Illustrated Physics
physics, drawn
Concept

Normal modes — where it appears

The motions of a bounded system that keep their shape as they oscillate, each at its own frequency, and into which any motion can be decomposed. They are independent only while the system is linear, and their independence is exactly what makes a mode able to store energy without sharing it.

Named by 26 essays across 5 fields — each of them below, with the objects they name alongside it.

Harmonics on a fixed string. Standing-wave patterns on a string clamped at both ends, at n = 1, 2, 3, 4. Only whole numbers of half-wavelengths fit, which is why the allowed frequencies are discrete.

Only some notes fit, and that is where discreteness comes from

A string clamped at both ends can vibrate at some frequencies and not others. A continuous object producing a whole-number list is the oldest quantisation in physics.

waves · Standing waves Response against driving frequency. The steady-state amplitude of a driven oscillator against driving frequency, at three damping ratios. Lighter damping gives a taller and narrower peak, and the peak sits slightly below the natural frequency.

The frequency that gets an answer, and the quarter cycle nobody mentions

Push an oscillator at its own frequency and the response grows enormously. The reason is not that the push is in step with the motion — at resonance it is a quarter cycle out, and that is precisely why it works.

waves · Resonance The blackbody spectrum, against what classical physics predicted. Spectral exitance against wavelength for a blackbody at 3000, 4000, 5000 kelvin, in kilowatts per square metre per nanometre. Each curve peaks at the wavelength Wien's displacement law gives — 966 nm at 3000 K, 724 nm at 4000 K, 580 nm at 5000 K — and falls to nothing at short wavelengths.

The curve that would not come down

Classical physics predicted that a warm object radiates infinite power at short wavelengths. Every step of the derivation was correct, the prediction was absurd, and closing the gap required assuming that energy comes in lumps.

quantum · Blackbody The states a box allows, drawn on their energies. A particle confined between two walls one unit apart. States 1, 2, 3 are drawn, each riding on a line at its own energy — 1E₁, 4E₁, 9E₁ — because the energies go as n². Each wavefunction has n − 1 places where it crosses zero inside the box: 0 for n = 1, 1 for n = 2, 2 for n = 3. Nothing about the particle's mass or the depth of the well appears in the shapes; only the count of half-wavelengths that fit does.

The box that allows only some energies

Confine a wave between two walls and only the shapes that fit survive. That is a fact about strings, organ pipes and drumheads, and applying it to a matter wave produces quantisation with no new assumption at all.

quantum · Standing waves One well, two wells, and the band they become. The energy levels of a chain of identical wells, for 1, 2, 3, 6, 12, 40 of them, with an on-site energy of -4 eV and a coupling of -0.9 eV between neighbours. One well has one level. Two split it into two, 1.80 eV apart. By 40 the levels have filled a band 3.59 eV wide, which is closing on the limit of four times the coupling, 3.60 eV — and no further widening happens however many more wells are added. The count of levels grows with the number of wells; the width of the band does not.

What happens when the wells get close

Two atoms brought together split one level into two. A thousand split it into a thousand, packed into a band whose width stops growing after the third. Whether that band is full or half full is the whole difference between a wire and a window.

quantum · Bands Which wavelength wins. The growth rate of a disturbance on a liquid thread against kR, the circumference divided by the wavelength. Everything to the right of one decays; the maximum sits at kR = 0.697, which is a wavelength of 9.01 radii or 4.51 diameters. That number, and not a property of any particular liquid, is what sets the spacing of the drops a tap breaks into.

The thread that cannot stay a thread

A stream of water from a tap breaks into drops, and it does so at a spacing that is always about four and a half diameters. Nothing chooses that number — it is the wavelength that grows fastest out of a competition between all of them, and it can be computed before any water is poured.

fluids · Surface tension One parabola, several wells. Unlike potential wells, each divided by its own curvature at the bottom, against the single parabola ½x² drawn through all of them. They agree near the minimum because a function with a minimum has no linear term there, so the quadratic term is the first thing it has. The labels give where each well departs from the parabola by more than 1% of the parabola's own value there: a pendulum at 0.35, a chemical bond at 0.01, a pair of atoms at 0.0015. A symmetric well has no cubic term and stays close for a long way; a well that is steeper on one side than the other has one, and leaves the parabola almost at once — which is why those numbers differ by factors of hundreds and not by a few per cent.

Every minimum is a parabola

A pendulum, a bond between two atoms and a ship rolling in a swell obey the same equation, and the reason is not that they are alike. It is that a function with a minimum has no linear term there, so the first thing every potential well looks like is the same well.

mechanics · Harmonic approximation The two normal modes of a coupled pair at kc/k = 0.1. Two equal masses, each held to a wall by a spring of stiffness k and to each other by a coupling spring of 0.1k. Above: the in-phase mode, in which both masses move the same way by the same distance, the coupling spring never changes length, and the frequency is therefore 1.0000√(k/m) — the coupling does not appear in it at all. Below: the out-of-phase mode, in which the coupling spring changes length by twice the displacement, so each mass feels k + 2kc and the frequency rises to 1.0954√(k/m), a ratio of 1.0954. Both displacement patterns are the eigenvectors of the pair's stiffness matrix, obtained from its trace and determinant and checked against those two square roots. The red arrows are the force each mass is pulled back by, computed as −Kx: 1.00kA in the first mode against 1.20kA in the second, a factor of 1.20, which is the square of the frequency ratio because ω² is a stiffness over a mass. Any motion of the pair whatsoever is a sum of these two and nothing else.

The two pendulums that will not stop swapping

Coupled oscillators joined by a weak spring appear to hand energy back and forth. Nothing is handed anywhere: the system has only a pair of motions that keep their shape, at √(k/m) and √((k+2kc)/m), and the apparent traffic is the beat between them — 51 swings from one handover to the next at a coupling of one part in fifty. Extend the same arithmetic to N masses and it produces a dispersion relation with a hard ceiling, near 7 THz in copper.

mechanics · Harmonic approximation 6 modes of a drum, and their frequency ratios. Nodal-line diagrams for 6 modes of a circular membrane, each labelled with its frequency as a multiple of the lowest mode's. A mode (m, n) has m nodal diameters and n − 1 nodal circles, and the circles are drawn at the radii where the computed radial function J_m(j(m,n)·r/R) crosses zero — not at guessed fractions of the radius. The two tints are the two directions the head is moving in at that instant, and the lines between them are the parts of it that never move. The ratios are 1.000, 1.593, 2.136, 2.295, 2.653, 2.917: each one is a quotient of two zeros of Bessel functions, computed here from the power series and checked against their published values to 4.4e-7. Not one is a whole number, which is why a drum has no harmonic series and no pitch in the sense a string has one — and why these same ratios belong to every circular membrane ever stretched, whatever it is made of and however tightly it is pulled.

The drum that has no harmonics

A string's allowed frequencies are 1, 2, 3, 4 times its lowest, because counting half-wavelengths is arithmetic. Clamp a membrane round a circle and the same reasoning returns 1.000, 1.593, 2.136, 2.295 instead — zeros of Bessel functions, not integers. And those numbers belong to the shape of the boundary alone, which raises a question nobody could answer until 1992.

waves · Standing waves The response of a machine with a 10% absorber bolted to it. The amplitude of the driven mass, in units of the deflection the same force would cause if applied slowly, against drive frequency in units of the machine's own natural frequency. Dashed: the machine alone, with its single resonance. Solid: the same machine with a second mass and spring attached, weighing 10 per cent of it and tuned to the same frequency. At that frequency the driven mass does not move at all — the amplitude is zero rather than small, and the absorber is moving 10.0 static deflections to make it so. What the device costs is the two new resonances it creates, at 0.854 and 1.171 times the original frequency, which are infinite in this undamped calculation and straddle the frequency the machine was protected at.

The mass that makes another stand still

Bolt a small mass on a spring to a machine that is shaking itself apart, tune it to the frequency that is doing the damage, and the machine stops moving. Not moves less — stops, exactly, at that one frequency. The price is two new resonances either side of it, and the whole device is a bet that the drive stays where it was put.

waves · Resonance Indistinguishable for 10.2 seconds, then not. The path of the lower bob for two double pendulums released 1e-8° apart, over 11 seconds, with the arms drawn at the final instant. The two traces lie on top of each other for the first 10.2 seconds — the point at which they are two pixels apart on this canvas — and after that they have nothing to do with one another. Neither is more correct: both are exact solutions of the same equations, differing only in a release angle that no apparatus could set apart. The separation is growing at 2.05 per second the whole time, including during the stretch where the picture shows one curve.

The error that doubles on a schedule

Two double pendulums released a hundred-millionth of a degree apart follow one curve for ten seconds and then have nothing to do with each other. The separation grows exponentially the whole time, including while the picture shows a single trace — which turns unpredictability into a rate, and makes the length of a forecast the logarithm of the precision rather than anything proportional to it.

mechanics · Chaos How many modes a square has below a given wavenumber. The number of vibration modes of a square with wavenumber below k, counted exactly — every eigenvalue of this region is a closed form, so the staircase is the true count and not an estimate. There are 265 of them below k = 60. The smooth curves are what Weyl's law predicts. The upper one is the leading term alone, the area times k² over 4π, and it is too high by 21.5 modes at the right-hand edge; the lower one subtracts the perimeter term, the perimeter times k over 4π, and is out by 2.4. The content of the law is that the count depends on the region through its area and its perimeter and — to this order — through nothing else at all: not through its shape, not through where its corners are, not through whether it is convex. The staircase's steps are the individual modes, and they cluster where two different pairs of indices give the same wavenumber. That the count is smooth in the large while being a staircase in the small is what makes a mode count usable in thermodynamics, where it appears as a density of states and never as a list.

How many ways there are to vibrate

A drum has infinitely many modes, and below any given frequency it has a finite number of them. That number turns out to depend on the drum's area and the length of its rim and — to the accuracy anybody uses — on nothing else about its shape. Almost every result in thermal physics that involves waves is an application of that count.

waves · Standing waves Four modes of a string that is heavier in one place. The first four modes of a string whose mass per unit length rises to 5 times the light end's over a smooth bump centred 62 per cent of the way along, drawn beneath them. Nothing is symmetric any more: the shapes bunch up over the heavy region, where the local wavelength is shorter, and the amplitudes there are smaller. The frequencies are 1.70, 4.01, 6.10, 8.12, which are in the ratios 1.000, 2.361, 3.590, 4.779 rather than 1, 2, 3, 4 — this string has no harmonics and would sound like a bell rather than a violin. What has not changed is the one thing an ordering needs: the interior zeros, marked, run 0, 1, 2, 3 exactly as they do on a uniform string. That is Sturm's theorem, and the nodes here are counted on the computed shapes rather than assumed.

The count that cannot be cheated

A string with a lump in it has no harmonics, no symmetry and no obvious order to its modes. It has one thing left: the nth mode crosses the axis exactly n−1 times, whatever the string is made of. Ordering by frequency and ordering by node count turn out to be the same operation, and in two dimensions the equality quietly becomes an inequality.

waves · Standing waves The chain's dispersion, and the frequency it stops at. Frequency against wavenumber for a chain of equal masses joined by equal springs, in units where the spacing, the mass and the spring are one. At long wavelength the curve is a straight line through the origin — the chain behaves as a continuous string with a sound speed, and the departure from the line is second order in the wavenumber, which is why a lattice is invisible until the wavelength approaches the spacing. At the zone edge, where neighbouring masses move in exact opposition, the curve flattens: the frequency stops rising, the group velocity falls to zero, and the mode is a standing wave that carries nothing. Above that frequency there is no travelling solution at all.

The frequency a lattice cannot carry

A continuous string carries every note. A row of masses joined by springs does not: there is a highest frequency, set by nothing but the time one mass takes to be pushed back by its neighbours, and above it a disturbance does not travel at all.

waves · Periodic media A bend of 5 mm, and where the field has to give up. The transverse profile of the guided mode, with the core shaded and the radius at which a bend of 5 millimetres would require the field to outrun the cladding marked. Inside the core the field is a cosine; outside it decays, and the decay is what keeps the mode together. Bending the guide imposes a rigid rotation, so the field a distance x from the axis must travel faster in proportion to x — and past 13.5 micrometres it would have to travel faster than the cladding permits. There the field can no longer be evanescent, and what is there radiates away. The amount there is 1.63e-2 of the peak, which is why the loss is negligible until the caustic moves in, and then is not.

The mode that will not turn a corner

Bend a waveguide and the field has to go round with it, which means the part furthest from the centre has to travel faster. Past a certain distance it would have to travel faster than the surrounding medium allows, and everything out there radiates away — which is why bend loss is exponential in the radius and arrives all at once.

waves · Guided waves One width that falls to nothing. The decay rates of the two modes of a pair of resonances that leak into one shared channel at rates 0.1 and 0.05 and are coupled to each other with strength 0.2, against the detuning of the first from the second. The two rates always add to 0.15, the trace of the leak matrix, to 10⁻¹². Where the resonances are far apart each mode keeps roughly its own resonance's leak. Near them the leaks interfere, and at a detuning of 0.1414 — κ(γ₁ − γ₂)/√(γ₁γ₂) — the slower mode's decay rate is zero to within 10⁻¹², and the faster carries all 0.15. That mode sits at a frequency where the channel is open and does not leak into it: the two routes by which it could escape cancel.

The resonance that refuses to leak

A resonance that sits at a frequency where waves can escape has a width, because it leaks. Put two such resonances into the same channel and let them talk to each other, and at one precise detuning one of their combinations stops leaking altogether — a mode with no width, surrounded by a continuum it could escape into and does not. It cannot be seen from outside, it traps whatever energy lands in it, and if a symmetry is what forbids the leak, it survives any change that keeps the symmetry.

waves · Resonance Two guides, and the two solutions they have. The transverse field of the two modes a pair of identical slab guides supports, against position across them, for guides half a micrometre wide separated by a gap of 300 nanometres at a wavelength of 1550 nanometres. A single guide has one fundamental mode; two guides side by side have two, and neither of them lives in one guide. The symmetric one is a single hump spanning both, the antisymmetric one has a node exactly between them — checked here to be exactly zero rather than nearly so — and they have slightly different propagation constants because the symmetric one has more of its field in the high-index gap region. That difference, computed from the slab's own dispersion condition, is 4.60e-2 per micrometre. Everything else about a coupler follows from it: a wave launched into one guide alone is the sum of the two supermodes in equal parts, they run at different speeds, and the interference between them moves the power from one guide to the other and back. There is no leakage in the account anywhere — only two solutions beating.

Two tails that swap everything

Bring two guides close enough for their evanescent tails to overlap and they do not leak a little power into each other. They exchange all of it, and then exchange it back, over a length fixed by the splitting between two modes that belong to neither guide — so a coupler is cut to a length rather than tuned to a ratio, and the length depends exponentially on a gap of a few hundred nanometres.

waves · Guided waves Modes added in step: a train of pulses. The intensity of the sum of N waves of equal amplitude at equally spaced frequencies, all starting in phase, against time in units of the round trip — the inverse of the spacing — for N = 5, 10, 20. The sum is a train of pulses, one per round trip, each of peak intensity N² times one wave's and width about 1/N of the round trip. Between the pulses the waves cancel almost completely. The average intensity is N, as it must be — 5.01 for N = 5, 10.04 for N = 10, 20.19 for N = 20 — so the pulses do not create energy; they move it all into a fraction 1/N of the time, where it is N times more intense than it would be if spread evenly.

The phases that turn a glow into pulses

A laser oscillates on many frequencies at once, spaced by the time light takes to go round its cavity. Add those waves with random phases and the output is a steady glow with noise in it. Add exactly the same waves with their phases equal and the output is a train of pulses, each N² times brighter than one wave and a fraction 1/N of the round trip long. Nothing about the frequencies, the amplitudes or the average power changes — only the phases, and they decide everything a detector fast enough would see.

waves · Superposition One mode that loses less the faster it is driven. The wall loss, in decibels per kilometre, of three modes in a copper pipe 60 mm across, against frequency on logarithmic axes: TE₁₁, the lowest mode, cut off below 2.93 GHz; TM₀₁, cut off below 3.82 GHz; and TE₀₁, cut off below 6.09 GHz. Each loss is infinite at its cutoff and falls just above it. Then TE₁₁ and TM₀₁ turn and rise with frequency, as the square root, the way every conductor's loss does. TE₀₁ keeps falling: 0.51 dB/km at 60 GHz and 0.20 dB/km at 110 GHz, against 21 dB/km for TE₁₁ at 60 GHz.

The loss that falls as the frequency rises

Every metal conductor loses more at higher frequencies, because the current crowds into a thinner skin and meets more resistance. One wave in a round metal pipe breaks the rule. The TE₀₁ mode's electric field circles the axis and touches the wall nowhere. As the frequency rises it becomes more and more nearly a plane wave running down the middle of the pipe, and it drives less and less current in the metal. Its loss falls with frequency, to about half a decibel per kilometre at 60 GHz in a pipe 60 mm across. In the 1960s it was the best way anyone knew to send a hundred thousand telephone calls across a continent. The catch was that the pipe also carried seven hundred other modes.

waves · Guided waves The band in which radiation feeds a mode instead of draining it. The speed of a mode's pattern as seen from far away, in units of the star's own spin rate, against the spin rate as a fraction of the fastest a star can turn. The shaded band, between zero and one, holds patterns that run forward across the sky and backward through the star; gravitational radiation drives any mode inside it and damps any mode outside. The r-modes with ℓ = m = 2, 3 and 4 sit at 0.667, 0.833 and 0.900 at every spin, inside the band however slowly the star turns. The dashed curves are schematic modes whose crests run backward on the star at a fixed 0.2, 0.5 and 0.9 of the Kepler rate: each is dragged into the band only once the star turns faster than its crest runs, which is why an ordinary oscillation needs a star near break-up and an r-mode does not.

The wobble that radiating makes larger

Radiation normally drains a vibration: a struck bell goes quiet because it sends its energy away as sound, and a ringing black hole settles in milliseconds because it sends its energy away as gravitational waves. A spinning star can have the opposite. A pattern that runs backward through the star's material, while the star carries it forward across the sky, has less angular momentum than the star would have without it, and every wave it radiates takes away angular momentum it did not have to give. The pattern grows. One family of such patterns, the r-modes, runs backward on every rotating star, so every spinning neutron star is unstable unless friction wins — and that may be why none has been found spinning faster than 716 turns a second.

astrophysics · Gravitational waves The frequency of a mass on a spring that weighs something. The angular frequency of a mass M on a spring of stiffness k and mass mₛ, in units of √(k/M), against the ratio mₛ/M on a logarithmic axis: exact, from β tan β = mₛ/M (solid); Rayleigh's estimate √(k/(M + mₛ/3)) (dashed); and the textbook answer that ignores the spring's mass, 1 (dotted). For a spring a tenth of the mass the exact value is 0.9836, Rayleigh's 0.9837 and the textbook's 1 — 1.7 per cent too fast. With spring and mass equal the exact frequency is 0.8603, Rayleigh's 0.8660: the third is good to 0.66 per cent where the textbook answer is 16 per cent out.

The third of itself a spring carries

Every textbook gives the frequency of a mass on a spring as the square root of stiffness over mass, and every real spring has mass of its own, which the formula leaves out. The repair is famous and oddly specific: add a third of the spring's mass to the load. The third is not a fudge. It is what the spring's kinetic energy comes to if each coil moves in proportion to its distance from the fixed end, and it is good to two-thirds of a per cent even when the spring weighs as much as the mass. Where it fails, it fails because the spring has stopped being a spring and become a column of waves, with notes of its own above the bounce — the notes that make the valve springs of an engine lose control.

mechanics · Harmonic approximation A ring turns, and its vibration turns less. Four snapshots of a thin ring vibrating in its lowest flexural mode, squashing alternately into two ellipses, while the whole ring is turned steadily about its axis; the deformation is exaggerated. The dot marks one point of the ring; the dashed line, the axis of the vibration's greatest swing. As the ring turns through 30°, 60° and 90°, the pattern of vibration turns through 18°, 36° and 54° — 60 per cent of the ring's angle. Measured from the ring, the pattern has slipped back 36° by the quarter turn. The vibration is neither carried along with the material nor left fixed in space. It follows the turning by a fixed fraction that depends only on the mode's shape, which is the effect G. H. Bryan described in 1890 from the note of a turning wine glass.

The vibration that turns part of the way

Strike a wine glass and turn it on the table while it rings, and the note beats: the pattern of the vibration does not turn with the glass. Nor does it stay still in the room. For a ring vibrating in its lowest mode it follows exactly three-fifths of the rotation and slips back the other two-fifths, a fraction that depends only on the shape of the vibration. That fraction, worked out by George Bryan in 1890, is how some of the most precise gyroscopes ever built measure an angle directly, with nothing spinning in them at all.

mechanics · Rotation What a station hears from one thunderstorm region. The power spectrum of the vertical field from a single region of thunderstorms, received 20° and 90° of arc away, from a sum of the first seven resonant patterns, each with the measured peak frequency and a quality factor of 4 to 6, both taken as inputs. Measured spectra fall off with frequency rather more than this one, because real strokes and the real ionosphere are not as simple. The resonances are so broad that neighbouring peaks run into each other; a quality factor of five means a peak about a fifth as wide as its frequency, and the ringing dies in a few cycles. At 20° every tone is present, the higher ones merging. At 90° the first, third and fifth have vanished, because those patterns have a node a quarter of the world away, and the spectrum is made of the even tones alone. Real stations receive the sum of the world's storms, which move with the time of day, and the heights of the peaks drift with them.

The tone the whole Earth rings at

The ground conducts and so does the ionosphere, eighty kilometres up, and between them lies a waveguide the size of the planet. Two walls, unlike a pipe, carry waves of any length, so the guide takes radio waves forty thousand kilometres long — longer than the world is round — and, closed on itself, it rings. Lightning strikes fifty times a second and keeps it ringing at about 7.8, 14.3 and 20.8 hertz. Perfect walls would put the tones at 10.6, 18.4 and 25.9; the difference is the ionosphere, a ceiling that leaks.

waves · Guided waves How a hanging chain can swing. The shapes of the first four ways a uniform hanging chain can swing, side by side, against height from its free end: J₀(j₀,ₙ √(y/L)), fixed at the top and largest at the free bottom end. For a 2-metre chain they sound at 0.424, 0.973, 1.525, 2.078 hertz. The first swings as a whole, like a pendulum, with its bottom moving most. The higher modes have nodes, and the nodes are not evenly spaced: they crowd towards the bottom, where the tension is least and waves travel slowest, so a stretch near the free end holds more of each wave than a stretch of the same length near the top.

The chain that swings in Bessel's notes

Hang a chain from a hook and set it swinging. It can swing as a whole, like a pendulum, or in shapes with still points along it, as a guitar string does — but its higher notes are not two, three and four times its lowest. They are 2.295, 3.598 and 4.903 times it, the zeros of a function that Daniel Bernoulli met for the first time in exactly this problem in 1732, a century before Bessel's name was attached to it. The reason is that a hanging chain is a string whose tension grows from nothing at its free end to its whole weight at the top, and a wave crossing it changes speed all the way.

waves · Standing waves Every wave a plate can carry, at every frequency. The phase velocities of the guided waves in an aluminium plate (longitudinal speed 6.32 km/s, shear 3.13), against the product of frequency and thickness in MHz·mm, from the Rayleigh–Lamb equations: symmetric modes, in which the two faces bulge and thin together (light), and antisymmetric ones, in which they bend together (dark). At low frequency only two exist. The symmetric S₀ is a stretching wave at the plate velocity, 5.44 km/s; the antisymmetric A₀ is a bending wave whose speed falls towards zero, 0.96 km/s at 0.1 MHz·mm. At high frequency both approach the 2.92 km/s of a Rayleigh wave on each face. Higher modes enter at cut-off frequencies where the plate's thickness holds a whole number of half-wavelengths of one bulk wave — the first at 1.56 MHz·mm — arriving from infinite phase velocity and slowing as the frequency rises.

The plate that carries two kinds of sound

Tap the edge of a sheet of aluminium and the sound that runs along it is not one wave but two, and neither is the sound of the metal. One stretches the sheet and travels at 5.44 km/s, faster than a shear wave and slower than a compression wave in the same metal. The other bends it, travels at a speed that depends on how high the note is and how thick the sheet is, and at low frequency crawls at a few hundred metres a second. Both are made of the same two bulk waves bouncing between the faces, and the two faces together decide everything.

waves · Guided waves How much of a wave of gravity a bar can catch. A resonant bar's cross-section for gravitational waves, integrated over its resonance, (8/π)(GM/c)(v/c)², against its mass, on logarithmic axes, for sound speeds of 3.0 km/s, 5.4 km/s, 10.0 km/s (aluminium is about 5.4). It is proportional to the mass, which is why the bars weighed tonnes, and to the square of the speed of sound over the speed of light — the bar answers a strain only by being stretched against its own stiffness, and (v/c)² is about 3 × 10⁻¹⁰. Weber's first bar caught 2.3·10⁻²⁵ m² Hz: to absorb a single joule, a burst must carry 4.4·10²⁴ J per square metre per hertz at the bar's frequency. Dots: Weber's bar, and the two cryogenic aluminium bars that ran longest, ALLEGRO in Louisiana and NAUTILUS near Rome.

The bar a wave of gravity barely touches

The first gravitational-wave detectors were tonnes of aluminium, hung in a vacuum and listened to for ringing. A passing wave stretches and squeezes the bar, and the bar can answer only by vibrating against its own stiffness, so the fraction of the wave it catches is set by the speed of sound over the speed of light, squared: about three parts in ten billion. That one ratio explains why the bars had to be enormous and cold, why Joseph Weber's reported detections of 1969 would have required the Galaxy to be converting dozens of suns a day into gravitational waves, and why the waves were finally found by light instead.

astrophysics · Gravitational waves

Named alongside it

The objects these essays reach for when they reach for this one.

Standing waveBoundary conditionsGuided wavesQuantisationResonanceCutoffDispersion relationQuality factorSimple harmonic motionSuperpositionDampingEigenvalue

All concepts