SoundGo deeperabout 45 min

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.

Start at chapter 1

In this part you’ll

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.

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?

You 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.

Chapter 01

Free vibration, forced vibration and damping

Physics separates vibration into three related ideas, and mixing them up is the root of most confusion about resonance.

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.

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.

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.

Worked example

0 / 5 steps shown

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?

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Chapter 02

The bridge everyone blames on resonance

On 7 November 1940, the Tacoma Narrows Bridge in Washington state, in the USA, twisted itself apart in a 65 km/h 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.

It is a wonderful, dramatic story. Large parts of it are wrong.

Common mix-up

"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.

As 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.

It 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.

Model limits

"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.

First, 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.

Second, 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.

Example

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.

The 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.

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.

Chapter 03

The pillars that sing: resonance carved in stone

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.

Example

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.

Each 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.

Aha

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.

Here 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.

Chapter 04

Harmonics: why the same note sounds different on every instrument

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.

Almost 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.

open pipe: f, 2f, 3f, 4f …

A pipe open at both ends (roughly, a bansuri) supports every whole-number multiple of its fundamental.

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.

string: f, 2f, 3f, 4f …

A string fixed at both ends behaves like an open pipe: every whole-number harmonic is possible.

Worked example

0 / 6 steps shown

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.

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Lab

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 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.

6 cards, 3 bins. Tap a card, then tap its bin. You can also drag, or press a bin’s number key.

Text version of this activity

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.

200 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.

400 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.

300 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.

The test behind every card: work out the multiple, then ask whether it is odd, even, or not a whole number at all.

Try it

It’s more subtle

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.

The 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.

Chapter 05

How one physicist and 300 cushions invented room acoustics

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.

Example

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.

He 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.

One 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.

It’s more subtle

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).

This 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.

Step through

How Sabine measured his way to the reverberation formula

Step 1 of 6

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.

All steps

: 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.
: Sabine sounded an organ pipe at a fixed note in the empty hall, then cut it off sharply and started a stopwatch.
: 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.
: At night, he and his assistants carried seat cushions from the nearby Sanders Theatre into the Fogg Hall to add absorption, then measured again.
: 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.
: 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.

Text version of this activity

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.

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.

Chapter 06

Decibels: combining sources and fading with distance

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.

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.

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.

Worked example

0 / 5 steps shown

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?

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Worked example

0 / 6 steps shown

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?

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Try it

Predict first

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?

The bathroom, because tiles are heavier than foamThe studio, because its soft padding absorbs sound quickly, adding little to the room's own reverberationThey are the same, since both rooms are smallThe bathroom, because water pipes absorb sound

Predict first

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?

80 − 20 × log₁₀(4) ≈ 68 dB80 ÷ 4 = 20 dB80 − 4 = 76 dB80 dB, unchanged

Chapter 07

Sonar and ultrasound: trading range for detail

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.

A 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.

TableThe range-versus-detail trade-off across several echo technologies
Technology	Roughly	Chooses	Because it needs
Ship sonar mapping the whole sea floor	tens of kHz	lower frequency	range across kilometres, not fine detail
Fish-finder sonar	100–200 kHz	higher frequency	enough detail to show a shoal of fish, at modest range
Deep abdominal medical scan	2–5 MHz	lower frequency	reaching an organ several centimetres inside the body
Eye or skin medical scan	10–20 MHz	higher frequency	fine detail; the target is only millimetres away
Bat hunting flying insects	20–120 kHz	higher frequency	resolving a target only millimetres to centimetres across, at short range

Worked example

0 / 5 steps shown

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/s in sea water. Find each wavelength, and say which system could reveal finer detail.

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Chapter 08

A short history of understanding sound

From string ratios to the sabin

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.
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.
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.
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.
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.
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.
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.

Chapter 09

Mix-ups worth clearing up

TableTen things people say about resonance and room sound, and what is actually true
People often say	What is actually true
"Resonance always makes something bigger, without limit"	Real objects are damped; the vibration grows only until energy in equals energy lost per cycle.
"Tacoma Narrows Bridge fell purely from resonance"	The dominant cause was aeroelastic flutter, a wind-driven feedback effect, not simple matched-frequency resonance.
"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.
"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.
"Two instruments playing the same note sound identical"	Only the fundamental matches; different harmonics give each instrument its own timbre.
"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.
"Sabine worked out his formula purely with mathematics"	He found it by years of measurement with an organ pipe, a stopwatch and borrowed cushions.
"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.
"Higher-frequency sonar and ultrasound are always better"	Higher frequency gives finer detail but shorter range; the right choice depends on the job.
"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.

Reflect

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.

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Chapter 10

Check what you know

Words to know

All maths vocabulary →

Deepen words to keep

Free vibration: How an object vibrates once disturbed and then left alone, at its own natural frequency.; Example: A plucked string ringing after release.
Forced vibration: Vibration driven continuously by an outside push at a frequency the pusher chooses.; Example: A washing machine's frame, driven by the motor.
Damping: Anything that removes energy from a vibration, such as friction or air resistance.; Example: Why a struck bell eventually falls silent.
Steady-state resonance: The balance point where energy added by resonance each cycle equals energy lost to damping.; Example: A pushed swing settling at a maximum height.
Aeroelastic flutter: A self-reinforcing feedback loop between wind and a flexible structure's own motion, distinct from simple resonance.; Example: The real cause of the Tacoma Narrows Bridge collapse.
Synchronous lateral excitation: A feedback loop where a crowd's footsteps unconsciously synchronise with a bridge's sway, amplifying it.; Example: London's Millennium Bridge in 2000.
Fundamental: The lowest, usually loudest, frequency a vibrating source produces; what your ear reports as pitch.; Example: A sitar string's main note.
Harmonic (overtone): An extra frequency present alongside the fundamental, at a whole-number multiple of it.; Example: 440 Hz and 660 Hz above a 220 Hz fundamental.
Timbre: The character or colour of a sound, created by the relative loudness of its harmonics.; Example: Why a sitar and a bansuri differ at the same pitch.
Sabin: The unit of sound absorption, roughly one square metre of open window, named after Wallace Sabine.; Example: One theatre cushion absorbs about 0.7 sabins.
Decibel combination: Adding sound levels by converting each to an energy ratio, summing, then converting back to decibels.; Example: 70 dB + 75 dB combine to about 76.2 dB, not 145 dB.
Resolution (of an echo system): The smallest detail a sonar or ultrasound system can distinguish, tied to its wavelength.; Example: A higher frequency gives finer resolution.

Lab

Connect six figures in the history of acoustics to what each one discovered or built.

Match each person to what they discovered or built.

6 pairs are hiding in two mixed-up columns. Pick one from each side to join them.

Text version of this activity

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).

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.

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Related to

Electricity

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.

Keep this

Cheat sheet

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.

Where this comes from

Sources

Standing Waves (opens another website) — HyperPhysics, Georgia State Universityawaiting check
Supports resonance and standing waves as constructive interference of waves travelling in opposite directions, nodes and antinodes, and resonance in strings and air columns such as wind instruments.
Speed of Sound in Air and Other Materials (opens another website) — HyperPhysics, Georgia State Universityawaiting check
Supports the value 343 m/s in dry air at 20 degrees C, the approximation v = 331.3 + 0.606 x temperature in Celsius, speeds in fresh water, wood and steel, and why stiffness rather than density decides the speed.
Ernst Chladni (opens another website) — Wikipediaawaiting check
Supports Chladni's 1787 technique of bowing a sand-covered metal plate to reveal its vibration patterns (Chladni figures), an early demonstration that a solid object vibrates in specific patterns at its natural frequencies.
Wallace Clement Sabine (opens another website) — Wikipediaawaiting check
Supports Sabine's work from 1895 on the Fogg Lecture Hall at Harvard, his development of the reverberation-time formula, and his role as acoustic consultant for Boston's Symphony Hall, which opened in 1900.
The Theory of Sound (opens another website) — Encyclopaedia Britannicaawaiting check
Supports Lord Rayleigh's two-volume The Theory of Sound (1877-1878), the first comprehensive mathematical treatise on acoustics and a foundation of modern acoustic theory.
Doppler effect (opens another website) — Encyclopaedia Britannicaawaiting check
Supports the Doppler effect for a passing train horn or ambulance siren, the formulas for an approaching and a receding source, and Christian Doppler's 1842 proposal.
Sound Waves and Music (opens another website) — The Physics Classroomawaiting check
Supports longitudinal versus transverse waves on a slinky and a rope, pitch and frequency, amplitude and loudness, resonance and standing waves on strings and in air columns, and the mathematics of echo and reverberation.
Sound (Science, Class 9, Chapter 12) (opens another website) — NCERTawaiting check
Supports sound as a vibration needing a medium, the bell-in-a-vacuum-jar experiment, compressions and rarefactions, longitudinal waves, v = f x wavelength, speeds in air, water and steel, echo timing t = 2d/v, reverberation, SONAR, the ear, and the 20 Hz to 20 kHz range.
Millennium Bridge, London (opens another website) — Wikipediaawaiting check
Supports the bridge's opening on 10 June 2000, about 90,000 pedestrians crossing that day, closure within three days after swaying up to 70 mm sideways, the synchronous-lateral-excitation explanation, and its 2002 reopening with dampers fitted.
Vitthala Temple Complex, Hampi (opens another website) — Incredible India, Ministry of Tourism, Government of Indiaawaiting check
Supports the 56 monolithic granite 'SaReGaMa' musical pillars of the 15th-16th century Vitthala Temple at Hampi, each producing a distinct musical tone when tapped, resembling different instruments.
Pillars that Sing! Architectural Marvels of Indian Temples (opens another website) — Sahapediaawaiting check
Supports musical pillars appearing in several Vijayanagar-era South Indian temples besides Hampi (including Madurai and Suchindram), carved from silica-rich stone, and pillars of the same natural frequency within a cluster ringing by sympathetic resonance when a neighbour is struck.
Tacoma Narrows Bridge (1940) (opens another website) — Wikipediaawaiting check
Supports the 1940 collapse in high wind, and the correction that many textbooks wrongly blame simple forced resonance, when the real cause was wind-driven aeroelastic flutter.

End of Go deeper

What you just read

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.

The web

Explore a connection

Contrasts with
Light
Both travel as waves and carry energy, but light needs no material and races a million times faster than sound.
Used inanother area
Anatomy of the human body
The ear turns shaking air into signals a nerve can carry: a drum, three tiny bones and a spiral of fluid.
Related to
Electricity
Microphones and speakers turn sound into current and current back into sound.

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Revision 1 · release preview-7e1cbbcc4f · accepted 20/09/2026

Two resonance boxes, one physical reason

Two harmonic series from the same fundamental

Three different classroom noises, combined

How much quieter is the back of the hall?

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