[{"data":1,"prerenderedAt":866},["ShallowReactive",2],{"layer:sound:deepen":3},{"layer":4,"contentHash":841,"dependencyHashes":842,"approval":859,"releaseId":865},{"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":836,"reviewStatus":837,"authoring":838},1,"sound","en","deepen","Why resonance, harmonics and reverberation work the way they do","Damping, aeroelastic flutter, singing granite pillars, harmonics and a physicist with 300 cushions","Find out why resonance cannot grow forever, why two famous bridge wobbles had different causes, and why 56 granite pillars at Hampi ring with different notes. Meet Wallace Sabine, who found the reverberation formula with borrowed cushions, and the arithmetic of combining decibels.",[13,14,15,16,17],"Distinguish free vibration, forced vibration, resonance and damping, and explain why resonance settles at a limit.","Compare the real causes of the Tacoma Narrows and Millennium Bridge wobbles, and explain why solid stone can ring at Hampi.","Explain timbre through harmonics, and predict which harmonics an open or closed pipe supports.","Describe how Sabine discovered the reverberation-time formula, and use it and decibel-combination arithmetic in worked examples.","Explain the range-versus-detail trade-off in sonar and ultrasound, and place milestones in the history of acoustics in order.",45,{"title":20,"rows":21},"Lesson plate",[22,25,28,31,34,37],{"label":23,"value":24},"Depth","Go deeper",{"label":26,"value":27},"Reading time","≈ 45 minutes",{"label":29,"value":30},"Prior knowledge","Discover, Understand, Investigate",{"label":32,"value":33},"Chapters","10",{"label":35,"value":36},"Labs","Two sort\u002Fmatch games, one history animation",{"label":38,"value":39},"Units used","Hz, dB, m, s, m³",[41,45,51,57,60,65,70,83,88,93,96,101,106,110,113,118,121,125,130,134,139,142,146,159,172,216,231,235,240,243,247,251,290,293,298,301,310,322,335,349,367,380,385,388,420,432,436,441,474,478,483,520,524,529,581,608,795,798,804,820],{"id":42,"type":43,"markdown":44},"d-intro","prose","You now know *what* resonance, reverberation and echoes are, and you have tried them out. This layer asks the harder question: **why do they work the way they do, and where do the tidy rules of a textbook start to bend?**\n\nYou will find out why a swing cannot be pushed to infinite height, why the most famous \"resonance\" story in engineering is actually told wrong, why a sitar and a flute playing the identical note still sound utterly different, and how one Harvard physicist, several hundred borrowed theatre cushions and a stopwatch created the entire science of room acoustics in five years.",{"id":46,"type":47,"variant":48,"title":49,"markdown":50},"d-how-to-read","callout","observation","How to use this lesson","This layer leans harder on reasoning than on new facts. When a callout says a common story is wrong, read it twice: it is very often the version you already believed.",{"id":52,"type":53,"title":54,"eyebrow":55,"navLabel":56},"d-ch01","chapter","Free vibration, forced vibration and damping","Chapter 01","1 Three kinds of shake",{"id":58,"type":43,"markdown":59},"d-ch1-p1","Physics separates vibration into three related ideas, and mixing them up is the root of most confusion about resonance.\n\n**Free vibration** is what an object does when you disturb it once and then leave it alone: a plucked string, a struck bell, a flicked ruler. It vibrates at its own **natural frequency** (or frequencies) and gradually dies away.\n\n**Forced vibration** is what happens when something else keeps pushing the object, over and over, at a frequency of the pusher's choosing, not the object's own. A washing machine's frame is forced to vibrate at whatever speed the motor spins.\n\n**Resonance** is the special case where the forcing frequency happens to match the natural frequency. Then, instead of fighting the object's own tendency, each push adds neatly to the last, and the vibration grows far larger than the same push would produce at any other frequency.",{"id":61,"type":47,"variant":62,"title":63,"markdown":64},"d-def-damping","definition","Damping","**Damping** is anything that removes energy from a vibration: friction, air resistance, internal stretching losses in a material. Every real vibration is damped to some degree, which is why a struck bell eventually falls silent instead of ringing forever.\n\nDamping is also what stops a resonating object from growing without limit. As the swing goes higher, air resistance and friction at the pivot remove more energy per swing, until they remove exactly as much as the pushes put in. The swing then holds a steady maximum height: that balance point is called **steady-state resonance**.",{"id":66,"type":47,"variant":67,"title":68,"markdown":69},"d-aha-not-infinite","aha","Why resonance does not grow forever","It is tempting to think that pushing at exactly the natural frequency should make an object shake more and more without limit. In an imaginary world with zero damping, that is even mathematically true.\n\nBut every real object loses some energy to damping on every cycle, and losses grow **faster** than the input as the amplitude increases (more movement usually means more friction and more air resistance). The vibration grows only until the energy lost per cycle catches up with the energy added per cycle, and then it levels off. Heavily damped systems (a door on a stiff, oiled hinge) barely resonate at all; lightly damped systems (a wine glass, a tuning fork, a well-oiled swing) can build up to a dramatic, long-lasting resonance.",{"id":71,"type":72,"title":73,"problem":74,"steps":75,"help":81},"d-we-tuning-fork-box","worked_example","Two resonance boxes, one physical reason","A workshop keeps two tuning forks, one stamped 256 Hz and one stamped 440 Hz, each with its own closed-tube resonance box. Which fork needs the longer box, and by how much?",[76,77,78,79,80],"Closed-pipe resonance: L = v ÷ (4f), so a **lower** frequency needs a **longer** column.","For 256 Hz: L = 343 ÷ (4 × 256) = 343 ÷ 1,024 = **0.335 m**.","For 440 Hz: L = 343 ÷ (4 × 440) = 343 ÷ 1,760 = **0.195 m**.","Difference: 0.335 − 0.195 = **0.140 m**, 14 centimetres longer for the lower fork.","The general rule behind both boxes, and behind every wind instrument in this topic: a longer resonating air column has a lower natural frequency, because a longer column takes longer to complete each cycle of the standing wave inside it.",{"simplerExplanation":82},"Lower notes need more air to slosh back and forth in, so their matching box has to be longer.",{"id":84,"type":47,"variant":85,"title":86,"markdown":87},"d-example-ringing-times","example","Long ringers and short ringers","Damping varies enormously between materials and shapes, and you can hear it directly. Tap a large temple bell or a good tuning fork and the note can be heard for many seconds, sometimes close to a minute for a big bell, because very little energy leaks out of the metal on each cycle.\n\nTap a block of foam, a cushion, or a piece of soft wood wrapped in cloth, and there is barely a note at all: a dull thud that dies in a fraction of a second, because the material absorbs almost all the vibration's energy immediately as internal friction, rather than letting it ring. Between these extremes sits almost everything else you have met in this topic: a ghatam somewhere in the middle, a tabla deliberately damped by the player's palm on request.",{"id":89,"type":53,"title":90,"eyebrow":91,"navLabel":92},"d-ch02","The bridge everyone blames on resonance","Chapter 02","2 Tacoma Narrows",{"id":94,"type":43,"markdown":95},"d-ch2-p1","On 7 November 1940, the Tacoma Narrows Bridge in Washington state, in the USA, twisted itself apart in a 65 km\u002Fh wind and collapsed into the water below, four months after it opened. Film of the deck twisting violently before the fall is one of the most famous pieces of engineering footage ever recorded, and for decades it was the textbook example of resonance destroying a structure.\n\nIt is a wonderful, dramatic story. Large parts of it are wrong.",{"id":97,"type":47,"variant":98,"title":99,"markdown":100},"d-misconception-tacoma","misconception","\"The wind blew at the bridge's natural frequency\"","Many older physics textbooks state that the wind happened to gust at exactly the bridge deck's natural frequency, driving simple forced resonance. Engineering historians now describe the real cause differently: **aeroelastic flutter**, a self-reinforcing feedback loop between the wind and the twisting motion of the deck itself.\n\nAs the bridge deck twisted slightly in the wind, that twisting changed how the wind flowed around it, which pushed the deck to twist further, which changed the airflow again, and so on: an unstable, growing feedback loop rather than a case of an outside frequency matching a fixed natural one. The wind did not need to gust in any special rhythm at all; a fairly steady wind above a critical speed was enough to trigger the runaway effect.\n\nIt is still, loosely, a story about a structure being driven into a large, destructive vibration by wind, which is why it survives as a teaching example. But calling it simple textbook resonance skips the real and more interesting physics, and gets the mechanism wrong.",{"id":102,"type":47,"variant":103,"title":104,"markdown":105},"d-model-limit-resonance","model_limit","\"Matching frequency\" is not the whole resonance story","Everything in this topic so far treats resonance as one clean rule: match the frequency, get a big response. Two refinements matter once you look closer, even at this level.\n\nFirst, real objects rarely have only one natural frequency; a drum skin, a bell or a bridge deck has many, and any of them can be excited.\n\nSecond, some of the most dramatic real-world \"vibration disasters\", including Tacoma Narrows, involve feedback between the vibrating object and the force acting on it (the wind, in that case), which is a richer and less tidy phenomenon than the simple matched-frequency picture used for a swing or a wine glass. The simple picture is not wrong, exactly; it is the easy case, and real engineering has to plan for the harder ones too.",{"id":107,"type":47,"variant":85,"title":108,"markdown":109},"d-example-millennium","A real feedback wobble: London's Millennium Bridge","None of this makes vibration feedback a myth; it just means the mechanism is often richer than \"matched frequency\". On **10 June 2000**, London's new Millennium footbridge opened, and about **90,000 people** crossed it that first day, with up to 2,000 on the deck at once. Within hours it was swaying sideways by as much as **70 millimetres**, enough that pedestrians had to plant their feet wide like ice-skaters. It was closed within three days.\n\nThe cause, named **synchronous lateral excitation**, was a feedback loop between the bridge and the crowd, not a matched outside frequency either: once the deck swayed even slightly sideways, people unconsciously adjusted their footsteps to keep balance, and doing so in a slightly swaying crowd tends to fall into step together, which pushed the deck to sway more, which synchronised more footsteps, and so on. Engineers fitted **91 dampers** (devices that absorb vibration energy, adding deliberate damping) and reopened the bridge in 2002, since when it has stood steady under any crowd.",{"id":111,"type":43,"markdown":112},"d-ch2-p2","Two different bridges, two different feedback mechanisms, one shared lesson. Marching soldiers really are still ordered to break step on some footbridges as a sensible precaution, and wine glasses really can be shattered by a matched, sustained, sufficiently loud note: resonance and vibration feedback are completely real. The lesson is not that these effects are fake; it is that a famous headline example can be simplified past the point of being true, and a careful scientist checks the actual mechanism rather than repeating the good story.",{"id":114,"type":53,"title":115,"eyebrow":116,"navLabel":117},"d-ch03","The pillars that sing: resonance carved in stone","Chapter 03","3 Hampi's pillars",{"id":119,"type":43,"markdown":120},"d-ch3-p1","Every resonance example so far has involved something obviously flexible: a string, a skin, a column of air, a swaying bridge deck. Solid stone looks like the last place to find a musical note. At the Vitthala Temple in Hampi, Karnataka, built in the 15th and 16th centuries, it is exactly where you find one.",{"id":122,"type":47,"variant":85,"title":123,"markdown":124},"d-example-hampi","The SaReGaMa pillars of Hampi","Inside the temple's main hall stand **56 tall pillars, each carved from a single block of granite**, no joints, no hollow parts. Tap one gently with a finger or a soft mallet and it rings with a clear, sustained musical note; tap its neighbour and you hear a different note entirely. Visitors and musicians have long grouped the sounds into families resembling different instruments: some pillars ring like a **mridangam** or **tabla**, others like a **ghatam**, giving the group its popular name, the **SaReGaMa pillars**, after *sa, re, ga, ma*, the first four notes of Indian classical music.\n\nEach solid granite column has its own natural frequency, fixed the same way a ghatam's is: by its exact size, shape and the density of the stone, no strings or skins required. The temple's builders, working entirely by testing and listening, selected and shaped pillars until they had a whole hall full of different, pleasing natural frequencies, centuries before resonance was described mathematically anywhere in the world.",{"id":126,"type":47,"variant":127,"title":128,"markdown":129},"d-nuance-hampi-mechanism","nuance","The same rule as every other solid in this topic","It can feel like the Hampi pillars must involve something exotic. They do not: a struck granite pillar behaves exactly like the steel tumbler from Discover, the ghatam from Understand, or Chladni's bowed metal plates from later in this chapter. Any solid object, tapped or bowed, rings at its own natural frequency, set by its stiffness, its density and its shape.\n\nWhat makes Hampi remarkable is not new physics; it is that master builders, centuries ago, controlled that ordinary physics precisely enough, across 56 separate blocks of stone, to build a hall that could be played like an instrument.",{"id":131,"type":47,"variant":67,"title":132,"markdown":133},"d-aha-hampi-sympathetic","Sympathetic resonance, carved in stone","Musical pillars are not unique to Hampi: similar clusters exist at temples including Meenakshi Amman in Madurai and Thanumalayam in Suchindram, all built in a similar era, typically arranged as a group of pillars around one central support. Within a cluster, only the outer pillars were shaped to ring; the central one is usually structural, left plain.\n\nHere is the detail worth savouring: strike one pillar, and **another pillar tuned to nearly the same natural frequency can be heard answering it faintly**, entirely untouched. That is the Investigate layer's sitar sympathetic strings and this chapter's wine-glass-to-wine-glass experiment, playing out in solid granite. Whoever carved these halls centuries ago had, by patient listening alone, discovered and used sympathetic resonance in stone.",{"id":135,"type":53,"title":136,"eyebrow":137,"navLabel":138},"d-ch04","Harmonics: why the same note sounds different on every instrument","Chapter 04","4 Harmonics and timbre",{"id":140,"type":43,"markdown":141},"d-ch4-p1","Play the note A at 220 Hz on a bansuri and on a sitar. Both instruments vibrate 220 times a second, so both give you exactly the same pitch. Yet nobody would mistake one sound for the other. What is different is not the pitch: it is the **timbre** (pronounced \"tam-ber\"), the particular colour or character of a sound, and timbre comes from harmonics.\n\nAlmost nothing vibrates at only one pure frequency. A plucked string or a blown pipe vibrates simultaneously at its **fundamental** frequency and at a whole series of higher frequencies layered on top, called **overtones** or **harmonics**, each quieter than the last. The exact mixture of how loud each harmonic is, called the **spectrum**, is what your ear reads as timbre.",{"id":143,"type":47,"variant":62,"title":144,"markdown":145},"d-def-harmonics","Fundamental, harmonics and timbre","The **fundamental** is the lowest, usually loudest, frequency a vibrating source produces, and the one your ear reports as the pitch.\n\n**Harmonics** (or overtones) are extra frequencies present at the same time, at whole-number multiples of the fundamental: 2×, 3×, 4× and so on.\n\n**Timbre** is the character of a sound created by the relative loudness of its different harmonics. Two sounds can share a fundamental frequency (the same pitch) and still sound completely different because their harmonics differ.",{"id":147,"type":148,"items":149},"d-formulas-harmonics","formulas",[150,153,156],{"expression":151,"caption":152},"open pipe: f, 2f, 3f, 4f …","A pipe open at both ends (roughly, a bansuri) supports every whole-number multiple of its fundamental.",{"expression":154,"caption":155},"closed pipe: f, 3f, 5f, 7f …","A pipe closed at one end (roughly, blowing across a bottle) supports only odd multiples: the even harmonics are missing.",{"expression":157,"caption":158},"string: f, 2f, 3f, 4f …","A string fixed at both ends behaves like an open pipe: every whole-number harmonic is possible.",{"id":160,"type":72,"title":161,"problem":162,"steps":163,"help":170},"d-we-harmonic-series","Two harmonic series from the same fundamental","A sitar string's fundamental is 220 Hz. A closed drone pipe is tuned to the same 220 Hz fundamental. List the first four harmonics each one can produce, and say what is missing from the pipe's list.",[164,165,166,167,168,169],"The string behaves like an open system: every whole-number multiple is present.","String harmonics: 220, 220×2, 220×3, 220×4 = **220, 440, 660, 880 Hz**.","The closed pipe supports only odd multiples of its fundamental.","Pipe harmonics: 220, 220×3, 220×5, 220×7 = **220, 660, 1,100, 1,540 Hz**.","Missing from the pipe's list compared with the string: 440 Hz, 880 Hz and every other even multiple.","This missing-even-harmonics pattern is a real, audible difference: instruments and sung notes that are closer to a closed-pipe pattern (clarinets are the classic example) sound noticeably hollower than instruments with a full harmonic series.",{"simplerExplanation":171},"Both start at 220 Hz. The string adds every multiple (×2, ×3, ×4…); the closed pipe skips every even one and only adds ×3, ×5, ×7…",{"id":173,"type":174,"component":175,"componentVersion":5,"config":176,"objective":214,"textAlternative":215},"d-lab-sort-harmonics","interactive","sort-game",{"prompt":177,"bins":178,"items":188,"seconds":213},"A string's fundamental is 200 Hz. Sort each frequency by whether an open pipe or string could produce it, a closed pipe could produce it, both, or neither.",[179,182,185],{"id":180,"label":181},"both","Both open and closed",{"id":183,"label":184},"open-only","Open\u002Fstring only",{"id":186,"label":187},"neither","Neither (not a whole-number multiple)",[189,193,197,201,205,209],{"id":190,"label":191,"bin":180,"why":192},"f200","200 Hz (the fundamental)","Every vibrating system produces its own fundamental.",{"id":194,"label":195,"bin":183,"why":196},"f400","400 Hz (2 × 200)","An even multiple: only an open pipe or a string produces this harmonic.",{"id":198,"label":199,"bin":180,"why":200},"f600","600 Hz (3 × 200)","An odd multiple: both open and closed systems can produce this one.",{"id":202,"label":203,"bin":183,"why":204},"f800","800 Hz (4 × 200)","Another even multiple, missing from a closed pipe's series.",{"id":206,"label":207,"bin":180,"why":208},"f1000","1,000 Hz (5 × 200)","Odd multiple again: present in both series.",{"id":210,"label":211,"bin":186,"why":212},"f300","300 Hz (1.5 × 200)","Not a whole-number multiple of 200 Hz, so it is not part of either harmonic series.",0,"Sort six frequencies by which kind of resonating system, open or closed, could produce each one as a harmonic of a 200 Hz fundamental.","A sorting game with six frequency cards, given a 200 Hz fundamental, and three bins: **both open and closed**, **open or string only**, and **neither**.\n\n200 Hz (×1), 600 Hz (×3) and 1,000 Hz (×5) are odd multiples, present in both an open system's full series and a closed pipe's odd-only series, so they sort to **both**.\n\n400 Hz (×2) and 800 Hz (×4) are even multiples, present only in an open pipe or a vibrating string's series, so they sort to **open or string only**.\n\n300 Hz is not a whole-number multiple of 200 Hz at all, so it belongs to neither series: it simply is not one of this fundamental's harmonics.\n\nThe test behind every card: work out the multiple, then ask whether it is odd, even, or not a whole number at all.",{"id":217,"type":218,"itemId":219,"prompt":220,"check":221,"hints":225,"feedback":228},"d-prac-harmonic","practice","sound.deepen-harmonic-3rd","An open pipe's fundamental is 150 Hz. What is its third harmonic, in hertz?",{"kind":222,"answer":223,"tolerance":213,"unit":224},"number",450,"Hz",[226,227],"An open pipe supports every whole-number multiple of its fundamental.","Third harmonic means 3 × the fundamental.",{"correct":229,"incorrect":230},"Right: the third harmonic is 3 × 150 = **450 Hz**.","The third harmonic of an open pipe or string is simply 3 times the fundamental: 3 × 150 = 450 Hz.",{"id":232,"type":47,"variant":127,"title":233,"markdown":234},"d-nuance-syahi-again","The tabla's syahi, revisited with harmonics","Earlier layers mentioned that a tabla's black syahi patch tunes its overtones so the drum gives a real musical pitch. Here is the fuller reason: an untreated drum skin's natural harmonics are **not** whole-number multiples of its lowest mode, the way a string's are, so a plain struck skin sounds like a thud with no clear note, an effect you can hear on a bare frame drum.\n\nThe carefully weighted syahi paste changes the mass and stiffness across the skin so that several of its vibration modes are pulled into, or close to, a whole-number relationship with each other, mimicking a string's harmonic series closely enough for your ear to hear a genuine pitch. It is centuries-old practical acoustics engineering, done entirely by ear, generations before harmonics were understood mathematically.",{"id":236,"type":53,"title":237,"eyebrow":238,"navLabel":239},"d-ch05","How one physicist and 300 cushions invented room acoustics","Chapter 05","5 Sabine's cushions",{"id":241,"type":43,"markdown":242},"d-ch5-p1","In 1895, Harvard's newly built Fogg Lecture Hall was, by every account, useless for lectures: a word spoken at the front blurred into the next word before it reached the back. A young physics instructor named **Wallace Sabine** was asked to fix it, with no existing science of room acoustics to draw on. He built one, by brute-force experiment.",{"id":244,"type":47,"variant":85,"title":245,"markdown":246},"d-example-sabine-experiment","An organ pipe, a stopwatch and 300 borrowed cushions","Sabine's method was almost absurdly direct. He would sound an organ pipe at a steady note in the empty Fogg Lecture Hall, stop it suddenly, and time with a stopwatch how many seconds the sound took to fade to inaudibility: the reverberation time, measured as around **5 seconds** in the bare hall, far too long for clear speech.\n\nHe then borrowed **seat cushions from the nearby Sanders Theatre**, sometimes hundreds of them, carried them into the Fogg Hall at night, and repeated the timing. More cushions meant more sound absorbed on each bounce, and a shorter, more measurable reverberation time. Night after night, for several years, he added and removed cushions and re-measured, until he had enough data to find that reverberation time depends on the room's volume divided by its total absorption, exactly the T = 0.161 × V ÷ A relationship you used in Investigate.\n\nOne of Sabine's own cushions absorbed sound about as well as **0.7 square metres of open window** (an open window absorbs essentially all the sound that reaches it, since none bounces back), which is why the standard unit of absorption used in acoustics today is still called the **sabin**, in his honour.",{"id":248,"type":47,"variant":127,"title":249,"markdown":250},"d-nuance-empirical","A formula found by measuring, not by deriving from first principles","Sabine did not sit down and derive his equation from the physics of individual sound rays bouncing around a room, at least not at first; he found the *pattern* by patient, repeated measurement, and only later did the mathematics behind it become well understood (it can be pictured, roughly, as sound doing many short, random hops between absorbing surfaces before its energy runs out, with a bigger room giving each hop more room to travel and more absorption removing more energy per hop).\n\nThis is a genuine and respectable way to do science: notice a reliable pattern in careful data even before you can fully explain *why* the pattern holds, publish it, use it, and let the deeper explanation catch up. Sabine's formula was designing better concert halls years before anyone had worked out a complete mathematical justification for it.",{"id":252,"type":253,"component":254,"componentVersion":5,"config":255,"textAlternative":289},"d-anim-sabine-method","animation","process-steps",{"title":256,"diagram":257,"steps":258},"How Sabine measured his way to the reverberation formula","none",[259,264,269,274,279,284],{"id":260,"label":261,"description":262,"highlight":263},"problem","A hall nobody could understand","The new Fogg Lecture Hall at Harvard blurred speech into unintelligibility. Sabine was asked to fix it with no existing science of room acoustics to draw on.",[],{"id":265,"label":266,"description":267,"highlight":268},"pipe","Sound a steady note, then stop it","Sabine sounded an organ pipe at a fixed note in the empty hall, then cut it off sharply and started a stopwatch.",[],{"id":270,"label":271,"description":272,"highlight":273},"time","Time the fade to silence","He timed how many seconds the note took to fade to inaudibility: about 5 seconds in the bare hall, far too long for clear speech.",[],{"id":275,"label":276,"description":277,"highlight":278},"cushions","Borrow hundreds of cushions","At night, he and his assistants carried seat cushions from the nearby Sanders Theatre into the Fogg Hall to add absorption, then measured again.",[],{"id":280,"label":281,"description":282,"highlight":283},"repeat","Repeat for years","Adding and removing cushions and re-timing, again and again, for several years, revealed a reliable pattern linking reverberation time to the room's volume and its total absorption.",[],{"id":285,"label":286,"description":287,"highlight":288},"formula","T = 0.161 × V ÷ A","The pattern became a formula, now used to design concert halls, classrooms and studios everywhere, and the sabin, the unit of absorption, is named after him.",[],"A six-step account of Sabine's method: the unusable lecture hall, sounding an organ pipe and timing its fade with a stopwatch, borrowing hundreds of cushions from a nearby theatre at night to change the room's absorption, repeating the timed measurements for years, and finally the formula, T = 0.161 × V ÷ A, that resulted. It is a story about finding a reliable pattern through patient, repeated measurement before the full mathematical reason for it was worked out.",{"id":291,"type":43,"markdown":292},"d-ch5-p2","Sabine went on to be the acoustic consultant for **Boston's Symphony Hall**, which opened in 1900 and is still regarded as one of the finest-sounding concert halls in the world, designed from his cushion-and-stopwatch data rather than guesswork. Every concert hall, cinema, classroom, mosque, temple and recording studio built since, anywhere in the world, that has been deliberately shaped for good sound owes something to those late nights moving borrowed cushions across Harvard's campus.",{"id":294,"type":53,"title":295,"eyebrow":296,"navLabel":297},"d-ch06","Decibels: combining sources and fading with distance","Chapter 06","6 Decibel arithmetic",{"id":299,"type":43,"markdown":300},"d-ch6-p1","Investigate showed that two identical machines combine to about 3 dB louder than one, and that decibels add by energy rather than by simple counting. Here is the general tool behind that fact, and the matching rule for how loudness fades as you walk away from a source.",{"id":302,"type":148,"items":303},"d-formulas-db",[304,307],{"expression":305,"caption":306},"L = 10 × log₁₀(Σ 10^(Lᵢ÷10))","Turn each decibel level back into an energy ratio, add the energies, then convert the sum back to decibels.",{"expression":308,"caption":309},"L(d₂) = L(d₁) − 20 × log₁₀(d₂ ÷ d₁)","For a small source in open air, loudness falls by about 6 dB every time the distance doubles.",{"id":311,"type":72,"title":312,"problem":313,"steps":314,"help":320},"d-we-classroom-combine","Three different classroom noises, combined","A classroom has a humming fan measured at 60 dB, chattering students at 65 dB and traffic through the window at 70 dB, each measured alone at the same spot. What is the combined level with all three happening together?",[315,316,317,318,319],"Convert each level to an energy ratio: 10^(60÷10) = 1,000,000; 10^(65÷10) ≈ 3,162,278; 10^(70÷10) = 10,000,000.","Add the three energy ratios: 1,000,000 + 3,162,278 + 10,000,000 ≈ 14,162,278.","Convert back to decibels: 10 × log₁₀(14,162,278) ≈ 10 × 7.151.","Combined level ≈ **71.5 dB**.","Notice how little the two quieter sounds add to the loudest one alone (70 dB): only about 1.5 dB. **The loudest single source usually dominates the total.**",{"simplerExplanation":321},"Turn each dB number back into 'how much energy', add the energies, then turn the total back into dB. The loudest source barely gets nudged up by the quieter ones.",{"id":323,"type":72,"title":324,"problem":325,"steps":326,"help":333},"d-we-distance-drop","How much quieter is the back of the hall?","A loudspeaker measures 110 dB for someone standing 1 metre away. How loud does it seem to someone standing 10 metres away, assuming open air with nothing reflecting the sound?",[327,328,329,330,331,332],"Use level(d₂) = level(d₁) − 20 × log₁₀(d₂ ÷ d₁).","d₂ ÷ d₁ = 10 ÷ 1 = 10.","log₁₀(10) = 1, so 20 × 1 = 20.","level(10 m) = 110 − 20 = **90 dB**.","Compare with doubling the distance instead, to 2 m: 110 − 20 × log₁₀(2) = 110 − 6.0 = **104 dB**.","The pattern behind the numbers in earlier layers' safety advice: **every doubling of distance costs about 6 dB**, which is why simply stepping back from a firecracker or a loudspeaker helps far more than it might seem to.",{"simplerExplanation":334},"Going ten times further away costs 20 dB. Going twice as far costs about 6 dB. Distance is one of the cheapest, most effective ways to protect your hearing.",{"id":336,"type":218,"itemId":337,"prompt":338,"check":339,"hints":343,"feedback":346},"d-prac-combine","sound.deepen-combine-two","A washing machine measures 65 dB alone. A vacuum cleaner measures 68 dB alone, at the same spot. Use L = 10 × log₁₀(10^(65÷10) + 10^(68÷10)) to find the combined level, to one decimal place.",{"kind":222,"answer":340,"tolerance":341,"unit":342},69.8,0.2,"dB",[344,345],"Convert each level to an energy ratio (10 to the power of level ÷ 10), add them, then take 10 log₁₀ of the sum.","10^6.5 ≈ 3,162,278 and 10^6.8 ≈ 6,309,573.",{"correct":347,"incorrect":348},"Right: 3,162,278 + 6,309,573 ≈ 9,471,851, and 10 × log₁₀(9,471,851) ≈ **69.8 dB**.","Add the two energy ratios (about 3,162,278 and 6,309,573) to get about 9,471,851, then take 10 × log₁₀ of that: about 69.8 dB.",{"id":350,"type":351,"prompt":352,"options":353,"explanation":366},"d-predict-studio-vs-bathroom","prediction","A heavily padded recording studio and a small bare-tiled bathroom are clapped in, one after another. Which one has the shorter reverberation time, and why?",[354,357,360,363],{"id":355,"label":356},"a","The bathroom, because tiles are heavier than foam",{"id":358,"label":359},"b","The studio, because its soft padding absorbs sound quickly, adding little to the room's own reverberation",{"id":361,"label":362},"c","They are the same, since both rooms are small",{"id":364,"label":365},"d","The bathroom, because water pipes absorb sound","**The studio.** Its walls, ceiling and floor are deliberately covered in absorbing foam and fabric, giving a high total absorption A, and by T = 0.161 × V ÷ A a high A means a short reverberation time.\n\nA small bathroom, despite being tiny, is usually the *opposite* case: hard tiles reflect almost all the sound reaching them (low absorption), so its reverberation time can be surprisingly long for such a small volume, which is exactly why singing in the shower sounds so rich and full to the singer.",{"id":368,"type":351,"prompt":369,"options":370,"explanation":379},"d-predict-distance","A generator measures 80 dB at 2 metres. Someone wants to know the level at 8 metres, four times further away. Which calculation gives the right answer?",[371,373,375,377],{"id":355,"label":372},"80 − 20 × log₁₀(4) ≈ 68 dB",{"id":358,"label":374},"80 ÷ 4 = 20 dB",{"id":361,"label":376},"80 − 4 = 76 dB",{"id":364,"label":378},"80 dB, unchanged","**About 68 dB.** The distance ratio is 8 ÷ 2 = 4, so the drop is 20 × log₁₀(4) ≈ 20 × 0.602 ≈ 12 dB, giving 80 − 12 = 68 dB.\n\nThe wrong answers reveal common slips: dividing decibels like plain numbers (b), or subtracting the distance ratio itself instead of 20 × its logarithm (c). Decibels only ever combine through logarithms, never through ordinary arithmetic on the numbers themselves.",{"id":381,"type":53,"title":382,"eyebrow":383,"navLabel":384},"d-ch07","Sonar and ultrasound: trading range for detail","Chapter 07","7 Range vs detail",{"id":386,"type":43,"markdown":387},"d-ch7-p1","Investigate showed that a bat's high-pitched call reveals a small insect and that a medical scanner's frequency choice trades detail against depth. The underlying rule is the same everywhere echoes are used on purpose, and it is worth stating precisely.\n\nA wave can only clearly reveal a feature at least about as large as its own wavelength; smaller detail simply blurs. So **higher frequency (shorter wavelength) buys sharper detail.** At the same time, higher-frequency sound loses more of its energy to the medium it travels through, mainly as heat, so it cannot be usefully detected after travelling as far. **Higher frequency costs range.** Every echo-based technology sits somewhere on this trade-off, chosen for its job.",{"id":389,"type":390,"caption":391,"columns":392,"rows":397},"d-table-echo-tech","table","The range-versus-detail trade-off across several echo technologies",[393,394,395,396],"Technology","Roughly","Chooses","Because it needs",[398,403,408,412,416],[399,400,401,402],"Ship sonar mapping the whole sea floor","tens of kHz","lower frequency","range across kilometres, not fine detail",[404,405,406,407],"Fish-finder sonar","100–200 kHz","higher frequency","enough detail to show a shoal of fish, at modest range",[409,410,401,411],"Deep abdominal medical scan","2–5 MHz","reaching an organ several centimetres inside the body",[413,414,406,415],"Eye or skin medical scan","10–20 MHz","fine detail; the target is only millimetres away",[417,418,406,419],"Bat hunting flying insects","20–120 kHz","resolving a target only millimetres to centimetres across, at short range",{"id":421,"type":72,"title":422,"problem":423,"steps":424,"help":430},"d-we-sonar-wavelength","Choosing a sonar frequency: two real jobs, two wavelengths","A mapping sonar aimed at the whole sea floor uses 40,000 Hz. A fish-finder uses 200,000 Hz. Sound travels at about 1,500 m\u002Fs in sea water. Find each wavelength, and say which system could reveal finer detail.",[425,426,427,428,429],"Use λ = v ÷ f for each system.","Mapping sonar: λ = 1,500 ÷ 40,000 = **0.0375 m**, 3.75 centimetres.","Fish-finder: λ = 1,500 ÷ 200,000 = **0.0075 m**, 0.75 centimetres.","The fish-finder's wavelength is exactly one-fifth of the mapping sonar's (200,000 is five times 40,000), so it can reveal detail about **five times finer**, such as an individual fish rather than only the shape of the sea floor.","The trade-off is real: the higher-frequency fish-finder loses more energy over long distances and would not usefully map a whole ocean basin, which is exactly why the two jobs use two different frequencies rather than one system doing both.",{"simplerExplanation":431},"Five times the frequency gives one-fifth the wavelength, which reveals about five times finer detail, but does not reach nearly as far.",{"id":433,"type":47,"variant":127,"title":434,"markdown":435},"d-nuance-not-only-frequency","Frequency is not the only lever engineers pull","Real sonar and ultrasound systems also shape the *pulse* itself: a shorter pulse in time gives sharper range resolution (telling two close objects apart along the beam direction), and a wider or narrower beam trades a broad search area against a precisely aimed one.\n\nSo two systems at the *same* frequency can still differ enormously in what they reveal, depending on pulse shape, beam width and how cleverly the returning echoes are processed by a computer. Frequency sets the basic trade-off; engineering can still push the result further in either direction.",{"id":437,"type":53,"title":438,"eyebrow":439,"navLabel":440},"d-ch08","A short history of understanding sound","Chapter 08","8 History of acoustics",{"id":442,"type":443,"title":444,"items":445},"d-timeline-acoustics","timeline","From string ratios to the sabin",[446,450,454,458,462,466,470],{"time":447,"title":448,"text":449},"c. 530 BCE","Pythagoras and the string ratios","According to long-standing tradition, Pythagoras and his followers in ancient Greece noticed that plucked strings whose lengths form simple whole-number ratios, such as 2:1 or 3:2, sound pleasingly related notes: the first known link between mathematics and musical sound.",{"time":451,"title":452,"text":453},"1636","Mersenne's laws of strings","The French scholar Marin Mersenne worked out how a string's frequency depends on its length, tension and thickness, close to the rules used for sitar and veena strings in this topic.",{"time":455,"title":456,"text":457},"1787","Chladni's sand patterns","Ernst Chladni bowed sand-covered metal plates and showed the sand collecting along still lines between vibrating regions, giving the first clear, visible picture of an object's natural vibration patterns.",{"time":459,"title":460,"text":461},"1842","Doppler's proposal","Christian Doppler proposed that a moving source changes the pitch or colour an observer receives, an idea confirmed for sound within a few years and used throughout the next layer of this topic.",{"time":463,"title":464,"text":465},"1877-1878","Rayleigh's Theory of Sound","Lord Rayleigh published a two-volume mathematical treatise that gathered and organised the acoustics known at the time, becoming the standard reference for decades.",{"time":467,"title":468,"text":469},"1895-1900","Sabine's reverberation formula","Wallace Sabine measured his way to the reverberation-time formula using an organ pipe, a stopwatch and hundreds of borrowed theatre cushions, then used it to help design Boston's Symphony Hall.",{"time":471,"title":472,"text":473},"1900s onward","Sonar and ultrasound","Underwater echo ranging, developed urgently during and after the First World War for finding submarines, matured into both sonar and, from the 1950s, medical ultrasound imaging.",{"id":475,"type":47,"variant":127,"title":476,"markdown":477},"d-nuance-pythagoras-story","A famous story, told with a pinch of salt","The tale of Pythagoras discovering these ratios by listening to blacksmiths' hammers of different weights is popular, memorable, and almost certainly not literally true as usually told (hammer pitch does not actually follow simple weight ratios the story requires). What is well attested is that the whole-number string-length relationship itself is real and became the ancient world's clearest link between mathematics and music, whatever the exact original discovery story was.\n\nIt is a useful habit for any student of history and science: enjoy a vivid story, but hold its details more loosely than the underlying fact it illustrates.",{"id":479,"type":53,"title":480,"eyebrow":481,"navLabel":482},"d-ch09","Mix-ups worth clearing up","Chapter 09","9 Mix-ups",{"id":484,"type":390,"caption":485,"columns":486,"rows":489},"d-table-mixups","Ten things people say about resonance and room sound, and what is actually true",[487,488],"People often say","What is actually true",[490,493,496,499,502,505,508,511,514,517],[491,492],"\"Resonance always makes something bigger, without limit\"","Real objects are damped; the vibration grows only until energy in equals energy lost per cycle.",[494,495],"\"Tacoma Narrows Bridge fell purely from resonance\"","The dominant cause was aeroelastic flutter, a wind-driven feedback effect, not simple matched-frequency resonance.",[497,498],"\"The Millennium Bridge wobbled because of wind, like Tacoma Narrows\"","It was pedestrians' footsteps synchronising with the bridge's sway, a different feedback loop, with no wind involved.",[500,501],"\"Stone cannot ring like a drum or a string\"","Hampi's granite pillars ring at their own natural frequency when tapped, exactly like any other solid.",[503,504],"\"Two instruments playing the same note sound identical\"","Only the fundamental matches; different harmonics give each instrument its own timbre.",[506,507],"\"A closed pipe and an open pipe of the same length sound the same note\"","A closed pipe of the same length sounds an octave lower and is missing its even harmonics.",[509,510],"\"Sabine worked out his formula purely with mathematics\"","He found it by years of measurement with an organ pipe, a stopwatch and borrowed cushions.",[512,513],"\"Doubling the distance from a speaker halves the loudness in decibels\"","It costs about 6 dB, not half the number: decibels never combine by ordinary division.",[515,516],"\"Higher-frequency sonar and ultrasound are always better\"","Higher frequency gives finer detail but shorter range; the right choice depends on the job.",[518,519],"\"Adding a quiet noise to a loud one meaningfully increases the total\"","A source much quieter than the dominant one barely changes the combined decibel level.",{"id":521,"type":522,"prompt":523},"d-reflect-hampi","reflection","The Hampi pillars, the Tacoma Narrows Bridge and the Millennium Bridge all involve a solid structure's own natural frequency, but only two of the three involve a feedback loop rather than simple resonance. Identify which two, and in one sentence each, say what was feeding back into what.",{"id":525,"type":53,"title":526,"eyebrow":527,"navLabel":528},"d-ch10","Check what you know","Chapter 10","10 Check yourself",{"id":530,"type":531,"title":532,"terms":533},"d-glossary","glossary","Deepen words to keep",[534,538,542,545,549,553,557,561,565,569,573,577],{"term":535,"meaning":536,"example":537},"Free vibration","How an object vibrates once disturbed and then left alone, at its own natural frequency.","A plucked string ringing after release.",{"term":539,"meaning":540,"example":541},"Forced vibration","Vibration driven continuously by an outside push at a frequency the pusher chooses.","A washing machine's frame, driven by the motor.",{"term":63,"meaning":543,"example":544},"Anything that removes energy from a vibration, such as friction or air resistance.","Why a struck bell eventually falls silent.",{"term":546,"meaning":547,"example":548},"Steady-state resonance","The balance point where energy added by resonance each cycle equals energy lost to damping.","A pushed swing settling at a maximum height.",{"term":550,"meaning":551,"example":552},"Aeroelastic flutter","A self-reinforcing feedback loop between wind and a flexible structure's own motion, distinct from simple resonance.","The real cause of the Tacoma Narrows Bridge collapse.",{"term":554,"meaning":555,"example":556},"Synchronous lateral excitation","A feedback loop where a crowd's footsteps unconsciously synchronise with a bridge's sway, amplifying it.","London's Millennium Bridge in 2000.",{"term":558,"meaning":559,"example":560},"Fundamental","The lowest, usually loudest, frequency a vibrating source produces; what your ear reports as pitch.","A sitar string's main note.",{"term":562,"meaning":563,"example":564},"Harmonic (overtone)","An extra frequency present alongside the fundamental, at a whole-number multiple of it.","440 Hz and 660 Hz above a 220 Hz fundamental.",{"term":566,"meaning":567,"example":568},"Timbre","The character or colour of a sound, created by the relative loudness of its harmonics.","Why a sitar and a bansuri differ at the same pitch.",{"term":570,"meaning":571,"example":572},"Sabin","The unit of sound absorption, roughly one square metre of open window, named after Wallace Sabine.","One theatre cushion absorbs about 0.7 sabins.",{"term":574,"meaning":575,"example":576},"Decibel combination","Adding sound levels by converting each to an energy ratio, summing, then converting back to decibels.","70 dB + 75 dB combine to about 76.2 dB, not 145 dB.",{"term":578,"meaning":579,"example":580},"Resolution (of an echo system)","The smallest detail a sonar or ultrasound system can distinguish, tied to its wavelength.","A higher frequency gives finer resolution.",{"id":582,"type":174,"component":583,"componentVersion":5,"config":584,"objective":606,"textAlternative":607},"d-lab-match-history","match-pairs",{"prompt":585,"mode":586,"pairs":587},"Match each person to what they discovered or built.","connect",[588,591,594,597,600,603],{"a":589,"b":590},"Pythagoras (c. 530 BCE)","Simple whole-number ratios of string length give related musical notes",{"a":592,"b":593},"Marin Mersenne (1636)","Worked out how a string's frequency depends on length, tension and thickness",{"a":595,"b":596},"Ernst Chladni (1787)","Sand on a bowed metal plate reveals its vibration patterns",{"a":598,"b":599},"Christian Doppler (1842)","A moving source shifts the pitch or colour an observer receives",{"a":601,"b":602},"Lord Rayleigh (1877-1878)","Wrote the first full mathematical textbook of acoustics",{"a":604,"b":605},"Wallace Sabine (1895-1900)","Measured his way to the reverberation-time formula with cushions and a stopwatch","Connect six figures in the history of acoustics to what each one discovered or built.","A matching game with six pairs, in chronological order: Pythagoras and string-length ratios (about 530 BCE), Marin Mersenne and the laws of vibrating strings (1636), Ernst Chladni and sand patterns on vibrating plates (1787), Christian Doppler and the shifting pitch of a moving source (1842), Lord Rayleigh and the first full mathematical textbook of acoustics (1877-1878), and Wallace Sabine and the reverberation-time formula, found through years of measurement (1895-1900).",{"id":609,"type":610,"title":611,"questions":612},"d-quiz","quiz","Fourteen questions on the reasoning behind resonance and room sound",[613,626,639,652,665,678,691,704,717,730,743,756,769,782],{"itemId":614,"prompt":615,"options":616,"correct":358,"why":625},"sound.deepen-q-hampi","The granite pillars of the Vitthala Temple at Hampi ring with different musical notes when tapped mainly because",[617,619,621,623],{"id":355,"label":618},"they are hollow and filled with air",{"id":358,"label":620},"each solid pillar has its own natural frequency, set by its size, shape and stone",{"id":361,"label":622},"they contain hidden metal strings",{"id":364,"label":624},"the temple's location amplifies any sound","Like the ghatam and the steel tumbler earlier in this topic, a solid object rings at its own natural frequency when struck; the pillars' different shapes and stone give each one a different note.",{"itemId":627,"prompt":628,"options":629,"correct":358,"why":638},"sound.deepen-q-millennium","London's Millennium Bridge swayed dangerously in 2000 mainly because of",[630,632,634,636],{"id":355,"label":631},"strong wind, exactly like Tacoma Narrows",{"id":358,"label":633},"pedestrians' footsteps unconsciously synchronising with the bridge's sway, reinforcing it",{"id":361,"label":635},"a design fault unrelated to vibration",{"id":364,"label":637},"an earthquake in London","Synchronous lateral excitation: as the deck swayed slightly, pedestrians adjusted their steps to keep balance, which tended to synchronise the crowd and reinforce the sway further, a feedback loop between people and bridge.",{"itemId":640,"prompt":641,"options":642,"correct":358,"why":651},"sound.deepen-q-damping","A struck bell eventually falls silent because",[643,645,647,649],{"id":355,"label":644},"the bell runs out of vibration",{"id":358,"label":646},"damping removes energy from the vibration each cycle",{"id":361,"label":648},"the air stops carrying sound after a while",{"id":364,"label":650},"the bell's frequency drops to zero","Friction and internal losses (damping) remove a little energy every cycle, so the vibration's amplitude shrinks until it is too small to hear.",{"itemId":653,"prompt":654,"options":655,"correct":358,"why":664},"sound.deepen-q-resonance-limit","A resonating swing settles at a steady maximum height instead of growing forever because",[656,658,660,662],{"id":355,"label":657},"the pusher gets tired",{"id":358,"label":659},"energy lost to damping each cycle eventually matches the energy added",{"id":361,"label":661},"gravity switches off",{"id":364,"label":663},"the chains stretch permanently","Growth stops once losses per cycle catch up with the energy the pushes add per cycle: a steady balance, not an infinite climb.",{"itemId":666,"prompt":667,"options":668,"correct":358,"why":677},"sound.deepen-q-tacoma","The Tacoma Narrows Bridge collapse of 1940 is now understood to have been caused mainly by",[669,671,673,675],{"id":355,"label":670},"wind gusting at exactly the bridge's natural frequency",{"id":358,"label":672},"aeroelastic flutter, a feedback loop between wind and the twisting deck",{"id":361,"label":674},"an earthquake",{"id":364,"label":676},"traffic vibration","Modern analysis points to aeroelastic flutter, a self-reinforcing feedback effect, rather than the simple forced-resonance story many older textbooks tell.",{"itemId":679,"prompt":680,"options":681,"correct":358,"why":690},"sound.deepen-q-timbre","A sitar and a bansuri play the same 220 Hz note. They sound different mainly because of differences in",[682,684,686,688],{"id":355,"label":683},"loudness",{"id":358,"label":685},"their harmonics (overtones), which give each its own timbre",{"id":361,"label":687},"the speed of sound",{"id":364,"label":689},"the room they are played in","Same fundamental, different mixture of harmonics: that mixture is timbre, the character that lets you tell instruments apart even at the same pitch.",{"itemId":692,"prompt":693,"options":694,"correct":358,"why":703},"sound.deepen-q-closed-harmonics","A pipe closed at one end, compared with an open pipe of the same fundamental, is missing",[695,697,699,701],{"id":355,"label":696},"its odd harmonics",{"id":358,"label":698},"its even harmonics",{"id":361,"label":700},"all its harmonics",{"id":364,"label":702},"nothing; they are identical","A closed pipe supports only odd multiples of its fundamental (f, 3f, 5f...), so all the even harmonics an open pipe would have are missing.",{"itemId":705,"prompt":706,"options":707,"correct":358,"why":716},"sound.deepen-q-sabine-method","Wallace Sabine found his reverberation-time formula mainly by",[708,710,712,714],{"id":355,"label":709},"pure mathematical derivation before any measurement",{"id":358,"label":711},"repeated measurement with an organ pipe, a stopwatch and borrowed cushions",{"id":361,"label":713},"copying an ancient Roman formula",{"id":364,"label":715},"computer simulation","Sabine measured reverberation time again and again while changing the absorption in Harvard's Fogg Lecture Hall, finding the pattern before it was fully explained mathematically.",{"itemId":718,"prompt":719,"options":720,"correct":358,"why":729},"sound.deepen-q-sabin-unit","The unit of sound absorption named after Sabine, the sabin, is defined as roughly equivalent to",[721,723,725,727],{"id":355,"label":722},"one seat cushion",{"id":358,"label":724},"one square metre of perfectly absorbing open window",{"id":361,"label":726},"one decibel",{"id":364,"label":728},"one hertz","One sabin is the absorption of about one square metre of open window, which absorbs essentially all the sound reaching it.",{"itemId":731,"prompt":732,"options":733,"correct":361,"why":742},"sound.deepen-q-combine-loudest","Three noise sources measure 60 dB, 65 dB and 70 dB alone. Combined, the level is closest to",[734,736,738,740],{"id":355,"label":735},"195 dB",{"id":358,"label":737},"70 dB, unchanged",{"id":361,"label":739},"71.5 dB",{"id":364,"label":741},"65 dB, the average","Adding the three energy ratios and converting back gives about 71.5 dB: only a little above the loudest source alone.",{"itemId":744,"prompt":745,"options":746,"correct":358,"why":755},"sound.deepen-q-distance-db","Doubling your distance from a small open-air sound source changes the level by about",[747,749,751,753],{"id":355,"label":748},"−3 dB",{"id":358,"label":750},"−6 dB",{"id":361,"label":752},"−12 dB",{"id":364,"label":754},"no change","level(2d) = level(d) − 20 × log₁₀(2) ≈ level(d) − 6 dB: the well-known 'double the distance, lose about 6 dB' rule.",{"itemId":757,"prompt":758,"options":759,"correct":358,"why":768},"sound.deepen-q-tradeoff","A sonar system needs to map an entire sea floor across several kilometres. It should probably use",[760,762,764,766],{"id":355,"label":761},"the highest possible frequency, for maximum detail",{"id":358,"label":763},"a lower frequency, trading detail for range",{"id":361,"label":765},"no sound at all, only light",{"id":364,"label":767},"a single very loud click and nothing else","Lower frequency loses less energy over long distances, giving the range needed to cover kilometres, at the cost of fine detail.",{"itemId":770,"prompt":771,"options":772,"correct":355,"why":781},"sound.deepen-q-history-order","Put these in the correct historical order: Sabine's reverberation formula, Chladni's sand patterns, Rayleigh's Theory of Sound.",[773,775,777,779],{"id":355,"label":774},"Chladni, Rayleigh, Sabine",{"id":358,"label":776},"Sabine, Chladni, Rayleigh",{"id":361,"label":778},"Rayleigh, Chladni, Sabine",{"id":364,"label":780},"Chladni, Sabine, Rayleigh","Chladni (1787), then Rayleigh's Theory of Sound (1877-1878), then Sabine's reverberation work (1895-1900).",{"itemId":783,"prompt":784,"options":785,"correct":358,"why":794},"sound.deepen-q-pythagoras","The Pythagoras hammer-weight story about discovering musical ratios is best described as",[786,788,790,792],{"id":355,"label":787},"confirmed exactly as told, with precise records",{"id":358,"label":789},"a popular story whose details are doubtful, though the underlying string-ratio fact is real",{"id":361,"label":791},"completely invented with no basis at all",{"id":364,"label":793},"about Chladni, not Pythagoras","The whole-number string-length relationship is genuine and historically important; the specific blacksmith's-hammer anecdote is a much later, probably embellished, retelling.",{"id":796,"type":522,"prompt":797},"d-reflect","Choose one popular \"science fact\" you have heard told as a dramatic story (Tacoma Narrows and resonance is one; you may pick another). Write down the simplified version people usually tell, then research or reason out what is actually more accurate, and explain in two or three sentences why the simplified version spread so widely anyway.",{"id":799,"type":800,"conceptId":801,"relation":802,"explanation":803},"d-conn-electricity","connection","electricity","related_to","Resonance is not unique to sound: a radio tuning circuit is deliberately built to resonate at one station's electrical frequency and ignore the rest, the same matched-frequency idea used throughout this layer.",{"id":805,"type":806,"title":807,"points":808},"d-cheat","summary","Cheat sheet",[809,810,811,812,813,814,815,816,817,818,819],"**Free vibration** happens once and dies away at the object's own natural frequency; **forced vibration** is driven by something else; **resonance** is forced vibration matched to the natural frequency.","**Damping** removes energy every cycle and is why resonance settles at a steady maximum instead of growing forever.","**Tacoma Narrows Bridge (1940)** is usually blamed on simple resonance; the real cause was wind-driven aeroelastic flutter, a feedback effect.","**London's Millennium Bridge (2000)** swayed from a different feedback loop: pedestrians' footsteps synchronising with its sway, fixed with 91 dampers.","**Hampi's granite pillars** ring at their own natural frequency when tapped, exactly like a ghatam or a steel tumbler, just carved from solid stone.","**Timbre** comes from harmonics layered on the fundamental. Open pipes and strings get every whole-number harmonic; closed pipes get only odd ones.","**Wallace Sabine** found T = 0.161 × V ÷ A by measuring reverberation with an organ pipe, a stopwatch and hundreds of borrowed cushions (1895-1900); the sabin unit honours him.","**Decibels combine by energy**: convert each level to a ratio, add, convert back. The loudest source usually dominates.","**Distance costs about 6 dB per doubling**: level(d₂) = level(d₁) − 20 × log₁₀(d₂ ÷ d₁).","**Higher frequency buys detail and costs range**, in sonar, ultrasound and echolocation alike; engineers choose frequency to match the job.","**Acoustics grew over 2,500 years**: Pythagoras's ratios, Chladni's patterns, Rayleigh's textbook, Sabine's formula, then sonar and medical ultrasound.",{"id":821,"type":822,"sourceIds":823},"d-sources","sources",[824,825,826,827,828,829,830,831,832,833,834,835],"sound-hyperphysics-standing-waves","sound-hyperphysics-speed","sound-wikipedia-chladni","sound-wikipedia-sabine","sound-britannica-theory-of-sound","sound-britannica-doppler","sound-physicsclassroom-sound","sound-ncert-class9-sound","sound-wikipedia-millennium-bridge","sound-incredibleindia-hampi","sound-sahapedia-musical-pillars","sound-wikipedia-tacoma-narrows",[824,825,826,827,828,829,830,831,832,833,834,835],"needs_review",{"generatedBy":839,"notes":840},"claude-code","Draft generated locally; pending owner review. Harmonic series, decibel combination, distance attenuation and sonar wavelengths computed and asserted in scratchpad\u002Fsound\u002Fnumbers.py; historical dates cross-checked against Wikipedia, Britannica and Incredible India.","0cc5ecebf75bc7fd6725560e29253ecf4b7d5d76730b307d358c154eda51b353",{"component:sort-game@1":843,"logic:practice":844,"component:process-steps@1":845,"component:match-pairs@1":846,"source:sound-britannica-doppler":847,"source:sound-britannica-theory-of-sound":848,"source:sound-hyperphysics-speed":849,"source:sound-hyperphysics-standing-waves":850,"source:sound-incredibleindia-hampi":851,"source:sound-ncert-class9-sound":852,"source:sound-physicsclassroom-sound":853,"source:sound-sahapedia-musical-pillars":854,"source:sound-wikipedia-chladni":855,"source:sound-wikipedia-millennium-bridge":856,"source:sound-wikipedia-sabine":857,"source:sound-wikipedia-tacoma-narrows":858},"b164f45a2c8ca08f26c450768ff0231e113e9fe45381eddb34dc6d0548596c38","3d6b0fe1b15255975a32b0fcd94e8019bc959ad45cbf12e136e86149549c6878","c2f918c426383c50d52939054780add1488f3f282c9d1a965c3a38345ddcb265","2a8ee4ac87460b4e1175a4bb13c96b03d577db06dde95670eb7fcfe4ad787899","838644206b59946eaed329bdfe25d227ac6dbb22279805052fee24175ed3ce31","cba75c58d9699758907d12617b5bb93fd35808e6103bbfde0e0ed7e6b1df79d2","4cb248d8beea032b112212f11d2a3d10751a3a68da1b26d2ef93b11835467932","aceb27d2545768b5ef61862c7c358c3c7c317288d7963dbd8712f8acc8af3529","3916ef6e0bb1454bf1760e0e6aa3f157db894496d902fc037c46e7f14af9c3bb","615e6ca7b7252ccb2523eea00746ef69215849397defe43c853a9fcb617799dd","967196792bc122ee73ed66cefbf9d62ac069b2c688af723b4f4eb52fcdd8174a","5d76b2a974c675cc5f3f01752c9c973f20f67755c6f44b54e67168e73bb823c6","9733a477f6f82245da6dd15c71225014e5550e2e8cdf0b50bd484824975ff25e","ae4374a4cfe24f389b893c7e05c95a3d831f7e74e18fec15f68911c7c81c5a9b","cbdb59e24046f047beb3a4063a9ca9ce41e626755e64ea1d8c73b3c518f9b2df","ce03a07117b6faa18180a8eb4bcc939f5789c12f2f2fb273245ac2f1e4d80934",{"state":860,"reviewer":861,"selfReview":862,"reviewedAt":863,"method":864},"approved","The library owner",true,"2026-09-20T10:18:37.581Z","owner_bulk","preview-7e1cbbcc4f",1789899597230]