Class 9 Science Chapter 10: Sound Waves: Characteristics and Applications

The periodic to and fro motion, or oscillation, of an object. It is what produces sound, and the sound lasts exactly as long as it does.

The object that produces the sound. It may be a string, a membrane, a column of air, or in a human being the vocal cords.

A U-shaped metal bar with a stem, of steel or aluminium, whose two sides are the prongs. Struck on a rubber pad it vibrates, which a water surface makes visible.

The material through which sound propagates. It may be a solid, a liquid or a gas, and without one no sound travels at all.

A space in which there is no medium, that is, no matter. Sound cannot propagate through it, as the bell jar experiment shows.

A rubber band is stretched across the open side of a cardboard box, the box is held steady, and the band is plucked. Sound is heard, and the band can be seen vibrating. Waiting until the vibration stops, the sound stops too. So as long as the band is stretched and vibrating sound is produced, and once the vibration ceases so does the sound, which establishes that sound is produced by vibrations.

Vibration is the periodic to and fro motion, or oscillation, of an object. Sound can be produced by vibrating strings, by vibrating membranes and by vibrating air columns, as in a bansuri where blowing sets the air inside the hollow pipe vibrating. Most musical instruments involve more than one vibrating part.

It comes from the shaking of the vocal cords, taut flaps of muscle set inside the larynx, the voice box in the throat. The vibration can be felt by touching the throat gently while speaking or singing. The tongue, lips, mouth and nasal cavity then help in converting that sound into speech or music.

A metal bar bent into a U and given a stem, steel usually, or aluminium; the two arms of the U go by the name of prongs, or tines. Held by the stem and struck gently against a rubber pad, it gives a sound that can be heard when brought near the ear. When a vibrating prong is touched to a water surface, waves form on the water. Both observations point to vibrating objects as the source of sound.

For the solid, a friend knocks gently on the far side of a desk while the listener hears it first with the ear in the air and then with the ear pressed against the desk and the other ear closed. The sound is heard through the table. For the liquid, two metal spoons are tapped together in the air and then submerged in a bucket of water, touching neither the sides nor the bottom, and tapped again. The sound is still heard, having travelled through water and then air.

The material through which sound propagates is called a medium, and it may be a solid, a liquid or a gas. A space in which there is no medium, that is, no matter at all, is referred to as a vacuum.

Put an electric bell under a bell jar, set it going, and note how loud it is. Draw the air off with a pump and the ringing grows weaker; pump long enough and next to nothing reaches you, though the clapper can plainly be seen still going. When air is let back in the sound can be heard again and gradually becomes as loud as before. The conclusion is that sound cannot propagate in a vacuum, and needs a medium.

In outer space there is a near vacuum, so sound cannot propagate there. Astronauts in spacesuits doing spacewalks therefore cannot directly hear each other speak, nor hear sounds such as two metal objects clanking together. They communicate through special devices fitted into their spacesuits. Shouting would not help, since the difficulty is the absence of a medium rather than the faintness of the sound.

Two activities make the case, and they make it in different ways.

The first uses a rubber band stretched across the open side of a cardboard box. Holding the box steady, the band is plucked with a finger. A sound is heard, and the band can be seen vibrating. The decisive step is to let it come to rest: once the band stops vibrating, the sound stops with it. As long as the band is stretched and vibrating, sound is produced, and no longer.

From this we conclude that sound is produced by vibrations, where vibration means the periodic to and fro motion, or oscillation, of an object.

The second uses a tuning fork, a U-shaped metal bar with a stem whose sides are called prongs. Held by the stem and struck gently on a rubber pad, it gives a sound when brought near the ear. Then a vibrating prong is touched to a water surface, and waves form on the water.

That second observation matters because it is independent of hearing. The ear says there is a sound; the water says there is a vibration; and the two agree.

The same account covers every other source. A pulled string or a struck metal object vibrates, and blowing through a bansuri sets the air inside the hollow pipe vibrating. In humans it is the vocal cords, tightly stretched muscular flaps inside the larynx, that vibrate, which can be felt by touching the throat while speaking.

Sound travelling through air is a matter of everyday experience, so the two activities set out to test the other two states.

For a solid, two people stand on opposite sides of a desk. One knocks or scratches gently on the desk while the other listens with the ear in the air. Then the listener places an ear against the desk, closes the other ear, and listens again. The sound is heard through the table, which shows that sound can travel through solids. Closing the other ear matters, since it removes the path through the air and leaves only the path through the wood.

For a liquid, a large tub or bucket is filled to the brim and two metal spoons are tapped together in the air. Then both spoons are submerged, touching neither the sides nor the bottom of the bucket, and tapped again. The sound is still heard, having reached the listener through the water and then the air. Keeping the spoons clear of the bucket matters, since otherwise the sound might have travelled through the bucket instead.

Put these beside the everyday experience of hearing through air, and it is clear that solid, liquid or gas will each carry sound.

Whatever it makes its way through is called the medium, and there must be one between a source and a listener.

An electric bell is kept inside a bell jar and switched on, and the loudness of its sound is noted. A vacuum pump then begins to draw the air out of the jar.

Draw the air off and the ringing grows weaker. Pump long enough and next to nothing reaches you. Then air is let back into the jar, and the sound can be heard again, gradually becoming as loud as before.

The experiment shows that sound cannot propagate in a vacuum. Sound needs a medium to propagate, and that medium can be a solid, a liquid or a gas.

Two details make the argument tight. The first is that the bell can still be seen ringing while almost nothing is heard. That rules out the obvious alternative explanation, namely that the bell had stopped working, since the eye reports that it has not. The second is that the change is reversible: letting the air back in brings the sound back. A change that comes and goes with the one thing being altered points firmly at that thing.

It also settles a question about space, which is nearly empty: two astronauts outside their craft cannot hear one another's voices, nor the clank of metal on metal, without radio. They communicate through special devices fitted into their spacesuits, because no amount of loudness can compensate for having nothing at all to carry the sound.

The topic answers two questions in turn: what produces a sound, and how it gets to a listener.

On the first, a rubber band stretched across a box gives sound only while it is vibrating, and the sound stops the moment the vibration does. A tuning fork confirms it twice over, since a struck prong is both heard by the ear and seen to make waves when touched to water. So sound is produced by vibrations, vibration being periodic to and fro motion. The vibrating thing may be a string, a membrane or a column of air, and in a human being it is the vocal cords inside the larynx. Whatever it is, it is called the source.

On the second, sound plainly travels through air, and two activities extend that. A knock on a desk is heard with an ear pressed to the wood, so solids carry sound. Two spoons tapped together under water are still heard, so liquids do too. The material that carries it is called a medium.

The last step removes the medium altogether. In the bell jar, as the air is pumped away the bell grows fainter until almost nothing is heard, although it can still be seen ringing, and the sound returns when air is let back in. So sound cannot propagate in a vacuum.

Hence two astronauts on a spacewalk, surrounded by near vacuum, must speak through devices in their suits.

A region of the medium whose density is higher than the average. It moves forward by collisions between particles, although the particles themselves do not travel with it.

A region of the medium whose density is lower than the average. Like a compression it moves ahead while the particles only oscillate about their mean positions.

A travelling disturbance of crowded and thinned regions following one another through a material, with none of that material carried along with it.

A wave in which the particles vibrate parallel to the direction of wave propagation. Sound is one. In a transverse wave they vibrate perpendicular to it.

A wave that requires a material medium to propagate. Sound is a mechanical wave; light, which travels through vacuum, is not.

A mark is made on one turn and the slinky is laid out horizontally, one end held fixed while the other is pushed sharply forward and pulled quickly back, repeatedly. Regions appear where the turns are closer together and others where they are more spread out, and these regions travel along the slinky. The mark, though, goes nowhere: it merely swings to and fro about the place it started from, along the same line the disturbance runs. Every coil, then, does the same about its own place, while the disturbance itself crosses from end to end.

Drive the piston forward and it shoves the air just ahead of it along, squeezing that air together so that a thin region next to the piston becomes denser than the rest. That region of higher than average density is a compression. The crowded particles knock into those in front, which hands the squeeze along; those knock into their own neighbours, and so on — so the squeeze itself advances through the air, while no particle of the air goes along with it.

A sound wave is squeezes and thinnings following one another along through a medium, with nothing of the medium itself flowing anywhere. What travels is the disturbance, and with it the energy. What does not travel is the particles, which only vibrate about their mean positions.

In a longitudinal wave the particles vibrate along the same line as the wave travels, while in a transverse wave they vibrate perpendicular to it. Sound is longitudinal, its shaking of the medium running along the very line the disturbance is travelling on.

A wave that requires a material medium for propagation is called a mechanical wave, and a sound wave is one type. Light fails the test: its shaking runs across the line of travel, and it needs no medium at all — which is how the Sun and the stars are visible from here.

A cellophane or rubber sheet is stretched tightly over a wide-mouthed container and fixed, grains are sprinkled evenly over it, and a loud sound is produced near the container without touching it. The grains move or jump. This happens because as sound propagates through the air it reaches the sheet and makes it vibrate, and that vibration moves the grains. The activity shows that sound is a form of energy.

A microphone converts sound energy into electrical energy: when one speaks or sings into it, the sound waves make a thin membrane called a diaphragm vibrate, and those vibrations are converted into an electrical signal. A speaker runs the process backwards: an electrical signal shakes a cone or diaphragm within it, and sound comes out. Provided nothing in the chain misbehaves, what you hear is very close to what went in.

Freeze the wave at one instant and the density rises and falls again and again as you move away from the source; the graph puts distance along the x-axis and density up the y-axis, with a dashed horizontal line to mark the average. In a compression the density rises above that average, and the highest point, called the crest, represents the maximum density. In a rarefaction it falls below, and the lowest point, called the trough, represents the minimum density.

The slinky comes first, as an analogy for the medium. A mark is made on one turn, the slinky is laid out horizontally with one end held fixed, and the other end is given a sharp push and quickly pulled back, repeatedly.

What is seen is that some regions have their turns closer together than usual and others more spread out, and these regions travel along the slinky. Everything turns on what the mark does: it goes nowhere along the slinky, but swings to and fro about the place it started from, along the same line the disturbance runs.

Air behaves the same way, with the closeness or spreading of the turns standing for higher or lower density. Picture a long tube of air with a piston at one end, open at the other, the piston being an idealised model of a source.

With the piston still, the air has a uniform average density. As the piston moves forward it pushes the nearby particles together, raising the density in a small region close to it, which is a compression. Those compressed particles collide with the particles further ahead, passing the compression forward, and so on, so that the compression moves through the air although the particles do not travel with it. As the piston moves back, the particles move back too and a region of lower density forms close to it, a rarefaction, which likewise moves ahead by collisions.

As the piston oscillates, compressions and rarefactions are produced alternately and travel away. That series of alternating compressions and rarefactions, propagating without any actual flow of the medium's particles, is a sound wave.

Sound belongs to two classifications at once, and both are worth stating carefully.

The first concerns the direction in which the particles move. In sound, the shaking of the medium runs along the very line the disturbance is travelling on. Waves in which the particles vibrate parallel to the direction of wave propagation are known as longitudinal waves, so sound is a longitudinal wave. The contrast is with a transverse wave, in which the particles vibrate perpendicular to the direction of wave propagation.

The second concerns whether a medium is needed. Without a material medium sound waves cannot propagate, and waves that require a material medium are called mechanical waves. A sound wave is a type of mechanical wave, and mechanical waves themselves are of two types, longitudinal and transverse.

Both kinds occur together in nature. Earthquakes produce seismic waves travelling through the Earth which may be longitudinal or transverse, and the longitudinal ones are the first to be detected by seismographs.

Not every wave is mechanical, however. Light is a transverse wave that can travel through vacuum, which is exactly why light from the Sun and other stars reaches the Earth. So light and sound sit at opposite corners of the classification: one transverse and needing no medium, the other longitudinal and stopping dead without one.

The evidence comes from an activity in which nothing touches the thing that moves.

A cellophane or rubber sheet is stretched tightly over the mouth of a wide container and fixed with tape or a rubber band. Grains such as rice, semolina, salt or chalk powder are sprinkled evenly over it, not clumped together. A loud sound is then produced near the container, without touching either the sheet or the container.

The grains move or jump. Since the source never touched anything, whatever made them move must have arrived through the air. As the sound propagates it reaches the sheet and makes it vibrate, and that vibration is what moves the grains.

So sound is a form of energy. A shaking source hands energy to the medium around it, and as the wave runs on, the shaking of the medium and the knocking of one part against another carry that energy along.

One point must be stated precisely. What a sound wave carries from place to place is energy; the medium itself stays where it is. The particles only vibrate about their mean positions, as the mark on the slinky showed. So when sound travels from a tuning fork to your ear, what reaches your ear is the energy carried by the wave, and not air particles from beside the fork.

Microphones put this to work, converting sound energy into electrical energy through a vibrating diaphragm.

Freeze a travelling sound wave at one instant and look along it: the density rises and falls again and again as you move away from the source. That is the fact the graph records.

Distance is plotted on the x-axis and the density of the medium on the y-axis. The average density, which is the uniform density the medium would have with no wave passing, is marked as a horizontal dashed line. The curve then varies above and below that line as distance changes.

Each feature answers to something real. Where the curve rises above the average line, the medium is denser than average, which is a compression. The highest point of that rise represents the maximum density, and that highest point is called the crest. Where the curve falls below the average line, the medium is thinner than average, which is a rarefaction. The lowest point represents the minimum density, and it is called the trough.

So reading the graph is a matter of translating: crest for the compression, trough for the rarefaction, and the dashed line for the undisturbed state.

One caution is worth keeping. A crest is a point on a drawing while a compression is a region of the medium, so the two are related but not identical.

There is also a second way of graphing the same wave: plotting the variation of density with time, at one particular location in the medium, rather than with distance at one instant.

The distance between two consecutive crests, or two consecutive troughs. Written $\lambda$ and measured in metres.

How many such oscillations a fixed spot gets through in a second. Written $\nu$ and measured in hertz, which is one per second.

However long one full oscillation takes at a fixed spot. Written T and measured in seconds, with $\nu = \frac{1}{T}$.

How far the densest point of a squeeze, or the thinnest point of a thinning, stands from the average density. Larger amplitude means more energy carried.

The sound energy passing through unit area perpendicular to the propagation direction in unit time. It falls with distance because the same energy spreads over a larger area.

The wavelength of a wave is the distance between two consecutive crests, or between two consecutive troughs. It is usually represented by $\lambda$, the Greek letter lambda, and its SI unit is the metre.

How many such oscillations a fixed spot gets through in a second is the frequency, written $\nu$ and counted in hertz, one per second. However long one full oscillation takes at a fixed spot is the time period, written T and measured in seconds. The two are inversely related, so that $\nu = \frac{1}{T}$ and a shorter time period corresponds to a higher frequency.

It is the density at one spot going from highest down to lowest and back up to highest, or the other way about. The detail matters because half of that, a change from maximum to minimum alone, is not one oscillation, and counting it as one would double every frequency calculated.

The frequency is the number of oscillations divided by the time taken, $\frac{10}{2} = 5$ Hz. The time taken for a single oscillation is $\frac{2}{10} = 0.2$ s, which is the time period. As a check, $\frac{1}{0.2} = 5$ Hz, which agrees with the frequency found.

The amplitude is how far the densest point of a squeeze, or the thinnest point of a thinning, stands from the average density; the bigger that gap, the bigger the amplitude. A wave with a larger amplitude carries more energy than one with a smaller amplitude.

Hit the plate more heavily and the medium around it receives more energy, so each part of it is driven further from where it normally sits. The sheet therefore swings further too, and the grains leap higher than they did. A harder strike therefore produces a larger amplitude and visibly more energy delivered.

The sound energy through one square metre held square across the line of travel, in a second.

The further a sound wave gets from its source, the wider the front it has to cover. No energy is lost on the way, so the same quantity has to stretch across that wider front; less goes through any one patch, and the intensity falls off with distance. It follows that sounds produced with a larger initial amplitude, carrying more energy, can travel a greater distance before their intensity reduces to zero.

The first is a distance. The wavelength of a wave is the distance between two consecutive crests, or two consecutive troughs. It is usually represented by $\lambda$, the Greek letter lambda, and its SI unit is the metre. The word consecutive matters, since crest to trough is only half a wavelength.

The other two concern time, and both depend on what happens at one fixed position as the wave goes past. There the density changes alternately between a maximum, which is a crest, and a minimum, which is a trough. Following the density from highest down to lowest and back up to highest, or the other way about, is one full oscillation.

How many such oscillations a fixed spot gets through in a second is the wave's frequency, written $\nu$ — the Greek nu — and counted per second, a unit also called the hertz. However long one full oscillation takes at a fixed spot is the time period, written T and measured in seconds.

Those two are inversely related, since a shorter time period corresponds to a higher frequency:

$$\nu = \frac{1}{T}$$

A worked case checks both at once. If ten density oscillations occur in 2 seconds, the frequency is $\frac{10}{2} = 5$ Hz and the time period is $\frac{2}{10} = 0.2$ s, and indeed $\frac{1}{0.2} = 5$.

Sound propagates as density oscillations of the medium, through compressions and rarefactions. Amplitude asks how large those oscillations are.

The amplitude is how far the densest point of a squeeze, or the thinnest point of a thinning, stands from the average density. The bigger that gap, the bigger the amplitude. Notice that it is defined as a change measured from the average, not as a density in its own right.

Amplitude and energy go together: widen the amplitude and the wave is carrying more energy, narrow it and it is carrying less. The grains activity shows this directly. Striking the plate harder transfers more energy to the particles of the surrounding medium, causing larger displacements from their mean positions, so the sheet vibrates further and the grains jump higher.

Intensity then asks how much of that energy arrives at a given place. The sound energy through one square metre held square across the line of travel, in a second, is called the intensity.

The further the wave gets from its source, the wider the front it has to cover. No energy is lost on the way, so the same quantity has to stretch across that wider front, and the intensity falls off as you go further out. Nothing is destroyed; it is shared out.

Hence a sound of larger initial amplitude, carrying more energy, travels a greater distance before its intensity reduces to zero.

The activity is a classroom demonstration, best run as a group with the teacher facilitating.

An app such as Phyphox, which can identify the frequencies of sounds, is opened at its Audio Spectrum option, which displays the frequency graphically or in hertz. The notes Sa, Re, Ga, Ma, Pa, Dha, Ni, Sa are then sung one after another, or produced with a tone-generating app on another phone, and the frequency is watched as each note is produced.

The approximate frequency of each note is recorded. The notes are then compared by taking the ratio of each frequency to that of Sa, to see whether a pattern appears. If both voice and app-generated notes are used, the frequencies for the same note can be compared between them.

What is found is that the frequency is lowest for Sa and gradually increases for the other notes, and that each musical note has a distinct frequency, which is what makes it sound different from the rest.

The activity also sets the context for a point made just before it. Everyday sounds usually contain a mixture of many frequencies, and nearly single frequency sounds can be made only by particular means, such as striking a tuning fork or whistling carefully. Indeed, careful whistling into the app's Audio Scope often produces the graph of almost a single frequency sound.

Being the same note, they share the quantities that describe the pattern. Each musical note has a distinct frequency, so both have the same frequency, the same number of density oscillations each second at any fixed point. Since frequency and time period are inversely related by $\nu = \frac{1}{T}$, both also have the same time period. And their wavelengths, the distances between consecutive crests, are alike too.

What differs is amplitude. The amplitude is how far the densest point of a squeeze, or the thinnest point of a thinning, stands from the average density, and the louder sound is the one with the bigger gap. Since a wave of larger amplitude carries more energy than one of smaller amplitude, the loud sound sets out with more energy than the soft one.

As both travel, each spreads over a larger area. Energy must be conserved, so for each the same amount of energy comes to be spread over a growing surface, and the intensity, which is the energy passing through unit area perpendicular to the propagation in unit time, decreases with distance for both.

But they do not fade alike. The loud sound started with more energy, so it can travel a greater distance before its intensity reduces to zero, while the soft one becomes inaudible sooner.

At no point is energy destroyed in either case. Both simply spread it thinner.

The distance a point on a wave, such as a crest, travels in unit time. It equals wavelength times frequency, $v = \lambda\nu$.

How frequency is perceived by humans. High pitch sounds generally have higher frequency, though the exact relation is complicated.

How the amplitude of a sound wave is perceived. It differs from intensity, which is measurable, since loudness depends on the listener's hearing.

A tone is a sound of a single frequency. A musical note combines a fundamental with higher frequencies called overtones.

The quality that makes a flute, an ektara and a tabla sound different on the same note at the same loudness. It comes from the pattern and intensity of the overtones.

The speed of sound is the ground one point on the wave, a crest or a trough, covers in one second. Fix the frequency and the gap from crest to crest is one wavelength $\lambda$, crossed in one time period $T$. So $v = \frac{\lambda}{T}$, and since $\nu = \frac{1}{T}$, this gives $v = \lambda \times \nu$: speed equals wavelength times frequency.

It depends on the medium through which it travels, being fastest in solids, slower in liquids and slowest in gases. Sound is about 4 to 5 times faster in water and typically 15 to 20 times faster in solids than in air. In air it also depends on temperature and humidity, rising as either increases. In dry air it is about 331 m s⁻¹ at 0 °C and nearly 344 m s⁻¹ at 22 °C.

In most media such as air the speed depends only on the medium and not on the source or the frequency. So altering the source's frequency alters the wavelength in the medium and leaves the speed where it was. A few unusual materials — porous foams, purpose-built structures — do not follow this.

Light travels at about 300000 km s⁻¹ and is taken to arrive almost instantaneously, so the whole of the delay is the time the sound took. With the speed of sound in air as 340 m s⁻¹, the distance is $v \times t = 340 \times 5 = 1700$ m, or about 1.7 km.

Period, wavelength, count per second, the size of the density swings and the rate of travel are all set down precisely and can be read off an instrument. What a listener makes of a sound is another matter, described by words such as loudness and pitch which answer only loosely to those figures.

Pitch is what the ear makes of a sound's count per second. Anything striking us as shrill, a whistle or a siren, counts as high in it; a rumble of thunder or of an aircraft counts as low. Broadly the shrill run faster and the deep slower, but setting down the exact relation is no simple matter.

Human ears reach from 20 Hz up to 20000 Hz — 20 kHz — though no two people are quite alike in this and the upper end retreats as one gets older. Below the bottom of that band the waves are called infrasonic; above the top of it, ultrasonic. Humans cannot hear either, though dogs, cats, bats and dolphins detect ultrasound and elephants detect infrasound.

A tone is a sound of a single frequency, such as that of a tuning fork or oral whistling. Pluck a tanpura string, or sing, and what comes out is several frequencies at once: a lowest one, the fundamental, with overtones stacked above it, the mixture being what makes the sound full and agreeable. Timbre is what makes a flute, an ektara and a tabla each unmistakable on one note at one loudness. It traces back to how each is shaped, what it is made of and how it is put together, all of which fix which overtones appear and how strongly.

What the speed of sound measures is the rate at which the disturbance crosses a medium, or, put another way, how quickly the squeezes and thinnings get from one place to another. It is the ground one point on the wave, a crest or a trough, covers in one second.

The relation follows from that definition. For a wave of given frequency the distance between two consecutive crests is one wavelength $\lambda$, and the disturbance covers that distance in one time period $T$. Since speed is distance over time, $v = \frac{\lambda}{T}$, and since frequency is the reciprocal of the time period, this becomes $v = \lambda \times \nu$.

What the speed depends on is the medium. It is fastest in solids, slower in liquids and slowest in gases: about 4 to 5 times faster in water and typically 15 to 20 times faster in solids than in air. Within air it also depends on temperature and humidity, increasing as either rises, so that dry air carries sound at about 331 m s⁻¹ at 0 °C and nearly 344 m s⁻¹ at 22 °C.

What it does not depend on, in most media, is the source or the frequency. Alter the source's frequency and the wavelength in the medium alters, the speed staying where it was. A few unusual materials — porous foams, purpose-built structures — do not follow this.

That last point matters musically. Were the speed to depend on frequency, a singer's notes and an instrument's would reach a listener out of step.

The first asks for the wavelengths at the edges of human hearing, taking the speed of sound in air as 344 m s⁻¹. Rearranging $v = \lambda \times \nu$ gives $\lambda = \frac{v}{\nu}$. At 20 Hz this is $\frac{344}{20} = 17.2$ m. At 20 kHz, which is 20000 Hz, it is $\frac{344}{20000} = 0.0172$ m, or 1.72 cm. A thousandfold rise in frequency has brought a thousandfold fall in wavelength, exactly as an inverse relation demands.

The second asks how far away lightning struck, given a 5 s delay between the flash and the thunder and a speed of sound of 340 m s⁻¹. Light travels at about 300000 km s⁻¹ and is taken to arrive almost instantaneously, so the whole delay belongs to the sound. Distance is $v \times t = 340 \times 5 = 1700$ m, about 1.7 km.

The third gives a wave in steel of wavelength 50 m, with the speed of sound in steel 5000 m s⁻¹. Its frequency is $\nu = \frac{v}{\lambda} = \frac{5000}{50} = 100$ Hz, and its time period is $T = \frac{1}{\nu} = \frac{1}{100} = 0.01$ s.

One method runs through all three: identify which two of the three quantities are given, and rearrange the single relation to find the third.

The physical properties of sound are well defined and can be measured: time period, wavelength, frequency, amplitude and speed. How we experience sound, by contrast, is subjective, and the terms that describe that experience do not have simple relations with the physical properties.

Pitch is the first of them. How frequency is perceived by humans is called pitch. A whistle or a siren strikes the ear as thin and piercing, and we call such a sound high; thunder, or an aircraft going over, we call low. Broadly the high ones are the higher in frequency, though writing down the exact relation is not simple.

Loudness is the second. Humans perceive the amplitude of a sound wave as loudness, larger amplitude being heard louder, and the loudness decreases as one moves farther from the source. A sharp line is wanted here. The two get used for one another in ordinary speech, but intensity is something an instrument reads off, while loudness belongs to whoever is listening and how well they hear.

Perception is also limited. Humans can hear only from 20 Hz to 20 kHz, a range that varies between people and decreases with age. Below it lie infrasonic waves and above it ultrasonic ones, neither of which we hear, though dogs, cats, bats and dolphins detect ultrasound and elephants detect infrasound.

And two sounds may match on every measurable count and still be told apart. That is timbre.

Begin with the simplest case. A tone carries one frequency only, of the sort a tuning fork gives, or a whistle. Nothing else is present in it.

A musical note is not like that. A plucked tanpura string, or singing, produces a combination of a lowest frequency, called the fundamental, together with higher frequencies called overtones. It is that combination which creates a rich and pleasant sound, and Indian string instruments such as the sarangi, sitar and veena often use extra strings precisely to enrich it.

Now the puzzle. Take a flute, an ektara and a tabla, and have each play the same note at the same loudness. Every measurable comparison is matched, and yet each still sounds unique.

The quality responsible is called timbre. It comes from the instruments' shape, material and construction, which determine the pattern and intensity of their overtones. Two instruments may share a fundamental and differ entirely in what sits above it, and the ear hears the difference at once.

One further term completes the picture. An octave is the step between two notes whose fundamentals stand in the ratio two to one, 200 Hz against 400 Hz for instance.

Indian drums add their own device. The black patch at the centre of a tabla or mridangam head, the syaahi, alters the vibration of the membrane, giving a rich variety of sounds and a tonal control rarely found elsewhere.

Sound bouncing off an obstacle such as a solid or a liquid, following the same laws as light: equal angles with the normal, and all three directions in one plane.

A reflected sound heard separately from the original. It needs a gap of at least 0.1 s, which sets a minimum distance of 17 m to the reflecting surface.

The persistence of sound after the source has stopped, caused by multiple reflections arriving with a time difference of less than 0.05 s.

Locating objects using reflected sound waves. Bats emit short bursts of ultrasound and read the echoes; dolphins, whales and some birds do the same.

Sound navigation and ranging: ultrasonic waves sent into water and their reflections analysed to find the distance, direction and speed of underwater objects.

Put something solid or liquid in a sound wave's path and the wave comes back off it; that is reflection. It obeys the rules light obeys: measure from the normal where the sound strikes and the angle going in equals the angle coming out, with normal, incoming and outgoing all in one plane.

An echo is the sound heard again after it has reflected off the hard surface of a distant object and come back, as when one shouts near a mountain, a cliff or in a long corridor. Shout in a small room and what comes off the walls is back before the ear can tell it apart from the shout itself. Two sounds are heard as two only if a tenth of a second or more separates them.

The gap must be at least 0.1 s, and in that time sound travels $340 \times 0.1 = 34.0$ m. But that 34 m is the distance from the source to the reflecting surface and back again, so the minimum echo distance is half of it, which is 17 m.

Echoes are stronger from hard and smooth surfaces, which reflect sound well. Soft surfaces such as curtains tend to absorb sound, so little comes back, while rough surfaces scatter it in different directions rather than returning it as a beam. In those two cases echoes cannot be heard clearly.

In a large hall or auditorium the emitted sound can undergo multiple reflections from the walls, making it persist after the source has stopped emitting, which is called reverberation. It occurs when reflections arrive with a time difference of less than 0.05 s. An echo differs in being heard as a separate sound, which requires a gap of at least 0.1 s.

A modern auditorium or concert hall is built with its reverberation in mind, so that wherever anyone happens to be sitting, speech and music reach them clean and undistorted. What is reduced is the unwanted kind, using curtains, upholstered chairs, sound absorbing panels and other soft porous surfaces, since without them the sound could become garbled.

Infrasonic waves, below 20 Hz, are used to detect natural events like earthquakes and volcanic eruptions, and for detecting severe storms, since they travel long distances through air and through the Earth. Beyond 20 kHz, ultrasound earns its keep in several trades. It locates an object by the echo it sends back, finds a flaw buried inside a metal block, and cleans or welds delicate machine parts. It also shatters a kidney stone into pieces small enough to pass, and produces an image of an organ with no incision at all.

Most bats emit short bursts of ultrasonic waves which reflect from nearby objects, and by sensing the echoes a bat determines the position of obstacles and prey. It can fly and hunt in the dark without colliding. Finding things by listening to your own sound come back is called echolocation, and dolphins, whales and certain birds do it as well. Humans have adapted the same principle in sonar, which stands for sound navigation and ranging, sending ultrasonic waves into water and analysing the reflections to find the distance, direction and speed of underwater objects.

Sound waves can bounce off obstacles like solids or liquids, which is the reflection of sound, and it follows the same laws as the reflection of light. The most common example is the echo.

Shout near a cliff, a mountain or down a long corridor and the voice may come back after some time. It happens because your sound has been turned around by some hard surface a good way off.

But echoes are not heard everywhere. Shout in a small room and what comes off the walls is back before the ear can tell it apart from the shout itself. The rule is that two sounds are heard as two only if a tenth of a second or more separates them, and that inside that interval they run together.

That 0.1 s fixes a distance. Taking the speed of sound as 340 m s⁻¹, in 0.1 s sound travels $340 \times 0.1 = 34.0$ m. But 34 m is the distance travelled from the source to the reflecting surface and back, so the minimum echo distance is half of it, 17 m.

The surface matters too. Echoes are stronger from hard and smooth surfaces, which reflect sound. Anything soft, curtains for instance, swallows sound rather than returning it, and anything rough throws it off every which way; neither gives a clean echo.

In a large hall or auditorium, the emitted sound can undergo multiple reflections from the walls. Those multiple reflections make the sound persist after the source has stopped emitting, and that phenomenon is called reverberation. It occurs when sound reflected from surfaces returns with a time gap of less than 0.05 s.

The contrast with an echo is a matter of timing, and the two thresholds must be kept apart. An echo needs a gap of at least 0.1 s, because only then can the brain separate the reflection from the original sound, and it is heard as a distinct second sound. Reverberation involves gaps shorter than 0.05 s, so nothing is heard as separate; instead the original sound seems to go on.

The design consequence is not what one might expect. A modern auditorium or concert hall is built with its reverberation in mind, so that wherever anyone happens to be sitting, speech and music reach them clean and undistorted. The aim is control rather than abolition.

What is removed is the unwanted kind, which would otherwise lead to a garbled sound. Curtains, upholstered chairs, sound absorbing panels and other soft, porous surfaces all reduce it.

Medieval architects understood these matters too. The Whispering Gallery of the Gol Gumbaz in Bijapur, Karnataka, is designed so that even a faint whisper is heard multiple times across the large dome.

Frequencies that fall beyond what the ear can register turn out to be of real use in science, in medicine and in engineering, and they fall into two sets.

Infrasonic waves, below 20 Hz, are used to detect natural events like earthquakes and volcanic eruptions, and for detecting severe storms. What makes them suitable is that such waves travel long distances through air and through the Earth, so an event far away can still be registered.

Ultrasonic waves, above 20 kHz, have a longer list. Medicine uses them to look at organs without cutting anyone open — ultrasonography — and to shatter a kidney stone into fragments small enough to leave the body on their own. In industry they are used for ultrasonic welding and for cleaning delicate machine parts, and to find flaws inside metal blocks in construction and industrial testing. And they are used for locating objects, by reflecting the sound waves off them.

That last application is the basis of echolocation and of sonar. Most bats emit short bursts of ultrasonic waves which reflect from nearby objects, and by sensing those echoes a bat determines the position of obstacles and prey. Dolphins, whales and some birds do the same.

Humans adapted it as sonar, sound navigation and ranging: ultrasonic waves sent into water, and the reflections analysed for the distance, direction and speed of submarines or shipwrecks.

Every calculation here involves a sound that travels out to a reflecting surface and comes back, so the measured time covers a double journey. Either the time must be halved before multiplying, or the distance halved after. That single step is what all of them share.

The first is the minimum echo distance. Two sounds are heard separately only if they arrive at least 0.1 s apart, and at 340 m s⁻¹ sound covers $340 \times 0.1 = 34$ m in that time. Halving gives 17 m as the closest a reflecting surface may be.

The second is a clap in a corridor with an echo after 0.5 s. The distance to the wall is $\frac{340 \times 0.5}{2} = 85$ m.

The third sets the requirement the other way round: echoes must arrive at least 0.2 s after emission, with the speed 343 m s⁻¹, so the surface must be at least $\frac{343 \times 0.2}{2} = 34.3$ m away.

The fourth is a naval sonar signal returning after 0.90 s in seawater at 1530 m s⁻¹. Halving the time gives 0.45 s each way, and $1530 \times 0.45 = 688.5$ m.

The fifth finds an ocean depth from a signal taking 4 s to return at 1500 m s⁻¹: 2 s each way gives $1500 \times 2 = 3000$ m.

Forget the halving and every answer comes out twice too large.

A rubber band is stretched across the open side of a cardboard box, the box is held steady, and the band is plucked. Sound is heard, and the band can be seen vibrating. Waiting until the vibration stops, the sound stops too. So as long as the band is stretched and vibrating sound is produced, and once the vibration ceases so does the sound, which establishes that sound is produced by vibrations.

Vibration is the periodic to and fro motion, or oscillation, of an object. Sound can be produced by vibrating strings, by vibrating membranes and by vibrating air columns, as in a bansuri where blowing sets the air inside the hollow pipe vibrating. Most musical instruments involve more than one vibrating part.

It comes from the shaking of the vocal cords, taut flaps of muscle set inside the larynx, the voice box in the throat. The vibration can be felt by touching the throat gently while speaking or singing. The tongue, lips, mouth and nasal cavity then help in converting that sound into speech or music.

A mark is made on one turn and the slinky is laid out horizontally, one end held fixed while the other is pushed sharply forward and pulled quickly back, repeatedly. Regions appear where the turns are closer together and others where they are more spread out, and these regions travel along the slinky. The mark, though, goes nowhere: it merely swings to and fro about the place it started from, along the same line the disturbance runs. Every coil, then, does the same about its own place, while the disturbance itself crosses from end to end.

Drive the piston forward and it shoves the air just ahead of it along, squeezing that air together so that a thin region next to the piston becomes denser than the rest. That region of higher than average density is a compression. The crowded particles knock into those in front, which hands the squeeze along; those knock into their own neighbours, and so on — so the squeeze itself advances through the air, while no particle of the air goes along with it.

A sound wave is squeezes and thinnings following one another along through a medium, with nothing of the medium itself flowing anywhere. What travels is the disturbance, and with it the energy. What does not travel is the particles, which only vibrate about their mean positions.

The wavelength of a wave is the distance between two consecutive crests, or between two consecutive troughs. It is usually represented by $\lambda$, the Greek letter lambda, and its SI unit is the metre.

How many such oscillations a fixed spot gets through in a second is the frequency, written $\nu$ and counted in hertz, one per second. However long one full oscillation takes at a fixed spot is the time period, written T and measured in seconds. The two are inversely related, so that $\nu = \frac{1}{T}$ and a shorter time period corresponds to a higher frequency.

It is the density at one spot going from highest down to lowest and back up to highest, or the other way about. The detail matters because half of that, a change from maximum to minimum alone, is not one oscillation, and counting it as one would double every frequency calculated.

The speed of sound is the ground one point on the wave, a crest or a trough, covers in one second. Fix the frequency and the gap from crest to crest is one wavelength $\lambda$, crossed in one time period $T$. So $v = \frac{\lambda}{T}$, and since $\nu = \frac{1}{T}$, this gives $v = \lambda \times \nu$: speed equals wavelength times frequency.