no information about the way thisdisturbance travels, or any of its characteristics other than the requirement for amedium in order for it to propagate. What is required is a more accuratedescription which can be used to make predictions of the behaviour of sound in a

variety of contexts.

1.1.1 The nature of sound wavesConsider the simple mechanical model of the propagation of sound through some physical medium,shown in Figure 1.1. This

Figure 1.1 Golf ball and spring model of a sound propagating material.

shows a simple one-dimensional model of a physical medium, such as air, which we call the golf balland spring model because it consists of a series of masses, e.g. golf balls, connected together by springs.

The golf balls represent the point masses of the molecules in a real material, and the springs representthe intermolecular forces between them. If the golf ball at the end is pushed toward the others then thespring linking it to the next golf ball will be compressed and will push at the next golf ball in the linewhich will compress the next spring, and so on.

Because of the mass of the golf balls there will be a time lag before they start moving from the action ofthe connecting springs. This means that the disturbance caused by moving the first golf ball will takesome time to travel down to the other end. If the golf ball at the beginning is returned to its originalposition the whole process just described will happen again, except that the golf balls will be pulledrather than pushed and the connecting springs will have to expand rather than compress. At the end of

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all this the system will end up with the golf balls having the same average spacing that they had beforethey were pushed and pulled.

The region where the golf balls are pushed together is known as a compression whereas the regionwhere they are pulled apart is known as a rarefaction, and the golf balls themselves are the propagatingmedium. In a real propagating medium, such as air, a disturbance would naturally consist of either acompression followed by a rarefaction or a rarefaction followed by a compression in order to allow the

medium to return to its normal state. A picture of what happens is shown in Figure 1.2. Because of theway the disturbance moves - the golf balls are pushed and pulled in the direction of the disturbance'stravel - this type of propagation is known as a longitudinal wave. Sound waves are therefore longitudinalwaves which propagate via a series of compressions and rarefactions in a medium, usually air.

There is an alternative way that a disturbance could be propagated down the golf ball and spring system.If, instead of being pushed and pulled toward each other, the golf balls were moved from side to sidethen a lateral disturbance would be propagated, due to the forces exerted by the springs on the golf ballsas described earlier. This type of wave is known as a transverse wave and is often found in the vibrationsof parts of musical instruments, such as strings or membranes.

Figure 1.2 Golf ball and spring model of a sound pulse propagating in a material.

The velocity of sound waves1.1.2 The velocity of sound wavesThe speed at which a disturbance, of either kind, moves down the 'string' of connected golf balls willdepend on two things:

The mass of the golf balls: the mass affects the speed of disturbance propagation because a golfball with more mass will take longer to start and stop moving. In real materials the density of the

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material determines the effective mass of the golf balls. A higher density gives a higher effectivemass and so the propagation will travel more slowly.

The strength of the springs: the strength of the springs connecting the golf balls together will alsoaffect the speed of disturbance propagation because a stronger spring will be able to push harderon the next golf ball and so accelerate it faster. In real materials the strength of the springs is

equivalent to the elastic modulus of the material, which is also known as the Young's modulus1 ofthe material. A higher elastic modulus in the material implies a stiffer spring and therefore a faster

speed of disturbance propagation.

For longitudinal waves in solids, the speed of propagation is only affected by the density and Young's

modulus of the material and this can be simply calculated from the following equation:2

v = (E/) (1.1)

where v = the speed in metres per second (ms-1)

= the density of the material (in kg m-3)

and E= the Young's modulus of the material (in N m-2)

However, although the density3 of a solid is independent of the direction of propagation in a solid, theYoung's modulus may not be. For example, brass will have a Young's modulus which is independent ofdirection because it is homogeneous whereas wood will have a different Young's modulus depending onwhether it is measured across the grain or with the grain. Thus brass will propagate a disturbance with avelocity which is independent of direction but in wood the velocity will depend on whether thedisturbance is travelling with the grain or across it. To make this clearer let us consider an example.

This variation of the speed of sound in materials such as wood can affect the acoustics of musicalinstruments made of wood and has particular implications for the design of loudspeaker cabinets, which

Example 1.1 Calculate the speed of sound in steel and in beech wood.

The density of steel is 7800 kg m-3, and its Young's modulus is 2.1 - 1011 Nm-2, so the speed of sound

in steel is given by:4

vsteel

= (2.1 x 1011 / 7800) = 5189 ms-1

The density of beech wood is 680 kg m-3, and its Young's modulus is 14 - 109 Nm-2 along the grain

and 0.88 - 109 Nm-2 across the grain. This means that the speed of sound is different in the twodirections and they are given by:

vbeech along the grain

= (14 x 109 / 680) = 4537 ms-1

and

vbeech across the grain

= (0.88 x 109 / 680) = 1138 ms-1

Thus the speed of sound in beech is four times faster along the grain than across the grain.

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are often made of wood. In general, loudspeaker manufacturers choose processed woods, such asplywood or MDF (medium density fibreboard), which have a Young's modulus which is independent ofdirection.

The velocity of sound in air1.1.3 The velocity of sound in air

So far the speed of sound in solids has been considered. However, sound is more usually considered assomething that propagates through air, and for music this is the normal medium for sound propagation.Unfortunately air does not have a Young's modulus so Equation 1.1 cannot be applied directly, eventhough the same mechanisms for sound propagation are involved. Air is springy, as any one who hasheld their finger over a bicycle pump and pushed the plunger will tell you, so a means of obtainingsomething equivalent to Young's modulus for air is required. This can be done by considering theadiabatic, meaning no heat transfer, gas law given by:

PV = constant (1.2)

where P = the pressure5 of the gas (in N m-2)

V= the volume of the gas (in m3)

and = is a constant which depends on the gas (1.4 for air)

The adiabatic gas law equation is used because the disturbance moves so quickly that there is no time forheat to transfer from the compressions or rarefactions. Equation 1.2 gives a relationship between thepressure and volume of a gas and this can be used to determine the strength of the air spring, or theequivalent to Young's modulus for air, which is given by:

Egas

= P (1.3)

The density of a gas is given by:

gas

= m/V= PM/RT (1.4)

where m = the mass of the gas (in kg)

M= the molecular mass of the gas (in kg mole-1)6

R = the gas constant (8.31 J K-1 mole-1)

and T= the absolute temperature (in K)

Equations 1.3 and 1.4 can be used to give the equation for the speed of sound in air, which is:

vgas = (Egas/gas) = (P/(PM/RT)) = (RT/M) (1.5)

Equation 1.5 is important because it shows that the speed of sound in a gas is not affected by pressure.Instead, the speed of sound is strongly affected by the absolute temperature and the molecular weight ofthe gas. Thus we would expect the speed of sound in a light gas, such as helium, to be faster than that ofa heavy gas, such as carbon dioxide, and air to be somewhere in between. For air we can calculate thespeed of sound as follows.

Example 1.2 Calculate the speed of sound in air at 0C and 20C.

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The reason for the increase in the speed of sound as a function of temperature is twofold. Firstly, asshown by Equation 1.4 which describes the density of an ideal gas, as the temperature rises the volumeincreases and providing the pressure remains constant, the density decreases. Secondly, if the pressuredoes alter, its effect on the density is compensated for by an increase in the effective Young's modulusfor air, as given by Equation 1.3. In fact the dominant factor other than temperature on the speed ofsound in a gas is the molecular weight of the gas. This is clearly different if the gas is different from air,for example helium. But the effective molecular weight can also be altered by the presence of watervapour, because the water molecules displace some of the air, and, because they have a lower weight,this slightly increases the speed of sound compared with dry air.

Although the speed of sound in air is proportional to the square root of absolute temperature we can

approximate this change over our normal temperature range by the linear equation:

v 331.3 + 0.6tms-1 (1.6)

where t= the temperature of the air in C

Therefore we can see that sound increases by about 0.6 ms-1 for each C rise in ambient temperature andthis can have important consequences for the way in which sound propagates.

The composition of air is 21% oxygen (O2), 78% nitrogen (N

2), 1% argon (Ar), and minute traces of

other gases. This gives the molecular weight of air as:

M= 21% x 16 x 2 + 78% x 14 x 2 + 1% x 18

= 2.87 x 10-2 kg mole-1

and

= 1.4

R = 8.31 J K-1 mole-1

which gives the speed of sound as:

v = ((1.4 x 8.31 x T) / 2.87 x 10-2)

v = 20.1T

Thus the speed of sound in air is dependent only on the square root of the absolute temperature, whichcan be obtained by adding 273 to the Celsius temperature; thus the speed of sound in air at 0C and 20C is:

v0C

= 20.1 (273 + 0) = 332 ms-1

v20C

= 20.1 (273 + 20) = 344 ms-1

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Table 1.1 gives the density, Young's modulus and corresponding velocity of longitudinal waves, for avariety of materials.

Table 1.1 Young's modulus, densities and speeds of sound for some common materials

The velocity of transverse waves

1.1.4 The velocity of transverse wavesThe velocities of transverse vibrations are affected by other factors. For example, the static springtension will have a significant effect on the acceleration of the golf balls in the golf ball and springmodel. If the tension is low then the force which restores the golf balls back to their original positionwill be lower and so the wave will propagate more slowly than when the

Figure 1.3 Some different forms of transverse wave.

tension is higher. Also there are several different possible transverse waves in three-dimensional objects.For example, there are different directions of vibration and in addition there are different forms,

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depending on whether opposing surfaces are vibrating in similar or contrary motion, such as transverse,torsional and others, see Figure 1.3. As all of these different ways of moving will have different springconstants, and will be affected differently by external factors such as shape, this means that for anyshape more complicated than a thin string the velocity of propagation of transverse modes of vibrationbecomes extremely complicated. This becomes important when one considers the operation ofpercussion instruments.

For transverse waves, calculating the velocity is more complex because for anything larger than atheoretical infinitely thin string the speed is affected by the geometry of the propagating medium and thetype of wave, as mentioned earlier. However, the transverse vibration of strings is quite important for anumber of musical instruments and the velocity of a transverse wave in a piece of string can becalculated by the following equation:

vtransverse

= T/ (1.7)

where = the mass per unit length (in kg m-1)

and T = the tension of the string (in N)

This equation, although it is derived assuming an infinitely thin string, is applicable to most strings thatone is likely to meet in practice. But it is applicable only to pure transverse vibration: it does not applyto torsional or other modes of vibration. However, this is the dominant form of vibration for thin strings.Its main error is due to the inherent stiffness in real materials which results in a slight increase invelocity with frequency. This effect does alter the timbre of percussive stringed instruments, like thepiano, and gets stronger for thicker pieces of wire. However, Equation 1.7 can be used for most practicalpurposes. Let us calculate the speed of a transverse vibration on a steel string.

The wavelength and frequency of sound waves1.1.5 The wavelength and frequency of sound wavesSo far we have only considered the propagation of a single disturbance through the golf ball and springmodel and we have seen that the disturbance travels at a constant velocity which is dependent only onthe characteristics of the medium. Thus any other type of disturbance, such as a periodic one, would alsotravel at a constant speed.

Example 1.3 Calculate the speed of a transverse vibration on a steel wire which is 0.8 mm in diameter(this could be a steel guitar string), and under 627 N of tension.

The mass per unit length is given by:

steel

= steel(r2) = 7800 x 3.14 x (0.8 x 10-3 / 2)2

= 3.92 x 10-3 kg m-1

The speed of the transverse wave is thus:

vsteel transverse

= (627 / 3.92 x 10-3) = 400 ms-1

This is considerably slower than a longitudinal wave in the same material and generally transversewaves propagate more slowly than longitudinal ones in a given material.

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Figure 1.4 shows the golf ball and spring model being excited by a pin attached to a wheel rotating at aconstant rate of rotation. This will produce a pressure variation as a function of time which isproportional to the sine of

Figure 1.4 Golf ball and spring model of a sine wave propagating in a material.

the angle of rotation. This is known as a sinusoidal excitation and produces a sine wave. It is importantbecause it represents the simplest form of periodic excitation. As we shall see later in the chapter, morecomplicated wave forms can always be described in terms of these simpler sine waves.

Sine waves have three parameters: their amplitude, rate of rotation or frequency, and their startingposition or phase. The frequency used to be expressed in units of cycles per second, reflecting the originof the waveform, but now it is measured in the equivalent units of hertz (Hz). This

Figure 1.5 The wavelength of propagating sine wave.

type of excitation generates a travelling sine wave disturbance down the model, where the compressionsand rarefactions are periodic. Because the sine wave propagates at a given velocity a length can beassigned to the distance between the repeats of the compressions or rarefactions, as shown in Figure 1.5.Furthermore, because the velocity is constant, the distance between these repeats will be inversely

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proportional to the rate of variation of the sine wave, known as its frequency. The distance between therepeats is an important acoustical quantity and is called the wavelength (). Because the wavelength andfrequency are linked together by the velocity, it is possible to calculate one of the quantities given theknowledge of two others using the following equation:

v = f (1.8)

where v = the velocity of sound in the medium (in ms-1)

f= the frequency of the sound (in Hz, 1 Hz = 1 cycle per second)and = the wavelength of the sound in the medium (in m)

This equation can be used to calculate the frequency given the wavelength, the wavelength given thefrequency, and even the speed of sound in the medium given the frequency and wavelength, and isapplicable to both longitudinal and transverse waves.

In acoustics the wavelength is often used as the 'ruler' for measuring length, rather than metres, feet orfurlongs, because many of the effects of real objects, such as rooms or obstacles, on sound waves aredependent on the wavelength.

The relationship between pressure, velocity and impedance in sound waves 1.1.6 The relationship between pressure, velocity and impedance in sound wavesAnother aspect of a propagating wave to consider is the movement of the molecules in the mediumwhich is carrying it. The wave can be seen as a series of compressions and rarefactions which aretravelling through the medium. The force required to effect the displacement, a combination of both

Example 1.4 Calculate the wavelength of sound, being propagated in air at 20C, at 20 Hz and 20 kHz.

For air the speed of sound at 20C is 344 ms

-1

(see Example 1.2), thus the wavelengths at the twofrequencies are given by:

= v/f

which gives:

= 344/20 = 17.2 m for 20 Hz

and

= 344/(20 x 103

) = 1.72 cm for 20 kHz

These two frequencies correspond to the extremes of the audio frequency range so one can see that therange of wavelength sizes involved is very large!

Example 1.5 Calculate the frequency of sound with a wavelength of 34 cm in air at 20C.

The frequency is given by:

f= v/ = 344/0.34 = 1012 Hz

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compression and acceleration, forms the pressure component of the wave.

In order for the compressions and rarefactions to occur, the molecules must move closer together orfurther apart. Movement implies velocity, so there must be a velocity component which is associatedwith the displacement component of the sound wave. This behaviour can be observed in the golf ballmodel for sound propagation described earlier.

In order for the golf balls to get closer for compression they have some velocity to move towards eachother. This velocity will become zero when the compression has reached its peak, because at this pointthe molecules will be stationary. Then the golf balls will start moving with a velocity away from eachother in order to get to the rarefacted state. Again the velocity towards the golf balls will become zero atthe trough of the rarefaction. The velocity does not switch instantly from one direction to another, due tothe inertia of the molecules involved, instead it accelerates smoothly from a stationary to a moving stateand back again. The velocity component reaches its peak between the compressions and rarefactions,and for a sine wave displacement component the associated velocity component is a cosine.

Figure 1.6 shows a sine wave propagating in the golf ball model with plots of the associatedcomponents. The force required to accelerate the molecules forms the pressure component of the wave.

This is associated with the velocity component of the propagating wave and therefore is in phase with it.That is, if the

Figure 1.6 Pressure, velocity and displacement components of a sine wave propagating in amaterial.

velocity component is a cosine then the pressure component will also be a cosine. Thus, a sound wave

has both pressure and velocity components that travel through the medium at the same speed.

Pressure is a scalar quantity and therefore has no direction; we talk about pressure at a point and not in aparticular direction. Velocity on the other hand must have direction; things move from one position toanother. It is the velocity component which gives a sound wave its direction.

The velocity and pressure components of a sound wave are also related to each other in terms of thedensity and springiness of the propagating medium. A propagating medium which has a low density andweak springs would have a higher amplitude in its velocity component for a given pressure amplitudecompared with a medium which is denser and has stronger springs. This relationship can be expressed,

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for a wave some distance away from the source and any boundaries, using the following equation:

Pressure component amplitude/Velocity component amplitude = Constant = Zacoustic

=

p/u (1.9)

where p = the pressure component amplitude

u = the volume velocity component amplitudeand Zacoustic

= the acoustic impedance

This constant is known as the acoustic impedance and is analogous to the resistance (or impedance) ofan electrical circuit.

The amplitude of the pressure component is a function of the springiness (Young's modulus) of thematerial and the volume velocity component is a function of the density. This allows us to calculate theacoustic impedance using the Young's modulus and density with the following equation:

Zacoustic

= E

However the velocity of sound in the medium, usually referred to as c, is also dependent on the Young's

modulus and density so the above equation is often expressed as:7

Zacoustic

= E= (2(E/)) = c = 1.21 x 344

= 416 kg m-2 s-1 in air at 20C (1.9a)

Note that the acoustic impedance for a wave in free space is also dependent only on the characteristics ofthe propagating medium.

However, if the wave is travelling down a tube whose dimensions are smaller than a wavelength, thenthe impedance predicted by Equation 1.9 is modified by the tube's area to give:

Zacoustic tube

= c/Stube

(1.9b)

where Stube

= the tube area

This means that for bounded waves the impedance depends on the surface area within the boundingstructure and so will change as the area changes. As we shall see later, changes in impedance can causereflections. This effect is important in the design and function of many musical instruments as discussed

in Chapter 4.

Coming up in Part 2: Sound intensity, power and pressure level

Footnotes:1. Young's modulus is a measure of the 'springiness' of a material. A high Young's modulus means the

material needs more force to compress it. It is measured in newtons per square metre (N m -2).2. A newton (N) is a measure of force.

3. Density is the mass per unit volume. It is measured in kilograms per cubic metre (kg m -3).

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4. s-1 means per second.

5. Pressure is the force, in newtons, exerted by a gas on a surface. This arises because the gas molecules

'bounce' off the surface. It is measured in newtons per square metre (N m -2).

6. The molecular mass of a gas is approximately equal to the total number of protons and neutrons in themolecule expressed in grams (g). Molecular mass expressed in this way always contains the same

number of molecules (6.022 x 1023). This number of molecules is known as a mole (mol).

7. m-2 means per square metre

Printed with permission from Focal Press, a division ofElsevier. Copyright 2006. "Acoustics andPsychoacoustics" by David Howard and Jamie Angus. For more information about this title, please visitwww.focalpress.com.

Related links:

Audio in the 21st Century - SoundSound focusing technologies make recent headlines'Acoustic cloak' makes objects invisible to sound waves

Audio Coding: An Introduction to Data CompressionHow audio codecs work - PsycoacousticsPrinciples of 3-D Audio

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Acoustics and Psychoacoustics - Introduction to Sound - Part 1

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