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Chapter 8: Propagation of Sound Waves

In Oxford, around the year 1660, Robert Boyle was working with a new machine. It was an air pump, built for him with the help of Robert Hooke, and it could slowly suck the air out of a sealed glass vessel. Nobody had been able to do that before, so nobody knew what happened to things left inside an emptied jar.

Boyle tried many things inside it. In one of these trials he hung a pocket watch inside the glass vessel, sealed it, and listened. The watch ticked away steadily, and he could hear it clearly through the glass.

Then he began to work the pump. Stroke by stroke, air was drawn out of the vessel. The ticking grew fainter. It did not stop suddenly, and the watch did not stop either — through the glass he could still see the hands moving, and he knew the balance inside was still swinging exactly as before. Yet the sound kept fading, until finally he could hear nothing at all.

Nothing had happened to the watch. Something had happened to the space around it. Boyle concluded that air itself was carrying the sound to his ear, and that with the air gone, the sound had no way to cross.

That single observation sits underneath this whole chapter.

Diagram

Fig. 8.0-1 — Boyle’s Watch In The Emptied Vessel Side-by-side pair of drawings of a sealed glass vessel standing on the plate of a hand-operated air pump, with a pocket watch hung by a thread inside. Left drawing labelled ‘vessel full of air’, with wavy lines spreading from the watch to the glass wall and out to a listener’s ear drawn at the side. Right drawing labelled ‘air pumped out’, with no wavy lines between the watch and the glass, the watch hands still shown moving. Label the glass vessel, the watch, the thread, the pump handle and the outlet to the pump.

NoteCuriosity Corner

Q1. The watch inside Boyle’s vessel never stopped ticking. So why did the ticking become fainter and fainter as the air was pumped out? Q2. Sound reached Boyle’s ear across the room, but the air in the room did not blow towards him. What, then, actually travelled from the watch to his ear? Q3. If Boyle had somehow filled the vessel with steel instead of air, would the sound have reached him faster or slower than through air? Why should the material make any difference at all? Q4. During a storm you see the flash of lightning first and hear the thunder several seconds later, even though both begin at the same instant. What does that delay tell you about sound? Q5. A bat flying in a dark room calls out constantly and never hits a wall, yet you hear almost nothing. What could be different about the sound a bat makes?

8.1 Wave Motion

Boyle’s watch sent something across the vessel to his ear. That something was energy. Before we can say how sound carries energy, we need to look at the general way in which energy travels without any material being carried along with it.

8.1.1 Two Ways Of Transferring Energy

Energy can be carried from one place to another in only two ways.

The first way is by the transfer of matter — the moving matter itself carries the energy from one place to the other. When you throw a ball at a wall, you give energy to the ball, the ball flies across the room, and it delivers that energy to the wall when it strikes. The energy travels because the ball travels.

The second way is by wave motion — energy is passed on from particle to particle of a material, while the material as a whole stays where it is. Nothing is thrown across; the disturbance is handed along. Wave motion is by far the more important of the two, and it is the method by which sound reaches you.

Diagram

Fig. 8.1-1 — Two Ways Of Carrying Energy From One Place To Another Two panels side by side. Left panel labelled ‘transfer of matter’: a hand throwing a ball, with a dashed curved path from the hand to a wall, and the ball drawn again striking the wall. Right panel labelled ‘wave motion’: a row of about eight identical particles drawn as circles joined by short springs; the first circle is pushed by a finger, arrows show each circle moving a little to and fro about its own position, and a large arrow above the row labelled ‘energy travels this way’ points from the first circle to the last. Mark ‘the ball itself moves across’ under the left panel and ‘no particle moves across’ under the right panel.

8.1.2 Mechanical And Electromagnetic Waves

Waves fall into two families, separated by one question: does the wave need a material to travel through?

Waves that do need a material to travel through are called mechanical waves — waves that can pass only through a substance such as a solid, a liquid or a gas. Vibrations sent along a rope, vibrations on a stretched string, ripples on the surface of water and sound waves are all mechanical waves. The material through which such a wave travels is called the medium.

Waves that need no material at all are called electromagnetic waves — waves that can travel through vacuum as well as through a substance. Light waves, X-rays, gamma rays and radio waves belong to this family. Sunlight crosses the empty space between the Sun and the Earth, and that is only possible because light does not need a medium.

NoteDo Not Mix Up

Mechanical wave and electromagnetic wave A mechanical wave must have a material to travel through — remove the material and it stops. An electromagnetic wave needs nothing to travel through and crosses vacuum quite happily. Sound is mechanical; light is electromagnetic. That single difference explains most of what this chapter has to say about sound.

8.1.3 How A Mechanical Wave Is Formed

A mechanical wave is produced when a large number of particles of the medium vibrate together. A vibration given to the first particle is passed on to the second, the second passes it to the third, and so on down the line. Energy therefore travels from the source right through the medium, while each particle only vibrates to and fro about its own mean position — the resting position it occupied before the wave arrived. The particles themselves are not carried forward.

Two familiar situations show this happening.

A Stone Dropped In A Pond

Drop a stone into the still water of a pond. You hear the sound of the stone striking the surface, and at the same time a disturbance is produced in the water at the point where it strikes. This disturbance spreads outwards in all directions as circular ripples, as shown in Fig. 8.1-2.

Diagram

Fig. 8.1-2 — Ripples Spreading On A Water Surface Top view of a still pond. A piece of stone shown entering the water at a point, with about five concentric circles of increasing radius drawn around that point to show the ripples spreading outwards. Small arrows on the circles point radially outwards. Label ‘piece of stone’ with a leader line to the centre, and label ‘ripples spread in all directions’.

Now float a small piece of cork on the surface, some distance away from where the stone struck. When the ripples reach it, the cork does not move ahead with them. It only bobs up and down while the wave moves on past it. The reason is that the cork rides with the water particles around it, and those water particles are themselves only moving up and down. Each particle, as it rises and falls, passes its energy on to its neighbours and then returns to its own mean position, so the disturbance keeps moving ahead across the surface, as shown in Fig. 8.1-3. The humps and hollows you see on the surface are given their proper names in Section 8.4.

Diagram

Fig. 8.1-3 — Propagation Of The Disturbance On A Water Surface Side view (section) of a water surface drawn as a smooth wavy line, with the still-water level shown as a horizontal dashed line. A piece of stone shown at the right-hand end at the point where it struck. A round cork floating on the wavy surface at the left, with a vertical double-headed arrow through it labelled ‘cork only moves up and down’. A horizontal arrow above the surface pointing from the stone towards the cork, labelled ‘direction of motion of the disturbance’. Label two humps above the dashed line and two hollows below it.

The ripples do not last forever. They die out as soon as the energy given by the stone is used up in the water. To keep a wave travelling continuously, the disturbance at the starting point must be repeated over and over again — that is, the source must go on vibrating.

NoteTry Yourself

Half-fill a wide bowl with water and let it settle completely. Drop in a single drop from a spoon held a little above the surface, and watch one small bit of dry leaf floating near the edge. Does the leaf travel towards the edge, or does it only rise and fall as the ripple passes under it? Try it again with a steady drip and see how the ripples now keep coming.

NoteCuriosity Corner

Q2. Sound reached Boyle’s ear across the room, but the air in the room did not blow towards him. What, then, actually travelled from the watch to his ear?

A. Energy travelled, not air. The ticking made the air particles nearest the watch vibrate to and fro about their mean positions. Each of those particles handed the vibration on to its neighbours and then returned to where it started, exactly as the water particles do under the floating cork. The disturbance moved all the way to Boyle’s ear; the air itself stayed put.

A Jerk Given To A String

Tie one end of a long string to a hook on a wall, hold the other end in your hand, and give it a single small jerk. The jerk does not stay where you made it — it runs along the string to the far end.

Each particle of the string is held to its neighbour by a tensile force, that is, by the pull along the stretched string. So when you displace one particle from its mean position, it pulls its neighbour along with it, that neighbour pulls the next, and the displacement is handed down the whole string. The travelling hump produced by a single jerk is called a pulse — a single disturbance travelling through a medium, as shown in Fig. 8.1-4.

Diagram

Fig. 8.1-4 — A Pulse Travelling Along A String Three horizontal strings drawn one below the other, each fixed at its right-hand end to a hatched wall. In the top string a small hump labelled A is near the left-hand end. In the middle string the same hump A has moved to the centre, with a short arrow labelled with an arrowhead pointing right. In the bottom string the hump A is close to the wall. Caption label ‘the hump moves along, the string does not’.

Notice what does not happen. The particles of the string end up exactly where they were before the pulse reached them. The pulse carries energy from your hand to the wall and leaves the string behind unchanged.

Now, instead of one jerk, move your hand steadily up and down, over and over. Every particle of the string then vibrates up and down in the same repeating way, and a continuous wave travels along the string from your hand to the hook, as in Fig. 8.1-5. This is the difference between a pulse and a wave: a pulse comes from one disturbance, a wave from a disturbance repeated again and again.

Diagram

Fig. 8.1-5 — Wave Motion In A String A hand at the left holding the end of a long string whose right-hand end is fixed to a hatched wall. The string is drawn as a continuous smooth wavy line with two full humps above the line and two hollows below it. Mark points A, B, C, D and E along the string at successive positions on the wave, with a vertical double-headed arrow at the hand showing that it is moved up and down, and a horizontal arrow near E pointing towards the wall to show the direction in which the wave travels.

NoteSide Note

You have probably seen a crowd in a stadium make a ‘wave’ — each person stands up, throws up their arms and sits down again. The wave races right round the ground, though not one person changes seat. It is a good picture of what a medium particle does: a quick movement about its own place, and the disturbance travels on.

8.1.4 Characteristics Of Wave Motion

Everything seen in the pond and in the string can now be gathered into five statements that are true of wave motion in general.

1. In a wave, the disturbance moves ahead with a definite speed, and it keeps its form as it goes.
2. The speed with which the disturbance moves through a medium depends on the elasticity and the density of that medium.
3. The particles of the medium do not travel along with the disturbance. Each one vibrates to and fro on either side of its mean position, taking the same fixed time for every complete vibration. This kind of regular repeating vibration is called simple harmonic motion.
4. Energy is transferred through the medium in the same direction in which the disturbance travels, that is, in the direction of wave motion.
5. Every particle of the medium vibrates in the same manner — to and fro, up and down, or forward and backward — but the particles do not all reach the same stage of their vibration at the same instant. One particle is a little behind its neighbour, and this difference in stage is called a difference of phase.

Point 2 is worth holding on to. Elasticity is the property by which a material returns to its original shape after being displaced, and density is the mass of the material contained in unit volume, measured in kg m⁻³. These two properties decide how quickly one particle can push its neighbour and how heavy that neighbour is to push. Section 8.8 returns to them when it explains why sound travels so much faster through steel than through air.

8.2 Some Terms Related To Wave Motion

A wave has a size, a rhythm and a speed, and each of these needs its own name and its own unit. The five terms that follow are used for every wave in this chapter, so it is worth meeting them carefully.

8.2.1 Amplitude

Watch the cork on the pond again. It rises a certain height above the still-water level and falls the same depth below it, and no further. That greatest distance from the resting position is the amplitude of the wave.

NoteDefinition

Amplitude When a wave passes through a medium, the maximum displacement of a particle of the medium on either side of its mean position is called the amplitude of the wave. It is denoted by the letter \(a\), and its SI unit is the metre (m).

The amplitude tells you how violently the particles are shaken, and so how much energy the wave carries. A gentle tap on a table sends out a wave of small amplitude; a hard blow sends out one of large amplitude.

8.2.2 Time Period

Each particle repeats the same to-and-fro journey over and over. One complete journey — up, back down through the mean position, down to the lowest point, and back to the mean position again — is called one vibration. The time it takes is the time period.

NoteDefinition

Time Period The time taken by a particle of a medium to complete one vibration is called the time period of the wave. It is denoted by the letter \(T\), and its SI unit is the second (s).

8.2.3 Frequency

The time period answers ‘how long for one vibration?’. Turn the question round — ‘how many vibrations in one second?’ — and you have the frequency.

NoteDefinition

Frequency The number of vibrations made by a particle of a medium in one second is called the frequency of the wave. It is the same as the number of complete waves passing a point in one second. It is denoted by the letter \(f\) (sometimes by \(n\) or by \(\nu\), the Greek letter nu), and its SI unit is the reciprocal second, written \(\text{s}^{-1}\) and named the hertz (Hz).

The two quantities are two ways of describing the same rhythm, so one must fix the other. In a time of \(T\) seconds the particle completes exactly one vibration. In one second, therefore, it completes \(1/T\) vibrations — and that number is the frequency. Hence

\[f = \frac{1}{T}\]

where \(f\) is the frequency in hertz (Hz) and \(T\) is the time period in seconds (s). The relation works both ways, so \(T = 1/f\) as well.

NoteExam Corner

Q. A particle takes \(0.02\ \text{s}\) to complete one vibration. What is its frequency?

A. \(f = 1/T = 1/0.02 = 50\ \text{Hz}\), that is, 50 complete vibrations every second.

The frequency of a wave is equal to the frequency of vibration of its source. It is a characteristic of the source that produces the disturbance, and of nothing else. It does not depend on the amplitude of the vibration, and it does not depend on the medium the wave travels through. A whistle blown in air and blown under water sends out the same frequency in both.

NoteExample 8.1

Time Period And Frequency Of A Vibrating Particle

1. Given: a particle of a medium completes 240 vibrations in one minute, that is, in \(60\ \text{s}\). To find: the frequency \(f\) in Hz and the time period \(T\) in s.

2. Frequency is the number of vibrations completed in one second, so divide the number of vibrations by the time taken in seconds. The time period then follows from \(T = 1/f\).

3. \(f = \dfrac{240}{60\ \text{s}} = 4\ \text{s}^{-1} = 4\ \text{Hz}\), and \(T = \dfrac{1}{f} = \dfrac{1}{4\ \text{s}^{-1}} = 0.25\ \text{s}\).

4. The frequency is \(4\ \text{Hz}\) and the time period is \(0.25\ \text{s}\). The check: four vibrations each lasting a quarter of a second do fill exactly one second, as they must.

NoteSide Note

The unit of frequency is named after Heinrich Hertz, the German physicist who in the 1880s first produced and detected radio waves in his laboratory — waves nobody could see or hear, whose existence had only been predicted on paper. One hertz simply means one vibration per second.

8.2.4 Wavelength

While a particle completes one vibration, the wave itself moves forward by a certain distance. That distance is the wavelength.

NoteDefinition

Wavelength The distance travelled by a wave in one time period of vibration of a particle of the medium is called its wavelength. It is denoted by the Greek letter \(\lambda\) (lambda), and its SI unit is the metre (m).

The value of the wavelength depends on the medium through which the wave travels, because the wave moves at different speeds in different media.

There is a second, more practical way to spot a wavelength: it is the distance between any two nearest points of the wave that are at the same stage of vibration. On a water surface or a string, the raised parts are the crests and the hollows are the troughs, so the distance between two consecutive crests, or between two consecutive troughs, is one wavelength. In a sound wave the medium is squeezed together in some regions, called compressions, and spread apart in others, called rarefactions; there the distance between two consecutive compressions, or between two consecutive rarefactions, is one wavelength. Both kinds of wave are described fully in Section 8.4.

8.2.5 Wave Velocity

The wave itself advances steadily through the medium, and we measure how fast it advances.

NoteDefinition

Wave Velocity The distance travelled by a wave in one second is called its wave velocity, or wave speed. It is denoted by the letter \(V\), and its SI unit is the metre per second (m s⁻¹).

Wave velocity is the speed at which energy is carried from one place to another by wave motion. It is not the velocity of any individual particle. A particle of the medium merely vibrates about its mean position and gets nowhere at all, while the energy travels on across the whole medium.

For a given medium the wave velocity is constant, since it is fixed by the elasticity and the density of that medium — the two properties met in Section 8.1. It changes only when the wave passes from one medium into another.

NoteDo Not Mix Up

Wave velocity and particle velocity Wave velocity is the speed at which the disturbance and its energy move forward through the medium, and it stays constant in a given medium. Particle velocity is the speed of one vibrating particle moving to and fro about its mean position; it keeps changing and averages out to nothing over a vibration.

8.2.6 Displacement–Time Graph

A wave is easiest to picture through its graphs, and there are two of them. The first fixes attention on one single particle of the medium and watches it as time passes.

Fig. 8.2-1 shows how the displacement of that one particle changes with time as a wave goes past it. Such a graph is called a displacement–time graph. On it, the amplitude \(a\) is the height of a peak above the time axis, and the time period \(T\) is the time from one peak to the next. Every particle of the medium goes through exactly this motion, but not all at the same instant — each takes its turn a little after the one before, as the wave moves along.

Diagram

Fig. 8.2-1 — Displacement–Time Graph Of A Particle Of A Wave Smooth sine-shaped curve drawn on axes. Vertical axis labelled ‘displacement (m)’ with the zero line marked as the mean position; horizontal axis labelled ‘time t (s)’ with an arrow to the right. Draw two complete cycles. Mark a vertical double-headed arrow from the zero line to the first crest labelled a, and another from the zero line down to the first trough labelled a. Mark a horizontal double-headed arrow from one crest to the next crest labelled T, and a second horizontal double-headed arrow along the base from the start of one cycle to the start of the next, also labelled T.

NoteInterpret Data

Fig. 8.2-1 — Reading A Displacement–Time Graph 1. Check the axes first. Time in seconds runs along the horizontal axis; displacement in metres runs up the vertical axis, measured from the mean position of the particle. The graph is about one particle only. 2. Find the mean position. It is the horizontal zero line. Every time the curve crosses it, the particle is passing through its resting position. 3. Read the amplitude. Measure straight up from the zero line to the highest point of the curve. That vertical distance, in metres, is \(a\). The lowest point lies the same distance below, which is why the amplitude is described as the maximum displacement on either side. 4. Read the time period. Measure horizontally from any point on the curve to the next point at the same stage of the motion — crest to crest is easiest. That horizontal distance, in seconds, is \(T\). 5. Get the frequency from it. Once \(T\) has been read off, the frequency follows from \(f = 1/T\), in hertz. So this one graph gives you both the amplitude and the frequency of the wave.

8.2.7 Displacement–Distance Graph

The second graph does the opposite. Instead of following one particle through time, it freezes the whole medium at one instant and shows where all the particles are at that moment. For this reason it is also called a snapshot of the wave.

Fig. 8.2-2 is the displacement–distance graph of a transverse wave at an instant. The curve shows the displaced positions of the particles of the medium from their mean positions at that instant. The amplitude \(a\) is again the greatest displacement, but now the horizontal axis carries distance in metres, so the repeating distance measured along it is the wavelength \(\lambda\), not the time period.

Diagram

Fig. 8.2-2 — Displacement–Distance Graph Of A Wave Smooth sine-shaped curve drawn on axes. Vertical axis labelled ‘displacement (m)’ with the zero line marked as the mean position; horizontal axis labelled ‘distance x (m)’ with an arrow to the right. Draw two complete cycles. Mark a vertical double-headed arrow from the zero line up to the first crest labelled a, and one from the zero line down to the first trough labelled a. Mark a horizontal double-headed arrow from one crest to the next crest labelled λ, and a second horizontal double-headed arrow along the base from the start of one cycle to the start of the next, also labelled λ.

The two graphs look almost identical, and that is exactly the trap. Always read the horizontal axis before saying what the repeating length stands for: if that axis carries time, the repeat is the time period \(T\) in seconds; if it carries distance, the repeat is the wavelength \(\lambda\) in metres.

NoteExtra Practice 8.1

A wave passing through a medium has a time period of \(0.4\ \text{s}\) and an amplitude of \(0.05\ \text{m}\).

(a) Find its frequency.

(b) Through what total distance does one particle of the medium move in going from its highest position to its lowest position?

(c) How many complete vibrations does a particle make in one minute?

8.3 Relationship Between Wavelength, Wave Velocity And Frequency

Three of the quantities just defined — wave velocity, frequency and wavelength — are not independent of one another. Fix any two and the third is settled. The link between them is the single most useful relation in this chapter.

Start from plain words. The wave moves forward steadily. In the time taken for one complete vibration, it moves forward by exactly one wavelength — that is what the wavelength was defined to be. So the faster the wave travels, and the longer each vibration lasts, the further it gets in one vibration, and the longer the wavelength must be.

Fig. 8.3-1 shows this happening.

Diagram

Fig. 8.3-1 — A Wave Advances One Wavelength In One Time Period Two identical sine-shaped wave profiles drawn one above the other on a common horizontal distance axis labelled ‘distance (m)’. The upper profile is labelled ‘wave at the instant t = 0’ with a crest marked P. The lower profile is the same curve shifted to the right by exactly one full cycle, labelled ‘wave one time period later, t = T’, with the corresponding crest marked P’. A horizontal double-headed arrow between P and P’ labelled λ, and a right-pointing arrow above it labelled ‘wave travels a distance λ in a time T’.

Deriving The Relation

Let the wave velocity be \(V\), the time period \(T\), the frequency \(f\) and the wavelength \(\lambda\). By the definition of wavelength,

\[\lambda = \text{distance travelled by the wave in one time period, that is, in } T \text{ second}\]

A distance travelled is always the speed multiplied by the time taken, so

\[\lambda = \text{wave velocity} \times \text{time period} = V \times T\]

which is usually written as

\[VT = \lambda\]

where \(V\) is the wave velocity in m s⁻¹, \(T\) the time period in s, and \(\lambda\) the wavelength in m.

Now bring in the frequency. From Section 8.2, the time period and the frequency are reciprocals of each other, \(T = \dfrac{1}{f}\). Substituting this value of \(T\),

\[V \times \left(\frac{1}{f}\right) = \lambda\]

Multiplying both sides by \(f\) gives the relation in its usual form,

\[V = f\lambda\]

where \(V\) is the wave velocity in m s⁻¹, \(f\) the frequency in Hz, and \(\lambda\) the wavelength in m. In words,

\[\text{wave velocity} = \text{frequency} \times \text{wavelength}\]

This relation holds for every kind of wave — waves on a string, ripples on water, sound waves in air, and light waves too.

NoteFormula in Words

V = fλ Each second, the source sends out \(f\) complete waves, and each of them is \(\lambda\) metre long. So the length of wave laid down in one second is \(f\) multiplied by \(\lambda\) — and that is exactly how far the wave has advanced in a second, which is its speed.

Before substituting into \(V = f\lambda\), always check that the quantities are in matching units — the wavelength in metre and the frequency in hertz, so that the velocity comes out in metre per second. A frequency given in kilohertz must first be converted, using \(1\ \text{kHz} = 1000\ \text{Hz}\).

NoteExample 8.2

Wavelength Of A Sound In Air

1. Given: frequency \(f = 500\ \text{Hz}\), wave velocity in air \(V = 330\ \text{m s}^{-1}\). To find: the wavelength \(\lambda\) in m.

2. The three quantities are linked by \(V = f\lambda\). Since the velocity and the frequency are known, rearrange it to make the wavelength the subject: \(\lambda = \dfrac{V}{f}\).

3. \(\lambda = \dfrac{330\ \text{m s}^{-1}}{500\ \text{s}^{-1}} = 0.66\ \text{m}\).

4. The wavelength is \(0.66\ \text{m}\). The check: the sound covers \(330\ \text{m}\) in a second and fits 500 whole waves into that distance, so each wave must be a little over half a metre long — which it is.

NoteExam Corner

Q. A sound made in air passes into water, where it travels much faster. Does its frequency change?

A. No. The frequency is fixed by the source, which is still vibrating at the same rate. Since \(V\) increases while \(f\) stays the same, the wavelength \(\lambda\) must increase instead.

NoteExample 8.3

Speed Of Ripples On A Pond

1. Given: the source dips into the water 8 times each second, so the frequency \(f = 8\ \text{Hz}\); the distance between two consecutive crests, which is the wavelength, \(\lambda = 0.25\ \text{m}\). To find: the wave velocity \(V\) in m s⁻¹ and the time period \(T\) in s.

2. The velocity follows directly from \(V = f\lambda\), and the time period from \(T = \dfrac{1}{f}\).

3. \(V = f\lambda = 8\ \text{s}^{-1} \times 0.25\ \text{m} = 2\ \text{m s}^{-1}\), and \(T = \dfrac{1}{8\ \text{s}^{-1}} = 0.125\ \text{s}\).

4. The ripples travel at \(2\ \text{m s}^{-1}\), with a time period of \(0.125\ \text{s}\). The check: in one time period the wave should advance one wavelength, and \(2\ \text{m s}^{-1} \times 0.125\ \text{s} = 0.25\ \text{m}\), which is the wavelength given.

NoteExtra Practice 8.2

(a) A wave on a stretched string has a frequency of \(25\ \text{Hz}\) and a wavelength of \(1.2\ \text{m}\). Find its velocity.

(b) A sound wave of wavelength \(2.2\ \text{m}\) travels in air at \(330\ \text{m s}^{-1}\). Find its frequency and its time period.

8.4 Types Of Mechanical Waves

Every mechanical wave carries its energy forward in one particular direction. The particles of the medium, meanwhile, vibrate in some direction of their own. Comparing those two directions splits mechanical waves neatly into two kinds, namely transverse waves and longitudinal waves.

8.4.1 Transverse Waves

Think again of the string held in your hand. Your hand moves up and down, so every particle of the string vibrates up and down — yet the wave travels horizontally, away from your hand and towards the wall. The vibration is at right angles to the travel.

NoteDefinition

Transverse Wave A wave in which the particles of a medium vibrate about their mean positions in a direction perpendicular to the direction of propagation of the wave is called a transverse wave.

A transverse wave is made up of crests and troughs. The position of maximum upward displacement of the medium is called a crest, and the position of maximum downward displacement is called a trough, as shown in Fig. 8.4-1. At any one place the two keep alternating in time: a crest arrives, then a trough, then a crest again. The distance between two consecutive crests, or between two consecutive troughs, is the wavelength of the wave.

Diagram

Fig. 8.4-1 — A Transverse Wave On A String A long smooth sine-shaped curve drawn horizontally, with a horizontal dashed line through it showing the mean positions of the particles. Two humps above the dashed line labelled ‘crest’ and two hollows below it labelled ‘trough’. A large horizontal arrow above the curve labelled ‘direction of propagation of the wave’. Two short vertical double-headed arrows drawn on particles of the string, one at a crest and one at a mean position, labelled ‘direction of vibration of the particles’. Mark a horizontal double-headed arrow from one crest to the next crest labelled λ.

Transverse waves cannot be produced in every medium. To make a particle vibrate sideways and drag its neighbour sideways with it, the medium must resist a change of shape — it must have rigidity. Solids have rigidity, and the free surface of a liquid behaves in much the same way because of the forces acting along the surface. So transverse waves can be produced in solids and on the surface of liquids only. They cannot be produced inside liquids or in gases, which simply flow aside instead of dragging their neighbours along.

NoteTry Yourself

Stretch a long soft spring toy along a smooth floor with a friend holding the far end. Flick your end quickly up and down once, and watch the hump run along the spring. Notice that your hand ends up exactly where it started, while the hump travels the whole length. Try flicking sideways instead of upwards, and see whether it makes any difference.

8.4.2 Longitudinal Waves

Now suppose the particles vibrate not across the direction of travel, but along it — forward and backward, in the same line in which the wave advances.

NoteDefinition

Longitudinal Wave A wave in which each particle of the medium executes vibrations about its mean position in the direction of propagation of the wave is called a longitudinal wave.

Since a longitudinal wave asks the particles only to move to and fro along a line, and not to be dragged sideways, it needs no rigidity. Longitudinal waves can therefore be produced in solids, in liquids and in gases alike. Sound travels through air in the form of longitudinal waves.

NoteDo Not Mix Up

Crest and trough, compression and rarefaction Crests and troughs belong to a transverse wave: they are places of maximum displacement, upward and downward. Compressions and rarefactions belong to a longitudinal wave: they are places where the medium is crowded together and where it is thinned out. Never mark a crest on a sound wave in air.

8.4.3 Propagation Of A Longitudinal Wave In Air

The following simple arrangement shows how a longitudinal wave is set up in air. Take a thin metal strip and fix its lower end so that the strip stands vertically. Push its upper end to one side and release it. The strip vibrates, moving alternately to the right and to the left, and a sound is heard. Fig. 8.4-2 follows what happens to the layers of air on the right-hand side of the strip.

Diagram

Fig. 8.4-2 — Propagation Of A Disturbance In Air Five panels drawn one below the other, each showing a thin vertical metal strip clamped at its lower end on the left, and a row of about twelve vertical lines to its right representing layers of air. Panel (a) labelled ‘normal condition’: the strip upright at position a, air layers evenly spaced. Panel (b): the strip moved right from a to b, the nearest few air layers crowded close together, that crowded region labelled C. Panel (c): the strip returning from b to a, the crowded region C now further to the right with evenly spaced layers behind it. Panel (d): the strip moved left from a to c, the layers just to its right spread widely apart, that region labelled R, with C still further right. Panel (e): the strip back at a, showing R travelling behind C. Label the strip, the air layers, C for compression and R for rarefaction, and mark the movement of the strip in each panel with a small horizontal arrow.

Fig. 8.4-2(a) shows the strip and the air layers near it undisturbed, evenly spaced, in their normal positions.

As the strip swings to the right from \(a\) to \(b\) in Fig. 8.4-2(b), it pushes the layers of air in front of it. The particles of air in these layers are driven closer to one another, so the air there is squeezed together. A region in which the particles of the medium are crowded closer than normal is called a compression, marked C in the figure. The crowded particles, moving to the right, push and squeeze the layers next to them, which squeeze the next layers in turn — and so the compression travels forward. The particles themselves are only displaced a little; they do not move along with the compression.

In Fig. 8.4-2(c) the strip is on its way back from \(b\) to \(a\), having already pushed the particles ahead of it. The air particles near the strip now return towards their mean positions, because the medium is elastic and springs back.

When the strip swings past its normal position and moves to the left, from \(a\) to \(c\) in Fig. 8.4-2(d), it draws away from the air on its right and leaves a space of very low pressure there. The air layers on that side spread apart into this space. A region in which the particles of the medium are spread further apart than normal is called a rarefaction, marked R in the figure.

In Fig. 8.4-2(e) the strip returns from \(c\) to its normal position \(a\), pushing the rarefaction R forward, and the air layers near the strip once again pass through their mean positions because of the elasticity of the medium.

As the strip goes on swinging right and left, compressions and rarefactions are produced one after another. They travel away from the strip, carrying the disturbance with them at a definite speed that depends on the nature of the medium. Gradually, friction takes the strip’s energy away into the medium and the disturbance dies out.

Flowchart

FC 8.1 — One Complete Vibration Of The Strip Makes One Complete Wave Strip at rest at a → strip swings right, a to b, air ahead squeezed, compression C formed → strip returns, b to a, air near it springs back to mean positions → strip swings left, a to c, air ahead spreads out, rarefaction R formed → strip returns, c to a, rarefaction pushed forward → back to the start, ready to form the next compression

One complete to-and-fro motion of the strip therefore forms one compression and one rarefaction, and together these two make one complete wave. This is exactly the pattern described in Section 8.4.2: the air particles vibrate to and fro along the same line in which the disturbance travels. Thus sound travels in air in the form of longitudinal waves.

One point is worth stating plainly. At a compression the particles are crowded, so both the density and the pressure of the medium are maximum there. At a rarefaction they are spread out, so both the density and the pressure of the medium are minimum. A sound wave passing a point is, in effect, a rapid rise and fall of the pressure at that point, and it is this changing pressure that your eardrum responds to.

NoteSide Note

An earthquake sends out both kinds of wave through the rock. The faster ones are longitudinal and reach a recording station first; the slower ones are transverse and arrive later. Because transverse waves cannot pass through a liquid, the pattern of which waves arrive where is one of the ways scientists worked out that part of the Earth’s core is molten.

8.5 Difference Between Transverse And Longitudinal Waves

Both kinds of wave carry energy forward without carrying the medium with it, and both obey \(V = f\lambda\). What separates them is the direction in which the particles vibrate, and almost every other difference follows from that one. Fig. 8.5-1 puts the two side by side, and Table 8.1 sets out the differences point by point.

Diagram

Fig. 8.5-1 — Comparing A Transverse Wave With A Longitudinal Wave Two panels one above the other, sharing a large horizontal arrow at the right of each labelled ‘direction in which the wave travels’. Upper panel labelled ‘transverse wave’: a row of about fifteen dots joined by a smooth sine-shaped curve, with short vertical double-headed arrows on three of the dots labelled ‘particles vibrate at right angles to the wave’; mark one hump ‘crest’ and one hollow ‘trough’. Lower panel labelled ‘longitudinal wave’: a straight horizontal row of about thirty dots, crowded closely together in two regions and widely spaced in two regions in between, with short horizontal double-headed arrows on three of the dots labelled ‘particles vibrate along the wave’; mark a crowded region ‘compression’ and a spread-out region ‘rarefaction’.

NoteCompare Table

Table 8.1 — Transverse Wave Compared With Longitudinal Wave | Basis of difference | Transverse wave | Longitudinal wave | |—|—|—| | Form of the wave | It is formed of crests and troughs. | It is formed of a series of compressions and rarefactions. | | Direction of vibration of the particles | The particles vibrate at right angles to the direction of propagation of the wave. | The particles vibrate in the same direction in which the wave advances. | | Media in which it can be produced | It is possible only in solids and at the surface of liquids, which have rigidity. | It is possible in all types of media — solid, liquid and gas. | | Pressure and density of the medium | There is no variation of pressure. | The pressure and density are maximum at the compressions and minimum at the rarefactions. | | Examples | Waves on a rope, waves on a water surface, light waves. | Sound waves in air. |

The last row needs one word of care. Light waves are indeed transverse, but they are not mechanical waves at all — they are electromagnetic, and they need no medium, as Section 8.1.2 explained. They appear in the table only as an example of the transverse pattern of vibration, not as an example of a mechanical wave.

NoteExtra Practice 8.3

Classify each of the following as transverse or longitudinal, and give the reason in one line.

(a) The wave that travels along a stretched clothes line when it is flicked.

(b) The wave that carries a shout across a room.

(c) A wave travelling through the water deep below the surface of a lake.

8.6 Sound And Its Production From Vibrations

Every day you hear sounds from all sorts of sources — a morning alarm, a school bell, a car horn, a dog barking, music from different instruments. You never see the sound coming, yet it reaches your ears in the form of waves, produced by the vibrations of the particles of the medium around you, which for us is usually air. These waves carry the mechanical energy of the vibrating source along with them, and on arriving at your ear they produce the sensation of hearing.

NoteDefinition

Sound Sound is a form of energy that produces the sensation of hearing in our ears.

So sound is energy, and like all energy it must come from somewhere. The question this section settles is: what exactly does a source have to do in order to make sound?

8.6.1 Experiments Showing That A Vibrating Body Produces Sound

The answer is that the source must vibrate. Each of the following simple experiments makes the point in a different way, and each makes the vibration visible or felt rather than merely assumed.

Experiment 1 — A Plucked Rubber String

Take a rubber string. Hold one end in your mouth between your teeth and the other end in one hand, so that the string is stretched taut, as shown in Fig. 8.6-1. With the other hand, pluck it near the middle.

Diagram

Fig. 8.6-1 — A Vibrating Rubber String Produces Sound Side view of a person’s head, holding one end of a stretched rubber string between the teeth, the other end held taut in one hand at the left. The middle of the string drawn as a blurred spindle shape of dotted outlines to show it vibrating, with short vertical double-headed arrows above and below it. A second hand shown plucking the string near the middle. Label ‘rubber string’, ‘string held between the teeth’ and ‘plucked here’.

You notice that the string starts vibrating, and at the same moment a sound is heard. After some time the string stops vibrating, and then no sound is heard.

Experiment 2 — A Wire With A Paper Rider

Take a thin wire and stretch it between two nails about a metre apart. Fold a small bit of paper over the wire near its middle, so that it sits astride the wire. Such a bit of paper is called a rider, and its only job is to show up a movement too small and too fast for the eye to catch. Now pluck the wire near the rider, as shown in Fig. 8.6-2.

Diagram

Fig. 8.6-2 — A Vibrating Wire Produces Sound Front view of a horizontal wire stretched tightly between two nails driven into two vertical supports about a metre apart. A small folded bit of paper sitting astride the wire near the middle, labelled ‘paper rider’. The wire drawn with dotted outlines above and below its straight position to show it vibrating, with short vertical arrows at the middle. A hand below plucking the wire, and a curved arrow near the rider labelled ‘rider flies off’.

The rider flies off as the wire starts vibrating, and a sound is heard at the same time. When the wire stops vibrating, the sound is no longer heard.

Experiment 3 — A Tuning Fork And A Suspended Ball

A tuning fork is a rectangular rod of steel, or of aluminium, bent into a U shape, with a metallic stem at the bend. Strike one of its arms on a rubber pad and bring the vibrating arm close to a small table tennis ball hung by a thread, as shown in Fig. 8.6-3.

Diagram

Fig. 8.6-3 — A Vibrating Tuning Fork Producing Sound A tuning fork held upright by its stem, its two prongs at the top drawn with dotted outlines on either side to show them vibrating, with small horizontal arrows at the prong tips. A light table tennis ball hanging from a thread fixed to a support above, touching the outer side of one prong. A curved arrow showing the ball being knocked away from the prong. Label ‘tuning fork’, ‘vibrating prongs’, ‘table tennis ball’ and ‘the ball is pushed away’.

As the vibrating arm of the fork is brought close to the ball, the ball is knocked to and fro, and the sound of the tuning fork is heard. When the arm stops vibrating, the ball becomes stationary and no sound is heard.

Experiment 4 — A Beaten Drum

Beat a drum. The stretched membrane vibrates — you can feel this by touching it lightly with your fingertips — and the sound of the drum is heard. As the membrane comes to rest, the sound stops. A vibrating drum head is producing the sound.

Experiment 5 — A Plucked String And A Blown Whistle

When the string of a sitar or a guitar is plucked, the string starts vibrating and its sound is heard. In the same way, when a whistle is blown, the column of air inside it is set vibrating, and a sound is heard.

NoteExam Corner

Q. In every one of these experiments, what happens to the sound at the moment the vibration stops?

A. It stops as well. The vibration is not merely present alongside the sound — it is the cause of it.

From these experiments it is concluded that sound is produced when a body vibrates, and that when the body stops vibrating, the sound produced by it ceases. Thus a vibrating body is a source of sound.

NoteTry Yourself

Rest two fingers lightly on the front of your throat and hum a long steady note. You will feel the vibration under your fingers. Stop humming and the trembling stops at the same instant. Try a low note and then a high one, and see whether the feel of the vibration changes.

8.6.2 Sound Is A Form Of Energy

Follow the energy through, from the source to the listener, and the claim that sound is a form of energy becomes easy to justify.

Mechanical energy is needed in the first place to start the vibrations in a body — the energy of your plucking hand, of the drumstick, of the blow on the tuning fork. The vibrations of the body are then handed on to the medium in the form of waves, from one particle to the next, exactly as described in Section 8.1.3. These waves travel outwards, and on reaching your ears they set the eardrum vibrating. Those vibrations are perceived by you as sound.

Flowchart

FC 8.2 — The Path Of The Energy From The Source To The Listener Muscular energy of the hand → mechanical energy of the vibrating body → energy of the vibrating particles of the medium, passed on as a wave → vibration of the eardrum of the listener → sensation of hearing

Energy has therefore been carried from the source to the listener without anything material passing between them. That is precisely why sound is counted as a form of energy.

NoteExtra Practice 8.4

A stretched wire is plucked and sounds clearly for a few seconds, then falls silent although nothing has touched it.

(a) Why does the sound stop?

(b) Where has the energy of the wire gone?

(c) Suggest one way of making the sound last longer.

8.7 Sound Propagation Requires A Material Medium

Sound produced by a vibrating body travels outwards through the mechanical vibrations of the particles of a medium, in the form of waves. If that is so, then taking the medium away should take the sound away with it. The following experiment tests exactly that.

8.7.1 The Bell Jar Experiment

Take an electric bell and an airtight glass bell jar. Suspend the electric bell inside the jar, seal the jar on its base plate, and connect the jar to a vacuum pump — a pump used to draw air out of a closed vessel. The bell is connected through the wall of the jar to a battery and a key outside, so that the circuit can be completed from outside without opening the jar. The arrangement is shown in Fig. 8.7-1.

Diagram

Fig. 8.7-1 — Sound Requires A Medium: The Bell Jar Experiment A glass bell jar standing on a base plate, sealed at the rim. Inside the jar, an electric bell hangs from the top by two wires, with its gong and hammer drawn clearly. The two wires pass out through a sealed plug at the top of the jar and run to a battery and a key connected in series outside the jar. A side tube from the base plate leads to an arrow labelled ‘to vacuum pump’. Label ‘bell jar’, ‘electric bell’, ‘gong’, ‘battery’, ‘key’ and ‘to vacuum pump’.

Press the key to complete the circuit. The hammer of the electric bell is seen striking the gong repeatedly, and the sound of the bell is clearly heard outside the jar.

Now, keeping the key pressed, start the vacuum pump so that the air is gradually withdrawn from the jar. As more and more air is taken out, the loudness of the sound goes on decreasing. Finally, when the whole of the air has been drawn out, no sound is heard at all — although the hammer of the electric bell is still seen striking the gong repeatedly. The gong is certainly still vibrating and still producing sound; that sound simply is not reaching you.

Explanation

When the hammer hits the gong, sound is produced by the vibrations of the gong. While air is present, this sound travels through the air inside the jar to the wall of the jar. The wall is set vibrating, and it in turn sets the air outside the jar vibrating, so the sound reaches your ears and is heard.

Once the air has been removed from the jar, the sound produced by the vibrating gong has nothing to travel through, so it cannot reach the wall of the jar. The wall is not set vibrating, the outside air is not disturbed, and no sound is heard. The experiment clearly demonstrates that sound requires a material medium for its transmission, and that it cannot travel through vacuum.

NotePrinciple / Law

A material medium is necessary for the propagation of sound from one place to another.

NoteCuriosity Corner

Q1. The watch inside Boyle’s vessel never stopped ticking. So why did the ticking become fainter and fainter as the air was pumped out?

A. Because the air was the only thing carrying the sound out. The ticking set the air particles nearest the watch vibrating, and they passed the disturbance on to the glass and to Boyle’s ear. Each stroke of the pump left fewer particles inside to do the passing on, so less and less of the energy got out and the ticking grew fainter. With the air gone, the sound had no medium at all — exactly what the bell jar shows.

NoteReal-World Application

Two astronauts standing side by side on the surface of the Moon cannot talk to each other directly, however loudly they shout. There is no atmosphere on the Moon, so there is no medium to carry the sound between their helmets. They speak instead by radio. Radio waves are electromagnetic, so they need no medium and cross the vacuum without difficulty — which is also how their voices reach a control room on the Earth.

8.7.2 Requisites Of The Medium

Not every material carries sound equally well. A medium fit for the propagation of sound must possess the following three properties.

1. The medium must be elastic, so that its particles can come back to their original positions after being displaced to either side. Only then are the particles capable of vibrating about their mean positions, and only then can a disturbance be handed on and passed back.
2. The medium must have inertia, so that its particles have mass and can store the mechanical energy given to them. A particle with inertia keeps moving once it has been set moving, and so carries the energy forward instead of stopping dead.
3. The medium should be frictionless, so that there is no loss of energy as the sound travels through it. Friction between the particles converts some of the wave’s energy into heat, which is why a sound grows fainter as it travels.

Sound can propagate not only in gases, but in solids and in liquids as well. Materials such as air, water and iron transmit sound readily from one place to another. On the other hand, materials like a blanket or a thick curtain absorb most of the sound falling on them and transmit or reflect only a small fraction of it. This is why a room hung with heavy curtains sounds quieter than a bare one.

Sound cannot travel in vacuum. On the Moon there is no medium, and so on the Moon one cannot hear the sound produced by others.

Light, by contrast, does not require any material medium for its propagation and can therefore travel through vacuum as well. That single difference between sound and light runs through the rest of this chapter.

NoteTry Yourself

Set an alarm on a clock and let it ring on a bare table; listen from a few steps away. Now switch it off, wrap the clock in a folded blanket, and set the alarm again from the same distance. Is the sound weaker? Which of the three properties above has the blanket interfered with?

NoteExtra Practice 8.5

In the bell jar experiment of Fig. 8.7-1, the pump is stopped and air is slowly let back into the jar while the key stays pressed.

(a) Describe what a listener outside the jar hears as the air returns.

(b) State what is being shown by the fact that the hammer can still be seen striking the gong throughout.

8.8 Speed Of Sound In Different Media

Sound takes time to get from one place to another, and how much time depends on what it is travelling through. This section fixes both points — first that the speed is finite, then what the speed actually depends on.

8.8.1 Sound Travels With A Finite Speed

Stand at some distance from a blacksmith at work. When the hammer comes down on the iron, you see the blow land, and only a short while later do you hear the clang. The same thing happens with a cracker burst high up: the flash is seen first, and the bang follows a little later, as shown in Fig. 8.8-1.

Diagram

Fig. 8.8-1 — The Sound Arrives After The Event Is Seen A blacksmith at an anvil at the left, hammer shown at the moment of striking the iron. An observer standing well to the right. Two labelled arrows run from the anvil to the observer: an upper straight arrow labelled ‘flash of the blow is seen at once’ and a lower wavy arrow labelled ‘sound arrives a moment later’. Mark the separation between the blacksmith and the observer as ‘distance’.

This tells you something important: sound does not reach us instantaneously. It travels through a medium with a finite speed, and it takes a definite time to cover the distance from the place where it is produced.

The speed of sound in a medium depends on the following two factors:

1. the elasticity \(E\) of the medium, and
2. the density \(\rho\) of the medium.

8.8.2 Newton’s Formula And Laplace’s Correction

The speed of sound in a medium is given by the relation

\[V = \sqrt{\frac{E}{\rho}}\]

where \(V\) is the speed of sound in m s⁻¹, \(E\) is the modulus of elasticity of the medium in N m⁻², and \(\rho\) (rho) is the density of the medium in kg m⁻³. The modulus of elasticity is Young’s modulus in the case of solids and bulk modulus in the case of fluids, that is, of liquids and gases. It is a measure of how stiffly a material resists being deformed.

NoteFormula in Words

V = √(E/ρ) Elasticity is the springiness with which one particle shoves the next; density is how heavy that next particle is to shove. A springier medium passes the disturbance on faster, and a heavier one slows it down — so \(E\) sits on top of the fraction and \(\rho\) underneath.

Newton’s Assumption

Newton applied this relation to a gas. He assumed that when sound travels through a gas, the temperature of the gas does not change — that is, that the propagation of sound is an isothermal change, one that takes place at constant temperature. For an isothermal change the modulus of elasticity of a gas is equal to its pressure, so \(E = P\). Putting this into the relation above, the speed of sound in a gas becomes

\[V = \sqrt{\frac{P}{\rho}}\]

where \(P\) is the pressure of the gas in N m⁻² and \(\rho\) its density in kg m⁻³.

For air at normal temperature and pressure, \(P = 1.01 \times 10^{5}\ \text{N m}^{-2}\) and \(\rho = 1.293\ \text{kg m}^{-3}\), so this formula gives \(V = 279.5\ \text{m s}^{-1}\). But the speed of sound in air is found by experiment to be nearly \(330\ \text{m s}^{-1}\). The value calculated from Newton’s formula is thus much lower than the measured value, and something in the assumption must be wrong.

Laplace’s Correction

Later, the scientist Laplace supplied the missing piece. He pointed out that when sound travels through a gas, the compressions and rarefactions form and disappear so rapidly that there is no exchange of heat with the surrounding medium at all. A compression is warmed slightly and has no time to lose that heat; a rarefaction is cooled slightly and has no time to gain any. The propagation of sound is therefore an adiabatic change, one in which no heat enters or leaves.

For an adiabatic change, the modulus of elasticity of a gas is \(E = \gamma P\), where \(\gamma\) (gamma) is the ratio of the specific heat of the gas at constant pressure to its specific heat at constant volume, \(\gamma = C_p / C_v\). Being a ratio of two like quantities, \(\gamma\) has no unit. Hence the speed of sound in a gas is

\[V = \sqrt{\frac{\gamma P}{\rho}}\]

where \(\gamma\) is the ratio of the specific heats of the gas, \(P\) its pressure in N m⁻², and \(\rho\) its density in kg m⁻³. The value of \(\gamma\) depends on the nature of the medium; for air it is \(1.4\).

This modified formula gives the correct value. At normal temperature and pressure in air, taking \(\gamma = 1.4\), \(P = 1.01 \times 10^{5}\ \text{N m}^{-2}\) and \(\rho = 1.293\ \text{kg m}^{-3}\), the speed \(V\) comes out to be nearly \(330.7\ \text{m s}^{-1}\), which agrees with the experimental value.

NoteReal Incident / Discovery

A Correction That Took More Than A Century Isaac Newton published his calculation of the speed of sound in the Principia in 1687 — the first time anyone had worked out a speed of sound from the properties of the air rather than measuring it with a stopwatch. His answer was about one-sixth too small, and he knew it; the measured value was already known to be higher, and he tried various ways of accounting for the gap. The real reason had to be waited for. In 1816 the French mathematician Pierre-Simon Laplace showed that the compressions of a sound wave are formed too quickly for heat to flow out of them, and that the elasticity to be used is therefore the adiabatic one, larger than Newton’s by the factor \(\gamma\). With that single change, theory and measurement agreed.

8.8.3 Speed Of Sound In Some Media

Because the speed depends on the elasticity and density of the medium, it has a different value in every material. Table 8.2 gives measured values at \(0^\circ\text{C}\).

Table 8.2 — Speed of sound in some media at 0 °C

Medium	Speed of sound (in m s⁻¹)
Gases
Air	330
Carbon dioxide	260
Hydrogen	1270
Liquids
Alcohol	1210
Turpentine	1325
Water	1450
Solids
Copper	3560
Steel	5100
Glass	5500
Granite	6000

Read down the table and one pattern stands out.

NotePrinciple / Law

The speed of sound is different in different media. It is greatest in solids, less in liquids, and least in gases, because solids are much more elastic than liquids and gases. The speed of sound is nearly \(5100\ \text{m s}^{-1}\) in steel, \(1450\ \text{m s}^{-1}\) in water and \(330\ \text{m s}^{-1}\) in air at \(0^\circ\text{C}\).

At first sight this looks wrong. A solid is far denser than a gas, and density sits in the denominator of \(V = \sqrt{E/\rho}\), so a solid ought to slow sound down. The point is that both quantities change from one medium to another, and they do not change by the same amount. Going from air to steel, the density rises by a few thousand times, but the elasticity rises by very much more — a steel bar springs back from a squeeze far more vigorously than a pocket of air does. The elasticity wins, and the speed goes up.

NoteCuriosity Corner

Q3. If Boyle had somehow filled the vessel with steel instead of air, would the sound have reached him faster or slower than through air? Why should the material make any difference at all?

A. Much faster — about fifteen times as fast, since sound travels at nearly \(5100\ \text{m s}^{-1}\) in steel against \(330\ \text{m s}^{-1}\) in air. The material matters because the speed is set by the elasticity and the density of the medium, through \(V = \sqrt{E/\rho}\). Steel’s particles are locked tightly to their neighbours and spring back hard when displaced, so each one passes the disturbance on almost at once, while the loosely spaced particles of air take far longer to do the same job.

NoteExample 8.4

Sound Through A Steel Rail And Through Air

1. Given: length of the rail, which is the distance travelled, \(d = 1020\ \text{m}\); speed of sound in steel \(V_{\text{steel}} = 5100\ \text{m s}^{-1}\); speed of sound in air \(V_{\text{air}} = 330\ \text{m s}^{-1}\). To find: the time taken by the sound along the rail and through the air, in s.

2. Sound covers the distance at a steady speed, so use \(\text{time} = \dfrac{\text{distance}}{\text{speed}}\), applied separately to each medium since the distance is the same for both.

3. Through the rail, \(t_{\text{steel}} = \dfrac{1020\ \text{m}}{5100\ \text{m s}^{-1}} = 0.2\ \text{s}\). Through the air, \(t_{\text{air}} = \dfrac{1020\ \text{m}}{330\ \text{m s}^{-1}} = 3.09\ \text{s}\).

4. The sound takes \(0.2\ \text{s}\) through the steel and \(3.09\ \text{s}\) through the air, arriving about \(2.89\ \text{s}\) earlier through the rail. The check: steel carries sound roughly fifteen times faster than air, and \(3.09\ \text{s}\) is indeed about fifteen times \(0.2\ \text{s}\).

8.8.4 Evidence That Sound Travels Faster In Steel Than In Air

Two everyday observations show the difference in speed directly.

1. If a sound is produced at one end of a very long steel bar, two sounds are heard at the other end. The one that reaches the listener first has travelled through the steel; the one heard a moment later has travelled through the air. Fig. 8.8-2 shows the two paths.
2. A person living near a railway track often presses an ear against the steel rail to judge whether a train is coming. The vibrations produced by the moving wheels travel much faster through the steel rail than through the air, so the sound is heard through the track well before it is heard through the air. In this way the person gets an indication of the approaching train before its sound arrives.

Diagram

Fig. 8.8-2 — Two Sounds From One Blow On A Long Steel Bar A long horizontal steel bar drawn in section, with a hammer striking its left-hand end and a listener’s ear at the right-hand end. Two labelled paths run from the struck end to the ear: a straight arrow drawn inside the bar labelled ‘through steel, 5100 m s⁻¹, heard first’, and a wavy arrow drawn in the space above the bar labelled ‘through air, 330 m s⁻¹, heard later’. Label the steel bar and the hammer.

NoteExtra Practice 8.6

Take the speed of sound in air as \(330\ \text{m s}^{-1}\) and in water as \(1450\ \text{m s}^{-1}\).

(a) Find the time taken by sound to travel \(2900\ \text{m}\) in water.

(b) Find the distance travelled by sound in air in the same time.

(c) A sound is produced at one end of a copper wire \(712\ \text{m}\) long. Using Table 8.2, find the time it takes to reach the other end through the copper.

8.9 Factors Affecting The Speed Of Sound In A Gas

The value \(330\ \text{m s}^{-1}\) quoted for air is not fixed for all time. It is the value for dry, still air at \(0^\circ\text{C}\). Change the gas, warm it, add moisture to it or set it moving, and the speed changes. Four factors alter the speed of sound in a gas: density, temperature, humidity and the direction of the wind.

8.9.1 Effect Of Density

From the relation established in Section 8.8.2,

\[V = \sqrt{\frac{\gamma P}{\rho}}\]

it is clear that if the pressure and the ratio of specific heats are unchanged, then

\[V \propto \frac{1}{\sqrt{\rho}}\]

That is, the speed of sound is inversely proportional to the square root of the density of the gas. A light gas carries sound quickly; a heavy one carries it slowly.

The comparison of hydrogen with oxygen shows how much difference this makes. The density of oxygen is 16 times the density of hydrogen. Taking the ratio of the two speeds,

\[\frac{V_{\text{H}_2}}{V_{\text{O}_2}} = \sqrt{\frac{\text{density of oxygen}}{\text{density of hydrogen}}} = \sqrt{\frac{16}{1}} = 4\]

so the speed of sound in hydrogen is four times the speed of sound in oxygen. Notice that the denser gas has the smaller speed, and that the ratio of speeds is the square root of the inverse ratio of densities — not the ratio itself.

8.9.2 Effect Of Temperature

The speed of sound in a gas increases with an increase in the temperature of the gas. The reason follows from what has just been said about density: when a gas is warmed it expands, so the same mass now occupies a larger volume and its density decreases. A smaller density means a higher speed.

More exactly, the speed of sound is directly proportional to the square root of the temperature of the medium,

\[V \propto \sqrt{T}\]

where \(T\) is the temperature of the gas on the Kelvin scale — the absolute scale, on which \(0^\circ\text{C}\) is \(273\ \text{K}\). The temperature must be taken on this scale and not in degrees celsius, since it is the absolute temperature that governs the density of the gas.

For air, this leads to a very convenient working rule. The speed of sound in air increases by about \(0.61\ \text{m s}^{-1}\), that is by \(61\ \text{cm}\) per second, for each degree celsius rise in temperature. Hence

\[V_t = V_0 + 0.61\,t\]

where \(V_t\) is the speed of sound in air at \(t\ ^\circ\text{C}\) in m s⁻¹, \(V_0\) is the speed of sound in air at \(0^\circ\text{C}\) in m s⁻¹, and \(t\) is the temperature in degrees celsius. This relation is an approximation, and it holds only provided the rise in temperature is not very large.

NoteFormula in Words

V_t = V₀ + 0.61 t Start from the speed at \(0^\circ\text{C}\) and add on \(0.61\ \text{m s}^{-1}\) for every degree celsius the air is above that. Here \(t\) is in degrees celsius, not kelvin, because it is a rise in temperature that is being counted. It is a good approximation only for moderate temperature changes.

Diagram

Fig. 8.9-1 — Speed Of Sound In Air Against Temperature Straight-line graph. Horizontal axis labelled ‘temperature t (°C)’, marked from 0 to 40 in steps of 10. Vertical axis labelled ‘speed of sound V (m s⁻¹)’, marked from 320 to 360 in steps of 10, with the scale break shown at the origin. Draw a straight line beginning at the point (0, 330) and rising steadily to about (40, 354.4). Mark the intercept on the vertical axis as 330 m s⁻¹, and draw a small right-angled triangle on the line with its horizontal side labelled ‘rise of 10 °C’ and its vertical side labelled ‘increase of 6.1 m s⁻¹’.

NoteInterpret Data

Fig. 8.9-1 — Reading The Speed–Temperature Graph For Air 1. Check the axes. Temperature in degrees celsius runs along the horizontal axis; the speed of sound in air, in metre per second, runs up the vertical axis. Note that the vertical scale does not start at zero — it starts at 320 — so the line must not be read as passing through the origin. 2. Read the intercept. Where the line meets the vertical axis, the temperature is \(0^\circ\text{C}\) and the speed is \(330\ \text{m s}^{-1}\). That intercept is \(V_0\). 3. Find the gradient. Take the small triangle drawn on the line: the speed rises by \(6.1\ \text{m s}^{-1}\) for a temperature rise of \(10\ ^\circ\text{C}\), so the gradient is \(6.1 \div 10 = 0.61\ \text{m s}^{-1}\) per degree celsius. The gradient of this line is exactly the constant in the formula. 4. Read off any value. To find the speed at \(30^\circ\text{C}\), go up from 30 on the horizontal axis to the line and across to the vertical axis; the reading is about \(348\ \text{m s}^{-1}\), which agrees with \(330 + 0.61 \times 30\). 5. Note what the straight line means physically. Equal rises in temperature give equal increases in speed over this range — which is the whole content of the approximation \(V_t = V_0 + 0.61\,t\).

NoteExample 8.5

Speed Of Sound In Air On A Hot Afternoon

1. Given: the speed of sound in dry still air at \(0^\circ\text{C}\), \(V_0 = 330\ \text{m s}^{-1}\); temperature of the air, \(t = 40\ ^\circ\text{C}\). To find: the speed of sound \(V_t\) at this temperature, in m s⁻¹.

2. For a moderate temperature change in air, use \(V_t = V_0 + 0.61\,t\), taking \(t\) in degrees celsius.

3. \(V_t = 330\ \text{m s}^{-1} + 0.61\ \text{m s}^{-1} \times 40 = 330\ \text{m s}^{-1} + 24.4\ \text{m s}^{-1} = 354.4\ \text{m s}^{-1}\).

4. The speed of sound in the air is \(354.4\ \text{m s}^{-1}\). The check: the answer is a little above the \(0^\circ\text{C}\) value, as it should be for warmer air, and the increase of about \(24\ \text{m s}^{-1}\) is a small fraction of \(330\ \text{m s}^{-1}\), which is what the approximation expects.

8.9.3 Effect Of Humidity

The speed of sound in air increases with an increase in the humidity of the air, that is, with the amount of water vapour present in it.

The reason is once again density. The density of water vapour is about \(\tfrac{5}{8}\) times the density of dry air at ordinary temperature. So when moisture is added to air, lighter water vapour takes the place of some of the heavier dry air, and the density of the air as a whole falls. A smaller density gives a greater speed. Hence the speed of sound in humid air is greater than the speed of sound in dry air — in other words, sound travels faster in humid air than in dry air.

8.9.4 Effect Of Direction Of Wind

The speed of sound increases or decreases according to the direction in which the wind is travelling. This is not a change in the speed of sound through the air; it is the moving air carrying the sound wave bodily along with it, in the same way that a swimmer is carried by a river current.

Let \(V\) be the speed of sound in still air and \(W\) the speed of the wind, both in m s⁻¹. When the wind blows in the direction of travel of the sound, the speed of the sound becomes \(V + W\). When the wind blows in the direction opposite to the direction of travel of the sound, the speed becomes \(V - W\). If the wind blows at some other angle, only a part of it helps or hinders, so in general the speed lies in between these two values.

Diagram

Fig. 8.9-2 — Effect Of The Direction Of The Wind On The Speed Of Sound Two panels one above the other, each showing a source of sound at the left and a listener at the right, with a wavy arrow running from source to listener labelled ‘sound travels this way’. Upper panel: a broad straight arrow above the wavy arrow pointing from source to listener, labelled ‘wind blowing towards the listener’, and the result marked as ‘speed becomes V + W’. Lower panel: the broad arrow reversed, pointing from listener to source, labelled ‘wind blowing away from the listener’, and the result marked as ‘speed becomes V − W’.

This is why the sound of distant music is heard quickly and loudly when it is coming with the wind, while the same music is hard to hear when the wind is blowing in the opposite direction.

NoteDo Not Mix Up

Speed of sound and speed of wind The wind speed \(W\) is the speed of the air itself moving from place to place. The sound speed \(V\) is the speed of the disturbance passing through the air. They are different quantities that simply add or subtract: sound riding on moving air travels at \(V + W\) or \(V - W\) relative to the ground.

NoteExtra Practice 8.7

Take the speed of sound in dry air at \(0^\circ\text{C}\) as \(330\ \text{m s}^{-1}\).

(a) Find the speed of sound in air at \(22\ ^\circ\text{C}\).

(b) At this temperature a wind of \(12\ \text{m s}^{-1}\) blows from the source towards the listener. What is the speed with which the sound now reaches the listener?

(c) How would your answer to (b) change if the wind reversed its direction?

8.10 Factors Not Affecting The Speed Of Sound In A Gas

Some quantities that seem as though they ought to matter turn out not to. The speed of sound in a gas is not affected by a change in (i) the pressure of the gas, (ii) the amplitude of the wave, or (iii) the wavelength or frequency of the wave. Knowing what does not affect the speed is as useful as knowing what does.

8.10.1 Effect Of Pressure

The speed of sound in a gas is independent of pressure. This is surprising at first, because pressure appears openly in the formula

\[V = \sqrt{\frac{\gamma P}{\rho}}\]

The resolution is that pressure does not appear alone. What the formula actually contains is the ratio \(\dfrac{P}{\rho}\), and that ratio remains unchanged when the pressure changes at a constant temperature.

Here is why. Take a fixed mass of gas and increase its pressure. Its volume decreases, but its mass remains unchanged, so its density increases — the same mass now packed into a smaller space. The two changes go together in step, so the ratio \(P/\rho\) stays constant. For example, if the pressure \(P\) of a gas is doubled, its volume becomes half, so its density \(\rho\) gets doubled, the mass being constant. As a result the ratio \(P/\rho\) does not change, and neither does the speed of sound. Fig. 8.10-1 shows the two changes side by side.

Diagram

Fig. 8.10-1 — Doubling The Pressure Also Doubles The Density Two panels side by side, each showing a cylinder fitted with a movable piston and containing a gas drawn as evenly scattered dots. Left panel: the piston high up, the gas filling a tall column with dots widely spaced; label ‘pressure P, volume V, density ρ’. Right panel: the same cylinder with the piston pushed down so the gas fills half the height, the dots twice as closely packed; label ‘pressure 2P, volume V/2, density 2ρ’. Beneath both panels write ‘the same mass of gas throughout, so the ratio P/ρ is unchanged’. Mark a downward arrow on the piston of the right panel labelled ‘gas compressed at constant temperature’.

NoteExample 8.6

Checking That The Ratio Stays Constant

1. Given: air at normal pressure, \(P = 1.01 \times 10^{5}\ \text{N m}^{-2}\), with density \(\rho = 1.293\ \text{kg m}^{-3}\). The pressure is now doubled at constant temperature. To show: the speed of sound is unchanged.

2. The speed depends on the pressure and density only through the ratio \(P/\rho\) in \(V = \sqrt{\gamma P/\rho}\), and \(\gamma\) is a property of the gas, unaffected by compression. So it is enough to compare the two values of \(P/\rho\).

3. Before: \(\dfrac{P}{\rho} = \dfrac{1.01 \times 10^{5}\ \text{N m}^{-2}}{1.293\ \text{kg m}^{-3}}\). After doubling, the pressure is \(2.02 \times 10^{5}\ \text{N m}^{-2}\) and, the mass being unchanged in half the volume, the density is \(2.586\ \text{kg m}^{-3}\), so \(\dfrac{P}{\rho} = \dfrac{2.02 \times 10^{5}\ \text{N m}^{-2}}{2.586\ \text{kg m}^{-3}}\). Both the numerator and the denominator have been multiplied by 2, so the two ratios are equal.

4. The ratio \(P/\rho\), and therefore the speed of sound, is the same before and after. The check: this is why sound travels at much the same speed at sea level and on a hilltop, where the air pressure is far lower but the air is correspondingly thinner as well.

8.10.2 Effect Of Amplitude Of The Wave

The speed of sound does not depend on the amplitude of the sound wave. A shout and a whisper made at the same place reach a listener at exactly the same instant. The amplitude decides how much energy the wave carries, and so how loud the sound is; it has nothing to do with how fast the disturbance is handed on from one particle to the next, which is settled by the medium alone.

NoteDo Not Mix Up

Louder and faster A louder sound is not a faster sound. Loudness comes from a larger amplitude, which means more energy in the wave. Speed comes from the elasticity and density of the medium. Increasing the amplitude changes what arrives, never when it arrives.

8.10.3 Effect Of Wavelength Or Frequency

The speed of sound does not depend on the wavelength or the frequency of the sound wave. A high note and a low note produced together in the same air travel at the same speed.

Note carefully that this does not contradict the relation \(V = f\lambda\) of Section 8.3. In a given medium the speed \(V\) is fixed. Raising the frequency of the source therefore does not raise the speed — it shortens the wavelength instead, so that the product \(f\lambda\) comes out to the same value as before.

NoteReal-World Application

Why A Band Still Sounds Like A Band From The Back Of The Field Listen to a marching band from two hundred metres away and the drum, the trumpet and the flute all stay together, exactly as they were played. That only works because sounds of every frequency travel through air at the same speed. If low notes ran even slightly slower than high ones, the notes played in one chord would arrive spread out in time, and the further away you stood the worse the smearing would be. Music heard across a stadium would arrive as nonsense.

NoteExtra Practice 8.8

Two sounds are produced at the same instant at the same place in still air — a soft high-pitched whistle and a loud low-pitched horn.

(a) Which reaches a distant listener first? Give a reason.

(b) State which of the two has the longer wavelength, and why.

(c) If the air pressure were halved without any change of temperature, would either answer change? Give a reason.

8.11 Comparison Of The Speed Of Sound With The Speed Of Light

Sound and light are the two ways in which most information reaches us about a distant event. They behave quite differently, and the difference between their speeds is so large that it shows up in ordinary life again and again.

8.11.1 Points Of Distinction

There are the following four points of distinction between the propagation of sound and of light.

1. Light waves can travel in vacuum, but sound waves cannot travel in vacuum. This is the difference established by the bell jar experiment of Section 8.7.
2. The speed of light waves in air is \(3 \times 10^{8}\ \text{m s}^{-1}\), which is about a million times greater than the speed of sound waves in air, namely \(330\ \text{m s}^{-1}\) at \(0^\circ\text{C}\).
3. The speed of light waves decreases in an optically denser medium — the speed of light in water is \(2.25 \times 10^{8}\ \text{m s}^{-1}\) and in glass \(2 \times 10^{8}\ \text{m s}^{-1}\) — while the speed of sound waves is more in solids, less in liquids and still less in gases, being nearly \(5100\ \text{m s}^{-1}\) in steel, \(1450\ \text{m s}^{-1}\) in water and \(330\ \text{m s}^{-1}\) in air.
4. Light waves are transverse electromagnetic waves, while sound waves in air are longitudinal mechanical waves.

Point 3 is worth pausing on, because the two behave in opposite ways. Going from a gas into a solid slows light down but speeds sound up. Light is slowed because it interacts with the closely packed particles it must pass; sound is speeded up because those same closely packed particles are bound elastically to one another and hand the disturbance on far more quickly.

NoteCompare Table

Table 8.3 — Sound Compared With Light | Basis of difference | Sound | Light | |—|—|—| | Need for a medium | Cannot travel through vacuum; a material medium is necessary. | Travels through vacuum as well as through a medium. | | Speed in air | Nearly \(330\ \text{m s}^{-1}\) at \(0^\circ\text{C}\). | \(3 \times 10^{8}\ \text{m s}^{-1}\), about a million times greater. | | Change of speed with medium | Greatest in solids, less in liquids, least in gases. | Decreases on entering an optically denser medium, such as water or glass. | | Nature of the wave | Longitudinal mechanical wave in air. | Transverse electromagnetic wave. |

8.11.2 Consequences Of The Large Difference In The Two Speeds

Because light covers any ordinary distance almost instantly while sound takes a measurable time, an event that produces both is always seen before it is heard.

Thunder And Lightning

In a thunderstorm, the light of the lightning is seen much earlier than the sound of the thunder is heard, although the two are produced at the same instant. The reason is that light takes almost negligible time in comparison to sound in reaching us from the place of the thunder, because the speed of light, \(3 \times 10^{8}\ \text{m s}^{-1}\), is so much greater than the speed of sound, \(330\ \text{m s}^{-1}\). Fig. 8.11-1 shows why the gap opens up.

Diagram

Fig. 8.11-1 — Why The Flash Arrives Before The Thunder A storm cloud at the top left with a jagged lightning stroke drawn beneath it, and an observer standing at the lower right at a distance. Two arrows run from the lightning stroke to the observer: a straight arrow labelled ‘light, 3 × 10⁸ m s⁻¹, arrives almost at once’ and a wavy arrow labelled ‘sound, 330 m s⁻¹, arrives seconds later’. Mark the separation as ‘distance d’ and add a small stopwatch symbol beside the observer labelled ‘time counted from flash to thunder’.

NoteCuriosity Corner

Q4. During a storm you see the flash of lightning first and hear the thunder several seconds later, even though both begin at the same instant. What does that delay tell you about sound?

A. It tells you that sound has a definite and rather modest speed. Light covers a few kilometres in a tiny fraction of a second, so the flash may be treated as arriving instantly; the sound of the same stroke crawls along at about \(330\ \text{m s}^{-1}\) and needs roughly three seconds for every kilometre. The delay is not a property of the storm but of the air — it is the time the disturbance takes to be handed from particle to particle all the way to your ear.

Two More Everyday Examples

1. When the starter in an athletic event fires a gun, a spectator sitting at a distance sees the smoke instantly but hears the sound of the shot a little later, because the speed of light is very much greater than the speed of sound.
2. Spectators watching a cricket match hear the sound of the ball being struck a little after they see the batter actually make the stroke, for the same reason.

NoteExample 8.7

How Far Away Was The Lightning?

1. Given: the time interval between seeing the flash and hearing the thunder, \(t = 4\ \text{s}\); speed of sound in air, \(V = 330\ \text{m s}^{-1}\). To find: the distance \(d\) of the flash from the observer, in m.

2. The sound travels at a steady speed over this distance, so \(d = V \times t\). The light is taken to arrive at the instant the flash occurs, so the whole of the measured \(4\ \text{s}\) is treated as the travel time of the sound.

3. \(d = 330\ \text{m s}^{-1} \times 4\ \text{s} = 1320\ \text{m}\).

4. The flash occurred \(1320\ \text{m}\), or about \(1.3\ \text{km}\), from the observer. The check on the approximation: light would cross \(1320\ \text{m}\) in about \(0.0000044\ \text{s}\), which is far too small to matter beside \(4\ \text{s}\), so ignoring it was justified.

NoteTry Yourself

The next time a thunderstorm passes, watch it from indoors through a window and count the seconds between a flash and its thunder. Divide the count by three to get a rough distance in kilometres. Keep counting over several flashes and see whether the storm is coming closer or moving away.

NoteExtra Practice 8.9

Take the speed of sound in air as \(340\ \text{m s}^{-1}\) and the speed of light as \(3 \times 10^{8}\ \text{m s}^{-1}\).

(a) A spectator sees the smoke of a starter’s gun and hears the report \(1.5\ \text{s}\) later. How far is the spectator from the starter?

(b) Find the time the light took to cover that same distance.

(c) Use your answer to (b) to explain why the light’s travel time is ignored in part (a).

8.12 Experimental Determination Of The Speed Of Sound In Air

The fact that light travels in air about a million times faster than sound is not merely a curiosity. It can be put to work, because it allows the flash of a distant event to be treated as the exact instant at which the sound set out. That is the whole idea behind the following method of measuring the speed of sound in air.

The Method

Choose two places, A and B, at high altitudes facing each other, a known distance \(d\) apart — say about \(1\ \text{km}\) — in still air. Measure and note the distance \(d\). At each place there is an observer with a gun and a stop watch. The arrangement is shown in Fig. 8.12-1.

Diagram

Fig. 8.12-1 — Measuring The Speed Of Sound In Air Between Two Hilltops Two hilltops drawn facing each other across a valley, with a clear line of sight between them. An observer standing on the left hilltop labelled A, holding a gun with a flash drawn at its muzzle and a stop watch in the other hand. An observer standing on the right hilltop labelled B, also with a gun and a stop watch. A straight arrow from A to B labelled ‘flash seen at once’ and a wavy arrow below it from A to B labelled ‘sound arrives after time t₁’. Mark the separation between the two hilltops with a horizontal double-headed arrow labelled ‘distance d’.

First the observer at place A fires the gun. The observer at place B starts the stop watch immediately on seeing the flash of fire at A, and stops it when the sound of the fire is heard. The watch at B thus records the time interval \(t_1\) taken by the sound to travel from A to B.

Now the observer at place B fires the gun. The observer at place A starts the stop watch on seeing the flash of fire at B and stops it on hearing the sound. The watch at A thus records the time interval \(t_2\) taken by the sound to travel from B to A.

The average of the two time intervals is

\[t = \frac{t_1 + t_2}{2}\]

where \(t_1\) and \(t_2\) are the two measured intervals in s. This average \(t\) is taken as the time taken by sound to travel the distance \(d\) between the places A and B.

The speed of sound is then calculated from

\[V = \frac{\text{distance}}{\text{time}} = \frac{d}{t}\]

where \(d\) is the distance between the two places in m, \(t\) the average time in s, and \(V\) the speed of sound in m s⁻¹.

Flowchart

FC 8.3 — Steps In Determining The Speed Of Sound In Air Measure and note the distance d between the two places → A fires; B starts the watch at the flash and stops it at the sound, giving t₁ → B fires; A starts the watch at the flash and stops it at the sound, giving t₂ → take the average time t = (t₁ + t₂)/2 → calculate V = d/t

Why The Measurement Is Made Twice

The time interval is measured twice, once for sound travelling from A to B and once for sound travelling from B to A, in order to eliminate the error arising from the flow of wind. As Section 8.9.4 showed, a wind blowing from A to B would add its speed to that of the sound in one direction and subtract it in the other. Averaging the two intervals cancels that effect out.

Even so, the speed of sound determined in this way is still not very accurate. Two sources of error remain: the personal error of the two observers, who cannot start and stop a watch at the exact instants, and the variation in temperature and humidity of the air in between the places A and B, both of which alter the speed along the way.

NoteSide Note

Why hilltops facing each other? The two stations must be in clear view of one another so that each observer sees the flash without obstruction, and the ground between them must not block or reflect the sound. A clear line across a valley gives both conditions at once, together with a distance long enough for the time interval to be worth measuring.

NoteExample 8.8

Working Out The Speed From The Two Readings

1. Given: distance between the two places, \(d = 1020\ \text{m}\); time recorded at B when A fired, \(t_1 = 3.2\ \text{s}\); time recorded at A when B fired, \(t_2 = 2.8\ \text{s}\). To find: the speed of sound in air, in m s⁻¹.

2. The two readings differ because a wind was blowing along the line joining the places, so first take the average time, \(t = \dfrac{t_1 + t_2}{2}\), and then use \(V = \dfrac{d}{t}\).

3. \(t = \dfrac{3.2\ \text{s} + 2.8\ \text{s}}{2} = \dfrac{6.0\ \text{s}}{2} = 3.0\ \text{s}\), so \(V = \dfrac{1020\ \text{m}}{3.0\ \text{s}} = 340\ \text{m s}^{-1}\).

4. The speed of sound in the air is \(340\ \text{m s}^{-1}\). The check: the value is close to the accepted speed of sound in air at ordinary room temperature, and it lies between the two speeds the separate readings alone would have given, as an average must.

NoteExtra Practice 8.10

In an experiment of this kind the two stations are \(1360\ \text{m}\) apart. The time recorded in one direction is \(3.8\ \text{s}\) and in the other direction \(4.2\ \text{s}\).

(a) Find the speed of sound in air from these readings.

(b) State which reading was taken with the wind and give a reason.

(c) Suggest one change to the method that would reduce the personal error.

8.13 Infrasonic, Sonic And Ultrasonic Frequencies

Everything so far has been true of sound waves of any frequency. Whether you actually hear such a wave is another matter altogether, and it depends on the ear that is listening.

8.13.1 The Audible Range Of Frequency

The human ear is able to hear sound in a frequency range of about \(20\ \text{Hz}\) to \(20{,}000\ \text{Hz}\), that is, to \(20\ \text{kHz}\). This is called the audible range of frequency. We cannot hear sounds of frequencies less than \(20\ \text{Hz}\) or more than \(20\ \text{kHz}\).

The audible range varies from person to person, and it also varies with the age of the person. It decreases as a person grows older, since the ears lose their sensitivity, particularly to the high frequencies. Children can hear sounds of somewhat higher frequencies, say up to \(30\ \text{kHz}\), while an old person may hear only up to about \(12\ \text{kHz}\). Hence the audible range of frequency for an average person is taken to be from \(20\ \text{Hz}\) to \(20\ \text{kHz}\).

Even within the audible range the human ear is not equally sensitive to all frequencies. The human ear is most sensitive in the range \(2000\ \text{Hz}\) to \(3000\ \text{Hz}\), where it can hear even a very feeble sound.

Sounds outside the audible range are given their own names, according to which end of the range they fall beyond.

NoteDefinition

Sonic, Infrasonic And Ultrasonic Sound Sound of frequency in the range \(20\ \text{Hz}\) to \(20\ \text{kHz}\) is called sonic sound, or audible sound. Sound of frequency less than \(20\ \text{Hz}\) is known as infrasonic sound, or simply infrasonic. Sound of frequency greater than \(20\ \text{kHz}\) is known as ultrasound, or ultrasonic.

Note that infrasonic sound and ultrasound are perfectly ordinary sound waves. They obey \(V = f\lambda\), they need a medium, and they travel at the same speed as audible sound in the same medium. The only thing that sets them apart is that our ears do not respond to them. Fig. 8.13-1 sets the three ranges out on a single scale.

Diagram

Fig. 8.13-1 — The Frequency Scale Of Sound A single long horizontal axis labelled ‘frequency (Hz)’, marked with the values 1, 10, 20, 100, 1000, 20 000 and 100 000 spaced out along it, with the spacing increasing towards the right to fit the large range. Two vertical division lines drawn on the axis, one at 20 Hz and one at 20 000 Hz. Label the region to the left of 20 Hz ‘infrasonic’, the region between the two lines ‘sonic, or audible’, and the region to the right of 20 000 Hz ‘ultrasonic’. Beneath the audible band, mark a shaded strip from 2000 Hz to 3000 Hz labelled ‘ear most sensitive here’. Place small labelled markers beneath the axis showing the ranges for humans, dogs and bats.

NoteCompare Table

Table 8.4 — Infrasonic, Sonic And Ultrasonic Sound | Basis of difference | Infrasonic sound | Sonic sound | Ultrasonic sound | |—|—|—|—| | Frequency | Less than \(20\ \text{Hz}\) | From \(20\ \text{Hz}\) to \(20\ \text{kHz}\) | Greater than \(20\ \text{kHz}\) | | Heard by a human ear | No | Yes | No | | Wavelength in air | Longest of the three | Intermediate | Shortest of the three | | Produced by | Elephants and whales, among others | The human voice and most everyday sources | Bats and dolphins, among others |

The third row follows from \(V = f\lambda\) and needs no new physics. In a given medium the speed is fixed, so the lowest frequencies must have the longest wavelengths, and the highest frequencies the shortest.

8.13.2 Frequency Ranges Of Humans And Animals

Animals can produce and hear sounds of frequencies below \(20\ \text{Hz}\) as well as above \(20\ \text{kHz}\), and different animals have different ranges of audible frequency.

Elephants and whales can produce infrasonic sounds of frequencies less than \(20\ \text{Hz}\). Some fishes can hear sounds of frequencies in the range of \(1\ \text{Hz}\) to \(25\ \text{Hz}\). Some animals can produce ultrasonic sounds and communicate in them: a dog can hear sounds of frequencies up to nearly \(50\ \text{kHz}\), a bat up to about \(100\ \text{kHz}\), while dolphins can hear sounds of even higher frequencies, up to \(150\ \text{kHz}\). Table 8.5 collects these ranges together.

Table 8.5 — Frequency ranges for hearing by humans and some animals

Object	Frequency range of hearing
Bat	10 Hz – 100 kHz
Cat	80 Hz – 60 kHz
Dog	20 Hz – 50 kHz
Dolphin	200 Hz – 150 kHz
Grasshopper	90 Hz – 1.0 kHz
Humans	20 Hz – 20 kHz

Read the table by comparing the upper limits. The grasshopper’s range stops well inside ours; the dog’s, the cat’s, the bat’s and the dolphin’s all run far beyond it. A sound can therefore be deafening to one animal and completely silent to another standing beside it.

NoteCuriosity Corner

Q5. A bat flying in a dark room calls out constantly and never hits a wall, yet you hear almost nothing. What could be different about the sound a bat makes?

A. Its frequency. A bat’s calls run far above \(20\ \text{kHz}\) — its hearing reaches to about \(100\ \text{kHz}\) — so most of what it produces is ultrasound, entirely outside the human audible range. The calls are real sound waves travelling through the same air at the same speed as your own voice; your ear simply does not respond to them. Section 8.14 explains what the bat does with the sound that comes back.

NoteSide Note

Elephants call to one another at frequencies too low for us to hear. Because these infrasonic calls have very long wavelengths, they spread across several kilometres of grassland without dying away, and herds far out of sight of each other stay in touch. Researchers first identified these calls in the 1980s, after noticing a faint throbbing in the air near elephants.

NoteExtra Practice 8.11

Take the speed of sound in air as \(340\ \text{m s}^{-1}\).

(a) Find the wavelength of a sound of frequency \(17\ \text{Hz}\), and name the class of sound it belongs to.

(b) Find the wavelength of a sound of frequency \(34\ \text{kHz}\), and name its class.

(c) Using Table 8.5, state which of the two a cat could hear.

8.14 Ultrasound And Its Applications

We have learnt that sound of frequency above \(20{,}000\ \text{Hz}\) is called ultrasound. Being inaudible is not the interesting thing about it. What makes ultrasound useful is that it behaves in ways audible sound does not, and those differences have been turned into a remarkable range of tools.

Ultrasound can travel quite freely in solids and in liquids, but in gases its intensity falls. In a medium it travels with the same speed with which the audible sound travels — the speed of a wave is fixed by the medium, as Section 8.10.3 established, not by its frequency. In air, therefore, the speed of ultrasound is \(330\ \text{m s}^{-1}\).

8.14.1 Properties Of Ultrasound

Ultrasound has all the properties of ordinary sound. Because of its high frequency, it has in addition the following two properties, which audible sound does not possess.

1. The energy carried by ultrasound is very high.
2. Ultrasound can travel along a well-defined straight path. It does not bend appreciably at the edges of an obstacle, because of its small wavelength — that is, it has high directivity.

The second property follows from the first relation of this chapter. Since \(V = f\lambda\) and the speed in a given medium is fixed, a very high frequency must go with a very short wavelength. A wave bends round an obstacle only when its wavelength is comparable with the size of that obstacle, so the short waves of ultrasound sweep past in a narrow beam instead of spreading out. It is this pair of properties — a concentrated beam, carrying plenty of energy — that makes ultrasound so useful.

8.14.2 Applications Of Ultrasound

A few of its applications are given below.

1. Bats avoid obstacles in their path by producing and hearing ultrasound. A bat produces ultrasound, which returns after striking an obstacle in its way. By hearing the reflected sound, the bat judges the direction of the obstacle, and from the time interval between producing the ultrasound and receiving it back, it judges the distance of the obstacle. Fig. 8.14-1 shows the idea.
2. Drilling and cutting. Ultrasound is used for drilling holes, or for making cuts of desired shapes, in materials like glass.
3. Cleaning minute objects such as the parts of watches and electronic components. The objects are placed in a cleaning solution and ultrasonic waves are sent into the solution. This causes high-frequency vibrations in the solution, which shake the dirt out of corners a brush could never reach, and makes the cleaning efficient.
4. Detection of defects in metals. Ultrasound passes straight through the object if there is no defect, such as a crack or a cavity, in it. But if there is some defect, a part of the ultrasound gets reflected back from it, and the reflection is picked up by a detector. Fig. 8.14-2 shows both cases.
5. Imaging the human organs. Ultrasonography, or ultrasound scanning, is widely used to obtain images of a patient’s organs, such as the liver, the gall bladder and the uterus. It helps to detect stones and tumours in them. Echo cardiography is used to obtain an image of the heart.
6. Surgery. Ultrasound is used in surgery to remove a cataract, and in the kidneys to break small stones into fine grains, which the body can then pass out naturally. Both are done by trained doctors with equipment designed to focus the beam exactly where it is needed.
7. SONAR, the abbreviated form of sound navigation and ranging, uses ultrasound to detect objects under water and to find their distance.

Diagram

Fig. 8.14-1 — A Bat Judging An Obstacle By Ultrasound A bat in flight at the left, with a cone of short wavy lines spreading from its mouth towards a wall at the right, labelled ‘ultrasound sent out’. A second set of short wavy lines returning from the wall towards the bat’s ears, labelled ‘ultrasound reflected back’. Label the bat’s ears, the obstacle, and add a small clock symbol beside the bat labelled ‘time between sending and receiving gives the distance’.

Diagram

Fig. 8.14-2 — Detecting A Defect In A Metal Block Two panels side by side, each showing a rectangular metal block in section with a small ultrasound transmitter placed on its upper surface and a detector beside the transmitter. Left panel labelled ‘no defect’: a straight arrow runs from the transmitter down through the block to the lower face and is marked ‘ultrasound passes through’. Right panel labelled ‘defect present’: the block contains a small cavity drawn part way down; a straight arrow runs from the transmitter to the cavity, and a second arrow returns from the cavity up to the detector, marked ‘part of the ultrasound is reflected back’. Label the transmitter, the detector, the metal block and the cavity.

Diagram

Fig. 8.14-3 — SONAR Finding An Object Under Water A ship on the surface of the sea at the top, with a SONAR transmitter and receiver drawn beneath its hull. A beam of short wavy lines travelling downwards from the ship to a large object resting on the sea bed, labelled ‘ultrasound sent into the water’. A second beam of wavy lines returning from the object up to the receiver on the ship, labelled ‘reflected ultrasound received’. Label the ship, the transmitter, the receiver, the object and the sea bed.

NoteReal Incident / Discovery

How Echolocation Was Discovered That bats can fly in the dark without colliding with anything had been known since the 1790s, when the Italian scientist Lazzaro Spallanzani found that a blinded bat still flew perfectly well, while a bat whose ears were blocked blundered into things. The result made no sense at the time — the bats were plainly silent. The puzzle was settled only in 1938 at Harvard, when Donald Griffin and Robert Galambos brought bats to a laboratory that had just acquired a microphone able to pick up frequencies above human hearing. The room turned out to be full of sound. The bats were shouting all along, far above \(20\ \text{kHz}\), and listening for the echoes.

NoteSide Note

An ultrasound scan of a patient is a sound wave, not a ray. Nothing is being ionised, and no radiation dose is being given, which is why scanning is considered safe where an X-ray would need care. The image is built up from the echoes returning from the boundaries between one tissue and the next.

8.14.3 Ultrasonic Compared With Supersonic

These two words sound alike and are constantly confused, yet they do not even describe the same kind of quantity. The word ultrasonic is used for ultrasound, that is, for sound of frequency above \(20\ \text{kHz}\). The word supersonic is used for objects which travel with a speed greater than the speed of sound in air, that is, greater than \(330\ \text{m s}^{-1}\), or Mach 1 — fighter planes, for example, and the former Concorde jet airliners.

NoteCompare Table

Table 8.6 — Ultrasonic Compared With Supersonic | Basis of difference | Ultrasonic | Supersonic | |—|—|—| | What the word describes | A sound wave. | A moving object. | | Quantity referred to | Frequency of the sound. | Speed of the object. | | Meaning | Frequency greater than \(20\ \text{kHz}\). | Speed greater than the speed of sound in air, about \(330\ \text{m s}^{-1}\), or Mach 1. | | Example | The calls of a bat; the beam used in a scan. | A fighter plane flying faster than sound. |

8.14.4 Infrasonic Compared With Subsonic

The same trap is set by the pair infrasonic and subsonic. Infrasonic refers to sound waves with a frequency below \(20\ \text{Hz}\), which are too low to be heard by humans; these are produced by animals such as elephants and whales. Subsonic, on the other hand, refers to the speed of an object that is less than the speed of sound in air, about \(340\ \text{m s}^{-1}\). Thus infrasonic is related to the frequency of a sound, while subsonic is related to a speed of motion less than the speed of sound.

NoteCompare Table

Table 8.7 — Infrasonic Compared With Subsonic | Basis of difference | Infrasonic | Subsonic | |—|—|—| | What the word describes | A sound wave. | A moving object. | | Quantity referred to | Frequency of the sound. | Speed of the object. | | Meaning | Frequency less than \(20\ \text{Hz}\). | Speed less than the speed of sound in air, about \(340\ \text{m s}^{-1}\). | | Example | The calls of an elephant or a whale. | An ordinary passenger aircraft in level flight. |

A single sentence holds both comparisons together: ultrasonic and infrasonic are about how rapidly something vibrates, while supersonic and subsonic are about how fast something moves.

NoteExtra Practice 8.12

(a) State which two properties of ultrasound allow a defect deep inside a metal casting to be located from the outside, and say what each contributes.

(b) A jet flies at \(400\ \text{m s}^{-1}\) while emitting a whine of frequency \(8\ \text{kHz}\). Name the jet’s speed and the sound’s frequency using the correct terms from this section.

(c) Explain why ultrasound, and not audible sound, is used for cleaning the parts of a watch.

8.15 Summary

1. Energy can be transferred from one place to another in two ways: by the transfer of matter, and by wave motion.
2. Waves that require a material medium for their transmission are called mechanical waves; waves that need no material medium and can travel through vacuum as well are called electromagnetic waves.
3. In wave motion the disturbance moves ahead with a definite speed, keeping its form, and energy travels in the direction of the wave; the particles of the medium only vibrate about their mean positions and are not carried forward.
4. The speed of propagation of a disturbance in a medium depends on the elasticity and the density of that medium.
5. The amplitude of a wave is the maximum displacement of a particle of the medium on either side of its mean position. It is denoted by \(a\) and measured in metre (m).
6. The time period is the time taken by a particle of the medium to complete one vibration. It is denoted by \(T\) and measured in second (s).
7. The frequency is the number of vibrations made by a particle in one second, equal to the number of waves passing a point in one second. It is denoted by \(f\) and measured in hertz (Hz). The frequency of a wave equals the frequency of its source, and does not depend on the amplitude or on the medium.
8. Frequency and time period are related by \[f = \frac{1}{T}\] where \(f\) is the frequency in Hz and \(T\) the time period in s.
9. The wavelength is the distance travelled by the wave in one time period. It is denoted by \(\lambda\) and measured in metre (m). It equals the distance between two consecutive crests or troughs in a transverse wave, and between two consecutive compressions or rarefactions in a longitudinal wave.
10. The wave velocity is the distance travelled by a wave in one second, and is the speed at which energy is carried by the wave. It is denoted by \(V\) and measured in m s⁻¹. It is constant for a given medium and changes when the medium changes.
11. The three are related by \[V = f\lambda\] where \(V\) is the wave velocity in m s⁻¹, \(f\) the frequency in Hz and \(\lambda\) the wavelength in m; equivalently \(VT = \lambda\), with \(T\) in s.
12. In a displacement–time graph of one particle, the amplitude is the greatest height above the axis and the repeat along the horizontal axis is the time period. In a displacement–distance graph, a snapshot of the whole wave, the repeat along the horizontal axis is the wavelength.
13. Mechanical waves are of two types: transverse waves, in which the particles vibrate perpendicular to the direction of propagation, and longitudinal waves, in which they vibrate along the direction of propagation.
14. A transverse wave is made of crests and troughs, and can be produced only in solids and at the surface of liquids. A longitudinal wave is made of compressions and rarefactions, and can be produced in solids, liquids and gases alike.
15. At a compression the density and pressure of the medium are maximum; at a rarefaction they are minimum. Sound travels in air in the form of longitudinal waves.
16. Sound is a form of energy that produces the sensation of hearing in our ears. Sound is produced when a body vibrates, and ceases when the body stops vibrating; a vibrating body is a source of sound.
17. A material medium is necessary for the propagation of sound from one place to another. Sound cannot travel through vacuum, as the bell jar experiment shows; light, being electromagnetic, can.
18. A medium fit to carry sound must be elastic, must have inertia, and should be frictionless.
19. The speed of sound in a medium is given by \[V = \sqrt{\frac{E}{\rho}}\] where \(V\) is the speed in m s⁻¹, \(E\) the modulus of elasticity in N m⁻², and \(\rho\) the density in kg m⁻³.
20. Newton assumed the propagation of sound in a gas to be isothermal, giving \(E = P\) and \(V = \sqrt{P/\rho}\), which yields \(279.5\ \text{m s}^{-1}\) for air at N.T.P. — well below the measured value of nearly \(330\ \text{m s}^{-1}\).
21. Laplace showed the propagation to be adiabatic, giving \(E = \gamma P\) and \[V = \sqrt{\frac{\gamma P}{\rho}}\] where \(\gamma = C_p/C_v\) has no unit and equals \(1.4\) for air, \(P\) is in N m⁻² and \(\rho\) in kg m⁻³. This gives nearly \(330.7\ \text{m s}^{-1}\) for air at N.T.P., agreeing with experiment.
22. The speed of sound is greatest in solids, less in liquids and least in gases: nearly \(5100\ \text{m s}^{-1}\) in steel, \(1450\ \text{m s}^{-1}\) in water and \(330\ \text{m s}^{-1}\) in air at \(0^\circ\text{C}\).
23. The speed of sound in a gas is affected by density (\(V \propto 1/\sqrt{\rho}\)), by temperature (\(V \propto \sqrt{T}\), with \(T\) in kelvin), by humidity, and by the direction of the wind.
24. For air over a moderate range, \[V_t = V_0 + 0.61\,t\] where \(V_t\) is the speed at \(t\ ^\circ\text{C}\) in m s⁻¹, \(V_0\) the speed at \(0^\circ\text{C}\) in m s⁻¹, and \(t\) the temperature in °C.
25. Sound travels faster in humid air than in dry air, because water vapour is lighter than dry air and lowers its density. With a wind of speed \(W\) in m s⁻¹, the speed of sound becomes \(V + W\) along the wind and \(V - W\) against it.
26. The speed of sound in a gas is not affected by pressure, since the ratio \(P/\rho\) stays constant, nor by the amplitude, nor by the wavelength or frequency of the wave.
27. Light travels in air at \(3 \times 10^{8}\ \text{m s}^{-1}\), about a million times faster than sound; light slows down in an optically denser medium, while sound speeds up in solids. Light is a transverse electromagnetic wave, sound in air a longitudinal mechanical wave.
28. Because of this large difference, an event is always seen before it is heard — lightning before thunder, the smoke of a starter’s gun before its report, a cricket stroke before its sound.
29. The speed of sound in air can be measured by firing a gun at each of two places a known distance \(d\) apart, timing from flash to sound in each direction, and using \[V = \frac{d}{t}, \qquad t = \frac{t_1 + t_2}{2}\] where \(d\) is in m and \(t_1\), \(t_2\) and \(t\) are in s. Timing in both directions cancels the effect of wind; personal error and changes of temperature and humidity remain.
30. Sound of frequency between \(20\ \text{Hz}\) and \(20\ \text{kHz}\) is sonic or audible; below \(20\ \text{Hz}\) it is infrasonic; above \(20\ \text{kHz}\) it is ultrasound. The human ear is most sensitive between \(2000\ \text{Hz}\) and \(3000\ \text{Hz}\), and its upper limit falls with age.
31. Ultrasound travels at the same speed as audible sound in the same medium, \(330\ \text{m s}^{-1}\) in air. It carries very high energy and has high directivity, travelling in a straight beam because of its small wavelength.
32. Ultrasound is used by bats to judge obstacles, and by us for drilling and cutting, cleaning small components, detecting defects in metals, ultrasonography and echo cardiography, surgery for cataract and kidney stones, and SONAR.
33. Ultrasonic and infrasonic describe the frequency of a sound; supersonic and subsonic describe the speed of a moving object relative to the speed of sound in air.

Flowchart

FC 8.4 — The Chapter In One Chain A body vibrates → the particles of the medium next to it are set vibrating about their mean positions → the vibration is handed on from particle to particle as a longitudinal wave of compressions and rarefactions → the wave travels at a speed fixed by the elasticity and density of the medium → the wave reaches the ear and sets the eardrum vibrating → if the frequency lies between 20 Hz and 20 kHz, sound is heard; if it lies outside, the wave arrives unheard as infrasonic or ultrasound

8.16 Key Terms

1. Amplitude — the maximum displacement of a particle of the medium on either side of its mean position; SI unit: metre (m).
2. Audible range of frequency — the band of frequencies an ear can respond to, about 20 Hz to 20 kHz for an average person; SI unit: hertz (Hz).
3. Compression — a region of a longitudinal wave in which the particles of the medium are crowded closer than normal, so that the density and pressure there are maximum.
4. Crest — the position of maximum upward displacement of the medium in a transverse wave.
5. Density — the mass of a material contained in unit volume; SI unit: kilogram per cubic metre (kg m⁻³).
6. Displacement–distance graph — a snapshot of a wave at one instant, showing the displacements of all the particles against distance; the repeat along the distance axis is the wavelength.
7. Displacement–time graph — a graph showing how the displacement of one particle of the medium changes with time; the repeat along the time axis is the time period.
8. Elasticity — the property by which a material returns to its original shape and position after being displaced.
9. Electromagnetic wave — a wave that needs no material medium and can travel through vacuum, such as light, X-rays, gamma rays and radio waves.
10. Frequency — the number of vibrations made by a particle of the medium in one second, equal to the number of waves passing a point per second; SI unit: hertz (Hz).
11. Infrasonic sound — sound of frequency less than 20 Hz, below the audible range; SI unit of frequency: hertz (Hz).
12. Longitudinal wave — a wave in which the particles of the medium vibrate about their mean positions along the direction in which the wave travels.
13. Mean position — the resting position a particle of the medium occupies before the wave arrives, and about which it vibrates.
14. Mechanical wave — a wave that can travel only through a material medium, such as a wave on a string, a ripple on water, or a sound wave.
15. Medium — the material substance through which a mechanical wave travels.
16. Pulse — a single disturbance travelling through a medium, produced by one sudden displacement of the source.
17. Rarefaction — a region of a longitudinal wave in which the particles of the medium are spread further apart than normal, so that the density and pressure there are minimum.
18. SONAR — sound navigation and ranging; a system that uses ultrasound to detect objects under water and find their distance.
19. Sonic sound — sound of frequency between 20 Hz and 20 kHz, which an average human ear can hear; also called audible sound.
20. Sound — a form of energy that produces the sensation of hearing in our ears.
21. Time period — the time taken by a particle of the medium to complete one vibration; SI unit: second (s).
22. Transverse wave — a wave in which the particles of the medium vibrate about their mean positions at right angles to the direction in which the wave travels.
23. Trough — the position of maximum downward displacement of the medium in a transverse wave.
24. Ultrasound — sound of frequency greater than 20 kHz, above the audible range; also called ultrasonic sound.
25. Wave motion — the transfer of energy from one place to another through the vibration of the particles of a medium about their mean positions, without the particles themselves being carried forward.
26. Wave velocity — the distance travelled by a wave in one second, that is, the speed with which energy is carried by the wave; SI unit: metre per second (m s⁻¹).
27. Wavelength — the distance travelled by a wave in one time period, equal to the distance between two consecutive crests, troughs, compressions or rarefactions; SI unit: metre (m).

8.17 Points to Ponder

1. A wave carries energy across a room, yet every particle of the air ends up exactly where it began. If nothing has been moved from one place to another, in what sense has anything actually travelled?

2. The speed of sound in a gas depends on its elasticity and its density, and both of these change when the gas is compressed. Is it a coincidence that they change in just such a way that the speed stays the same, or is there a reason behind it?

3. Newton’s formula gave an answer about one-sixth too small, and the error stood for well over a century before Laplace explained it. What does this episode suggest about what a physicist should do when a careful calculation disagrees with a careful measurement?

4. Sound needs a medium and light does not. If you had to design a warning system for a place where people cannot rely on hearing — a very noisy factory floor, for instance — which properties of sound and of light would decide your choice?

5. A bat’s calls and an elephant’s calls are both inaudible to us, but for opposite reasons. Which of the two do you expect to travel further across open country before fading out, and what property of the waves would you appeal to in arguing your case?

6. Our ears are most sensitive between about 2000 Hz and 3000 Hz, and lose the high frequencies first as we age. What might follow from this for the way an alarm, a doorbell or a public announcement system ought to be designed?

8.18 Objective Questions

1. A wave that cannot travel through vacuum is:
   1. a radio wave
   2. a light wave
   3. a sound wave
   4. an X-ray
2. In a longitudinal wave, the particles of the medium vibrate:
   1. at right angles to the direction of propagation
   2. along the direction of propagation
   3. in circles about their mean positions
   4. not at all
3. The speed of sound is greatest in:
   1. air
   2. water
   3. hydrogen
   4. granite
4. The speed of sound in a gas is not affected by a change in its:
   1. temperature
   2. density
   3. pressure
   4. humidity
5. The relation between the wave velocity \(V\), the frequency \(f\) and the wavelength \(\lambda\) is:
   1. \(V = f/\lambda\)
   2. \(V = \lambda/f\)
   3. \(V = f\lambda\)
   4. \(V = f\lambda^2\)
6. A sound of frequency \(12\ \text{Hz}\) is:
   1. sonic
   2. infrasonic
   3. ultrasonic
   4. supersonic
7. Laplace corrected Newton’s formula by assuming the propagation of sound in a gas to be:
   1. an isothermal change
   2. an adiabatic change
   3. a change at constant volume
   4. a change at constant density

For each of the following, choose from the codes below. (1 mark each) a. Both A and R are true and R is the correct explanation of A b. Both A and R are true but R is not the correct explanation of A c. A is false but R is true d. A is true but R is false

8. Assertion (A): A sound produced inside a sealed jar from which all the air has been removed cannot be heard outside. Reason (R): Sound requires a material medium for its propagation.

9. Assertion (A): The speed of sound in air increases when the air becomes more humid. Reason (R): Water vapour is denser than dry air.

10. Assertion (A): A loud sound reaches a distant listener earlier than a faint one produced at the same instant and place. Reason (R): The speed of sound does not depend on the amplitude of the wave.

11. Assertion (A): Ultrasound travels in a well-defined straight beam. Reason (R): The wavelength of ultrasound is very small.

12. Match each quantity in Column A with its correct SI unit in Column B.

Column A	Column B
a. Amplitude	i. hertz
b. Time period	ii. metre per second
c. Frequency	iii. metre
d. Wave velocity	iv. second

13. Fill in the blanks. (1 mark each)
    1. A region of a longitudinal wave where the density and pressure of the medium are maximum is called a ………… .
    2. The speed of sound in air increases by about ………… m s⁻¹ for each degree celsius rise in temperature.
    3. The human ear is most sensitive to frequencies in the range ………… to ………… .
14. Write ‘true’ or ‘false’ and correct the false statement. (1 mark each)
    1. Transverse waves can be produced inside a gas.
    2. The frequency of a wave changes when the wave passes from air into water.
15. Name the following. (1 mark each)
    1. The kind of wave in which sound travels through air.
    2. The scientist who corrected the formula for the speed of sound in a gas.
    3. The system that uses ultrasound to find the distance of objects under water.
    4. An animal that produces sound of frequency below 20 Hz.
16. Give one word or phrase for each of the following. (1 mark each)
    1. The distance travelled by a wave in one time period.
    2. The maximum displacement of a particle from its mean position.
    3. A single disturbance travelling through a medium.
17. State the SI unit of the following. (1 mark each)
    1. wavelength
    2. frequency
    3. wave velocity
    4. density.
18. Pick the odd one out and give a reason for your choice. (2 marks each)
    1. Radio wave, gamma ray, sound wave, X-ray.
    2. Steel, water, glass, granite.

8.19 Check Your Concepts

1. Define the amplitude of a wave and state its SI unit. (2 marks)

2. State the relation between the frequency and the time period of a wave, naming each symbol with its unit. (2 marks)

3. Name the two properties of a medium on which the speed of sound in it depends. (2 marks)

4. Give a reason: a person standing on the Moon cannot hear the sound of a hammer struck beside them. (2 marks)

5. State the three properties that a medium must possess if it is to carry a sound wave. (3 marks)

6. Differentiate between a mechanical wave and an electromagnetic wave, stating the basis of difference. (3 marks)

7. Explain, with an example, what is meant by saying that sound is a form of energy. (3 marks)

8. Why does a person living near a railway track press an ear to the rail to know whether a train is approaching? (3 marks)

9. Differentiate between a transverse wave and a longitudinal wave, stating four bases of difference. (4 marks)

10. Explain why the speed of sound in a gas is independent of the pressure of the gas. (4 marks)

11. Ultrasonic and supersonic are often confused with each other. Distinguish between them and give one example of each. (4 marks)

12. What would happen to the wavelength of a sound wave if the source vibrated twice as rapidly while the medium stayed the same? Justify your answer using a relation. (4 marks)

13. Describe the bell jar experiment and state clearly what it establishes. (5 marks)

14. Newton’s formula for the speed of sound in a gas gave a value well below the measured one. State the assumption Newton made, state the correction Laplace applied, and write the corrected relation with each symbol named. (5 marks)

15. How do each of the following affect the speed of sound in air, if at all: (i) a rise in temperature, (ii) an increase in humidity, (iii) an increase in the frequency of the sound, (iv) a wind blowing from the listener towards the source? Give a reason in each case. (5 marks)

8.20 Diagrams, Graphs & Constructions

1. Draw a displacement–time graph of a wave, of the kind shown in Fig. 8.2-1, and mark on it the amplitude and the time period. (3 marks)

2. Draw a displacement–distance graph of a wave, of the kind shown in Fig. 8.2-2, and mark on it the amplitude by the letter \(a\) and the wavelength by the letter \(\lambda\). State one way in which this graph differs from the graph in the previous question. (3 marks)

3. Copy the sketch of a transverse wave given in Fig. 8.4-1 and label on it a crest, a trough, one wavelength, and the direction in which the particles of the medium vibrate. (3 marks)

4. Draw a labelled diagram of the apparatus used to demonstrate that a material medium is necessary for the propagation of sound, as in Fig. 8.7-1, and label the bell jar, the electric bell and the connection to the vacuum pump. (4 marks)

5. Fig. 8.9-1 shows the speed of sound in air plotted against temperature. Using a copy of that graph, (a) read the speed at \(0^\circ\text{C}\), (b) find the gradient of the line, and (c) state in words what the gradient represents. (4 marks)

6. Sketch the shape you would expect for a graph of the speed of sound in air, plotted on the vertical axis, against the frequency of the sound on the horizontal axis. Give a reason for the shape you have drawn. (3 marks)

7. Draw a diagram to show a bat locating an obstacle by ultrasound, as in Fig. 8.14-1, marking the outgoing and the returning sound. State the two properties of ultrasound that make this possible. (4 marks)

8.21 Practice With Numericals

Take the speed of sound in air at \(0^\circ\text{C}\) as \(330\ \text{m s}^{-1}\) unless a different value is given in the question.

1. The time period of a vibrating particle of a medium is \(0.004\ \text{s}\). Find its frequency. (2 marks)

2. The heart of a person beats 120 times in a minute. Find (a) the frequency and (b) the time period of the beating. (2 marks)

3. A sound wave of wavelength \(1.5\ \text{m}\) travels in air at \(330\ \text{m s}^{-1}\). Find its frequency. (2 marks)

4. A source of waves produces 50 crests and 50 troughs in \(0.5\ \text{s}\). Find the frequency of the wave. (2 marks)

5. A longitudinal wave of frequency \(25\ \text{Hz}\) travels at \(0.4\ \text{m s}^{-1}\). Find the separation between two consecutive compressions. (2 marks)

6. A wave has a time period of \(0.05\ \text{s}\) and a wavelength of \(20\ \text{m}\). Find (a) its frequency and (b) its velocity. (3 marks)

7. The thunder of a lightning stroke is heard \(6\ \text{s}\) after the flash is seen. Taking the speed of sound in air as \(330\ \text{m s}^{-1}\), find the distance of the stroke from the observer. (3 marks)

8. A gun is fired at a place \(1700\ \text{m}\) from an observer. Taking the speed of sound in air as \(340\ \text{m s}^{-1}\), find the time after which the observer hears the report. (3 marks)

9. An ultrasound of frequency \(50\ \text{kHz}\) travels in air at \(330\ \text{m s}^{-1}\). Find its wavelength in millimetre. (3 marks)

10. Sound takes a certain time to travel \(7250\ \text{m}\) in water, in which its speed is \(1450\ \text{m s}^{-1}\). Find (a) that time and (b) the distance sound would travel in air in the same time, taking the speed of sound in air as \(330\ \text{m s}^{-1}\). (4 marks)

11. A sound is produced at one end of an iron rail \(2640\ \text{m}\) long, and is heard twice at the other end. Taking the speed of sound in iron as \(5280\ \text{m s}^{-1}\) and in air as \(330\ \text{m s}^{-1}\), find the time interval between the two sounds. (4 marks)

12. On a day when the air temperature is \(20\ ^\circ\text{C}\), a wind of \(8\ \text{m s}^{-1}\) blows from a siren towards a listener. Find (a) the speed of sound in the still air at that temperature and (b) the speed with which the sound reaches the listener. (4 marks)