Chapter 7: Sound
In April 1912, a passenger liner struck an iceberg in the North Atlantic and sank. Ice floating in the sea keeps most of its bulk under water, and a lookout on deck can see almost nothing of it at night or in fog. Engineers began asking whether a ship could be given a sense that did not depend on light at all.
Reginald Fessenden, a Canadian-born inventor working with an American submarine signalling company, thought sound could do the job. He built an electrically driven oscillator that could push strong, low pulses of sound into the water and then listen for whatever came back.
In the spring of 1914, his device was tested from a revenue cutter working off the Grand Banks. The oscillator sent out a pulse. A moment later a returning pulse was picked up — sound that had travelled out, struck an iceberg nearly two miles away, and come back. Turned downwards, the same device brought back an echo from the sea floor.
Nothing had been seen. Nothing had been touched. A sound had gone out into the dark water and returned carrying information about where an object was. That single idea — sending sound out and timing what comes back — runs through this whole chapter.
Diagram
Fig. 7.0-1 — Locating An Iceberg By An Echo A ship on the sea surface at the left, with a large iceberg at the right showing a small tip above water and a much larger mass below the surface. Draw an outgoing sound pulse as a series of arcs travelling from the ship’s hull towards the submerged part of the iceberg, and a returning set of arcs travelling back from the iceberg to the ship; label the outgoing pulse, the reflected pulse, the distance d between ship and iceberg, and the sea floor at the bottom.
7.1 Sound Waves
7.1.1 How Sound Is Produced And The Audible Range
In Class IX you learnt that sound is produced when a body vibrates, and that it reaches us through the vibrations of the particles of the surrounding medium. A vibrating body sets the particles around it shaking, and that disturbance travels outwards as a wave. Sound therefore needs a material medium for its propagation; it cannot travel where there is no matter at all.
The speed with which the disturbance travels depends on the medium — on its elasticity and on its density. For a medium of elasticity \(E\) and density \(d\), the speed is
\[V = \\sqrt{\\frac{E}{d}}\]
where \(E\) is the elasticity of the medium in \(\\text{N m}^{-2}\), \(d\) its density in \(\\text{kg m}^{-3}\), and \(V\) the speed of the wave in \(\\text{m s}^{-1}\). You are not asked to use this relation in calculations; it is written here only to show that the speed belongs to the medium, not to the source.
When these vibrations reach the ear drum, the ear drum is set vibrating and sound is heard. But the ear responds only to a limited band of frequencies, roughly from \(20\\ \\text{Hz}\) to \(20{,}000\\ \\text{Hz}\). This band is called the audible range of frequency.
The audible range narrows as a person grows older, because the sensitivity of the ears falls at both the low and the high end. Sound of frequency above \(20{,}000\\ \\text{Hz}\) is called ultrasonic, and sound of frequency below \(20\\ \\text{Hz}\) is called infrasonic. Neither can be heard by human beings, although both travel in a medium with the same speed as audible sound.
7.1.2 Quantities That Describe A Wave
As a sound wave passes, each particle of the medium vibrates to and fro about a fixed point called its mean position. The amplitude \(a\) of the wave is the maximum displacement of the particle from its mean position, on either side. Amplitude is a length, and is measured in metre (m).
The time period \(T\) of the wave is the time taken by a particle of the medium to complete one full vibration, measured in second (s). The frequency \(f\) is the number of vibrations made by the particle in one second. Since the particles simply pass on the shaking they receive, the frequency of a wave is the same as the frequency of the source producing it.
The wave itself also moves forward. The distance travelled by the wave in one time period of vibration is called the wavelength \(\\lambda\) of the wave, measured in metre (m). Amplitude and wavelength are both shown in Fig. 7.1-1.
Diagram
Fig. 7.1-1 — Amplitude And Wavelength Of A Wave A smooth sine curve drawn on axes; horizontal axis labelled distance (in m), vertical axis labelled displacement (in m), with the horizontal axis drawn through the middle of the curve as the mean position. Mark a double-headed vertical arrow from the mean position to a peak, labelled amplitude a. Mark a double-headed horizontal arrow from one peak to the next peak, labelled wavelength λ. Label the mean position line.
Finally, the distance travelled by the wave in one second is called the wave velocity \(V\), measured in \(\\text{m s}^{-1}\). Notice that the wave velocity describes how fast the disturbance moves forward, while the particles themselves only vibrate about their own mean positions and do not travel with the wave.
7.1.3 The Wave Equation And The Time Period
These quantities are not independent of one another. In one time period the wave advances by exactly one wavelength, and the number of time periods packed into one second is the frequency. So the distance covered in one second is the wavelength multiplied by the frequency:
\[V = f\\lambda\]
where \(V\) is the wave velocity in \(\\text{m s}^{-1}\), \(f\) the frequency in Hz, and \(\\lambda\) the wavelength in m.
Since the time period is the time for one vibration, and the frequency is the number of vibrations in one second, the two are simply reciprocals of each other:
\[f = \\frac{1}{T}\]
where \(f\) is the frequency in Hz and \(T\) the time period in s. A wave that completes one vibration in \(0.01\\ \\text{s}\) therefore has a frequency of \(100\\ \\text{Hz}\).
Which Property Belongs To What
It is worth separating clearly what fixes each quantity. The frequency (and hence the time period) is decided by the vibrating source producing the sound. The velocity of the wave is decided by the medium through which it travels. The wavelength is then whatever the wave equation demands of the two, so it depends on the medium as well.
7.1.4 Mechanical Waves: Longitudinal And Transverse
A sound wave carries energy from one place to another through the vibrations of the medium particles about their mean positions. As a particle swings out and comes back, its kinetic energy changes into potential energy and back again. Because the wave needs a material medium and depends on its elastic properties, sound waves are called elastic or mechanical waves.
Mechanical waves are of two kinds, and they differ in the direction in which the particles vibrate.
If the medium particles vibrate along the direction in which the wave travels, the particles crowd together in some places and spread apart in others. The crowded regions are called compressions and the spread-out regions are called rarefactions, and the wave is called a longitudinal wave. Sound in air, in a solid and inside a liquid travels as a longitudinal wave. Longitudinal waves can travel in solids, liquids and gases alike. The pattern is shown in Fig. 7.1-2.
Diagram
Fig. 7.1-2 — A Longitudinal Wave A long horizontal row of dots representing medium particles, with the dots bunched closely together in three regions and spread widely apart in the regions between them. Label two of the bunched regions as compressions and two of the spread regions as rarefactions. Draw a short double-headed arrow above one particle, parallel to the row, labelled direction of vibration of particle, and a long arrow at the right, along the row, labelled direction of propagation of wave.
If instead the medium particles vibrate normal to the direction in which the wave travels, the medium is thrown into humps and hollows. The humps are called crests and the hollows troughs, and the wave is called a transverse wave. Waves on a stretched string, and waves on the surface of a liquid, are transverse. Fig. 7.1-3 shows the arrangement.
Diagram
Fig. 7.1-3 — A Transverse Wave A wavy curve drawn along a horizontal stretched string held between two fixed supports. Label two upward humps as crests and two downward hollows as troughs. Draw a short double-headed vertical arrow at one point on the curve labelled direction of vibration of particle, and a long horizontal arrow at the right labelled direction of propagation of wave.
Transverse waves need a medium that resists being twisted out of shape — that is, a medium with rigidity. Gases and the interiors of liquids have no rigidity, so transverse waves are formed only in solids and on the surface of liquids.
7.1.5 Factors Affecting The Speed Of Sound
A longitudinal wave travels through a gas so quickly that the compressed regions have no time to share heat with their surroundings. The gas is compressed and expanded without any change in the heat content of the medium. Under this condition, the speed \(V\) of a longitudinal wave — that is, of sound — in a gas of density \(d\) at pressure \(P\) is given by
\[V = \\sqrt{\\frac{\\gamma P}{d}}\]
where \(P\) is the pressure of the gas in \(\\text{N m}^{-2}\), \(d\) its density in \(\\text{kg m}^{-3}\), \(V\) the speed in \(\\text{m s}^{-1}\), and \(\\gamma\) is the ratio of the two specific heats of the gas, whose value is \(1.4\) for air.
This one relation explains three familiar facts. The density of a gas decreases as its temperature rises, so the speed of sound in a gas increases with temperature; for air, a rise of \(1^\\circ\\text{C}\) increases the speed by nearly \(0.61\\ \\text{m s}^{-1}\). The density also decreases when water vapour is present, so the speed of sound increases with humidity.
The third fact is less obvious. A change in pressure does not affect the speed of sound, because changing the pressure of a gas changes its density in the same proportion. The ratio \(P/d\) inside the square root is therefore unaltered, and so is \(V\).
The speed of a transverse wave in a stretched string is governed by a different pair of properties — how tightly it is stretched, and how heavy it is per unit of its length:
\[V = \\sqrt{\\frac{T}{m}}\]
where \(T\) is the tension in the string in newton (N), \(m\) the mass per unit length of the string in \(\\text{kg m}^{-1}\), and \(V\) the speed in \(\\text{m s}^{-1}\). Tightening a string makes the wave on it travel faster; using a thicker, heavier string makes it travel slower.
7.1.6 What Changes When A Wave Enters A New Medium
When a wave travelling in one medium passes into another medium — that is, when it is refracted — the medium changes but the source does not. The frequency of the wave therefore does not change. Its speed, its wavelength and its intensity all do change. The intensity changes because part of the wave is reflected at the boundary surface separating the two media, so less energy carries on forward.
The direction of travel of the wave also changes on entering the new medium. The one exception is normal incidence, that is, when the angle of incidence \(\\angle i = 0^\\circ\); a wave striking the boundary squarely carries straight on.
7.1.7 Sound Waves Compared With Electromagnetic Waves
Sound waves differ in almost every respect from electromagnetic waves, the family that includes gamma rays, X-rays, ultraviolet light, visible light, infrared rays, micro waves and radio waves. An electromagnetic wave is formed by the periodic vibrations of mutually perpendicular electric and magnetic fields, in a plane normal to the direction in which the wave travels. Electromagnetic waves are therefore transverse waves.
Unlike sound waves, they can also travel through vacuum. Their speed is different in different media and is maximum in vacuum (or air), where it equals \(3 \\times 10^{8}\\ \\text{m s}^{-1}\). Electromagnetic waves transfer energy in the form of photons. The main points of difference are collected in Table 7.1.
Reading A Wave From Its Graph
The vibration of a single particle of the medium can be plotted against time, giving a displacement–time graph. Such a graph is drawn in Fig. 7.1-4, and it carries more information than it first appears to.
Diagram
Fig. 7.1-4 — Displacement–Time Graph Of A Particle A smooth sine curve of two full cycles. Horizontal axis labelled time t (in s) with markings at 0, 0.01, 0.02, 0.03 and 0.04; vertical axis labelled displacement (in mm) with markings at +3, 0 and −3. The curve starts at the origin, rises to +3 mm, returns to zero at 0.01 s, falls to −3 mm, returns to zero at 0.02 s, and repeats the same shape up to 0.04 s.
7.2 Reflection Of Sound Waves
7.2.1 The Laws Of Reflection Applied To Sound
A sound wave travelling through air does not always continue forwards. When it strikes a hard surface, or the boundary of another medium, it turns back into the medium it came from. This turning back is what we mean by the reflection of a sound wave, and it obeys the same two laws that you met for the reflection of light.
The path of the sound can be drawn as a ray, exactly as a ray of light is drawn, and the normal is the line drawn at right angles to the reflecting surface at the point where the sound strikes it. Fig. 7.2-1 shows the two laws together.
Diagram
Fig. 7.2-1 — Reflection Of Sound At A Wall A vertical wall drawn on the right with hatching behind it. An incident ray arrow travelling from the lower left and striking a point O on the wall, and a reflected ray arrow leaving O towards the upper left. A dashed normal drawn horizontally at O, perpendicular to the wall. Mark the angle between the incident ray and the normal as angle of incidence i, and the angle between the normal and the reflected ray as angle of reflection r, with i = r. Label the incident ray, the reflected ray, the normal and the point of incidence O.
7.2.2 Condition For Reflection And Its Everyday Uses
Light needs a smooth, shining surface such as a mirror if it is to be reflected as a clear beam. Sound is far less fussy. Sound waves are reflected from any surface, whether it is smooth or rough, polished or unpolished.
The one requirement is a matter of size. The size of the reflecting surface must be bigger than the wavelength of the sound wave falling on it. A surface smaller than the wavelength does not turn the sound back; the wave simply spreads around it and carries on.
This is a demanding condition, because sound wavelengths in air are large — often a metre or more — while light wavelengths are millionths of a metre. It is the reason a reflector of sound has to be a wall, a board or a hillside rather than a small plate.
The reflection of sound waves is put to use in several familiar devices. A megaphone, also called a speaking tube, has a widening conical mouth whose walls reflect the speaker’s sound repeatedly forward, so that it is sent out in one direction as a beam instead of spreading in all directions. A sound board placed behind and above a speaker reflects sound towards the audience. An ear trumpet works the other way round, collecting sound over a wide mouth and reflecting it down a narrowing tube into the ear of a person who is hard of hearing.
7.3 Echo
7.3.1 Production Of An Echo
Stand at some distance from a wall or a hillside and produce a single sharp sound — a clap or a shout. You hear two distinct sounds. The first is the original, or direct, sound, which reaches your ears almost instantaneously because it travels only the short distance from your hands to your ears.
The second arrives a little later. It is the part of the sound that travelled all the way to the wall, was reflected there, and travelled back to you. This second sound, heard after reflection, is called an echo. Its path is drawn in Fig. 7.3-1.
Diagram
Fig. 7.3-1 — The Path Of Sound That Produces An Echo A person standing at the left facing a tall vertical wall at the right, with hatching behind the wall. A double-headed horizontal arrow between the person and the wall labelled distance d. Draw an outgoing arrow from the person’s mouth to the wall labelled original sound, and a returning arrow from the wall back to the person’s ear labelled reflected sound (echo). Below the figure, label the total path as 2d.
The last few words of that definition matter. If the reflected sound arrives while the original is still being heard, the two merge and nothing separate is noticed; that is not an echo. Only a reflected sound heard after the original sound has ceased, and so heard distinctly separate from it, is called an echo.
7.3.2 Condition For Hearing An Echo
Why should the reflected sound have to arrive late? The answer lies in the ear rather than in the wall. The sensation of a sound persists in our ears for about \(0.1\\ \\text{s}\) after the exciting stimulus has ceased to act. A sound arriving within this interval falls on ears that are still occupied with the first one, and the two are not told apart.
So for the echo to be heard distinctly, it must reach the ears at least \(0.1\\ \\text{s}\) after the original sound is heard. Since the reflected sound can only be delayed by making it travel further, this becomes a condition on distance: the person producing the sound must be far enough from the rigid obstacle, or reflector, to allow the reflected sound that extra tenth of a second.
Flowchart
FC 7.1 — How An Echo Reaches The Listener Listener produces a sharp sound → sound travels the distance d to the obstacle → sound is reflected at the obstacle → sound travels the distance d back → total path 2d covered in time t = 2d/V → echo heard distinctly if t is at least 0.1 s
7.3.3 Minimum Distance Of The Reflector
Let \(d\) be the distance of the observer from the obstacle and \(V\) the speed of sound in the medium. The sound must travel out to the obstacle and then come back, so the total distance travelled is \(2d\). The time taken to hear the echo is therefore
\[t = \\frac{\\text{total distance travelled}}{\\text{speed of sound}} = \\frac{2d}{V}\]
Rearranging this to make the distance the subject gives
\[d = \\frac{Vt}{2}\]
where \(d\) is the distance of the reflector from the listener in m, \(V\) the speed of sound in the medium in \(\\text{m s}^{-1}\), and \(t\) the time interval between producing the sound and hearing the echo, in s.
Now put in the smallest interval that still lets us hear the echo separately, \(t = 0.1\\ \\text{s}\), with \(V = 340\\ \\text{m s}^{-1}\) for air at ordinary temperature:
\[d = \\frac{340\\ \\text{m s}^{-1} \\times 0.1\\ \\text{s}}{2} = 17\\ \\text{m}\]
To hear an echo distinctly, the reflecting surface in air should therefore be at a minimum distance of \(17\\ \\text{m}\) from the listener. If the distance is less than \(17\\ \\text{m}\), the reflected sound reaches the ears before the original sound dies out, the two get mixed, and no echo is heard.
7.3.4 Reverberation
Sound does not always come back once and stop. If there are repeated reflections at the reflecting surfaces, the sound gets prolonged, and this effect is known as reverberation. Each reflection returns a little of the sound to the listener, so the sound seems to hang on and fade slowly instead of stopping cleanly.
Reverberation can easily be experienced in high tombs such as the Taj Mahal at Agra and the tomb at Sikandra, where a single sound rolls around the tall domed chamber for some seconds before dying away.
7.3.5 The Three Conditions For A Distinct Echo
Bringing together what this section has established, three conditions must be satisfied if an echo is to be heard distinctly.
- The minimum distance between the source of sound, or the observer, and the reflector must be \(17\\ \\text{m}\) in air. This minimum is different in different media, because it depends on the speed of sound in the medium. Inside sea water, where \(V = 1400\\ \\text{m s}^{-1}\), the same calculation gives \(d = \\dfrac{1400\\ \\text{m s}^{-1} \\times 0.1\\ \\text{s}}{2} = 70\\ \\text{m}\), so the obstacle in sea water should be at least \(70\\ \\text{m}\) from the listener.
- The size of the reflector must be large enough as compared to the wavelength of the sound wave, as you saw in Section 7.2.
- The intensity of the sound must be such that the reflected sound reaching the ear is still sufficiently loud to be audible.
7.4 Determination Of The Speed Of Sound By The Method Of Echo
The echo gives us more than a curiosity to listen to. Since the relation between distance, speed and time contains three quantities, measuring any two of them yields the third. In the last section the speed of sound was known and the distance was calculated. Here the distance is measured instead, and the speed of sound in air is found from it.
Setting Up The Measurement
Sound is produced from a place at a known distance \(d\) from a large reflecting surface, such as the wall of a tall building or a cliff face. The distance \(d\) is measured beforehand with a tape, and it is kept at least \(50\\ \\text{m}\), so that the returning sound is comfortably separate from the original and the interval is long enough to time. The arrangement is shown in Fig. 7.4-1.
Diagram
Fig. 7.4-1 — Finding The Speed Of Sound By The Echo Method An observer standing at the left holding a stop watch, and a tall reflecting wall at the right. A measured distance between them marked with a double-headed arrow labelled d (at least 50 m), with a measuring tape drawn along the ground. An arrow from the observer to the wall labelled sound produced, and an arrow from the wall back to the observer labelled echo received. Label the stop watch as least count 0.01 s.
The observer produces a sharp sound and at the same instant starts a stop watch, stopping it the moment the echo is heard. The watch used has a least count of \(0.01\\ \\text{s}\), because the whole interval being measured is itself only a fraction of a second, and a coarser watch would swamp the reading with error.
Flowchart
FC 7.2 — Steps In The Echo Method Measure the distance d from the source to the reflecting surface → produce a sharp sound and start the stop watch at the same instant → stop the watch on hearing the echo, giving the time interval t → calculate the speed from V = 2d/t → repeat the whole measurement several times → take the average of the values obtained
Calculating The Speed
The sound travels from the observer to the wall and back again before it is heard, so the path covered in the time interval \(t\) is \(2d\), not \(d\). The speed of sound is then
\[V = \\frac{\\text{total distance travelled}}{\\text{time interval}} = \\frac{2d}{t}\]
where \(V\) is the speed of sound in air in \(\\text{m s}^{-1}\), \(d\) the measured distance from the source to the reflecting surface in m, and \(t\) the time interval between producing the sound and hearing its echo, in s.
The experiment is repeated several times, and the average value of the speed of sound \(V\) is determined from the readings.
7.5 Use Of Echoes
Echoes find their application in sound ranging and in echo depth sounding, and in both of these the waves used are ultrasonic rather than audible.
7.5.1 Properties Of Ultrasonic Waves
Ultrasonic waves are waves of frequency above \(20\\ \\text{kHz}\), that is, above \(20{,}000\\ \\text{Hz}\). They are more energetic than audible sound, and they have three properties that make them suitable for a wide variety of uses.
- They can travel undeviated through a long distance.
- They can be confined to a narrow beam.
- They are not easily absorbed in a medium.
Audible sound waves, of frequency \(20\\ \\text{Hz}\) to \(20\\ \\text{kHz}\), do not possess these properties. A shout spreads out in all directions, is absorbed as it goes, and cannot be aimed. This is why an ordinary sound is of little use for finding a distant object, however loud it may be.
One thing does not change with frequency, and it is worth being clear about. Ultrasonic waves in a medium travel with the same speed as audible sound waves in that medium. Being ultrasonic makes a wave finer and more sharply directed, not faster.
7.5.2 Use Of Echoes By Bats, Dolphins And Fishermen
Different animals have different audible ranges of frequency. Bats, dolphins and dogs all have a much higher upper audible limit than human beings do, so sounds that are silent to us are ordinary sounds to them. Bats can produce as well as detect sound of very high frequency, up to about \(100\\ \\text{kHz}\).
A bat flies with a speed much lower than the speed of sound, which is what makes its method work: the sound it sends out races ahead, strikes an obstacle and returns while the bat has barely moved. From this echo the bat comes to know, even in complete darkness, where the obstacle lies, so it turns away in time and flies on safely without colliding. This process of detecting an obstacle by means of the reflected sound is called sound ranging. Fig. 7.5-1 shows the idea.
Diagram
Fig. 7.5-1 — A Bat Locating An Obstacle A bat in flight at the left with a tree trunk or a wall as an obstacle at the right. Draw arcs spreading from the bat’s mouth towards the obstacle labelled ultrasonic waves sent out, and arcs returning from the obstacle to the bat labelled reflected waves (echo). Show the bat’s flight path as a curved arrow turning aside before reaching the obstacle. Label the distance between bat and obstacle as d.
Dolphins detect their enemies and obstacles in the same way, by emitting ultrasonic waves and hearing their echo. They also use ultrasonic waves for hunting their prey.
A trawlerman or fisherman uses the same principle at sea. A source that is a very high frequency vibrator sends an ultrasonic pulse into the water, and a detector receives the pulse reflected from a shoal of fish. The total time \(t\) of the to-and-fro journey of the pulse is recorded, and the distance \(d\) of the shoal is calculated from
\[d = \\frac{Vt}{2}\]
where \(d\) is the distance of the shoal in m, \(t\) the total time of the to-and-fro journey in s, and \(V\) the speed of ultrasonic waves in sea water, which is nearly \(1400\\ \\text{m s}^{-1}\).
7.5.3 Use Of Echoes By SONAR
The word SONAR stands for sound navigation and ranging. Fig. 7.5-2 shows the principle of a sonar, in which ultrasonic waves are sent into sea water in all directions from a ship. These waves are received after reflection from an obstacle such as an enemy submarine, an iceberg or a sunken ship.
Diagram
Fig. 7.5-2 — Principle Of Sonar A ship drawn on the sea surface at the top, with a transmitter and a receiver marked side by side on its hull. Dashed horizontal lines below the ship to represent sea water. From the transmitter, draw arrows fanning downwards into the water labelled ultrasonic waves sent out; from an obstacle drawn at the bottom, draw an arrow returning up to the receiver labelled reflected waves. Mark the vertical distance from the ship to the obstacle with a double-headed arrow labelled d. Label the ship, the transmitter, the receiver and the obstacle.
To find the distance of the obstacle from the ship, the time interval \(t\) between the instant when the waves are sent and the instant when the waves are received after reflection is measured. The distance \(d\) of the obstacle from the source is then
\[d = \\frac{Vt}{2}\]
where \(V\) is the speed of ultrasonic waves in water in \(\\text{m s}^{-1}\), \(t\) the time interval in s, and \(d\) the distance of the obstacle in m. The factor of two is present for the same reason as in an ordinary echo: the waves cover the gap once going and once returning.
The depth of the sea can also be found by this method, by aiming the waves straight down at the sea bed instead of at a distant object. The process is then called echo depth sounding.
7.5.3.1 Radar Compared With Sonar
In radar, which stands for radio detection and ranging, the echo method is used to detect the presence of an obstacle and also to find its range. Here the signal sent out is not sound at all. A signal of electromagnetic waves, such as radio waves or micro waves, is sent into space, and after reflection from an object in its path — an aircraft, for instance — it returns to the radar itself.
In both sonar and radar the transmitter and the receiver are placed close to each other, since the signal must be sent and received at very nearly the same point for the timing to give the distance. In Fig. 7.5-2 they are drawn a little apart only for clarity.
7.5.4 Use Of Echoes In The Medical Field
In the medical field, the echo method of ultrasonic waves is used for imaging human organs such as the liver, the gall bladder and the uterus. The ultrasonic waves are sent into the body, and the echoes returning from the boundaries between different tissues are used to build up an image. This is called ultrasonography.
The same principle applied to the heart is called echo cardiography, and it is used to obtain an image of the human heart while it is beating.
7.6 Natural Vibrations
7.6.1 Meaning Of Natural Vibrations
Take a body that is clamped at one point — a metal blade held in a vice, a stretched wire fixed at its ends, a pendulum hanging from a support — and disturb it slightly from its rest position. It does not stay where you put it, and it does not simply return quietly. It starts vibrating.
The reason lies in the elasticity of the body. Once it is deformed, forces develop inside it that tend to restore it to its original position. These restoring forces pull it back, it overshoots the rest position, is pulled back again, and so a to-and-fro motion around the undisturbed position sets in. The vibrations so produced are called natural vibrations, also called free vibrations.
The period, or the frequency, of these vibrations depends on the shape and size — that is, on the structure — of the body itself. The time period of the body is called its natural period, measured in second (s), and the frequency is called its natural frequency, measured in hertz (Hz). A body capable of vibrating has a constant natural frequency of its vibrations, and in vacuum its amplitude of vibration also remains constant.
Two conditions are worth stating plainly. A body executes natural vibrations only when restoring forces are present; without them there is nothing to bring it back and no vibration at all. And natural vibrations persist only in vacuum, because the medium around the body offers resistance to its motion, due to which the amplitude of vibration does not remain constant but continuously decreases.
7.6.2 Examples Of Natural Vibrations
Natural vibrations are all around us, and in each case the frequency is decided by the structure of the vibrating body.
A Pendulum And A Loaded Spring
If the bob of a simple pendulum is displaced slightly from its mean position and released, it vibrates with its natural frequency. That frequency is determined by the length \(l\) of the pendulum and by the acceleration due to gravity \(g\) at that place:
\[f = \\frac{1}{2\\pi}\\sqrt{\\frac{g}{l}}\]
where \(f\) is the natural frequency in Hz, \(l\) the length of the pendulum in m, and \(g\) the acceleration due to gravity in \(\\text{m s}^{-2}\). The relation holds for a simple pendulum displaced only slightly from its mean position.
Different pendulums, having different lengths, vibrate with different frequencies — that is, they have different natural frequencies. A simple pendulum of length \(1.0\\ \\text{m}\) on the earth’s surface, where the acceleration due to gravity is \(9.8\\ \\text{m s}^{-2}\), has a natural frequency of \(0.5\\ \\text{Hz}\).
A load suspended from a spring behaves in the same way. When the spring is stretched, or compressed, and then released, the load starts vibrating with its natural frequency:
\[f = \\frac{1}{2\\pi}\\sqrt{\\frac{K}{m}}\]
where \(K\) is the force constant of the spring in \(\\text{N m}^{-1}\) and \(m\) the mass of the load in kg. The force constant \(K\) of a spring is the force needed to produce unit extension in it; it is more for a hard spring and less for a soft one. The frequency will therefore be different for the same load on different springs, and different for different loads on the same spring.
Diagram
Fig. 7.6-1 — Two Bodies Executing Natural Vibrations Two separate sketches side by side. On the left, a simple pendulum of length l hanging from a rigid support, with the bob shown at the mean position and dashed outlines at the two extreme positions on either side; label the support, the length l and the mean position. On the right, a spiral spring hanging from a rigid support with a load of mass m at its lower end, an arrow showing the load moving up and down, and dashed outlines of the load at its highest and lowest positions; label the spring, the load m and the mean position.
Forks, Air Columns And Strings
When a tuning fork is struck against a hard rubber pad, it vibrates with its natural frequency. Its vibrations form longitudinal waves in the air, and when these reach our ears, sound is heard. The sound is of a single frequency, that is, it is a pure note. Different tuning forks produce sounds of different frequencies.
When an air column in a flute, or in an organ pipe, is made to vibrate, it vibrates with its natural frequency, and that frequency is inversely proportional to the length of the air column, \(f \\propto 1/l\). In a flute, notes of different frequencies are produced by changing the effective length of the air column as different holes in it are closed. In an organ pipe of given length open at both ends, different modes of vibration are produced by blowing the air differently, and these are of frequencies in the ratio \(1 : 2 : 3 : \\ldots\), while in an organ pipe with one end closed the frequencies of the different modes are in the ratio \(1 : 3 : 5 : \\ldots\)
When a string in an instrument such as a sitar, a sonometer or a violin is plucked once, transverse vibrations of a definite natural frequency are produced in the string. The frequency \(f\) of vibration in a stretched string depends on three things: the length \(l\) of the string, with \(f \\propto 1/l\); the radius \(r\) of the string, with \(f \\propto 1/r\); and the tension \(T\) in the string, with \(f \\propto \\sqrt{T}\). Taken together, the frequency of the note produced by a string stretched between its ends is
\[f = \\frac{1}{2l}\\sqrt{\\frac{T}{\\pi r^{2} d}}\]
where \(l\) is the length of the string in m, \(T\) the tension in it in newton (N), \(r\) the radius of the string in m, and \(d\) the density of the material of the string in \(\\text{kg m}^{-3}\). Here the product \(\\pi r^{2} d = m\) is the mass of unit length of the string, in \(\\text{kg m}^{-1}\). Note that \(T\) stands for tension in this relation, not for time period.
The frequency \(f\) of the note can therefore be increased in three ways: by decreasing the length \(l\) of the string, by decreasing the radius \(r\), that is the thickness, of the string, and by increasing the tension \(T\) in the string. This is exactly why, when we strike the keys of a piano, the various strings are set in vibration each at its own natural frequency — they are of different thicknesses and are held under different tensions.
Modes Of Vibration Of A String
A string of a given length, stretched between its ends under a given tension, can be made to vibrate in different modes by plucking it at different points. The three modes are drawn in Fig. 7.6-2.
Diagram
Fig. 7.6-2 — Different Modes Of Vibration In A String Three separate sketches side by side, each showing a string stretched between two fixed supports, with the vibrating outline drawn as a solid curve and a mirrored dashed curve. Sketch (a): one loop spanning the whole length; label it frequency f, principal note. Sketch (b): two loops of equal size; label it frequency 2f, first subsidiary. Sketch (c): three loops of equal size; label it frequency 3f, second subsidiary. Mark the full length of the string as l under each sketch.
If the string is plucked in the middle, it vibrates in one loop as shown in Fig. 7.6-2(a). This vibration is called the principal note, of frequency \(f\). If the string is plucked at one-fourth of its length from one end, it vibrates in two loops, as in Fig. 7.6-2(b), and this vibration is called the first subsidiary vibration, of frequency \(2f\). Similarly, if it is plucked at one-sixth of its length from one end, it vibrates in three loops, as in Fig. 7.6-2(c), and this is called the second subsidiary vibration, of frequency \(3f\).
The different modes of vibration in a stretched string are thus of frequencies in the ratio \(1 : 2 : 3\). If \(l\) is the length of the string stretched between its ends, the wavelengths of the different modes in Fig. 7.6-2(a), (b) and (c) are \(2l\), \(2l/2\) and \(2l/3\) respectively, that is, they are in the ratio \(6 : 3 : 2\).
7.6.3 Nature Of Natural Vibrations
Natural vibrations are simple harmonic vibrations taking place under the restoring force, and both the amplitude and the frequency of these vibrations remain constant. Once such a body starts vibrating, it continues its vibrations with the same amplitude and the same frequency forever. The displacement–time graph for the natural, or free, vibrations of a body in this ideal condition is drawn in Fig. 7.6-3.
Diagram
Fig. 7.6-3 — Displacement–Time Graph For Natural Vibration In Vacuum A smooth sine curve of about five complete cycles drawn on axes. Vertical axis labelled displacement, marked +a at the top and −a at the bottom, with the horizontal line through the middle marked O as the mean position. Horizontal axis labelled t. Every peak reaches exactly +a and every trough exactly −a; draw dashed horizontal guide lines at +a and −a to show that the peaks do not fall in height.
Vibrations of constant amplitude can occur only in vacuum. Since in practice it is very difficult to have a vacuum, it is very difficult to realise such vibrations in practice. As a matter of fact, the surrounding medium offers resistance, or friction, to the motion, so the energy of the vibrating body continuously decreases, and with it the amplitude of vibration gradually decreases. What happens then is the subject of the next section.
7.7 Damped Vibrations
7.7.1 Meaning Of Damped Vibrations
It is our common experience that when a body is made to vibrate in a medium, the amplitude of the vibrating body continuously decreases with time, and ultimately the body stops vibrating. A struck tuning fork grows quieter; a swinging pendulum swings through smaller and smaller arcs. Such vibrations are called damped vibrations.
In damped vibrations, two forces act on the vibrating body: the restoring force, which keeps the vibration going, and the frictional, or resistive, force due to the surrounding medium, which works against it. The amplitude of motion decreases because of this frictional force.
The frictional force at any instant is proportional to the velocity of the body, and it has the tendency to resist motion. This proportionality is not an extra rule to be memorised. From \(F = ma\), if a body moving with velocity \(v\) is brought to rest by a resistive force \(F\) in time \(t\), then \(F = \\dfrac{mv}{t}\), so the resistive force needed to stop a moving body is proportional to the velocity \(v\) of the body.
Where The Energy Goes
As a result of this force, the energy of the vibrating body continuously dissipates in doing work against the force of friction, and so its amplitude gradually decreases. After some time, when it has lost all its energy, it stops vibrating.
The energy lost by the vibrating body changes continuously into heat energy, and this heat gets dissipated in the surrounding medium. Nothing is destroyed; the ordered to-and-fro energy of the body is simply spread out as heat among the particles of the medium.
How quickly this happens is not the same in every case. The rate at which the energy is lost to the surroundings — that is, the rate of decrease of amplitude — depends on the nature of the surrounding medium, such as its viscosity and density, and also on the shape and size of the vibrating body. The same pendulum dies away far sooner in water than in air.
Fig. 7.7-1 shows the displacement–time graph for a body executing damped motion.
Diagram
Fig. 7.7-1 — Displacement–Time Graph For Damped Vibrations A sine-shaped curve of about eight cycles whose peaks get steadily smaller from left to right and finally die away close to the axis. Vertical axis labelled displacement, marked +a at the top and −a at the bottom, with the horizontal line through the middle marked O as the mean position. Horizontal axis labelled t. Draw two dashed envelope curves, one touching the tops of successive peaks and one touching the bottoms of successive troughs, both converging towards the axis on the right.
7.7.2 Examples Of Damped Vibrations
Every vibration you have ever watched has in fact been a damped one, because every one of them took place in a medium.
- When a slim branch of a tree is pulled and then released, it makes damped vibrations.
- A tuning fork, when stroked on a rubber pad, executes damped vibrations in air.
- A simple pendulum oscillating in air, or in any other medium, executes damped vibrations.
- The vibrations of a spring having a mass at its end, in air, are damped vibrations.
Notice that the last three of these appeared in the previous section as examples of natural vibrations. There is no contradiction. The frequency of each is still its own natural frequency, decided by its structure; it is the amplitude that the surrounding air steadily eats away.
7.7.3 Natural Vibrations Compared With Damped Vibrations
The two kinds of vibration are set side by side in Table 7.2.
7.8 Forced Vibrations
7.8.1 Meaning Of Forced Vibrations
So far the vibrating body has been left to itself, with only the medium acting on it. Suppose instead that some outside agency keeps pushing it, over and over, in a regular rhythm. Such a repeating push is called an external periodic force, and the vibrations it produces are called forced vibrations.
A body executing forced vibrations is thus acted upon by three forces: the restoring force, the frictional or resistive or damping force, and the external periodic force. The external applied force is also called the driving force.
Something important happens to the frequency when the driving force is applied. The body no longer vibrates with its own natural frequency; it gradually acquires the frequency of the applied periodic force. Whatever rhythm the driver insists on, the driven body ends up following.
The amplitude of the forced vibrations depends on the frequency of the external force. If the frequency of the external force is different from the natural frequency of the body, the amplitude of oscillations is usually very small, and these oscillations die out soon due to the presence of damping forces. But if the frequency of the external force is exactly equal to the natural frequency of the vibrating body, the amplitude of oscillations is very large; in the absence of damping forces the amplitude of vibrations would be infinite. That special case is the subject of Section 7.9.
The amplitude of forced vibrations does not remain constant, because of the presence of damping forces of the surrounding medium. It is possible, however, to keep the amplitude of vibrations constant, by applying the external periodic force in such a way that it compensates for the loss of energy in each vibration due to the damping forces. The vibrations of the body are then called sympathetic vibrations.
7.8.2 Examples Of Forced Vibrations
- When the stem of a vibrating tuning fork is pressed against the top of a table, the tuning fork produces forced vibrations in the table top. The table top has a much larger vibrating area than the tuning fork, so the forced vibrations of the table top send forth a greater energy and produce a louder, or more intense, sound than that produced by the tuning fork alone. The arrangement is drawn in Fig. 7.8-1.
Diagram
Fig. 7.8-1 — A Tuning Fork Driving A Table Top A tuning fork held vertically with its two prongs at the top shown vibrating, indicated by dashed outlines on either side of each prong, and its stem pressed down on to the top of a table. Draw a downward arrow along the stem labelled vibrations passed to the table top, and short arrows spreading outwards from a wide region of the table top labelled forced vibrations of the table top. Label the tuning fork, the stem, the table top and the large vibrating area of the table top.
- The vibrations produced in the diaphragm of a microphone sound box, with frequencies corresponding to the speech of the speaker, are forced vibrations.
- In a guitar, when an artist plays on its strings with the fingers, the vibrations in them cause forced vibrations in the air enclosed in its hollow box.
- In stringed instruments provided with a hollow sound box containing air, the strings are made to vibrate by plucking, which produces forced vibrations in the air of the sound box. The surface area of the air in the sound box is large, so the forced vibrations of the air send forth a greater energy and cause a loud sound.
7.8.3 Natural Vibrations Compared With Forced Vibrations
Table 7.3 sets out the differences between vibrations a body performs on its own and vibrations forced on it from outside.
7.9 Resonance
7.9.1 Meaning And Condition For Resonance
At the end of the last section one case was set aside: the case in which the driving frequency happens to match the natural frequency of the body being driven. When a body vibrates in a medium under the influence of an external periodic force of frequency exactly equal to the natural frequency of vibration of the body, it is said to execute resonant vibrations. These vibrations are of large amplitude, and how large depends on the frictional forces present.
It is worth seeing why the match matters so much. Suppose a body is set to vibrate freely and it vibrates with a frequency \(f\), which is its natural frequency of vibration. Now let a periodic force of frequency \(n\) be applied on the vibrating body.
If \(n = f\), resonance takes place. Every push from the applied force arrives exactly in step with the motion already going on, so the body starts getting energy from the source of the applied periodic force, and its amplitude of vibration increases. In case \(n\) is greater or less than \(f\), that is, not equal to \(f\), there will be no resonance; the pushes fall out of step and partly cancel what the body was doing, so the body executes only forced vibrations of small amplitude.
At resonance a loud sound is heard, because the body vibrating with a very large amplitude sends forth a large amount of energy in the medium.
7.9.2 Demonstration Of Resonance
The phenomenon of resonance can be demonstrated by the following experiments.
7.9.2.1 Resonance With Tuning Forks
Mount two identical tuning forks A and B, of the same frequency, on two separate sound boxes such that their open ends face each other, as shown in Fig. 7.9-1. Strike the prong of one of the tuning forks, say A, on a rubber pad, so that it starts vibrating, and place fork A on its sound box. We find that the other tuning fork B also starts vibrating, and a loud sound is heard.
Diagram
Fig. 7.9-1 — Resonance With Tuning Forks Two identical tuning forks A and B, each mounted upright on its own rectangular wooden sound box, standing on a table with the open ends of the two boxes facing each other. Show the prongs of fork A vibrating with dashed outlines on either side, and the prongs of fork B also vibrating with lighter dashed outlines. Draw a series of arcs travelling through the air from the open end of box A to the open end of box B, labelled vibrations communicated through the air. Label fork A, fork B, the two sound boxes, and mark both forks as being of frequency n.
The reason runs in a chain. The vibrating tuning fork A produces forced vibrations in the air column of its sound box. These vibrations are of large amplitude, because of the large surface area of air in the sound box. They are communicated through the air to the sound box of fork B, and the air column of B starts vibrating with the frequency of fork A. Since the frequency of these vibrations is the same as the natural frequency of the fork B, the fork B readily picks up these vibrations and starts vibrating under resonance. The two sound boxes thus help in communicating the vibrations and in increasing the amplitude of vibrations.
7.9.2.2 Forced And Resonant Vibrations Of Pendulums
Take four pendulums A, B, C and D, and suspend them from the same elastic support — say from a piece of stretched rubber string PQ, as shown in Fig. 7.9-2. A and B are pendulums of the same length, so their natural frequency of vibration is the same. D is a pendulum which is longer than A or B, so its natural frequency is lower than that of A or B. C is shorter than A or B, so its natural frequency is higher than that of A or B. The two unequal ones, C and D, are suspended on either side of the pendulum B.
Diagram
Fig. 7.9-2 — Forced And Resonant Vibrations Of Pendulums A horizontal rubber string stretched between two fixed points P at the left and Q at the right. Four pendulums hang from it: A at the left, then C, then B, then D at the right. Pendulum C is drawn clearly shorter than A and B; pendulum D is drawn clearly longer than A and B; A and B are drawn exactly equal in length. Label each bob A, B, C and D, label the ends P and Q, and label the string as stretched rubber string. Show pendulum A displaced to one side with a curved arrow indicating that it has been set vibrating.
Pendulum A is set into vibration by displacing its bob to one side, normal to the length of the string PQ. Three things are then observed.
- Pendulum B also starts vibrating, initially with a small amplitude, but gradually it acquires the same amplitude as pendulum A initially had. When the amplitude of pendulum B becomes maximum, the amplitude of pendulum A becomes minimum, because of the sharing of energy by them. Thereafter the amplitude of pendulum B decreases and that of A increases, and this process continues.
- The vibrations of pendulum B are in phase with those of A, that is, they reach their extreme positions on either side simultaneously.
- The pendulums C and D also vibrate, but each of them vibrates with a very small amplitude.
The explanation is this. The vibrations produced in pendulum A are communicated as forced vibrations to all the other pendulums B, C and D through the rubber string PQ. The pendulums C and D remain in the state of forced vibrations, since their natural frequencies do not match the driving frequency. The pendulum B, however, comes into the state of resonance, because its natural frequency is equal to that of A, both being of the same length.
There is therefore an exchange of energy between the pendulums A and B. When the amplitude of pendulum B increases, the amplitude of pendulum A decreases, because energy is conserved, and vice versa. Pendulum B vibrates with the frequency of pendulum A and remains in phase with it.
7.9.2.3 Resonance In An Air Column
Fig. 7.9-3 shows an arrangement for studying resonance in an air column. It consists of a long cylindrical tube A and a cylindrical vessel B. The tube A is fixed, while the vessel B can be moved up or down and can be clamped at any desired position. Both the tube A and the vessel B are connected at their lower ends by a rubber tube, and they are partially filled with water.
Diagram
Fig. 7.9-3 — Resonance In An Air Column A tall narrow cylindrical glass tube A clamped upright on a stand at the left, with a vertical scale marked alongside it, partly filled with water. A wider cylindrical vessel B clamped on a stand at the right, also containing water, drawn higher than the water level in A. A flexible rubber tube joining the bottom of tube A to the bottom of vessel B. A vibrating tuning fork held horizontally just above the open mouth of tube A, its prongs shown vibrating with dashed outlines. Label tube A, vessel B (reservoir), the rubber tube, the scale, the water surface in tube A, the length of the air column between the water surface and the mouth, and the tuning fork. Show an up-and-down arrow beside vessel B.
The movable vessel B serves as a reservoir. The tube A serves as a closed end air pipe, with the water surface in it forming the closed end, that is, the reflecting surface. An air column is thus formed in the tube A between the water surface and its mouth. The natural frequency of vibrations of an air column depends on its length, since \(f \\propto 1/l\). By moving the reservoir B up or down, the length of the air column in tube A can be changed, and so the natural frequency of the air column of tube A changes.
While performing the experiment, the tube A is first filled with water up to the top. Then a vibrating tuning fork is placed just above the mouth of tube A, and the level of water in it is gradually lowered.
A loud sound is heard at a certain level of water. On further lowering the level of water in tube A, the loud sound ceases; but when the length of the air column in the tube becomes nearly three times the previous one, a loud sound is heard again.
The vibrating tuning fork held just above the mouth of the tube A produces forced vibrations in the air column of tube A. When the frequency of the air column is decreased by increasing the length of the air column — that is, by lowering the water level in the tube — a stage is reached at which the natural frequency of the air column becomes equal to the frequency of the tuning fork. The vibrations of the air column are then in resonance with those of the tuning fork, and a loud sound is heard.
On further lowering the water level, the frequency of the air column does not remain equal to the frequency of the fork, so the loud sound ceases. But on lowering the level still further, a stage is again reached when a loud sound is heard. At this stage the frequency of the air column again becomes equal to the frequency of the tuning fork, when the length of the air column becomes three times the previous length, so resonance occurs again.
7.9.3 Forced Vibrations Compared With Resonant Vibrations
Resonance is a special case of forced vibrations, so the two share a great deal. Table 7.4 shows where they part company.
7.10 Some Examples Of Resonance
Once you know what to look for, resonance turns up in swings and bridges, in machines, in musical instruments and inside a radio set. Six examples are described here.
Pendulums And Machine Parts
Resonant vibrations of pendulums. As discussed earlier in the experiment with four pendulums, if two pendulums of the same length are suspended from a rubber string and one pendulum is made to vibrate, the other pendulum also starts vibrating with a large amplitude and in the same phase, because of resonance.
Resonance in machine parts. When a vehicle is driven, the piston of the engine makes a to-and-fro motion at a frequency depending upon the speed of the vehicle. The vibrations caused by the movement of the piston are communicated to all parts of the vehicle. It is just possible that at a certain speed of the vehicle, some part or the frame of the vehicle may have a natural frequency of vibration equal to the frequency of the to-and-fro movement of the piston. At this particular speed, that part or frame starts vibrating vigorously due to resonance, and a rattling sound is heard. If this part is not tightly fixed, such vibrations may cause the part to drop out. To stop these vibrations, the speed of the vehicle is changed, so that the condition of resonance is disturbed.
Resonance can thus occur in all kinds of machines while they are operating in a particular condition. It is a good reason for taking a new rattle in a machine seriously rather than living with it.
Instruments, Air Columns And Bridges
Resonance in a stretched string and the sound box of musical instruments and of a sonometer. A vibrating string by itself produces a very weak sound which cannot be heard at a distance. Therefore all musical stringed instruments such as the sitar, the guitar and the sonometer are provided with a sound box, that is, a hollow chamber. The box is so constructed that the column of air inside it has a natural frequency which is the same as that of the string stretched on it. So when the string is made to vibrate, the air column inside the box is set into forced vibrations, the frequency of which is the same as that of the string. Since the sound box has a large area, it sets a large volume of air into vibrations, and so, due to resonance, a loud sound is produced.
Resonance in an air column and a tuning fork. As described in the resonance tube experiment, when the frequency of the air column becomes equal to the frequency of the tuning fork vibrating over its mouth, a loud sound is heard due to resonance.
Resonance in a bridge. When a troop crosses a suspension bridge, the soldiers are asked to break their steps. The reason is that when soldiers march in step, each soldier exerts a periodic force in the same phase, and therefore the bridge executes forced vibrations of frequency equal to the frequency of their steps. Incidentally, if the bridge happens to have its natural frequency equal to the frequency of the steps of the soldiers, the bridge will vibrate with a large amplitude due to resonance, and the suspension bridge may collapse. Fig. 7.10-1 shows the two ways of crossing.
Diagram
Fig. 7.10-1 — Why Soldiers Break Step On A Bridge Two sketches of the same suspension bridge, one above the other, each with towers and hanging cables drawn. In the upper sketch, a column of soldiers crossing with all feet placed together in step; draw the bridge deck with an exaggerated sagging curve and label it large amplitude vibrations, resonance possible. In the lower sketch, the same soldiers crossing with feet at different positions, out of step; draw the deck nearly straight and label it small, irregular vibrations, no resonance. Label the upper sketch marching in step and the lower sketch broken step.
Tuning A Radio Or Television Set
- Resonance in radio and TV receivers. The tuning of radio and TV receivers is based on resonance. These receivers have electronic circuits which produce electromagnetic vibrations, the frequency of which can be changed by varying the values of the electronic components — either the capacity of the condenser or the inductance of the coil, and usually it is the capacity of the condenser that is changed.
When we want to tune a radio or TV receiver, we merely adjust the values of the electronic components to produce vibrations of frequency equal to that of the radio waves which we want to receive. When both the frequencies match, resonance occurs, and only the energy of the signal of that particular frequency is received from the waves present in space, leaving the signals of other frequencies which do not match with the frequency of the receiver circuit. The signal received is then amplified in the receiver set.
Flowchart
FC 7.3 — Tuning A Radio Receiver By Resonance Radio waves of many different frequencies arrive at the aerial → the capacity of the condenser in the receiver circuit is adjusted → the natural frequency of the circuit is made equal to the frequency of the wanted station → resonance occurs for that frequency alone → energy of that one signal is picked up while the others are left out → the received signal is amplified in the receiver set
7.11 Characteristics Of Sound
Two sounds can be distinguished from one another by the following three different characteristics:
- loudness,
- pitch, or shrillness, and
- quality, or timbre.
7.11.1 Studying The Wave Pattern Of A Sound
All three characteristics of a given sound can be studied from the wave pattern of that sound, so we begin with the apparatus that lets us see a sound instead of only hearing it. Fig. 7.11-1 shows a set-up to obtain the wave pattern of a sound wave, using a microphone and a cathode ray oscilloscope, written C.R.O. for short.
Diagram
Fig. 7.11-1 — Set-Up To Study The Wave Pattern Of A Sound A cathode ray oscilloscope drawn as a box at the left with a rectangular screen on its front showing a wave trace, and control knobs below the screen. A microphone standing on the table in front of it, connected by a lead to a terminal on the oscilloscope marked Y-input. A vibrating tuning fork held to the right of the microphone with its prongs shown vibrating. Label the C.R.O., the screen, the microphone, the lead to the Y-input, and the vibrating tuning fork.
The sound to be studied is produced in front of the microphone, and the microphone is connected to the Y-input of the C.R.O. The wave pattern of the sound — that is, its displacement–time graph — is then obtained on the screen of the C.R.O. Every comparison that follows is a comparison between two such traces.
7.11.2 Loudness And Intensity
Take a tuning fork and strike it gently on a rubber pad. You will hear a feeble, or soft, sound. Bring it in front of the microphone and sketch the trace you observe on the screen of the C.R.O. Now strike the same tuning fork hard. A loud sound is heard. Bring the tuning fork near the microphone and again sketch the trace observed. The two sketches are shown in Fig. 7.11-2.
Diagram
Fig. 7.11-2 — Soft And Loud Notes Two displacement–time traces drawn one above the other, each on its own axes. In sketch (a), a sine curve of small height; mark a double-headed vertical arrow from the axis to a peak labelled small amplitude, label the horizontal axis time and the vertical axis displacement, and title the sketch soft note. In sketch (b), a sine curve of exactly the same wavelength along the time axis but of much greater height; mark a double-headed vertical arrow labelled large amplitude, label the axes in the same way, and title the sketch loud note.
In Fig. 7.11-2 the two waves have the same frequency and the same wave form, a sine wave, but they differ in amplitude. Obviously, the loud sound corresponds to the wave of large amplitude.
Something similar happens with any instrument. When the key of a piano is hit hard, or a pipe is blown hard, we impart more energy to the vibrating system, due to which the amplitude of vibrations is increased, and a loud sound of the same characteristics is produced.
Loudness tells us how a sound seems to us. The energy actually arriving is a separate, measurable thing, and it has its own name.
The intensity of ordinary sound is nearly \(10^{-6}\\ \\text{W m}^{-2}\), while the minimum intensity of sound audible to our ears is \(10^{-12}\\ \\text{W m}^{-2}\). The greater the energy carried by a sound wave, the greater is the intensity of the sound.
The intensity of a sound wave in air is proportional to 1. the square of the amplitude of vibrations, 2. the square of the frequency of vibrations, 3. the density of air, and 4. the velocity of sound in air.
7.11.2.1 Subjective Nature Of Loudness And Objective Nature Of Intensity
The loudness of a sound depends on two quite different things: on the intensity, that is, the energy conveyed by the sound wave near the eardrum of the listener, and on the sensitivity of the ears of that listener.
Because of the second of these, the loudness of a sound of a given intensity may differ from listener to listener. A sound of the same intensity may appear to be of different loudness to different persons, owing to differences in the sensitivities of their ears.
Further, two sounds of the same intensity, but of different frequencies, may differ in loudness even to the same listener, because the sensitivity of the ears of a person is different for different frequencies. For normal ears, the sensitivity is maximum at the frequency \(1\\ \\text{kHz}\).
Thus, for a sound wave, loudness has a subjective nature — it depends on the listener — while intensity, being a measurable quantity, has an objective nature.
7.11.2.2 Factors Affecting The Loudness Of Sound
The loudness of sound heard at a place depends on the following five factors.
- Loudness is proportional to the square of the amplitude. When a body vibrates with a greater amplitude, it sends forth a greater amount of energy, and hence the energy received by the eardrum is large, so the sound appears louder.
- Loudness varies inversely as the square of the distance from the source. If the listener is close to the source of sound, the person hears it quite loudly; but when the same person is far away, the sound becomes feeble, because the energy sent out has spread over a much larger area. If one moves further away from the source, a stage may be reached when the sound becomes inaudible. Thus, the closer the source, the louder is the sound.
- Loudness depends on the surface area of the vibrating body. A large vibrating area sends forth a greater amount of energy. Hence the larger the surface area of the vibrating body, the louder is the sound heard. For this reason the bell in a school or a temple is made big in size. The same reason explains why a tuning fork sounded in air is feeble, but becomes much louder when its stem is placed on a table top: the vibrations set up by the fork in the table top, because of its large surface area, set in vibration a large surface area of the air in contact with it, so the sound energy reaching our ears increases. In stringed instruments too, a sound chamber is provided to increase the volume of air set vibrating, and hence the loudness of the sound.
- Loudness depends on the density of the medium. The more the density of the medium, the more is the loudness.
- Loudness depends on the presence of resonant bodies. The presence of resonant bodies near the vibrating body increases the loudness of sound.
7.11.2.3 Relationship Between Loudness And Intensity
Although the loudness of sound depends on its intensity, loudness is not the same as intensity. Intensity is a measurable quantity, while loudness is a sensation. Experimentally, Weber and Fechner established a relationship between loudness \(L\) and intensity \(I\), which is given as
\[L = \\text{K} \\log\_{10} I\]
where \(L\) is the loudness, \(I\) the intensity of the sound in \(\\text{W m}^{-2}\), and \(\\text{K}\) is a constant of proportionality. Because the intensity enters through a logarithm, loudness increases with an increase in intensity, but not in the same proportion. Multiplying the intensity ten times does not make the sound ten times as loud.
The unit of loudness is the phon. It is used to indicate an individual’s perception of loudness. The level of sound is expressed in decibel (dB). The loudness of a sound in phon is the same as the loudness in decibel of an equally loud pure sound of frequency \(1\\ \\text{kHz}\), so both the phon and the decibel scales agree at the frequency \(1\\ \\text{kHz}\).
7.11.2.4 Sound Level And The Decibel
From the relation above, the loudness \(L\) is related to the intensity \(I\) as \(L = \\text{K} \\log\_{10} I\). If at a given frequency \(I\_1\) and \(I\_0\) are the intensities of two sounds whose loudness are \(L\_1\) and \(L\_0\) respectively, then
\[L\_1 = \\text{K} \\log\_{10} I\_1 \\qquad \\text{and} \\qquad L\_0 = \\text{K} \\log\_{10} I\_0\]
Hence the difference in loudness of the two sounds is
\[L = L\_1 - L\_0 = \\text{K},(\\log\_{10} I\_1 - \\log\_{10} I\_0)\]
\[L = \\text{K} \\log\_{10} \\frac{I\_1}{I\_0}\]
If \(I\_0\) is the minimum intensity of audible sound at frequency \(1\\ \\text{kHz}\), that is \(10^{-12}\\ \\text{W m}^{-2}\), then \(L\) is called the sound level. Taking \(\\text{K} = 1\), the sound level is written as
\[L = \\log\_{10}\\left(\\frac{I\_1}{I\_0}\\right)\\ \\text{bel} = \\left{10 \\log\_{10}\\left(\\frac{I\_1}{I\_0}\\right)\\right} \\times \\frac{1}{10}\\ \\text{bel}\]
If we write \(\\dfrac{1}{10}\) bel as one decibel (dB), then
\[L = 10 \\log\_{10}\\left(\\frac{I\_1}{I\_0}\\right)\\ \\text{decibel (dB)}\]
where \(I\_1\) is the intensity of the sound in \(\\text{W m}^{-2}\) and \(I\_0 = 10^{-12}\\ \\text{W m}^{-2}\) is the minimum audible intensity at \(1\\ \\text{kHz}\). Here the bel, and with it the decibel, is the unit of loudness level, named after the scientist Alexander Graham Bell, the inventor of the telephone.
Obviously, for \(I\_1 = I\_0\) the sound level is \(L = 0\\ \\text{dB}\); that is, a sound of intensity \(10^{-12}\\ \\text{W m}^{-2}\) at frequency \(1\\ \\text{kHz}\) has a loudness of zero decibel. If the intensity at a point increases from \(I\_0\) to \(I\_1 = 100,I\_0\), the increase in sound level will be \(10 \\log\_{10} 100 = 20\\ \\text{dB}\). For an increase in intensity from \(I\_0\) to \(I\_1 = 10{,}000,I\_0\), the increase in sound level will be \(10 \\log\_{10} 10{,}000 = 40\\ \\text{dB}\).
Now consider the smallest step on this scale. If \(L = 1\\ \\text{dB}\), then
\[10 \\log\_{10}\\left(\\frac{I\_1}{I\_0}\\right) = 1 \\qquad \\text{or} \\qquad \\log\_{10}\\left(\\frac{I\_1}{I\_0}\\right) = \\frac{1}{10}\]
\[\\frac{I\_1}{I\_0} = \\text{antilog}\\left(\\frac{1}{10}\\right) = \\text{antilog},(0.1) = 1.26\]
Thus one decibel is defined as the increase in level of loudness when the intensity of sound increases by \(26%\).
The sound levels of some familiar sources are listed in Table 7.5.
Table 7.5 — Level Of Sound Produced By Different Objects
| Object producing sound | Sound level (dB) |
|---|---|
| Jet at takeoff | 140 |
| Pop concert | 130 |
| Police car siren | 120 |
| Heavy hammering machine | 110 |
| Diesel lorry | 90 |
| Motorcycle or car | 80 |
| Vacuum cleaner | 60 |
| Normal conversation | 50 |
| Soft whisper | 30 |
| Recording studio | 20 |
| Leaves rustling | 10 |
| Dropping pin (minimum audible sound) | 0 |
7.11.2.5 Noise Pollution
The disturbance produced in the environment due to undesirable loud and harsh sound, of level above \(120\\ \\text{dB}\), from various sources such as loudspeakers, sirens and moving vehicles, is called noise pollution.
This is not merely a nuisance. A constant exposure to sound of level above \(120\\ \\text{dB}\) can cause headache and permanent damage to the ears of the listener. The safe limit of level of sound for hearing is from \(0\\ \\text{dB}\) to \(80\\ \\text{dB}\). Sound of level \(10\\ \\text{dB}\) to \(30\\ \\text{dB}\) has a soothing sensation, while the sound of level \(0\\ \\text{dB}\) represents the lower limit of hearing.
The limit of hearing should not be confused with the limit of audibility. The limit of audibility refers to the frequency range of sound, from \(20\\ \\text{Hz}\) to \(20\\ \\text{kHz}\), which a human can hear. The limit of hearing is the lowest level of loudness, \(0\\ \\text{dB}\), which a human can hear.
7.11.3 Pitch And Frequency
Pitch is considered only in reference to musical sounds. Each musical note has a definite pitch. If the pitch is high, the sound is shrill, and if the pitch is low, the sound is flat. In a tape recorder, or a television set, the terms bass and treble refer to the low and the high pitch respectively. At bass, the low pitch — that is, grave — sound produced by percussion instruments like the tabla or the dholak becomes predominant, while at treble the high pitch, or shrill, sound produced by the flute or the ghoonghroo becomes predominant.
Fig. 7.11-3 shows the traces of two waves obtained on the screen of the C.R.O. when two tuning forks A and B, of frequencies \(256\\ \\text{Hz}\) and \(512\\ \\text{Hz}\) respectively, are separately made to vibrate one after the other, by striking them with the same force on a rubber pad, in front of the microphone of the set-up of Fig. 7.11-1.
Diagram
Fig. 7.11-3 — Waves Of Different Pitch Two displacement–time traces drawn one above the other, each on its own axes with the horizontal axis labelled time and the vertical axis labelled displacement. In sketch (a), a sine curve of about two complete cycles across the width of the screen; mark its height with a double-headed arrow labelled a, and title it low pitch note, frequency f, fork A (grave sound). In sketch (b), a sine curve of exactly the same height a but with twice as many cycles, four across the same width; title it high pitch note, frequency 2f, fork B (shrill sound).
Both the traces have the same amplitude and the same wave form, that is, a sine wave, but they differ in frequency. On hearing the sounds of the two tuning forks, it is observed that the sound of tuning fork B, of frequency \(512\\ \\text{Hz}\), is shriller than the sound of tuning fork A, of frequency \(256\\ \\text{Hz}\). Thus, the more the frequency of a wave, the higher is its pitch, and the less the frequency of a wave, the lower is its pitch.
The pitch of a note therefore depends on its frequency. Two notes sounded on the same instrument with the same amplitude will differ in pitch when their vibrations are of different frequencies.
7.11.3.1 Examples Of Change In Pitch
- In string instruments such as the piano, the violin and the guitar, the frequency of vibrations of a string depends on the tension and the thickness of the string. They can produce notes of different pitch by changing the thickness and tension of the string. A note of higher pitch can be obtained by vibrating the string under high tension, or by vibrating a thinner string.
- In the case of a flute, a lower note is obtained by closing some more holes, so that the length of the vibrating air column increases.
- As the water level in a pitcher kept under a water tap rises, the length of the air column above the water level decreases, so the frequency of the sound produced increases; that is, the sound becomes shriller and shriller. Thus, by hearing the sound from a distance, one can get an idea of the water level in the pitcher.
- The voice of a woman is usually of a higher pitch than that of a man.
7.11.3.2 Subjective Nature Of Pitch And Objective Nature Of Frequency
Pitch is not the same as frequency. Pitch depends on the sensation as perceived by a listener. It may be different, for a sound of a particular frequency, to different listeners; that is, pitch is of a subjective nature.
Frequency, on the other hand, is a measurable quantity. It depends on the source producing the sound, and it has a definite value for a given sound, equal to the frequency of its source. It has nothing to do with the listener, so frequency is of an objective nature.
7.11.4 Quality And Wave Form
Fig. 7.11-4 shows the wave patterns of two sounds of the same loudness, that is, the same amplitude, and the same pitch, that is, the same frequency, but emitted from two different sources.
Diagram
Fig. 7.11-4 — Two Sounds Of The Same Amplitude And Frequency But Different Wave Forms Two traces drawn one above the other, each with the horizontal axis labelled time and the vertical axis labelled displacement. Sketch (a): a smooth sine curve, with its height marked by a double-headed arrow labelled a. Sketch (b): a triangular wave of exactly the same height a and exactly the same spacing between repeats, drawn with straight sloping sides and sharp peaks instead of smooth curves.
They produce different sensations in our ears, because they differ in wave form: one is a sine wave while the other is a triangular wave. The quality of a musical sound depends on the wave form.
7.11.4.1 Principal And Subsidiary Vibrations
Why should two instruments give different wave forms at all? Only a tuning fork emits sound of a single frequency, that is, a monotone. The sound from an instrument does not contain a note of a single frequency; it contains a combination of vibrations of different frequencies and different amplitudes.
Among these, the vibration of the lowest frequency and maximum amplitude is called the principal vibration, also called the fundamental vibration, and the vibrations of frequency that are integer multiples of it are called the subsidiary vibrations, also called secondary vibrations. The subsidiary vibrations are of small amplitudes. The resultant vibration obtained by the superposition of all these vibrations gives the wave form of the sound which we hear.
Now suppose that with the set-up of Fig. 7.11-1, first a tuning fork of frequency \(256\\ \\text{Hz}\) is struck and placed in front of the microphone, and next a note of frequency \(256\\ \\text{Hz}\) is produced by a piano in front of the microphone. The traces obtained on the screen of the C.R.O. for the two cases are shown in Fig. 7.11-5.
Diagram
Fig. 7.11-5 — Pure Note And Musical Note Two traces drawn one above the other, each with the horizontal axis labelled time and the vertical axis labelled displacement. Sketch (a): a smooth sine curve of two complete cycles, its height marked by a double-headed arrow labelled a; title it wave form of a tuning fork. Sketch (b): a complex repeating curve of the same overall height a and the same repeat spacing as in (a), but with several small extra humps and dips riding on each large cycle so that it is clearly not a sine curve; title it wave form of a piano.
The sketch (a) shows a pure note, a sine curve, of single frequency produced by the tuning fork. The sketch (b) is of the same amplitude and the same frequency, or same pitch, but it has a different wave form, due to the presence of a mixture of subsidiary vibrations alongwith the principal vibration. The subsidiary vibrations present in the musical note make the wave form complex; it is not a sine curve.
Thus, the quality of a musical sound depends on the number of subsidiary notes and their relative amplitudes present in it alongwith the principal note.
Different instruments produce different subsidiary notes. A note played on a piano has a large number of subsidiary notes, while the same note when played on a flute contains only a few subsidiary notes. Thus we can easily distinguish between the sounds of a piano and a flute by their different wave forms, though they may be of exactly the same loudness and the same pitch.
7.11.4.2 Recognition Of Sources By Their Quality
The sound from each vibrating body has its own characteristic wave form. This makes it possible for a person to recognise the vibrating body even without seeing it.
You generally recognise a person by hearing the voice on a telephone, without seeing him or her. It is because the vibrations produced by the vocal chord of each person have a characteristic wave form, which is different for different persons. Similarly, one can distinguish and recognise the sounds of two different musical instruments because of their different wave forms, even if they are of the same pitch and the same loudness.
To summarise, Table 7.6 gathers the three characteristics of sound together with the factor each one depends on.
7.12 Music And Noise
All sounds which produce the sensation of hearing can be divided into two categories: music and noise.
What Makes A Sound Musical
Music is a pleasant, continuous and uniform sound produced by regular and periodic vibrations. For example, the sounds produced by a violin, a piano, a flute or a tuning fork are musical sounds. The sound level of these musical sounds is usually between \(10\\ \\text{dB}\) and \(30\\ \\text{dB}\).
Noise is the sound other than a musical sound. It is a sound produced by an irregular succession of disturbances, and it is usually a discontinuous sound. It is discordant and unpleasant to the ears. For example, the sound produced when a stone is thrown on a tin shade is a noise. Usually all sounds of level above \(120\\ \\text{dB}\) are termed as noise.
The difference shows itself plainly in the wave pattern. Fig. 7.12-1 shows the wave forms of a musical sound and of noise.
Diagram
Fig. 7.12-1 — Wave Forms Of A Musical Sound And Of Noise Two traces drawn one above the other, each with the horizontal axis labelled time and the vertical axis labelled displacement. Sketch (a): a regular repeating curve in which every cycle has the same height and the same width as the one before it; title it music. Sketch (b): an irregular curve in which the humps are of clearly different heights and different widths, with sudden sharp changes and short flat gaps between bursts; title it noise.
Where The Line Is Drawn
There is no sharp difference between music and noise. The difference is subjective. A sound which is music to someone may be a noise to others; a pop song, for instance, is music for the young generation, while it may be a noise for the older generation.
What is not subjective is the level. Whether a sound is enjoyed or not, a sound level above the safe limit does the same damage to the ears of the listener who loves it as to the ears of the one who does not.
7.13 Summary
Sound is produced when a body vibrates, and it reaches us through the vibrations of the particles of the surrounding medium. Sound therefore requires a material medium for its propagation.
The audible range of frequency for human beings is \(20\\ \\text{Hz}\) to \(20{,}000\\ \\text{Hz}\). Sound of frequency above \(20{,}000\\ \\text{Hz}\) is ultrasonic and below \(20\\ \\text{Hz}\) is infrasonic; both are inaudible to us, but both travel in a medium with the same speed as audible sound.
The amplitude \(a\) of a wave is the maximum displacement of a particle of the medium on either side of its mean position, in m. The time period \(T\) is the time taken by a particle to complete one vibration, in s. The frequency \(f\) is the number of vibrations made in one second, in Hz. The wavelength \(\\lambda\) is the distance travelled by the wave in one time period, in m. The wave velocity \(V\) is the distance travelled by the wave in one second, in \(\\text{m s}^{-1}\).
The wave velocity, frequency and wavelength are related as \[V = f\\lambda\] where \(V\) is in \(\\text{m s}^{-1}\), \(f\) in Hz and \(\\lambda\) in m.
The time period and the frequency are related as \[f = \\frac{1}{T}\] where \(f\) is in Hz and \(T\) in s.
The frequency of a wave is fixed by the vibrating source; the wave velocity, and hence the wavelength, depends on the medium.
Sound waves are mechanical, or elastic, waves. They are of two kinds: longitudinal waves, in which the particles vibrate along the direction of propagation forming compressions and rarefactions, and transverse waves, in which the particles vibrate normal to the direction of propagation forming crests and troughs. Longitudinal waves travel in solids, liquids and gases; transverse waves are formed only in solids and on the surface of liquids.
The speed of a longitudinal wave in a gas of density \(d\) at pressure \(P\) is \[V = \\sqrt{\\frac{\\gamma P}{d}}\] where \(P\) is in \(\\text{N m}^{-2}\), \(d\) in \(\\text{kg m}^{-3}\), \(V\) in \(\\text{m s}^{-1}\), and \(\\gamma\) is the ratio of the two specific heats, equal to \(1.4\) for air.
The speed of sound in a gas increases with a rise in temperature and with an increase in humidity, but it is not affected by a change in pressure, because pressure and density change in the same proportion.
The speed of a transverse wave in a stretched string is \[V = \\sqrt{\\frac{T}{m}}\] where \(T\) is the tension in the string in N, \(m\) the mass per unit length in \(\\text{kg m}^{-1}\), and \(V\) in \(\\text{m s}^{-1}\).
When a wave passes from one medium into another, its speed, wavelength and intensity change, but its frequency does not change. Its direction of travel changes too, except at normal incidence.
Light waves are electromagnetic and travel in vacuum at \(3 \\times 10^{8}\\ \\text{m s}^{-1}\), with wavelengths of the order of \(10^{-6}\\ \\text{m}\), and are always transverse. Sound waves are mechanical, need a medium, travel in air at about \(330\\ \\text{m s}^{-1}\), have wavelengths from \(10^{-2}\\ \\text{m}\) to \(10\\ \\text{m}\), and in air are longitudinal.
Sound waves obey the laws of reflection: the angle of reflection is equal to the angle of incidence, and the incident ray, the reflected ray and the normal at the point of incidence all lie in one plane.
Sound is reflected from any surface, smooth or rough, polished or unpolished. The only condition is that the size of the reflecting surface must be bigger than the wavelength of the sound wave. Megaphones, sound boards and ear trumpets make use of this reflection.
An echo is the sound heard after reflection from a distant obstacle after the original sound has ceased. Since the sensation of sound persists in our ears for about \(0.1\\ \\text{s}\), the reflected sound must reach the ears at least \(0.1\\ \\text{s}\) after the original sound.
For an echo, the sound covers the distance to the obstacle and back, so \[d = \\frac{Vt}{2}\] where \(d\) is the distance of the reflector in m, \(V\) the speed of sound in the medium in \(\\text{m s}^{-1}\), and \(t\) the time interval in s. With \(t = 0.1\\ \\text{s}\) and \(V = 340\\ \\text{m s}^{-1}\), the minimum distance in air is \(17\\ \\text{m}\); in sea water, where \(V = 1400\\ \\text{m s}^{-1}\), it is \(70\\ \\text{m}\).
Repeated reflections at the reflecting surfaces prolong a sound; this effect is called reverberation, and it is heard in high tombs such as the Taj Mahal and the tomb at Sikandra.
Three conditions must be satisfied to hear an echo distinctly: the minimum distance of the reflector must be at least \(17\\ \\text{m}\) in air, the reflector must be large compared with the wavelength, and the reflected sound must be loud enough to be audible.
The speed of sound can be found by the echo method. Sound is produced at a measured distance \(d\) of at least \(50\\ \\text{m}\) from a reflecting surface, the time interval \(t\) is noted with a stop watch of least count \(0.01\\ \\text{s}\), and \[V = \\frac{2d}{t}\] where \(V\) is in \(\\text{m s}^{-1}\), \(d\) in m and \(t\) in s. The experiment is repeated and the average value is taken.
Ultrasonic waves, of frequency above \(20\\ \\text{kHz}\), travel undeviated through long distances, can be confined to a narrow beam, and are not easily absorbed in a medium. Audible waves do not have these properties.
Bats produce and detect sound of frequency up to about \(100\\ \\text{kHz}\) and use the echo to locate obstacles in the dark, a process called sound ranging. Dolphins use ultrasonic waves to detect enemies and obstacles and to hunt prey. Fishermen locate a shoal of fish from the time of the to-and-fro journey of an ultrasonic pulse, using \(d = Vt/2\) with \(V\) nearly \(1400\\ \\text{m s}^{-1}\) in sea water.
SONAR stands for sound navigation and ranging. Ultrasonic waves sent into sea water are reflected from a submarine, an iceberg or a sunken ship, and the distance is found from \(d = Vt/2\). Finding the depth of the sea this way is called echo depth sounding. Radar, standing for radio detection and ranging, uses the same echo method but with electromagnetic waves.
In the medical field, the echo method of ultrasonic waves is used for imaging organs such as the liver, the gall bladder and the uterus, called ultrasonography, and for imaging the heart, called echo cardiography.
Natural, or free, vibrations are the periodic vibrations of a body in the absence of any external force on it. Their frequency, called the natural frequency, depends on the shape and size of the body. Their amplitude stays constant only in vacuum.
The natural frequency of a simple pendulum is \[f = \\frac{1}{2\\pi}\\sqrt{\\frac{g}{l}}\] where \(l\) is the length in m and \(g\) the acceleration due to gravity in \(\\text{m s}^{-2}\); for a load on a spring it is \[f = \\frac{1}{2\\pi}\\sqrt{\\frac{K}{m}}\] where \(K\) is the force constant in \(\\text{N m}^{-1}\) and \(m\) the mass in kg.
For an air column, \(f \\propto 1/l\). For a stretched string, \(f \\propto 1/l\), \(f \\propto 1/r\) and \(f \\propto \\sqrt{T}\), so the frequency of a string is raised by shortening it, by using a thinner string, or by increasing its tension.
A stretched string vibrating in one, two and three loops gives the principal note of frequency \(f\), the first subsidiary of frequency \(2f\) and the second subsidiary of frequency \(3f\); the frequencies are in the ratio \(1 : 2 : 3\) and the wavelengths \(2l\), \(2l/2\) and \(2l/3\), that is, in the ratio \(6 : 3 : 2\).
Damped vibrations are the periodic vibrations of a body of decreasing amplitude in the presence of a resistive force. The energy lost is changed into heat and dissipated in the medium, and the rate of decrease of amplitude depends on the nature of the medium and on the shape and size of the body.
Forced vibrations are the vibrations of a body under an external periodic force, called the driving force. The body vibrates with the frequency of the applied force, and the amplitude is small unless the two frequencies match.
Resonance is a special case of forced vibrations, occurring when the frequency of the applied periodic force is exactly equal to the natural frequency of the body. The body then vibrates with a very large amplitude and a loud sound is heard.
Resonance is shown by two identical tuning forks on sound boxes, by pendulums of equal length hung from the same elastic support, and by an air column whose length is adjusted until it resonates with a tuning fork, the next resonance coming at nearly three times that length.
Resonance explains the rattling of machine parts at a particular speed, the loud sound from the sound box of a stringed instrument, the order to soldiers to break step on a suspension bridge, and the tuning of radio and TV receivers.
Two sounds are distinguished by three characteristics: loudness, pitch and quality. Their wave patterns can be studied with a microphone and a cathode ray oscilloscope.
Loudness is the characteristic by which a loud sound is distinguished from a faint one of the same pitch and quality, and it depends on the amplitude of the wave. Loudness is subjective.
The intensity of a sound wave at a point is the sound energy passing per second normally through unit area at that point, measured in \(\\text{W m}^{-2}\). Intensity is objective. Ordinary sound has an intensity of nearly \(10^{-6}\\ \\text{W m}^{-2}\), and the minimum audible intensity is \(10^{-12}\\ \\text{W m}^{-2}\).
Loudness depends on five factors: the square of the amplitude, the inverse square of the distance from the source, the surface area of the vibrating body, the density of the medium, and the presence of resonant bodies.
Loudness and intensity are related by \(L = \\text{K} \\log\_{10} I\), and the sound level is \[L = 10 \\log\_{10}\\left(\\frac{I\_1}{I\_0}\\right)\\ \\text{decibel (dB)}\] where \(I\_1\) is the intensity in \(\\text{W m}^{-2}\) and \(I\_0 = 10^{-12}\\ \\text{W m}^{-2}\). The unit of loudness is the phon, and one decibel is the increase in level of loudness when the intensity increases by \(26%\).
Noise pollution is the disturbance produced in the environment by undesirable loud and harsh sound of level above \(120\\ \\text{dB}\), which can cause headache and permanent damage to the ears. The safe limit of sound level for hearing is \(0\\ \\text{dB}\) to \(80\\ \\text{dB}\).
Pitch is the characteristic by which a shrill note is distinguished from a flat note of the same loudness and quality, and it depends on the frequency: the higher the frequency, the higher the pitch. Pitch is subjective while frequency is objective.
Quality, or timbre, is the characteristic which distinguishes two sounds of the same loudness and pitch given by two different instruments, and it depends on the wave form. A tuning fork gives a pure note of a single frequency, while an instrument adds subsidiary vibrations of frequencies that are integer multiples of the principal vibration; the number of these subsidiary notes and their relative amplitudes decide the quality.
Music is a pleasant, continuous and uniform sound produced by regular and periodic vibrations, with a regular wave form and a level usually between \(10\\ \\text{dB}\) and \(30\\ \\text{dB}\). Noise is produced by an irregular succession of disturbances, has an irregular wave form, and is usually of level above \(120\\ \\text{dB}\).
Flowchart
FC 7.4 — From Free Vibration To Resonance Body disturbed and left alone: natural vibrations at its own natural frequency → same body vibrating in a medium: damped vibrations, amplitude falling as energy turns to heat → an external periodic force applied: forced vibrations at the frequency of the driving force → driving frequency made exactly equal to the natural frequency: resonance, with very large amplitude and a loud sound
7.14 Key Terms
- Amplitude — the maximum displacement of a particle of the medium on either side of its mean position; SI unit: metre (m).
- Audible range of frequency — the range of frequencies a human ear can hear, from 20 Hz to 20,000 Hz.
- Compressions — the regions of a longitudinal wave in which the particles of the medium are crowded together.
- Crests — the humps of a transverse wave, where the particles are displaced furthest to one side.
- Damped vibrations — the periodic vibrations of a body of decreasing amplitude in the presence of a resistive force.
- Decibel — the unit in which the level of sound is expressed; one decibel is one-tenth of a bel; symbol dB.
- Driving force — the external periodic force applied to a body to make it execute forced vibrations.
- Echo — the sound heard after reflection from a distant obstacle, after the original sound has ceased.
- Echo depth sounding — finding the depth of the sea by sending ultrasonic waves downwards and timing the returning echo.
- Electromagnetic waves — transverse waves formed by periodic vibrations of mutually perpendicular electric and magnetic fields, which can travel through vacuum.
- Forced vibrations — the vibrations of a body which take place under the influence of an external periodic force acting on it.
- Frequency — the number of vibrations made by a particle of the medium in one second; SI unit: hertz (Hz).
- Infrasonic sound — sound of frequency below 20 Hz, which is inaudible to human beings.
- Intensity — the sound energy passing per second normally through unit area at a point of the medium; SI unit: watt per metre squared (W m⁻²).
- Longitudinal wave — a wave in which the particles of the medium vibrate along the direction of propagation, forming compressions and rarefactions.
- Loudness — the characteristic by which a loud sound is distinguished from a faint one of the same pitch and quality; it depends on the amplitude of the wave; unit of loudness: phon.
- Mechanical waves — waves that need a material medium and travel through the elastic vibrations of its particles; sound waves are mechanical waves.
- Music — a pleasant, continuous and uniform sound produced by regular and periodic vibrations, having a regular wave form.
- Natural frequency — the frequency with which a body vibrates freely, decided by its own shape and size; SI unit: hertz (Hz).
- Natural period — the time period of the free vibrations of a body; SI unit: second (s).
- Natural vibrations — the periodic vibrations of a body in the absence of any external force on it; also called free vibrations.
- Noise — a sound produced by an irregular succession of disturbances, usually discontinuous and unpleasant to the ears.
- Noise pollution — the disturbance produced in the environment by undesirable loud and harsh sound of level above 120 dB.
- Phon — the unit of loudness, used to indicate a person’s perception of loudness; it agrees with the decibel scale at 1 kHz.
- Pitch — the characteristic by which a shrill note is distinguished from a flat note of the same loudness and quality; it depends on the frequency.
- Principal vibration — the vibration of the lowest frequency and maximum amplitude present in the sound from an instrument; also called the fundamental vibration.
- Quality (timbre) — the characteristic which distinguishes two sounds of the same loudness and pitch given by two different instruments; it depends on the wave form.
- Rarefactions — the regions of a longitudinal wave in which the particles of the medium are spread apart.
- Resonance — the special case of forced vibrations in which the frequency of the applied periodic force is exactly equal to the natural frequency of the body, so that it vibrates with a greatly increased amplitude.
- Resonant vibrations — the vibrations of large amplitude produced in a body at resonance.
- Reverberation — the prolonging of a sound caused by repeated reflections at the reflecting surfaces.
- Sound — the form of energy produced by a vibrating body, which reaches us through the vibrations of the particles of the surrounding medium.
- Sound level — the loudness level of a sound measured against the minimum audible intensity of 10⁻¹² W m⁻² at 1 kHz; SI unit: decibel (dB).
- Sound ranging — the process of detecting an obstacle, and finding its distance, from the sound waves reflected back by it.
- Subsidiary vibrations — the vibrations of small amplitude, of frequencies that are integer multiples of the principal vibration, present in the sound from an instrument; also called secondary vibrations.
- Time period — the time taken by a particle of the medium to complete one vibration; SI unit: second (s).
- Transverse wave — a wave in which the particles of the medium vibrate normal to the direction of propagation, forming crests and troughs.
- Troughs — the hollows of a transverse wave, where the particles are displaced furthest to the other side.
- Ultrasonic sound — sound of frequency above 20,000 Hz, which is inaudible to human beings.
- Ultrasonography — the imaging of human organs by the echo method using ultrasonic waves.
- Wave velocity — the distance travelled by a wave in one second; SI unit: metre per second (m s⁻¹).
- Wavelength — the distance travelled by a wave in one time period of vibration of a particle of the medium; SI unit: metre (m).
7.15 Points To Ponder
The ear holds on to a sound for about \(0.1\\ \\text{s}\) after it has ceased, and that is why an echo must arrive late to be heard separately. This looks like a defect of the ear. Can you think of what would be lost, in listening to speech or to music, if the sensation stopped the instant the sound did?
A hall in which every surface absorbed sound completely would have no echo and no reverberation at all. Would such a hall be a good place to hear a speaker in, or a bad one? Argue your answer from what decides the loudness reaching a listener.
Resonance can bring down a suspension bridge, and the same phenomenon makes a guitar loud enough to be worth listening to. What, physically, is different between the two situations, if the condition for resonance is the same in both?
Damping is deliberately added to a vehicle’s suspension and deliberately kept small in a musical instrument. In each case, what is being gained and what is being given up? Is there any device you can think of in which the right amount of damping matters more than having a lot or a little?
The decibel scale rises by only \(20\\ \\text{dB}\) when the intensity becomes a hundred times greater. Why might a scale that compresses such a huge range of intensities be a more useful description of what we actually hear than the intensity itself?
Bats hunt in the dark and dolphins hunt in murky water, and both use echoes rather than eyes. What is it about their surroundings that makes sound a better sense to rely on than light there? Would the same reasoning hold for an animal living in clear, brightly lit water?
7.16 Objective Questions
Multiple Choice Questions
Choose the correct answer from the options given below.
- (MCQ) A sound wave passes from air into water. The quantity that does not change is: (1 mark)
- speed
- wavelength
- frequency
- intensity
- (MCQ) Transverse mechanical waves can be formed in: (1 mark)
- gases only
- the interior of a liquid only
- solids and on the surface of liquids
- all media, including vacuum
- (MCQ) The speed of sound in air is not affected by a change in: (1 mark)
- temperature
- humidity
- pressure
- the nature of the gas
- (MCQ) For an echo to be heard distinctly in air at \(340\\ \\text{m s}^{-1}\), the reflector must be at least: (1 mark)
- \(1.7\\ \\text{m}\) away
- \(17\\ \\text{m}\) away
- \(34\\ \\text{m}\) away
- \(70\\ \\text{m}\) away
- (MCQ) Ultrasonic waves are preferred to audible sound for sound ranging because they: (1 mark)
- travel faster in the medium
- can be confined to a narrow beam
- are heard clearly by the operator
- need no medium to travel
- (MCQ) A body vibrating in air under no external force executes: (1 mark)
- forced vibrations
- resonant vibrations
- damped vibrations
- vibrations of constant amplitude
- (MCQ) In forced vibrations, the body finally vibrates with: (1 mark)
- its own natural frequency
- the frequency of the applied periodic force
- half the frequency of the applied force
- a frequency that keeps changing
- (MCQ) The energy lost by a body executing damped vibrations appears mainly as: (1 mark)
- light energy
- heat energy in the surrounding medium
- electrical energy
- potential energy of the body
- (MCQ) Two notes of the same loudness and the same pitch played on a sitar and on a flute differ in: (1 mark)
- amplitude
- frequency
- wave form
- wave velocity
- (MCQ) If the amplitude of a sound wave is doubled, its loudness becomes: (1 mark)
- half
- twice
- four times
- unchanged
- (MCQ) The safe limit of sound level for our ears is up to about: (1 mark)
- \(20\\ \\text{dB}\)
- \(80\\ \\text{dB}\)
- \(120\\ \\text{dB}\)
- \(140\\ \\text{dB}\)
- (MCQ) A singer holding one steady note can crack a thin glass placed nearby. The best explanation is that: (1 mark)
- the sound heats the glass
- the note matches the natural frequency of the glass, so it vibrates with large amplitude
- glass absorbs all sound waves falling on it
- the sound pushes the glass with a steady force in one direction
Assertion–Reason Questions
For each of the following, choose from these four options: (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.
(Assertion–Reason) Assertion (A): A reflected sound heard \(0.05\\ \\text{s}\) after the original sound is not called an echo. Reason (R): The sensation of a sound persists in our ears for about \(0.1\\ \\text{s}\). (1 mark)
(Assertion–Reason) Assertion (A): Ultrasonic waves travel faster than audible sound waves in the same medium. Reason (R): The speed of a wave in a medium depends on the properties of the medium. (1 mark)
(Assertion–Reason) Assertion (A): A tuning fork sounds much louder when its stem is pressed on a table top. Reason (R): The table top has a much larger vibrating surface area than the prongs of the fork. (1 mark)
(Assertion–Reason) Assertion (A): Loudness and intensity are two names for the same physical quantity. Reason (R): The intensity of a sound wave is measured in watt per metre squared. (1 mark)
Match The Following
- (Match) Match each item in Column X with the correct item in Column Y. (1 mark)
| Column X | Column Y |
|---|---|
| a. A longitudinal wave | i. A tuning fork stroked on a rubber pad and left to sound in air |
| b. A transverse wave | ii. Sound travelling through air |
| c. Damped vibrations | iii. A wave travelling along a stretched string |
| d. Resonance | iv. A tuning fork held over an air column of exactly the right length |
Fill In The Blanks
(Fill in the blank) The distance travelled by a wave in one time period is called its ……………. (1 mark)
(Fill in the blank) The prolonging of a sound by repeated reflections is called ……………. (1 mark)
(Fill in the blank) Resonance is a special case of ……………. vibrations. (1 mark)
(Fill in the blank) The quality of a musical sound depends on its ……………. (1 mark)
True Or False
(True/False) Sound waves can travel through vacuum. (1 mark)
(True/False) A reflecting surface must be polished before it can reflect sound. (1 mark)
(True/False) The pitch of a note depends on its frequency. (1 mark)
Name The Following
(Name the following) Name the phenomenon by which a bat locates an obstacle in the dark. (1 mark)
(Name the following) Name the device used on a ship to find the depth of the sea. (1 mark)
(Name the following) Name the characteristic of sound that lets you recognise a friend’s voice on the telephone. (1 mark)
Give One Word Or Phrase For
(One word or phrase) The vibrations of a body of decreasing amplitude in the presence of a resistive force. (1 mark)
(One word or phrase) Sound of frequency below \(20\\ \\text{Hz}\). (1 mark)
(One word or phrase) The sound energy passing per second normally through unit area at a point. (1 mark)
State The SI Unit Of
(SI unit) State the SI unit of frequency. (1 mark)
(SI unit) State the SI unit of intensity of sound. (1 mark)
(SI unit) State the SI unit of wave velocity. (1 mark)
Odd One Out
(Odd one out) Amplitude, frequency, wave form, echo. Give a reason for your choice. (1 mark)
(Odd one out) Ultraviolet rays, micro waves, ultrasonic waves, X-rays. Give a reason for your choice. (1 mark)
7.17 Check Your Concepts
(Subjective) Define the time period and the frequency of a wave, and write the relation between them. (2 marks)
(Subjective) State the two laws of reflection of sound. (2 marks)
(Subjective) Give a reason why a sound wave cannot travel through vacuum while a light wave can. (2 marks)
(Subjective) State the condition that a surface must satisfy if it is to reflect a sound wave, and name one device that uses the reflection of sound. (2 marks)
(Subjective) Why is the minimum distance needed to hear an echo greater in sea water than in air? (2 marks)
(Subjective) Differentiate between natural vibrations and damped vibrations, stating the basis of difference in each case. (3 marks)
(Subjective) Differentiate between loudness and intensity of sound, stating the basis of difference in each case. (3 marks)
(Subjective) State three properties of ultrasonic waves that make them suitable for sound ranging. (3 marks)
(Subjective) State the condition for resonance, and explain why a loud sound is heard at resonance. (3 marks)
(Subjective) Explain why a vibrating string by itself gives a weak sound, while the same string mounted on a musical instrument gives a loud one. (3 marks)
(Subjective) State the three conditions that must be satisfied for an echo to be heard distinctly, and explain the first of them. (4 marks)
(Subjective) Name the three characteristics of sound, and state the factor on which each of them depends. (4 marks)
(Subjective) What would happen if a troop of soldiers marched in step across a suspension bridge whose natural frequency happened to equal the frequency of their steps? Explain the physics, and state the precaution taken. (4 marks)
(Subjective) Describe, with the necessary steps, how the speed of sound in air can be determined by the method of echo, and write the relation used. (5 marks)
(Subjective) Explain what is meant by noise pollution, state its effect on a listener, and give the safe limit of sound level for hearing. (5 marks)
7.18 Diagrams, Graphs & Constructions
(Figure) Draw a labelled diagram of a wave to show its amplitude and its wavelength, as in Fig. 7.1-1. (2 marks)
(Figure) Fig. 7.1-4 shows the displacement–time graph of a particle of a medium. From a graph of this kind, explain how you would find (a) the amplitude and (b) the time period, and state why the wavelength cannot be read directly from it. (3 marks)
(Figure) Sketch the displacement–time graphs for (a) a natural vibration in vacuum and (b) a damped vibration, as in Fig. 7.6-3 and Fig. 7.7-1, and state one difference between the two graphs. (3 marks)
(Figure) Fig. 7.6-2 shows a string vibrating in one, two and three loops. Name the vibration in each case and write the ratio of their frequencies. (3 marks)
(Figure) Draw a labelled diagram of the apparatus used to demonstrate resonance in an air column, as in Fig. 7.9-3, and mark the air column whose length is varied. (3 marks)
(Figure) Draw two displacement–time traces on the same time scale to represent (a) a soft note and (b) a loud note of the same pitch, and state clearly how the two traces differ. (3 marks)
(Figure) Fig. 7.5-2 shows the principle of sonar. Name the parts marked as the transmitter and the receiver, state why they are placed close together, and write the relation used to find the distance of the obstacle. (3 marks)
7.19 Practice With Numericals
(Numerical) A sound wave of frequency \(425\\ \\text{Hz}\) travels in air with a speed of \(340\\ \\text{m s}^{-1}\). Calculate its wavelength. (2 marks)
(Numerical) A wave has a time period of \(0.005\\ \\text{s}\) and travels with a speed of \(340\\ \\text{m s}^{-1}\). Calculate (a) its frequency and (b) its wavelength. (3 marks)
(Numerical) A person standing in front of a cliff hears the echo of a shout \(1.5\\ \\text{s}\) after producing it. Taking the speed of sound in air as \(340\\ \\text{m s}^{-1}\), find the distance of the cliff from the person. (2 marks)
(Numerical) Calculate the minimum distance from a reflector at which a distinct echo can be heard in a liquid in which the speed of sound is \(1500\\ \\text{m s}^{-1}\). (2 marks)
(Numerical) A man standing \(40\\ \\text{m}\) from a tall wall fires a shot. Taking the speed of sound in air as \(320\\ \\text{m s}^{-1}\), calculate the time after which he hears the echo, and state with a reason whether the echo will be distinct. (3 marks)
(Numerical) Waves sent from a ship by sonar are received back from the sea bed after \(3\\ \\text{s}\). If the speed of ultrasonic waves in sea water is \(1400\\ \\text{m s}^{-1}\), find the depth of the sea. (2 marks)
(Numerical) A radar sends a signal towards an aeroplane at a distance of \(150\\ \\text{km}\) from it. If the signal travels with a speed of \(3 \\times 10^{8}\\ \\text{m s}^{-1}\), calculate the time after which the reflected signal is received. (3 marks)
(Numerical) Waves on the surface of water have a wavelength of \(25\\ \\text{cm}\) and travel with a speed of \(20\\ \\text{m s}^{-1}\). Calculate the frequency of the waves. (3 marks)
(Numerical) A sonometer wire of length \(80\\ \\text{cm}\) gives a note of frequency \(210\\ \\text{Hz}\). Keeping the tension unchanged, find the length of the wire that will give a note of frequency \(280\\ \\text{Hz}\). (3 marks)
(Numerical) An air column of length \(15\\ \\text{cm}\) in a pipe closed at one end resonates with a tuning fork of frequency \(660\\ \\text{Hz}\). Find the length of the air column that will resonate with a tuning fork of frequency \(495\\ \\text{Hz}\). (3 marks)
(Numerical) The amplitudes of two sound waves of the same pitch are in the ratio \(1 : 4\). Find the ratio of their loudness. (2 marks)
(Numerical) The intensity of a sound at a point is \(10^{-6}\\ \\text{W m}^{-2}\). Taking the minimum audible intensity at \(1\\ \\text{kHz}\) as \(10^{-12}\\ \\text{W m}^{-2}\), calculate the sound level at that point in decibel. (3 marks)
(Numerical) A person standing between two vertical cliffs fires a gun and hears the first echo after \(2\\ \\text{s}\) and the second echo after \(3\\ \\text{s}\). Taking the speed of sound in air as \(340\\ \\text{m s}^{-1}\), calculate the distance between the two cliffs. (4 marks)