[{"data":1,"prerenderedAt":1150},["ShallowReactive",2],{"layer:sound:understand":3},{"layer":4,"contentHash":1129,"dependencyHashes":1130,"approval":1144,"releaseId":1149},{"schemaVersion":5,"conceptId":6,"locale":7,"depth":8,"revision":5,"title":9,"subtitle":10,"summary":11,"objectives":12,"estimatedMinutes":18,"plate":19,"blocks":40,"sourceIds":1124,"reviewStatus":1125,"authoring":1126},1,"sound","en","understand","Compressions, rarefactions and the wave equation","What is really travelling, how fast, and how the ear turns it into a signal","See what a sound wave actually is: a train of squashed and stretched air marching outwards. Meet longitudinal waves on a slinky, the equation v = f × λ, why steel beats air by seventeen times, how decibels multiply, and the engineering of the human ear.",[13,14,15,16,17],"Describe a sound wave as compressions and rarefactions, and explain why it is longitudinal rather than transverse.","Use v = f × λ in all three rearrangements, and say what stays the same when sound changes medium.","Explain the speed of sound from stiffness and density, and use v = 331.3 + 0.606 × T for air.","Read the decibel scale correctly: differences, not ratios, with +10 dB meaning ten times the energy.","Trace the ear as a machine: the middle-ear matching device and the cochlea's frequency place map.",40,{"title":20,"rows":21},"Lesson plate",[22,25,28,31,34,37],{"label":23,"value":24},"Depth","Understand",{"label":26,"value":27},"Reading time","≈ 40 minutes",{"label":29,"value":30},"Prior knowledge","Discover: sound is a vibration",{"label":32,"value":33},"Chapters","9",{"label":35,"value":36},"Labs","Wavelength lab, five-lane race, two games",{"label":38,"value":39},"Units used","Hz, kHz, m, m\u002Fs, s, dB, °C",[41,45,51,57,60,65,70,75,80,83,116,121,178,183,188,191,210,214,229,277,295,331,347,360,365,368,422,437,441,444,460,464,498,503,506,556,570,575,579,591,596,599,624,629,642,660,665,668,704,708,712,742,748,753,757,762,765,805,809,821,826,860,865,1024,1090,1094,1112],{"id":42,"type":43,"markdown":44},"u-intro","prose","In Discover you found the vibration behind every sound, and followed it to your ear. Good. But a sentence such as “the pattern of pushes travels through the air” is a promise, not an explanation.\n\nThis layer keeps the promise. You will see exactly what is squashed and what is stretched, learn why a sound wave is a completely different *shape* of wave from a wave on a rope, get the one equation that ties speed, frequency and wavelength together, find out why steel carries sound seventeen times faster than air, and take the ear apart properly.\n\nBy the end you should be able to take any sound and say three things about it with numbers: how fast it is travelling, how often it is vibrating, and how long each wave is.",{"id":46,"type":47,"variant":48,"title":49,"markdown":50},"u-how-to-read","callout","observation","How to use this lesson","Each chapter answers one question and hands the answer to the next. Read in order the first time.\n\nWhen a worked example appears, cover the steps and try it yourself first. Every number in this lesson was computed and checked, so if your answer differs, look for the reason rather than assuming a typo.",{"id":52,"type":53,"title":54,"eyebrow":55,"navLabel":56},"u-ch01","chapter","What is actually travelling","Chapter 01","1 What travels",{"id":58,"type":43,"markdown":59},"u-ch1-p1","Air is not empty. Every cubic centimetre around you holds an enormous number of molecules, flying about, bumping into each other billions of times a second. On average they are evenly spread, and the pressure is the same everywhere.\n\nNow put a loudspeaker cone in the middle of it and push the cone **forward**. The molecules right in front have nowhere to go instantly, so they crowd together. That crowded patch has slightly **higher pressure** than normal. We call it a **compression**.\n\nPull the cone **back** and it leaves a little extra room behind it. The molecules there spread out. That thinned patch has slightly **lower pressure** than normal. We call it a **rarefaction**.\n\nNow move the cone in and out steadily, hundreds of times a second. You make compression, rarefaction, compression, rarefaction, and each one shoves the next patch of air, so the whole striped pattern marches outwards. That marching pattern of high and low pressure **is** the sound wave.",{"id":61,"type":47,"variant":62,"title":63,"markdown":64},"u-def-compression","definition","Compression and rarefaction","A **compression** is a region where the particles of the medium are pushed closer together than normal, so the pressure is slightly above normal.\n\nA **rarefaction** is a region where the particles are further apart than normal, so the pressure is slightly below normal.\n\nA sound wave is an endless train of them, alternating. One compression plus the rarefaction that follows it makes **one complete wave**.",{"id":66,"type":47,"variant":67,"title":68,"markdown":69},"u-aha-tiny","aha","How small are these pressure changes?","Absurdly small. Normal air pressure at sea level is about 101,000 pascals. The quietest sound a healthy young ear can detect changes that by about **0.00002 pascals**, which is two parts in ten thousand million.\n\nA painfully loud sound, near 120 decibels, is still only about 20 pascals: two hundredths of one per cent of the background pressure.\n\nSo your ear is not a crude pressure gauge. It is one of the most sensitive detectors of any kind in nature, and it never stops working, even while you sleep.",{"id":71,"type":47,"variant":72,"title":73,"markdown":74},"u-misconception-air-travels","misconception","“The air travels from the speaker to me”","Follow a single molecule of air while music plays. It jiggles back and forth by a distance far smaller than the width of a hair, thousands of times a second, and ends up almost exactly where it started.\n\nNothing is delivered. The molecules pass the **push** along, like a queue of people pressed together: nudge the person at the front and the nudge reaches the back, without anybody changing their place in the queue.\n\nThis is why a fan blowing across a room does not carry music with it, and why your voice reaches a friend even when the wind is blowing the other way.",{"id":76,"type":53,"title":77,"eyebrow":78,"navLabel":79},"u-ch02","Longitudinal: the slinky, not the rope","Chapter 02","2 The slinky",{"id":81,"type":43,"markdown":82},"u-ch2-p1","Waves come in two shapes, and sound belongs firmly to one of them.\n\nStretch a long **rope** across the floor and flick one end **sideways**. A hump runs down the rope to the far end. But look at any one point of the rope: it moves **up and down**, at right angles to the direction the hump is travelling. That is a **transverse** wave. Water ripples and light behave this way.\n\nNow stretch a **slinky** spring along the floor and, instead of flicking it sideways, push one end sharply **forwards and back along its own length**. A bunched-up patch of coils races down the spring. Watch one coil: it slides **forwards and back**, in the *same* line as the travelling patch, and stays roughly in place. Between the bunched patches the coils are stretched apart.\n\nThat is a **longitudinal** wave, and a slinky is doing exactly what air does when a drum sounds. Bunched coils are compressions; stretched coils are rarefactions.",{"id":84,"type":85,"caption":86,"columns":87,"rows":91},"u-table-wave-types","table","Longitudinal and transverse waves compared",[88,89,90],"Feature","Longitudinal (sound)","Transverse (rope, water, light)",[92,96,100,104,108,112],[93,94,95],"Which way do the particles move?","Back and forth **along** the direction of travel","**Across** the direction of travel",[97,98,99],"What you see on the model","Slinky: coils bunch and spread","Rope: humps and dips",[101,102,103],"The parts have names","Compressions and rarefactions","Crests and troughs",[105,106,107],"Needs a medium?","Always. No medium, no sound","Ropes and water do; light does not",[109,110,111],"Can it be polarised?","No, because the motion is along one line only","Yes, which is how polarised sunglasses work",[113,114,115],"Everyday example","A tabla stroke crossing a room","A ripple crossing a village pond",{"id":117,"type":47,"variant":118,"title":119,"markdown":120},"u-tryit-slinky","try_it","Both waves on one slinky","You need a slinky, a friend, and a bit of floor.\n\n1. Stretch it to about three metres. Ask your friend to hold one end still.\n2. **Transverse:** flick your end quickly to the **side**. Watch the hump travel to the far end and bounce back. Tie a small paper tag to one coil and watch the tag: it goes side to side, not along.\n3. **Longitudinal:** now gather five or six coils at your end and release them **along** the spring. A tight bunch races away, and the tag now moves forwards and back along the line of the spring.\n\nThe second one is a sound wave, slowed down enough to see. Do it several times at different strengths: a bigger squeeze gives a denser bunch, which is exactly what a louder sound is.",{"id":122,"type":123,"component":124,"componentVersion":5,"config":125,"objective":176,"textAlternative":177},"u-lab-sort-waves","interactive","sort-game",{"prompt":126,"bins":127,"items":134,"seconds":175},"Longitudinal or transverse? Sort each wave, and each statement, into the right bin.",[128,131],{"id":129,"label":130},"long","Longitudinal",{"id":132,"label":133},"trans","Transverse",[135,139,143,147,151,155,159,163,167,171],{"id":136,"label":137,"bin":129,"why":138},"sound-air","Sound travelling through air","Air is squashed and stretched along the direction the sound is going.",{"id":140,"label":141,"bin":129,"why":142},"slinky-push","A slinky pushed along its own length","Coils bunch and spread along the spring: the model of a sound wave.",{"id":144,"label":145,"bin":132,"why":146},"rope-flick","A rope flicked sideways","Each point of the rope moves across the direction of travel.",{"id":148,"label":149,"bin":132,"why":150},"pond","A ripple on a pond","A floating leaf bobs up and down while the ripple moves outwards.",{"id":152,"label":153,"bin":132,"why":154},"light","Light from the Sun","Light is a transverse electromagnetic wave, and it needs no medium at all.",{"id":156,"label":157,"bin":129,"why":158},"compressions","Has compressions and rarefactions","Those are the names for the bunched and thinned regions of a longitudinal wave.",{"id":160,"label":161,"bin":132,"why":162},"crests","Has crests and troughs","Crest and trough are the high and low points of a transverse wave.",{"id":164,"label":165,"bin":129,"why":166},"particles-along","Particles move along the direction of travel","That is the definition of longitudinal.",{"id":168,"label":169,"bin":132,"why":170},"particles-across","Particles move across the direction of travel","That is the definition of transverse.",{"id":172,"label":173,"bin":129,"why":174},"sound-water","Sound travelling through sea water","Sound is longitudinal in every medium: gas, liquid or solid.",0,"Sort ten waves and statements into longitudinal and transverse, and fix the difference for good.","A sorting game with ten cards and two bins: **longitudinal** and **transverse**.\n\nLongitudinal: sound in air, a slinky pushed along its length, sound in sea water, “has compressions and rarefactions”, “particles move along the direction of travel”.\n\nTransverse: a rope flicked sideways, a ripple on a pond, light from the Sun, “has crests and troughs”, “particles move across the direction of travel”.\n\nThe single test that settles every card: **ask which way the stuff moves compared with which way the wave goes.** Same line means longitudinal. At right angles means transverse. Sound is always longitudinal, in every medium, with no exceptions at this level.",{"id":179,"type":47,"variant":180,"title":181,"markdown":182},"u-nuance-solids-both","nuance","Solids can do both","In a gas or a liquid the particles can be squashed together but cannot be sheared sideways in a way that springs back, so **only longitudinal** sound waves travel in air and water.\n\nA solid is different: it resists twisting and shearing too. So a solid can carry both a longitudinal wave and a transverse one, and they travel at different speeds.\n\nGeologists use exactly this. An earthquake sends out fast longitudinal **P-waves** and slower transverse **S-waves**. The gap between their arrivals tells a seismograph station how far away the quake was, using the same reasoning you use for lightning and thunder. And because S-waves cannot cross liquid, their shadow on the far side of the Earth is how we know the outer core is molten.",{"id":184,"type":53,"title":185,"eyebrow":186,"navLabel":187},"u-ch03","Frequency, wavelength and speed","Chapter 03","3 v = f × λ",{"id":189,"type":43,"markdown":190},"u-ch3-p1","Freeze a sound wave in mid-air and measure the distance from one compression to the next. That distance is the **wavelength**, written with the Greek letter lambda, λ.\n\nNow unfreeze it and stand still while it goes past. In one second, **f** complete waves sweep over you, where f is the frequency in hertz. Each of those waves is λ metres long. So in that one second, the front of the wave must have moved f × λ metres.\n\nThat is the speed. And so we have the single most useful equation in the whole of wave physics:\n\n**v = f × λ**\n\nSpeed equals frequency times wavelength. It is not a law about sound in particular; it is simply what the words mean, and it is true for every wave there is.",{"id":192,"type":193,"items":194},"u-formulas-wave","formulas",[195,198,201,204,207],{"expression":196,"caption":197},"v = f × λ","Speed (m\u002Fs) = frequency (Hz) × wavelength (m). True for every wave.",{"expression":199,"caption":200},"λ = v ÷ f","Rearranged to find the wavelength, when you know the speed and the frequency.",{"expression":202,"caption":203},"f = v ÷ λ","Rearranged to find the frequency from the speed and the wavelength.",{"expression":205,"caption":206},"T = 1 ÷ f","The time period: how many seconds one complete vibration takes. 440 Hz means T ≈ 0.0023 s.",{"expression":208,"caption":209},"v(air) = 331.3 + 0.606 × T°C","Speed of sound in dry air, with the temperature in degrees Celsius.",{"id":211,"type":47,"variant":62,"title":212,"markdown":213},"u-def-wavelength","Wavelength","The **wavelength** (λ) is the length of one complete wave: the distance from one compression to the next compression, or equally from one rarefaction to the next.\n\nIt is measured in metres. For everyday sounds in air it ranges from about **17 metres** (a 20 Hz rumble, longer than a bus) down to about **1.7 centimetres** (a 20,000 Hz hiss, the width of your thumb).\n\nImportant: wavelength is *not* a property of the source alone. Send the same 440 Hz note into water and its wavelength becomes more than four times longer, because the speed is more than four times greater while the frequency stays exactly the same.",{"id":215,"type":216,"title":217,"problem":218,"steps":219,"help":226},"u-we-wavelength","worked_example","The wavelength of the tuning note","An orchestra tunes to 440 Hz. How long is one wave of that note in air at 20 degrees Celsius, where sound travels at 343 m\u002Fs?",[220,221,222,223,224,225],"Write down what you know: v = 343 m\u002Fs, f = 440 Hz. You want λ.","Choose the right rearrangement: λ = v ÷ f.","λ = 343 ÷ 440.","λ = **0.78 metres**, or 78 centimetres.","Sense check: that is about the length of your arm, and a bansuri or a flute playing near this note is indeed of that order of size. Good.","Check the other way: f × λ = 440 × 0.7795 = 343 m\u002Fs. ✓",{"simplerExplanation":227,"anotherExample":228},"440 waves stream past you every second, and together they stretch 343 metres. So each one is 343 ÷ 440 = about 0.78 m long.","For a 1,000 Hz beep: λ = 343 ÷ 1,000 = 0.343 m, about 34 cm.",{"id":230,"type":85,"caption":231,"columns":232,"rows":236},"u-table-wavelengths","Wavelength in air at 343 m\u002Fs, from the lowest note we hear to the highest",[233,234,212,235],"Sound","Frequency","Roughly as long as",[237,242,247,252,257,262,267,272],[238,239,240,241],"Lowest audible rumble","20 Hz","17.15 m","A five-storey building is tall",[243,244,245,246],"A big dhol","50 Hz","6.86 m","A small bus",[248,249,250,251],"A low male voice","100 Hz","3.43 m","A room is wide",[253,254,255,256],"Middle C on a harmonium","256 Hz","1.34 m","A child is tall",[258,259,260,261],"The tuning note","440 Hz","0.78 m","Your arm",[263,264,265,266],"A phone ring","1,000 Hz","0.34 m","A school ruler and a half",[268,269,270,271],"A whistle","4,000 Hz","0.086 m","A matchbox",[273,274,275,276],"Highest audible hiss","20,000 Hz","0.017 m","Your thumb is wide",{"id":278,"type":279,"prompt":280,"options":281,"explanation":294},"u-predict-into-water","prediction","A 440 Hz tone made in air passes into water, where sound travels at 1,480 m\u002Fs instead of 343 m\u002Fs. Compared with in air, in the water the sound has",[282,285,288,291],{"id":283,"label":284},"a","a higher frequency and the same wavelength",{"id":286,"label":287},"b","the same frequency and a longer wavelength",{"id":289,"label":290},"c","the same frequency and a shorter wavelength",{"id":292,"label":293},"d","a lower frequency and a longer wavelength","**Same frequency, longer wavelength.** The frequency is set by the *source*: the thing shaking 440 times a second does not change its habits because the wave has moved into water. Every compression that arrives at the boundary must go on into the water, so 440 of them enter every second.\n\nBut they now travel at 1,480 m\u002Fs, so in one second they spread over 1,480 metres instead of 343 metres. So λ = 1,480 ÷ 440 = 3.36 m, more than four times longer than the 0.78 m it had in air.\n\nThe rule worth remembering: **when sound changes medium, frequency stays and wavelength changes.** Pitch is a property of the source; wavelength is a property of the journey.",{"id":296,"type":123,"component":297,"componentVersion":5,"config":298,"objective":329,"textAlternative":330},"u-lab-wave-lambda","sound-wave",{"presets":299,"hertz":327,"allowAudio":328,"showWaveform":328,"quiz":328},[300,304,308,312,315,319,323],{"label":301,"hertz":302,"amplitude":303},"20 Hz: λ = 17.15 m",20,0.8,{"label":305,"hertz":306,"amplitude":307},"100 Hz: λ = 3.43 m",100,0.6,{"label":309,"hertz":310,"amplitude":311},"256 Hz: λ = 1.34 m",256,0.5,{"label":313,"hertz":314,"amplitude":311},"440 Hz: λ = 0.78 m",440,{"label":316,"hertz":317,"amplitude":318},"1,000 Hz: λ = 0.34 m",1000,0.45,{"label":320,"hertz":321,"amplitude":322},"4,000 Hz: λ = 8.6 cm",4000,0.4,{"label":324,"hertz":325,"amplitude":326},"20,000 Hz: λ = 1.7 cm",20000,0.3,{"min":302,"max":325,"initial":314},true,"Watch the wavelength shrink as the frequency rises, and check every pair against λ = 343 ÷ f.","A tone lab with a wavelength readout. A slider sets the frequency from 20 Hz to 20,000 Hz; the wave picture is drawn against a **distance** scale in metres, not a time scale, so the humps really are wavelengths.\n\nStart at **440 Hz**: the readout says λ = 0.78 m, and one hump measures 78 cm on the scale.\n\nDouble the frequency to **880 Hz** and the humps halve to 0.39 m. Halve it to **220 Hz** and they double to 1.56 m. Frequency and wavelength are a see-saw: multiply one, divide the other, and their product is always 343.\n\nThe extremes are startling. At **20 Hz** a single wave is **17.15 m** long, taller than most houses. At **20,000 Hz** it is **1.7 cm**, narrower than your thumb. A thousand-fold change in frequency, a thousand-fold change in wavelength, and the speed never budges.\n\nThe quiz mode gives you a frequency and asks for the wavelength, or the reverse.",{"id":332,"type":333,"itemId":334,"prompt":335,"check":336,"hints":341,"feedback":344},"u-prac-lambda-100","practice","sound.understand-lambda-100","Sound travels at 343 m\u002Fs in air. What is the **wavelength**, in metres, of a 100 Hz hum? Give your answer to two decimal places.",{"kind":337,"answer":338,"tolerance":339,"unit":340},"number",3.43,0.02,"m",[342,343],"Use λ = v ÷ f.","343 ÷ 100 = ?",{"correct":345,"incorrect":346},"Right: λ = 343 ÷ 100 = **3.43 m**. One wave of a 100 Hz hum is about as long as a small room is wide.","Use λ = v ÷ f = 343 ÷ 100 = 3.43 m. Dividing by 100 just moves the decimal point two places left.",{"id":348,"type":333,"itemId":349,"prompt":350,"check":351,"hints":354,"feedback":357},"u-prac-freq-from-lambda","sound.understand-f-from-lambda","A sound in air has a wavelength of exactly **0.5 metres**. What is its frequency, in hertz? (Speed of sound = 343 m\u002Fs.)",{"kind":337,"answer":352,"tolerance":5,"unit":353},686,"Hz",[355,356],"Use f = v ÷ λ.","343 ÷ 0.5 is the same as 343 × 2.",{"correct":358,"incorrect":359},"Right: f = 343 ÷ 0.5 = **686 Hz**. Dividing by a half is the same as doubling.","Rearrange to f = v ÷ λ = 343 ÷ 0.5. Dividing by 0.5 doubles the number, so f = 686 Hz.",{"id":361,"type":53,"title":362,"eyebrow":363,"navLabel":364},"u-ch04","The speed of sound, properly","Chapter 04","4 How fast",{"id":366,"type":43,"markdown":367},"u-ch4-p1","“343 metres per second” is not one of those numbers that is simply true, like the number of sides of a triangle. It is the answer to a specific question: *how fast does sound travel in dry air at 20 degrees Celsius?*\n\nChange the material and the answer changes enormously. Change the temperature and it changes a little. Change the loudness, the pitch or the wavelength and it does not change at all, which is a surprise worth taking seriously.\n\nTwo properties of a material decide its answer:\n\n- **Stiffness:** how strongly the particles resist being squashed and how hard they spring back. More stiffness means a faster hand-off, so **faster** sound.\n- **Density:** how much mass has to be shifted each time. More density means more sluggishness, so **slower** sound.\n\nSpeed rises with stiffness and falls with density. In practice stiffness wins by a mile going from a gas to a solid, which is why steel beats air even though steel is thousands of times denser.",{"id":369,"type":85,"caption":370,"columns":371,"rows":377},"u-table-speeds","Speed of sound in different materials (values used throughout this topic)",[372,373,374,375,376],"Material","State","Speed (m\u002Fs)","Times faster than air","Why",[378,384,390,395,401,406,412,417],[379,380,381,382,383],"Vacuum","nothing","no sound","—","No particles, so no hand-off is possible at all",[385,386,387,388,389],"Air at 20 °C","gas","343","1","Particles far apart and free; each must fly across a gap",[391,386,392,393,394],"Helium","about 1,000","about 3","Same freedom, but far lighter particles, so they move faster",[396,397,398,399,400],"Fresh water at 20 °C","liquid","1,480","4.3","Particles touching and much harder to squash than a gas",[402,397,403,404,405],"Sea water","about 1,500","4.4","Salt makes it slightly stiffer than fresh water",[407,408,409,410,411],"Wood, along the grain","solid","about 3,800","11.1","Long stiff fibres pass the push straight down their length",[413,408,414,415,416],"Steel","5,960","17.4","Every atom locked to its neighbours; extremely stiff",[418,408,419,420,421],"Diamond","about 12,000","about 35","The stiffest common material there is",{"id":423,"type":123,"component":424,"componentVersion":5,"config":425,"objective":435,"textAlternative":436},"u-lab-speed-media","sound-speed",{"media":426,"distanceM":317,"modes":432},[427,428,429,430,431],"vacuum","air","water","wood","steel",[433,434],"race","echo","Race one sound down a kilometre of five different materials, then read the arrival times against v = d ÷ t.","A five-lane race over **1,000 metres**: vacuum, air, water, wood and steel. One sound is released into all five lanes at the same instant.\n\n- **Steel** arrives first, at **0.168 s** (1,000 ÷ 5,960).\n- **Wood** second, at **0.263 s** (1,000 ÷ 3,800).\n- **Water** third, at **0.676 s** (1,000 ÷ 1,480).\n- **Air** fourth, at **2.92 s** (1,000 ÷ 343). This is also exactly the thunder rule: about three seconds per kilometre.\n- **Vacuum** never arrives, however long you wait.\n\nThe order is not about density: steel is about 6,500 times denser than air and still wins, easily. It is about **stiffness**. A slow-motion view shows why: in steel the atoms are locked together and the nudge passes on at once, while in air each molecule must fly across a wide gap before it meets the next one.\n\nAn **echo** mode lets you send the sound to a wall and back in each material.",{"id":438,"type":47,"variant":72,"title":439,"markdown":440},"u-misconception-density","“Sound is slower in heavy materials”","It sounds obvious, and it is wrong. Steel is roughly 6,500 times denser than air, and sound goes through it about **17 times faster**.\n\nDensity really does slow sound down, but only when you compare materials of *similar stiffness*. Among gases, for example, the rule works: sound is about three times faster in light helium than in heavier air, which is why a lungful of helium makes your voice squeak.\n\nGoing from gas to solid, the stiffness rises far more than the density does, and stiffness wins. So the reliable order is **solids fastest, then liquids, then gases, and nothing in a vacuum** — not “lightest fastest”.",{"id":442,"type":43,"markdown":443},"u-ch4-p2","Now temperature. Warm air is air whose molecules are moving faster, and molecules that already move faster carry a push to their neighbours sooner. So **warm air carries sound faster than cold air**.\n\nThe school formula is a straight line and is accurate enough for anything you will meet:\n\n**v = 331.3 + 0.606 × (temperature in °C)**\n\nEvery extra degree adds about 0.6 metres per second. So:\n\n- At **0 °C**, a cold Shimla morning: **331.3 m\u002Fs**.\n- At **20 °C**, a pleasant room: **343.4 m\u002Fs**, which we round to 343 throughout this topic.\n- At **40 °C**, a May afternoon in Nagpur: **355.5 m\u002Fs**.\n\nThat is a change of about 7 per cent across the year, small enough to ignore in most problems and large enough to be real. Notice what is *not* in the formula: pressure. At ordinary pressures, sound travels at the same speed at sea level and on a hill. Only the temperature matters.",{"id":445,"type":216,"title":446,"problem":447,"steps":448,"help":455},"u-we-temperature","Sound on a hot afternoon","It is 45 degrees Celsius in Jaisalmer. How fast does sound travel, and how much further does it get in one second than it would at 20 degrees Celsius?",[449,450,451,452,453,454],"Use v = 331.3 + 0.606 × T, with T = 45.","0.606 × 45 = 27.27.","v = 331.3 + 27.27 = **358.6 m\u002Fs**.","At 20 °C the same formula gives 331.3 + 12.12 = 343.4 m\u002Fs.","Difference = 358.6 − 343.4 = **15.2 metres further every second**.","In perspective: over a kilometre the thunder would arrive 2.79 s after the flash instead of 2.91 s. The three-seconds-per-kilometre rule still works fine.",{"simplerExplanation":456,"hints":457},"Each degree adds about 0.6 m\u002Fs. From 20 °C to 45 °C is 25 degrees, so add about 25 × 0.6 = 15 m\u002Fs to 343.",[458,459],"Multiply the temperature by 0.606 first.","Then add 331.3.",{"id":461,"type":47,"variant":180,"title":462,"markdown":463},"u-nuance-independent","Speed does not depend on pitch or loudness","Stop and check this, because it is easy to assume otherwise.\n\nIf high notes travelled faster than low notes, then music from the far end of a stadium would arrive scrambled: all the flutes first, all the drums later. It does not. Everything arrives together, in order, from any distance.\n\nIf loud sounds travelled faster than soft ones, a shout would overtake a whisper. It does not.\n\nIn one medium at one temperature, **every sound travels at the same speed**. Frequency and loudness are decided by the source; speed is decided by the material. Keep those two jobs separate and most sound problems become easy.\n\n(The one genuine exception is a shock wave from an explosion or a supersonic aircraft, where the pressure change is so violent that the ordinary rules bend. That is far beyond anything you will meet here.)",{"id":465,"type":123,"component":466,"componentVersion":5,"config":467,"objective":496,"textAlternative":497},"u-lab-match-units","match-pairs",{"prompt":468,"mode":469,"pairs":470},"Match each quantity to its unit and to what decides it.","connect",[471,473,475,478,481,484,487,490,493],{"a":234,"b":472},"Hertz (Hz): decided by the source",{"a":212,"b":474},"Metres (m): decided by speed ÷ frequency",{"a":476,"b":477},"Speed","Metres per second (m\u002Fs): decided by the medium",{"a":479,"b":480},"Amplitude","How far the source swings: decides loudness",{"a":482,"b":483},"Loudness level","Decibels (dB): a squashed, ten-times-per-step scale",{"a":485,"b":486},"Time period","Seconds (s): one divided by the frequency",{"a":488,"b":489},"Pitch","What you hear when the frequency changes",{"a":491,"b":492},"Compression","A patch of higher-than-normal pressure",{"a":494,"b":495},"Rarefaction","A patch of lower-than-normal pressure","Fix the nine quantities of sound to their units and to what actually decides each one.","A matching game with nine pairs, which together form the vocabulary you need for every calculation in this topic.\n\n- **Frequency** is in hertz (Hz) and is decided by the source.\n- **Wavelength** is in metres and equals speed divided by frequency.\n- **Speed** is in metres per second and is decided by the medium and its temperature.\n- **Amplitude** is how far the source swings, and it decides loudness.\n- **Loudness level** is in decibels (dB), on a squashed scale where each 10 dB is ten times the energy.\n- **Time period** is in seconds and equals one divided by the frequency.\n- **Pitch** is what you hear when the frequency changes.\n- **Compression** is a patch of higher-than-normal pressure.\n- **Rarefaction** is a patch of lower-than-normal pressure.\n\nThe pair worth arguing about is speed: many people expect the source to decide it. It does not. **The medium decides the speed; the source decides the frequency; the wavelength is whatever is left over.**",{"id":499,"type":53,"title":500,"eyebrow":501,"navLabel":502},"u-ch05","Loudness and the decibel scale","Chapter 05","5 Decibels",{"id":504,"type":43,"markdown":505},"u-ch5-p1","Your ear can hear a pin drop and can also survive a thunderclap. Between those two the energy arriving differs by a factor of about **a million million**: 1,000,000,000,000.\n\nNo ordinary scale can show that. If a whisper were one millimetre on a ruler, a thunderclap would be a thousand kilometres along it.\n\nSo we squash the scale. The **decibel** scale counts *multiplications* rather than additions:\n\n- **+10 dB** means the sound is carrying **ten times** as much energy, and sounds about **twice as loud** to you.\n- **+20 dB** means a hundred times the energy.\n- **+30 dB** means a thousand times.\n\nZero decibels is not silence. It is the quietest sound a healthy young ear can just detect, and it is the point the whole scale is measured from.",{"id":507,"type":508,"title":509,"note":510,"scale":511,"rungs":512},"u-ladder-db","ladder","The decibel ladder, with what is happening at each step","0 dB is the quietest detectable sound, not silence. Every 10 dB step above it means ten times the energy and roughly twice the loudness.","linear",[513,517,520,524,528,532,536,540,544,548,552],{"label":514,"value":515,"display":516},"A whisper at 1 m",30,"30 dB",{"label":518,"value":18,"display":519},"A quiet room at night","40 dB",{"label":521,"value":522,"display":523},"Normal conversation",60,"60 dB",{"label":525,"value":526,"display":527},"Busy city traffic",80,"80 dB",{"label":529,"value":530,"display":531},"Damage begins with long exposure",85,"85 dB",{"label":533,"value":534,"display":535},"Motorbike with no silencer",95,"95 dB",{"label":537,"value":538,"display":539},"Personal music player, full",105,"105 dB",{"label":541,"value":542,"display":543},"Loudspeaker at a wedding",110,"110 dB",{"label":545,"value":546,"display":547},"Pain and immediate risk",120,"120 dB",{"label":549,"value":550,"display":551},"India's firecracker limit at 4 m",125,"125 dB(AI)",{"label":553,"value":554,"display":555},"A firecracker close up",150,"150 dB or more",{"id":557,"type":216,"title":558,"problem":559,"steps":560,"help":567},"u-we-decibels","How much stronger is traffic than a conversation?","A normal conversation measures about 60 dB. Busy traffic measures about 80 dB. How many times more sound energy is reaching your ear from the traffic, and roughly how many times louder does it seem?",[561,562,563,564,565,566],"Find the difference in decibels: 80 − 60 = **20 dB**.","Every 10 dB means ten times the energy, so 20 dB means 10 × 10.","Energy ratio = **100 times** as much.","Now loudness as your ear judges it: every 10 dB sounds about twice as loud.","20 dB is two such steps, so 2 × 2 = about **4 times as loud**.","That gap between 100 and 4 is the whole point of the decibel scale. Your ear compresses enormous changes in energy into modest changes in loudness, which is what lets one pair of ears handle both a whisper and a storm.",{"simplerExplanation":568,"anotherExample":569},"Count the 10 dB steps. Two steps up: multiply the energy by 10 twice (100 times), and the felt loudness by 2 twice (4 times).","A 110 dB speaker against 80 dB traffic is three steps: 1,000 times the energy and about 8 times as loud.",{"id":571,"type":47,"variant":572,"title":573,"markdown":574},"u-careful-85","careful","The 85 decibel line","Hearing damage is not only about how loud, but how loud **for how long**. The usual guideline:\n\n- Below about **85 dB**, you can listen all day without harm.\n- At **85 dB**, about eight hours is the limit for a working day.\n- Every 3 dB above that roughly **halves** the safe time, because 3 dB is double the energy. At 100 dB you are down to minutes.\n- Above about **120 dB**, damage can happen from a single short exposure, with no warning and no pain in some cases.\n\nA firecracker at arm's length is far beyond 120 dB, which is why India limits crackers to 125 dB(AI) measured 4 metres away and why children's ears are the ones most often hurt at festivals.\n\nThree simple protections: **distance** (every doubling of distance takes about 6 dB off), **time** (leave, or take breaks) and **barriers** (plugs, muffs, or your own fingers).",{"id":576,"type":47,"variant":72,"title":577,"markdown":578},"u-misconception-double-db","“Twice the decibels means twice as loud”","It does not, and the error is huge. 80 dB is not twice as loud as 40 dB; it is about **16 times** as loud and carries **ten thousand times** the energy.\n\nDecibels are not a counting scale. They count steps of multiplication, so you must always work with the **difference** between two readings, never the ratio of them.\n\nA second surprise from the same fact: put two identical loudspeakers side by side and you double the energy, but double energy is only **+3 dB**, barely a noticeable change. To sound genuinely twice as loud you would need about ten speakers.",{"id":580,"type":279,"prompt":581,"options":582,"explanation":590},"u-predict-two-speakers","One loudspeaker on a stage measures 100 dB where you stand. A second, identical speaker is switched on right beside it. What does the meter read now?",[583,585,586,588],{"id":283,"label":584},"200 dB",{"id":286,"label":543},{"id":289,"label":587},"103 dB",{"id":292,"label":589},"100 dB, no change","**About 103 dB.** Two identical sources deliver twice the energy, and doubling the energy is +3 dB, because 3 decibels is very close to a factor of two.\n\nTo reach 110 dB you would need **ten** speakers (ten times the energy). To reach 200 dB you would need a number with twenty zeros in it, and in fact ordinary sound in air cannot exceed about 194 dB at all: beyond that the rarefaction would need a pressure below zero, which does not exist. Louder than that is not sound any more; it is a shock wave.\n\nThis is why sound engineers say that adding speakers is an expensive way to get louder, and why moving the audience closer is a cheap one.",{"id":592,"type":53,"title":593,"eyebrow":594,"navLabel":595},"u-ch06","Pitch, instruments and what sets the note","Chapter 06","6 Setting the note",{"id":597,"type":43,"markdown":598},"u-ch6-p1","In Discover you found three ways to raise the pitch of a string: make it **shorter**, **tighter** or **thinner**. Now we can say why, and put numbers on the first one.\n\nA string clamped at both ends can only vibrate in whole patterns that fit exactly between the clamps. The simplest one has the string bulging in the middle and still at both ends, and that pattern makes the note you hear. Make the string half as long and the same wave has half the distance to cover on each swing, so it completes twice as many swings a second. **Half the length is exactly double the frequency**, a jump musicians call an **octave**.\n\nTension and thickness work through how sharply the string springs back. Tighter means a stronger restoring pull, so a faster spring-back, so a higher note: frequency rises with the square root of the tension, so you need **four times** the tension to double the pitch. Thicker means more mass to move, so a slower spring-back and a lower note.",{"id":600,"type":601,"title":602,"items":603},"u-steps-instrument-families","steps","The four families of Indian instruments, and what vibrates in each",[604,608,612,616,620],{"title":605,"tag":606,"text":607},"Tat: stringed, plucked","sitar, veena, tanpura","A stretched string vibrates. Length is set by the frets and the player's finger, tension by the tuning pegs, thickness by the choice of string. The hollow body and gourd make it loud.",{"title":609,"tag":610,"text":611},"Vitat: stringed, bowed","sarangi, esraj, violin","A bow drags across the string and keeps feeding it energy, so the note lasts as long as the bow moves instead of dying away like a pluck.",{"title":613,"tag":614,"text":615},"Sushir: blown","bansuri, shehnai, nadaswaram","A column of air vibrates. Opening a hole shortens the effective column and raises the note. Nothing solid is doing the singing.",{"title":617,"tag":618,"text":619},"Avanaddh: covered with skin","tabla, mridangam, dholak","A stretched membrane vibrates. Tighter straps mean a higher note; the tabla's syahi patch tunes the overtones so a drum gets a real pitch.",{"title":621,"tag":622,"text":623},"Ghan: solid","ghatam, manjira, jal tarang","The solid object itself rings. In jal tarang the pitch is set by how much water is in each china bowl: more water, lower note.",{"id":625,"type":47,"variant":626,"title":627,"markdown":628},"u-example-jal-tarang","example","Jal tarang: a scale made of water","A jal tarang is a set of china bowls, each with a different amount of water, struck with light bamboo sticks.\n\nAdd water to a bowl and the note goes **down**. The water has to be shoved along with the china every time the bowl flexes, so the whole vibrating system is heavier and springs back more slowly.\n\nYou can build one in five minutes with glass tumblers, water and a spoon. Fill eight tumblers with steadily less water and you can tune a rough scale by ear, adding a drop at a time.\n\nCareful with the opposite experiment: **blowing across the top** of a bottle works the other way round. There the vibrating thing is the air above the water, so more water means a *shorter* air column and a *higher* note. Same bottles, opposite answer, because a different thing is vibrating. Always ask first: what is shaking here?",{"id":630,"type":279,"prompt":631,"options":632,"explanation":641},"u-predict-bottle","You have two identical glass bottles. One is a quarter full of water, the other is three-quarters full. You **blow across the top** of each. Which gives the higher note?",[633,635,637,639],{"id":283,"label":634},"The quarter-full one",{"id":286,"label":636},"The three-quarters-full one",{"id":289,"label":638},"The same: the bottles are identical",{"id":292,"label":640},"It depends on how hard you blow","**The three-quarters-full one.** When you blow across the top, the thing that vibrates is the **column of air above the water**. The fuller bottle has less air above the water, so a shorter air column, so a faster vibration and a higher note.\n\nNow **tap the same two bottles with a spoon** and you get the opposite answer: the thing vibrating is now the glass together with the water it has to drag along, so the fuller bottle is the heavier, slower, **lower** one.\n\nSame bottles, opposite results, and the whole difference is which object you set shaking. Always identify the vibrator before predicting the pitch.",{"id":643,"type":193,"items":644},"u-formulas-pitch",[645,648,651,654,657],{"expression":646,"caption":647},"half the length → double f","Shortening a string to half its length raises it by exactly one octave.",{"expression":649,"caption":650},"4 × tension → double f","Frequency rises with the square root of the tension, so quadrupling it doubles the note.",{"expression":652,"caption":653},"f × 2 = one octave up","Doubling any frequency raises it an octave: 220, 440, 880 Hz are the same note in three registers.",{"expression":655,"caption":656},"open pipe: f = v ÷ (2L)","A bansuri with both ends effectively open: a 39 cm column gives about 440 Hz.",{"expression":658,"caption":659},"closed pipe: f = v ÷ (4L)","A pipe closed at one end sounds an octave lower than an open pipe of the same length.",{"id":661,"type":53,"title":662,"eyebrow":663,"navLabel":664},"u-ch07","The ear as a machine","Chapter 07","7 The ear machine",{"id":666,"type":43,"markdown":667},"u-ch7-p1","The ear has a hard engineering problem to solve, and the solution is worth admiring.\n\nSound arrives in **air**, which is easy to move. It must end up shaking **liquid** in the inner ear, which is much harder to move. Shout at a swimming pool and almost all the sound bounces off the surface; only about a thousandth of it gets in. If your ear worked that way you would be nearly deaf.\n\nThe middle ear is the fix. Three small bones take the movement of the fairly large eardrum and concentrate it onto the much smaller oval window, while acting as a lever at the same time. Area concentration multiplied by lever action gives a pressure gain of roughly **twenty times**.\n\nThat is exactly enough to push the liquid properly. The middle ear is a matching device, the same idea as the gears on a bicycle: it trades a big, weak movement for a small, strong one.",{"id":669,"type":601,"title":670,"items":671},"u-steps-ear-detail","The ear, part by part, with what each part is for",[672,676,680,684,688,692,696,700],{"title":673,"tag":674,"text":675},"Pinna","the visible flap","Gathers sound and funnels it inwards. Its ridges add a direction-dependent colouring that helps you tell front from behind and above from below.",{"title":677,"tag":678,"text":679},"Ear canal","about 2.5 cm","Carries sound to the eardrum and protects it. As a tube closed at one end it resonates near 3,000 Hz, which is why we are most sensitive around there: exactly the range of a baby's cry and of consonants in speech.",{"title":681,"tag":682,"text":683},"Eardrum","tympanic membrane","A taut cone of skin about a centimetre across. It moves in and out, following the arriving pressure pattern faithfully across the whole range of hearing.",{"title":685,"tag":686,"text":687},"Malleus, incus, stapes","hammer, anvil, stirrup","The three smallest bones in the body. They bridge the air-filled middle ear and focus the eardrum's motion onto the tiny oval window, roughly twenty times stronger in pressure.",{"title":689,"tag":690,"text":691},"Eustachian tube","pressure valve","A tube from the middle ear to the back of the throat. It opens when you swallow or yawn to equalise pressure, which is why ears pop on a hill road or in an aircraft.",{"title":693,"tag":694,"text":695},"Cochlea","the snail shell","A liquid-filled coil about the size of a pea and about 35 mm long if uncoiled. A travelling ripple inside peaks at a different place for every frequency.",{"title":697,"tag":698,"text":699},"Hair cells","about 16,000","Cells with fine bristles standing along the cochlea. A sway opens tiny channels and the cell fires. Loud noise breaks them and they never regrow.",{"title":701,"tag":702,"text":703},"Auditory nerve","to the brain","About 30,000 nerve fibres carry the pattern to the hearing cortex, which compares the two ears for direction and recognises the sound.",{"id":705,"type":47,"variant":67,"title":706,"markdown":707},"u-aha-place-map","The cochlea is a keyboard, rolled up","Here is the loveliest fact about hearing. The cochlea does not measure frequency by counting; it measures it by **position**.\n\nNear the entrance the membrane inside is narrow and stiff, so it responds best to **high** frequencies. As you go deeper into the coil it becomes wider and floppier, and responds best to **lower** ones. A 10,000 Hz tone shakes hair cells near the front; a 100 Hz tone shakes hair cells near the middle of the coil.\n\nSo your cochlea is a piano keyboard rolled into a spiral, and your brain reads the pitch from **which cells are shouting**, not from how fast they shout.\n\nThat is also why hearing loss usually starts with high notes: the cells nearest the entrance meet every sound that comes in and wear out first.",{"id":709,"type":47,"variant":180,"title":710,"markdown":711},"u-nuance-not-flat","Your ears are not equally good at all pitches","A 30 dB whisper at 1,000 Hz is easy to hear. A 30 dB tone at 50 Hz is nearly inaudible, and the same at 18,000 Hz probably is too.\n\nHuman hearing peaks between about **2,000 and 5,000 Hz**, helped by the resonance of the ear canal, and falls away steeply at both ends. This is not a fault; it is a fit. Speech carries most of its meaning in that band, and so does the cry of a child.\n\nOne consequence you can hear: turn music down very low and the bass seems to vanish before the voices do. Your ear loses the low notes first at low volumes, which is why some music players have a “loudness” setting that boosts bass when the volume is low.",{"id":713,"type":714,"tone":715,"items":716},"u-spec-ear-numbers","spec","copper",[717,721,724,728,732,735,738],{"label":718,"big":719,"value":720},"Ear canal length","≈ 2.5 cm","Resonates near 3,000 Hz, boosting the frequencies that matter most for speech.",{"label":681,"big":722,"value":723},"≈ 1 cm across","Thinner than a sheet of paper, and it follows the pressure faithfully from 20 Hz to 20 kHz.",{"label":725,"big":726,"value":727},"Smallest bone","stapes, 3 mm","The smallest bone in the human body, lighter than a grain of rice.",{"label":729,"big":730,"value":731},"Middle ear gain","≈ 20 ×","Pressure amplification from eardrum to oval window: area concentration plus lever action.",{"label":693,"big":733,"value":734},"≈ 35 mm","Uncoiled length of the spiral, all packed into a space the size of a pea.",{"label":697,"big":736,"value":737},"≈ 16,000","Per ear. Compare that with about 100 million light detectors in one eye.",{"label":739,"big":740,"value":741},"Quietest detectable","0 dB","The eardrum moves less than the width of a single atom.",{"id":743,"type":744,"conceptId":745,"relation":746,"explanation":747},"u-conn-anatomy","connection","human-body-anatomy","part_of","The ear is one of the sense organs: see where its parts sit inside the skull alongside the rest of the body's anatomy.",{"id":749,"type":744,"conceptId":750,"relation":751,"explanation":752},"u-conn-systems","body-systems","related_to","Hearing is a nervous-system job: hair cells make the signal, the auditory nerve carries it, the brain interprets it.",{"id":754,"type":47,"variant":72,"title":755,"markdown":756},"u-misconception-wax","“Ears need cleaning inside”","They do not. Earwax is made on purpose: it traps dust, repels water and carries dirt slowly outwards on its own, taking dead skin with it. The canal cleans itself.\n\nPushing a cotton bud in does the opposite of cleaning. It packs the wax **inwards** against the eardrum, where it can block hearing and cannot come out. Sharper objects can tear the eardrum, which is painful, and the damage is not always reversible.\n\nWash the outer flap with a cloth and leave the tunnel alone. If an ear feels blocked or hurts, that is a matter for a doctor, not for a bud, a pin or a matchstick.",{"id":758,"type":53,"title":759,"eyebrow":760,"navLabel":761},"u-ch08","The window we hear through","Chapter 08","8 20 Hz to 20 kHz",{"id":763,"type":43,"markdown":764},"u-ch8-p1","Human hearing runs from about **20 Hz to 20,000 Hz**. That is a range of a thousand to one, which musicians count as about **ten octaves**, since each octave is a doubling and ten doublings multiply by 1,024.\n\nIt is a generous window, but it is only a window. Sound exists above and below it, made by ordinary vibrations, obeying the same rules, and simply not detected by us.\n\n- Above 20,000 Hz is **ultrasound**. Bats shout up to about 120,000 Hz and listen for the echoes. Dolphins go higher still. A hospital scanner uses millions of hertz to draw a picture of a baby.\n- Below 20 Hz is **infrasound**. Elephants call at around 14 Hz and hear each other across kilometres. Volcanoes, earthquakes and large storms make it, and networks of infrasound microphones listen worldwide for distant explosions.\n\nAnd the window narrows as you age. Nearly everyone loses the top end, from about 20 kHz in childhood to perhaps 12–15 kHz by middle age. It is normal, it is gradual, and loud noise makes it happen sooner and go further.",{"id":766,"type":85,"caption":767,"columns":768,"rows":772},"u-table-ranges","Hearing ranges of some animals, and what they use them for",[769,770,771],"Animal","Range","What the extra range is for",[773,777,781,785,789,793,797,801],[774,775,776],"Human (child)","20 Hz – 20,000 Hz","Speech and music sit comfortably inside it",[778,779,780],"Human (adult, 50)","20 Hz – about 12,000 Hz","Nothing is lost that matters much for speech",[782,783,784],"Dog","67 Hz – 45,000 Hz","Hears the high squeaks of rodents, and dog whistles we cannot hear",[786,787,788],"Cat","48 Hz – 85,000 Hz","Hunting mice, which squeak far above our range",[790,791,792],"Bat","1,000 Hz – 120,000 Hz","Echolocation: short wavelengths give sharp detail on small insects",[794,795,796],"Dolphin","75 Hz – 150,000 Hz","Echolocation in murky water, where eyes are little use",[798,799,800],"Elephant","14 Hz – 12,000 Hz","Infrasound calls that travel kilometres across the ground and air",[802,803,804],"Mouse","1,000 Hz – 90,000 Hz","Ultrasonic squeaks that predators mostly cannot hear",{"id":806,"type":47,"variant":180,"title":807,"markdown":808},"u-nuance-bat-wavelength","Why bats shout so high","It is not for privacy. It is for **detail**.\n\nA wave can only reveal things that are bigger than about one wavelength. A 100 Hz sound has a wavelength of 3.43 metres and would flow round a mosquito without noticing it, the way a sea swell flows round a stick.\n\nA bat's 100,000 Hz call has a wavelength of only 3.4 millimetres, and a mosquito is bigger than that, so the echo comes back clearly.\n\nThe same rule governs every echo technique. Ship sonar looking for the sea floor uses low frequencies, which travel far. A hospital ultrasound scanner needs millimetre detail, so it uses millions of hertz and accepts that they do not reach deep. **High frequency buys detail and costs range.**",{"id":810,"type":333,"itemId":811,"prompt":812,"check":813,"hints":815,"feedback":818},"u-prac-octaves","sound.understand-octaves","A note of 110 Hz is played. Three octaves higher is a note of what frequency, in hertz? (One octave up means doubling.)",{"kind":337,"answer":814,"tolerance":175,"unit":353},880,[816,817],"Double it once for each octave.","110 → 220 → 440 → ?",{"correct":819,"incorrect":820},"Right: 110 → 220 → 440 → **880 Hz**. Three doublings multiply by 8, and 110 × 8 = 880.","Each octave doubles the frequency. Three octaves means × 2 × 2 × 2 = × 8, so 110 × 8 = 880 Hz.",{"id":822,"type":53,"title":823,"eyebrow":824,"navLabel":825},"u-ch09","Mix-ups worth clearing up","Chapter 09","9 Mix-ups",{"id":827,"type":85,"caption":828,"columns":829,"rows":832},"u-table-mixups","Nine things people say about sound, and what is actually true",[830,831],"People often say","What is actually true",[833,836,839,842,845,848,851,854,857],[834,835],"“Sound travels through space, just faintly”","It does not travel at all. No medium, no sound. Radio, which is a kind of light, is how astronauts talk.",[837,838],"“The air moves from the speaker to my ear”","Each molecule jiggles and stays put. Only the pressure pattern travels.",[840,841],"“Heavier materials slow sound down”","Stiffness matters more. Steel is far denser than air and 17 times faster.",[843,844],"“High notes travel faster”","Every frequency travels at the same speed in the same medium. Otherwise music would arrive scrambled.",[846,847],"“A louder sound travels faster”","Loudness changes how far a sound stays audible, never how fast it moves.",[849,850],"“Doubling the decibels doubles the loudness”","Decibels multiply. +10 dB is ten times the energy and about twice the loudness.",[852,853],"“The eardrum hears”","The eardrum only moves. Hearing happens in the brain, from hair-cell signals.",[855,856],"“Echoes need a cave”","Any hard surface about 17 m away or more will do: a school wall, a hill, a well.",[858,859],"“Ultrasound is a special kind of energy”","It is ordinary sound, simply above 20,000 Hz. Bats and scanners use the same physics you do.",{"id":861,"type":47,"variant":862,"title":863,"markdown":864},"u-model-limit","model_limit","What this lesson has simplified","Three honest simplifications, so that you know where the edges are.\n\n**The single-frequency picture.** Almost no real sound is one clean frequency. A tabla stroke or a spoken word is dozens of frequencies at once, and their mixture is what makes a sitar sound different from a flute playing the same note. Everything here still applies, one frequency at a time.\n\n**The straight-line picture.** Real sound bends round corners, reflects off every surface, and is absorbed differently at different frequencies. Rooms are complicated, which is why concert halls are designed with such care.\n\n**Speed as a single number.** 343 m\u002Fs assumes dry air at 20 °C. Humidity, wind and temperature layers all change it a little. On a cold clear night a warm layer overhead can bend sound back down to the ground, which is why distant trains sound closer at night.",{"id":866,"type":867,"title":868,"questions":869},"u-quiz","quiz","Twelve questions on how sound works",[870,883,894,907,920,933,946,959,972,985,998,1011],{"itemId":871,"prompt":872,"options":873,"correct":286,"why":882},"sound.understand-q-longitudinal","A sound wave in air is longitudinal. This means the air particles move",[874,876,878,880],{"id":283,"label":875},"across the direction of travel",{"id":286,"label":877},"along the direction of travel",{"id":289,"label":879},"in circles",{"id":292,"label":881},"not at all","In a longitudinal wave the particles oscillate back and forth in the same line as the wave travels, making compressions and rarefactions.",{"itemId":884,"prompt":885,"options":886,"correct":289,"why":893},"sound.understand-q-parts","The regions of a sound wave where particles are crowded together are called",[887,888,890,891],{"id":283,"label":160},{"id":286,"label":889},"troughs",{"id":289,"label":156},{"id":292,"label":892},"rarefactions","Crowded and higher pressure means a compression. Spread out and lower pressure means a rarefaction. Crests and troughs belong to transverse waves.",{"itemId":895,"prompt":896,"options":897,"correct":283,"why":906},"sound.understand-q-lambda","A 500 Hz sound travels in air at 343 m\u002Fs. Its wavelength is closest to",[898,900,902,904],{"id":283,"label":899},"0.69 m",{"id":286,"label":901},"1.46 m",{"id":289,"label":903},"171,500 m",{"id":292,"label":905},"0.0015 m","λ = v ÷ f = 343 ÷ 500 = 0.686 m, which rounds to 0.69 m.",{"itemId":908,"prompt":909,"options":910,"correct":289,"why":919},"sound.understand-q-medium-change","A 300 Hz sound passes from air into water. In the water its frequency is",[911,913,915,917],{"id":283,"label":912},"higher",{"id":286,"label":914},"lower",{"id":289,"label":916},"still 300 Hz",{"id":292,"label":918},"zero","The source still vibrates 300 times a second, so 300 compressions enter the water every second. The speed and the wavelength change; the frequency does not.",{"itemId":921,"prompt":922,"options":923,"correct":286,"why":932},"sound.understand-q-stiff","Sound is fastest in steel and slowest in air mainly because",[924,926,928,930],{"id":283,"label":925},"steel is denser",{"id":286,"label":927},"steel's particles are locked tightly to each other",{"id":289,"label":929},"air is transparent",{"id":292,"label":931},"steel is colder","Stiffness is what matters: tightly bonded particles pass the push on almost instantly. Density alone would predict the opposite, and would be wrong.",{"itemId":934,"prompt":935,"options":936,"correct":286,"why":945},"sound.understand-q-temp","On a hot day compared with a cold day, sound in air travels",[937,939,941,943],{"id":283,"label":938},"slower",{"id":286,"label":940},"faster",{"id":289,"label":942},"at the same speed",{"id":292,"label":944},"only if it is windy","v = 331.3 + 0.606 × T. Warmer molecules already move faster and hand on the push sooner: about 0.6 m\u002Fs more per degree Celsius.",{"itemId":947,"prompt":948,"options":949,"correct":289,"why":958},"sound.understand-q-speed-indep","Which of these changes the speed of a sound in a room?",[950,952,954,956],{"id":283,"label":951},"Making it louder",{"id":286,"label":953},"Making it higher in pitch",{"id":289,"label":955},"Warming the air",{"id":292,"label":957},"Using a bigger speaker","Speed belongs to the medium. Loudness and pitch belong to the source and change nothing about how fast the wave moves. Temperature changes the medium.",{"itemId":960,"prompt":961,"options":962,"correct":292,"why":971},"sound.understand-q-db","A sound rises from 50 dB to 80 dB. The energy arriving has gone up by a factor of",[963,965,967,969],{"id":283,"label":964},"30",{"id":286,"label":966},"1.6",{"id":289,"label":968},"100",{"id":292,"label":970},"1,000","The difference is 30 dB, which is three steps of 10 dB, so 10 × 10 × 10 = 1,000 times the energy. It sounds about 8 times as loud.",{"itemId":973,"prompt":974,"options":975,"correct":283,"why":984},"sound.understand-q-two-speakers","Two identical speakers instead of one raises the level by about",[976,978,980,982],{"id":283,"label":977},"3 dB",{"id":286,"label":979},"10 dB",{"id":289,"label":981},"double the dB",{"id":292,"label":983},"100 dB","Twice the energy is +3 dB, a barely noticeable change. You would need ten speakers for +10 dB, which sounds about twice as loud.",{"itemId":986,"prompt":987,"options":988,"correct":286,"why":997},"sound.understand-q-string","A vibrating string is pressed at its midpoint so only half can vibrate. The new note is",[989,991,993,995],{"id":283,"label":990},"half the frequency",{"id":286,"label":992},"double the frequency",{"id":289,"label":994},"unchanged",{"id":292,"label":996},"four times the frequency","Half the length means double the frequency: one octave higher. This is what a sitar player's finger does on every fret.",{"itemId":999,"prompt":1000,"options":1001,"correct":286,"why":1010},"sound.understand-q-cochlea","The cochlea tells high notes from low notes by",[1002,1004,1006,1008],{"id":283,"label":1003},"counting the vibrations",{"id":286,"label":1005},"which place along the coil responds most",{"id":289,"label":1007},"how loud they are",{"id":292,"label":1009},"which ear hears first","The cochlea is a rolled-up keyboard: stiff and narrow near the entrance for high notes, wide and floppy deeper in for low ones. Pitch is read from position.",{"itemId":1012,"prompt":1013,"options":1014,"correct":286,"why":1023},"sound.understand-q-bat","A bat uses 100,000 Hz calls rather than 1,000 Hz ones because",[1015,1017,1019,1021],{"id":283,"label":1016},"high sounds travel further",{"id":286,"label":1018},"the short wavelength reveals small insects",{"id":289,"label":1020},"high sounds are louder",{"id":292,"label":1022},"low sounds cannot echo","At 100,000 Hz the wavelength is about 3.4 mm, small enough to bounce off a mosquito. A 1,000 Hz wave is 34 cm long and flows straight past it.",{"id":1025,"type":1026,"title":1027,"terms":1028},"u-glossary","glossary","Precise words for this layer",[1029,1032,1035,1039,1043,1047,1051,1055,1059,1063,1067,1071,1075,1079,1082,1086],{"term":491,"meaning":1030,"example":1031},"A region of a sound wave where the particles are crowded and the pressure is above normal.","The bunched coils on a slinky.",{"term":494,"meaning":1033,"example":1034},"A region where the particles are spread out and the pressure is below normal.","The stretched coils between the bunches.",{"term":1036,"meaning":1037,"example":1038},"Longitudinal wave","A wave in which the particles move back and forth along the direction the wave travels. All sound is longitudinal.","A slinky pushed along its length.",{"term":1040,"meaning":1041,"example":1042},"Transverse wave","A wave in which the particles move across the direction of travel.","A flicked rope; ripples; light.",{"term":1044,"meaning":1045,"example":1046},"Wavelength (λ)","The distance from one compression to the next, in metres.","440 Hz in air has λ = 0.78 m.",{"term":1048,"meaning":1049,"example":1050},"Time period (T)","The time one complete vibration takes, in seconds. T = 1 ÷ f.","440 Hz means T = 0.0023 s.",{"term":1052,"meaning":1053,"example":1054},"Wave equation","v = f × λ: speed equals frequency times wavelength, for every wave.","343 = 440 × 0.78.",{"term":1056,"meaning":1057,"example":1058},"Medium","The material a wave travels through. Sound needs one; light does not.","Air, water, wood, steel.",{"term":1060,"meaning":1061,"example":1062},"Stiffness","How strongly a material resists being squashed and how fast it springs back. More stiffness means faster sound.","Steel is very stiff.",{"term":1064,"meaning":1065,"example":1066},"Density","How much mass is packed into a given volume. More density, on its own, means slower sound.","Helium is light, so sound is fast in it.",{"term":1068,"meaning":1069,"example":1070},"Decibel (dB)","The unit of sound level, on a scale where +10 dB means ten times the energy and about twice the loudness.","Traffic 80 dB, conversation 60 dB.",{"term":1072,"meaning":1073,"example":1074},"Octave","A doubling of frequency, heard as the same note higher up.","110, 220, 440 and 880 Hz.",{"term":1076,"meaning":1077,"example":1078},"Ossicles","The three small bones of the middle ear: malleus, incus and stapes.","The stapes is the body's smallest bone.",{"term":693,"meaning":1080,"example":1081},"The coiled, liquid-filled tube of the inner ear where position along the coil encodes pitch.","About 35 mm long uncoiled.",{"term":1083,"meaning":1084,"example":1085},"Ultrasound","Sound above 20,000 Hz. Bats, dolphins and medical scanners use it.","A bat call at 100,000 Hz.",{"term":1087,"meaning":1088,"example":1089},"Infrasound","Sound below 20 Hz. Elephants, volcanoes and storms make it.","An elephant call at 14 Hz.",{"id":1091,"type":1092,"prompt":1093},"u-reflect","reflection","A friend insists that sound must travel faster in air than in steel, because air is lighter and easier to push through. Write the clearest three-sentence reply you can, and then describe one experiment they could do at home, with no equipment, that would settle it.",{"id":1095,"type":1096,"title":1097,"points":1098},"u-cheat","summary","Cheat sheet",[1099,1100,1101,1102,1103,1104,1105,1106,1107,1108,1109,1110,1111],"**A sound wave is a travelling pattern of pressure:** compressions (crowded, higher pressure) alternating with rarefactions (spread out, lower pressure).","**Sound is longitudinal:** particles move back and forth **along** the direction of travel, like coils on a pushed slinky, not across it like a flicked rope.","**v = f × λ**, so λ = v ÷ f and f = v ÷ λ. In air, 20 Hz has λ = 17.15 m and 20,000 Hz has λ = 1.7 cm.","**The medium decides the speed; the source decides the frequency; the wavelength is whatever is left.** Change medium and frequency stays, wavelength changes.","**Speeds:** air 343 m\u002Fs at 20 °C, fresh water 1,480, wood about 3,800, steel 5,960, vacuum none at all.","**Stiffness beats density.** Solids fastest, liquids next, gases slowest, nothing in a vacuum.","**Temperature:** v = 331.3 + 0.606 × T°C. About 0.6 m\u002Fs more per degree. Pressure makes no difference.","**Speed does not depend on pitch or loudness.** If it did, distant music would arrive scrambled.","**Decibels multiply:** +10 dB is ten times the energy and about twice the loudness. Two identical speakers give only +3 dB.","**Strings:** half the length doubles the frequency (an octave); four times the tension doubles it; thicker is lower.","**The middle ear is a matching device**, turning a big weak movement of air into a small strong push on liquid, about twenty times stronger in pressure.","**The cochlea is a rolled-up keyboard:** high notes near the entrance, low notes deep inside. Pitch is read from *where*, not how fast.","**We hear 20 Hz to 20,000 Hz**, about ten octaves, best around 2–5 kHz. Above is ultrasound, below is infrasound.",{"id":1113,"type":1114,"sourceIds":1115},"u-sources","sources",[1116,1117,1118,1119,1120,1121,1122,1123],"sound-ncert-class9-sound","sound-britannica-sound","sound-hyperphysics-speed","sound-hyperphysics-ear","sound-nidcd-hearing","sound-nidcd-noise","sound-physicsclassroom-sound","sound-wikipedia-instruments",[1116,1117,1118,1119,1120,1121,1122,1123],"needs_review",{"generatedBy":1127,"notes":1128},"claude-code","Draft generated locally; pending owner review. Speeds, wavelengths, decibel ratios and temperature values computed and asserted in scratchpad\u002Fsound\u002Fnumbers.py.","ce965c08dd9f2abb5c1ffbc764f41e9a7afe2d4d625ad9a94072ef04c7e0f648",{"component:sort-game@1":1131,"component:sound-wave@1":1132,"logic:practice":1133,"component:sound-speed@1":1134,"component:match-pairs@1":1135,"source:sound-britannica-sound":1136,"source:sound-hyperphysics-ear":1137,"source:sound-hyperphysics-speed":1138,"source:sound-ncert-class9-sound":1139,"source:sound-nidcd-hearing":1140,"source:sound-nidcd-noise":1141,"source:sound-physicsclassroom-sound":1142,"source:sound-wikipedia-instruments":1143},"b164f45a2c8ca08f26c450768ff0231e113e9fe45381eddb34dc6d0548596c38","c04a20203101dfc3dcfa0440d2d1ec936289920891a98545909218b09a36ac4b","3d6b0fe1b15255975a32b0fcd94e8019bc959ad45cbf12e136e86149549c6878","22b6738152933cfccb190e600f1f387e7d7c3b1b36f37184363a3cd5f8dae8b6","2a8ee4ac87460b4e1175a4bb13c96b03d577db06dde95670eb7fcfe4ad787899","e63e8ac99c466c18f61ba209707bcf992538e2d87b16d8b6e7ceef147566ec7f","0a3d2c27986cca40d06b5de949d3c7a13272672e9766ecf33e53ec27ee28c886","4cb248d8beea032b112212f11d2a3d10751a3a68da1b26d2ef93b11835467932","615e6ca7b7252ccb2523eea00746ef69215849397defe43c853a9fcb617799dd","562aa906d72517f7a88b5d3bae2937ed3678a02844460b91738fc93e493f3490","6c2548bc22d1e37e553f1f05a7b1c919c097e49ad319523b186e766828b576cd","967196792bc122ee73ed66cefbf9d62ac069b2c688af723b4f4eb52fcdd8174a","0b24a438b7c5bb6e98dad1043334a28d32ba3aa4ef1eca60660933b537c66d0a",{"state":1145,"reviewer":1146,"selfReview":328,"reviewedAt":1147,"method":1148},"approved","The library owner","2026-09-20T10:18:37.581Z","owner_bulk","preview-7e1cbbcc4f",1789899598561]