Simply false — part 5 of 5

“A lens focuses light because of the glass it is made of, so a lens with most of its glass removed cannot focus as well.”

For rays the bending happens at the surface and depends only on the slope there. A 200 mm acrylic lens 100 mm across needs 12.2 mm of glass at its centre; cut into 13 rings 1 mm deep, it sends every ray to the same focus. The glass removed was crossed without any turning.

Tested in The lens with its glass taken out · the refraction reading path

“Two flat parallel mirrors are the simplest and best cavity for a laser, since they send every ray straight back.”

They sit exactly on the edge of stability, g1 g2 = 1: a ray that starts with the slightest tilt walks off sideways, the beam they hold would be infinitely wide, and the axis moves without limit for any tilt. Curving the mirrors slightly, to g = 0.99, traps a ray, though it takes 44 bounces to swing back once.

Tested in The two mirrors that hold a beam · the imaging reading path

“A particle much heavier than an electron can give it at most about 4m/M of its energy in one collision, a fifth of a per cent for a proton.”

Only while it is slow. The largest kick is 2mc²β²γ²/(1 + 2γm/M + (m/M)²), which grows as γ²: a 1 GeV proton can give 3.33 MeV, 1.5 times the slow figure, and a 1 TeV proton 538 GeV, more than half its energy.

Tested in The kick a fast particle can give an electron · the relativistic dynamics reading path

“A perfect gyroscope in free fall keeps pointing at the same distant star for ever.”

Not near a spinning mass. In a polar orbit 650 km above the Earth, the Earth's rotation turns a gyroscope's axis by 41 milliarcseconds a year, on top of a 6604 mas/yr geodetic drift from the orbit through curved space. Gravity Probe B measured 37.2 ± 7.2 and 6601.8 ± 18.3 against 39.0 and 6604 predicted for its guide star.

Tested in The gyroscope a spinning Earth turns · the field transformation reading path

“Supercooled water turns completely to ice the moment a crystal starts.”

Only the share whose latent heat can warm the rest back to 0 °C freezes at once: 19 per cent at −15 °C, 26 per cent at −20 °C, 51 per cent at −38 °C. The rest stays liquid at 0 °C until the surroundings remove its heat — 45 seconds for a 2 mm drop in −20 °C air.

Tested in The supercooled water that only partly freezes · the phase change reading path

“Adding insulation to a hot object always reduces the heat it loses.”

Not for a thin cylinder or a small ball. A 2 mm wire 40 K above still air loses 2.51 W per metre bare and 12.11 W per metre with PVC out to 19 mm radius, the critical radius k/h; it does not come back down to the bare value until the PVC is 178 km thick.

Tested in The insulation that makes a wire lose more heat · the diffusion reading path

“Adding noise to a weak signal can only make it harder to detect.”

Not in a system with a threshold or a barrier. A double well pushed at 0.12 — a third of the push needed to tip it — and given no noise never hops; with noise of intensity 0.085 it hops 30 times in eight cycles, almost always in step with the push; with noise 0.3 it hops 285 times at random. The response to the push peaks at a noise of 0.083.

Tested in The noise that helps a weak signal through · the resonance reading path

“The deep interior of the Earth has never been observed, so its structure is a guess.”

It is measured, by the timing of earthquake waves. Traced through a reference model of the Earth's speeds, direct P waves stop at 97° and P waves through the core resume at 144°; no direct S wave arrives beyond 103°. The edges of those shadows fix the core's radius, and the S shadow shows it is liquid.

Tested in The shadow that shows the core is liquid · the wave motion reading path

“The number of collisions between two blocks and a wall depends on their speeds and starting positions.”

It depends only on the mass ratio. Speeds scale the energy circle and positions only shift the times; the count is the number of steps of 2θ, θ = arctan √(m/M), that fit into half a turn: 3, 31, 314, 3141 and 31415 for mass ratios of 1, 10², 10⁴, 10⁶ and 10⁸, simulated collision by collision.

Tested in The collisions that count the digits of π · the momentum reading path

“A spinning coin slows down as it loses energy.”

The coin's face does — at 1° of tilt it turns 0.0059 times a second for a 75 mm disk — but the point where it touches the table goes round faster as the tilt falls, as one over the square root of the tilt: 12 times a second at 10°, 39 at 1°, 123 at 0.1°. The rising whirr is that point.

Tested in The disk that spins faster as it stops · the rotation reading path

“The elements are ordered in the periodic table by atomic weight.”

By nuclear charge, which their X-ray lines count. Argon (39.95) is heavier than potassium (39.10) and cobalt (58.93) heavier than nickel (58.69), yet in both pairs the Kα energy rises from the first to the second, as atomic number does: 2.96 to 3.31 keV and 6.93 to 7.48 keV.

Tested in The X-ray line that counts the protons · the atomic spectra reading path

“An atom struck by an electron absorbs whatever energy the electron happens to give it.”

A mercury atom takes either almost nothing — an elastic bounce, at most 4m/M of the electron's energy, a hundred-thousandth — or exactly 4.886 eV, its first excitation energy. The current through a mercury tube therefore collapses each time the electrons can just afford 4.886 eV, at intervals of 4.9 V, rather than falling smoothly.

Tested in The electrons that pay in fixed amounts · the atomic structure reading path

“Since a planet's orbit is an ellipse, its velocities, drawn from a common point, trace an ellipse too.”

They trace a circle, of radius GM/h, offset from the common point by e times that radius: the integrated velocities of an orbit of eccentricity 0.6 stay on it to one part in ten thousand, from 1.60 GM/h at closest approach to 0.40 at the furthest point.

Tested in The velocity that always lies on a circle · the orbit stability reading path

“A spinning fluid body is always symmetric about its axis of spin, flattening more the faster it turns.”

Past an eccentricity of 0.8127, where Ω²/πGρ = 0.374 and the angular momentum is 0.304 √(GM³R), a uniform spinning fluid can also be a triaxial ellipsoid, and for the same angular momentum that shape has less energy — 0.0065 GM²/R less at angular momentum 0.4. The axially symmetric spheroid stops being the preferred shape.

Tested in The spinning fluid that becomes a cigar · the Self-gravity reading path

“The ring jumps because, by Lenz's law, its induced current opposes the coil's field and is repelled by it.”

In an alternating field the induced voltage reverses every half-cycle, so a current simply in step with it would be pushed and pulled equally and the average force would be zero. The push survives only because the ring's inductance makes its current lag; with reactance equal to resistance it lags by 45° and the average force is a quarter of the peak product of flux and current.

Tested in The ring that jumps because its current is late · the induction reading path

“Separating electric charge needs friction, chemistry or a power supply.”

Kelvin's water dropper uses none. Drops forming inside a ring at potential V carry an induced charge −cV into the opposite can, and with the rings cross-connected the voltage between the cans grows as exp(nct/C): with 3 mm drops, 20 a second into 20 pF cans, from 1 V to a 10 kV spark in 55 s. The energy comes from the drops' fall.

Tested in The water drops that charge themselves to a spark · the potential reading path

“An electron moving towards a laser photon reflects it like a mirror moving at the electron's speed, so the photon's energy is multiplied by 4γ² however fast the electron is.”

Only while the photon is soft in the electron's frame. With the electron's recoil the edge is E·x/(1 + x), x = 4γε/mc². At 46.6 GeV and 2.35 eV the mirror's figure is 78 GeV, more than the electron has; the photon comes back with 29.2 GeV.

Tested in The photon a fast electron throws back · the length contraction reading path

“A light pulse intense enough to push an electron to a large fraction of the speed of light leaves it moving fast.”

In a plane wave the electron's transverse canonical momentum and γ − pz/mc are both conserved exactly, so pₓ/mc = a and γ = 1 + a²/2 at every instant. When the pulse has passed, a = 0 and the electron is at rest: an a₀ = 2 pulse raises γ to 3 in the middle and leaves it at 1, displaced forward by 2.25 wavelengths.

Tested in The light that moves an electron and pays it nothing · the relativistic dynamics reading path

“A body in equilibrium in its own frame feels no net torque in any frame, since a torque would make it turn.”

A right-angled lever balanced by two forces F at arm length L, seen moving along one arm at speed v, has torques FL/γ² and FL: the arm along the motion is contracted and the force across the motion is reduced by γ. The net torque is FLβ², 0.64 FL at 0.8c, and the lever does not turn.

Tested in The lever that is twisted and does not turn · the Mass-energy reading path

“A charge must be accelerated — turned, slowed or shaken — to radiate.”

In empty space, yes. A charge moving uniformly over a grating of period d radiates λ = d(1/β − cos θ) without any change in its own motion: 300 keV electrons over a 1.67-micrometre grating give visible light from 0° to 30°. The grating supplies a wavenumber 2π/d that lifts the charge's field into the light cone.

Tested in The rainbow a passing electron leaves on a grating · the radiating charge reading path

“Sound in a plasma travels at the speed of sound of its ions, set by the ions' temperature, as in any gas.”

The ions supply the inertia and the electrons the pressure: cₛ = √(kTₑ/mᵢ). A discharge with 3 eV electrons and room-temperature argon ions carries sound at 2.69 km/s, 8.3 times the speed of sound in argon gas at that ion temperature.

Tested in The sound a plasma carries on its electrons' heat · the plasma oscillation reading path

“A wire with no resistance can carry an unlimited current.”

The current's own field at the surface, I/2πa, cannot exceed the critical field. A lead wire 1 mm across carries at most 132 A in liquid helium at 4.2 K and 201 A at absolute zero; above that its surface stops superconducting.

Tested in The current a superconductor stops with its own field · the superconductivity reading path

“A magnetic field along a diode's axis gradually reduces its current as it bends the electrons' paths more and more.”

The current is unchanged until the cut-off and then stops completely. In a diode with a 4 mm cathode and an 8 mm anode at 4 kV, electrons still reach the anode at 0.98 of the cut-off field of 142 mT and none do at 1.02.

Tested in The field at which no electron reaches the anode · the magnetism reading path

“A flat plate and a ball pressed onto a wet surface meet the same resistance from the water between.”

A curved face squeezes liquid out of a small patch and its resistance grows as 1/h; a flat face squeezes it from its whole area through a slot narrowing everywhere at once, and its resistance grows as 1/h³. For a 10 cm disc against a ball of the same radius the ratio is R²/4h²: 625 at a millimetre, 625 million at a micrometre.

Tested in The wet plates that hold on when pulled · the viscosity reading path

“Water pushes on a submerged gate as if all of its force acted at the gate's centre.”

Pressure grows with depth, so the resultant acts below the centroid by I/Aȳ — a sixth of the gate's height below its middle when the water is level with its top, a third of the way up from its bottom when the water is lower. Only as the water deepens does the offset shrink, as one over the depth.

Tested in The floodgate that opens at a chosen depth · the hydrostatics reading path

“Pulling a spool's thread always unwinds it, so the spool must roll away from the hand or slide towards it.”

Taking moments about the contact point, the spool turns according to T(R cos θ − r): it rolls towards the hand when the thread's line meets the floor behind the contact point, stays still when it passes through it at cos θ = r/R — 60° for a hub half the rim's radius — and rolls away, winding the thread in, at steeper angles.

Tested in The spool that rolls towards the hand pulling it · the Free-body reading path

“A ladder sliding down a smooth wall keeps its top against the wall until it lies flat.”

The wall's push needed to keep the top on it is (3/4)mg sin θ (3 cos θ − 2 cos θ₀), which falls to zero when the top has come down to two-thirds of its starting height and would have to turn into a pull below that. A 4 m ladder released at 15° leaves the wall at 2.58 m, two-thirds of 3.86 m.

Tested in The ladder that lets go of the wall · the energy reading path

“Every lens has a focus somewhere, even if it is blurred by aberrations.”

A lens with focal lengths of 100 mm and 125 mm in perpendicular meridians brings a parallel beam to a horizontal line at 100 mm and a vertical line at 125 mm, and nowhere to a point. Its smallest spot is a circle at 111.1 mm, a ninth of the aperture's radius.

Tested in The focus that is two lines · the imaging reading path

“The spikes on bright stars in telescope photographs come from the camera's lens or sensor, or from the atmosphere.”

They are the diffraction pattern of the vanes holding the secondary mirror: each straight vane throws two spikes at right angles to itself. The figures compute them as the Fourier transform of the pupil alone, with no camera or atmosphere in the calculation, and the count follows the vanes — four from a cross, six from three vanes at 120°.

Tested in The spikes a bright star wears · the diffraction reading path

“An ideal amplifier multiplies its input by a gain and adds nothing; the noise of real amplifiers is a matter of engineering.”

Multiplying a mode by √G would break [b, b†] = 1. The least correction adds a second mode's vacuum fluctuations, at least ½(1 − 1/G) of a quantum referred to the input — 0.45 at a gain of 10, approaching ½ — so a coherent signal leaves with at most half its signal-to-noise ratio.

Tested in The noise every amplifier must add · the uncertainty reading path

“The Hall resistance of a conducting layer rises in proportion to the magnetic field, as B/ne.”

Classically. In a two-dimensional layer at low temperature it climbs in steps and stays flat at h/νe² — 25,813, 12,906, 8,604, 6,453 ohms — each plateau centred where the classical line crosses it, at integer filling ν = nh/eB.

Tested in The resistance counted in whole numbers · the bands reading path

“Atoms collect in the lowest state of a trap because they attract one another, as a vapour condenses into a liquid.”

The atoms drawn here do not interact at all. The trap's excited levels can hold at most ζ(3)(kT/ħω)³ atoms, and below kTc = 0.94 ħω N^⅓ that is fewer than there are; the rest must be in the ground state. A hundred thousand ideal atoms at 0.7 Tc put 63 per cent in that one state.

Tested in The condensate a trap makes · the superfluidity reading path

“A thin sheet of foil in a vacuum gap cannot insulate, since it is too thin to hold back heat and conducts it straight through.”

In a vacuum the heat crosses as radiation, and a foil adds a resistance 2/ε − 1 in series whatever its thickness. Between two aluminised walls at 293 K and 77 K, one aluminised foil halves the flux from 6.3 to 3.15 W/m²; thirty bring it to 204 mW/m².

Tested in The thin foils that stop radiant heat · the blackbody reading path

“Spontaneous combustion needs a spark or flame somewhere in the heap; a heap at the air's temperature cannot ignite on its own.”

A slow reaction makes heat at a rate growing as exp(−E/RT), and a heap loses it only by conduction to its surface. Past a critical value of δ = (QAEρr²/kRTₐ²)e^(−E/RTₐ) — 0.878 for a slab — there is no steady temperature and the centre runs away with no external source.

Tested in The heap that sets itself alight · the diffusion reading path

“A chain hanging from a hook is a string, so its modes of swinging are harmonics of its lowest.”

Its tension is not uniform: T = ρgy grows from zero at the free end. The modes are J₀(2ω√(y/g)) and the frequencies (j₀,ₙ/2)√(g/L) are in the ratios 1 : 2.295 : 3.598 : 4.903 — not harmonics.

Tested in The chain that swings in Bessel's notes · the standing waves reading path

“Heat put into a resonant tube always feeds its sound, since heat is energy.”

Only heat released while the pressure is high drives the sound. In the lowest mode the driving is ∝ sin(2πx/L)·sin(ωτ): positive with the gauze in the lower half, negative in the upper. A gauze at 3L/4 damps a disturbance to nothing within a few periods; the same gauze at L/4 grows it into a loud note.

Tested in The tube that sings when heated low down · the resonance reading path

“Measuring the shift in more massive clusters would separate gravity's redshift from the slowing of the galaxies' moving clocks.”

Both scale as GM/R, and so both rise as the two-thirds power of the mass. At a concentration of 4 the gravitational shift at half of R200 is 4.45 times the transverse Doppler shift for every mass from 10¹³ to 3 × 10¹⁵ solar masses.

Tested in The light that climbs out of a cluster of galaxies · the gravitational redshift reading path

“No method that keeps an atom scattering photons can cool it below the momentum of a single photon's kick.”

Only if the atom keeps scattering at every momentum. A dark state that stops scattering near zero momentum lets atoms collect there: in one dimension the cold peak is 0.010 ħk wide after 10,000 scattering times and still narrowing, as the time to the power −0.54 — about ten thousand times below the recoil temperature, read from the width.

Tested in The atoms the light stops touching · the radiation pressure reading path

“Relativity is a small correction to the radio emitted by electrons of a few kiloelectronvolts, which move at a fifth of the speed of light.”

For a wave crossing the field it is the whole mechanism. Without the mass shift, every electron resonates at exactly the cyclotron frequency and the sum over all of them, −2∫v⊥f, is negative for any distribution: the wave is always absorbed. With it, the resonant electrons lie on a circle of radius c√(1 − ω²/Ω²), and a shell of 10 keV electrons amplifies at 0.9835 of the cyclotron frequency.

Tested in The radio that only heavier electrons can make · the radiating charge reading path

“Single-electron effects need extreme cold.”

They need the charging energy to beat the thermal energy, and that depends on size. For an isolated sphere, e²/2C exceeds ten times kT for radii below 2.8 nm at room temperature, 199 nm at 4.2 K and 41.8 µm at 20 mK.

Tested in The island that takes electrons one at a time · the conductors reading path

“A permanent magnet reverses when the reverse field overcomes the anisotropy that holds its magnetisation along the easy axis.”

A perfect crystal would, at the anisotropy field 2K/μ₀Mₛ — 7.7 T for neodymium-iron-boron. Real magnets reverse at 1 to 3 T, because reversal starts at defects where the anisotropy is lower: a completely soft layer ten wall parameters wide, 12.5 nm in this material, would bring the nucleation field down to 0.068 of the anisotropy field.

Tested in The magnet that reverses where its crystal is weakest · the magnetisation reading path

“A magnet's field must appear on both sides of it, since every magnet has two poles.”

Every magnet in a Halbach row has two poles, but their fields add on one side and cancel on the other. With four blocks a wavelength the weak side, a tenth of a wavelength away, keeps 9.4 per cent of the strong side's field; with eight, 0.25 per cent; with a continuously turning magnetisation, none.

Tested in The magnets that put their whole field on one side · the magnetism reading path

“Rutherford's scattering formula is the classical approximation to a quantum result, accurate only when the particles' wavelengths are small.”

The exact quantum amplitude for a pure Coulomb force has a magnitude equal to Rutherford's at every angle and every energy; ħ enters only its phase, e^(−iη ln sin²(θ/2)). The lowest-order quantum approximation gives the same. A screened charge, which has a length in it, does not: at 5° a charge screened at 2d scatters 85 per cent of Rutherford's rate, one screened at d/2, 19 per cent.

Tested in The scattering quantum mechanics returns unchanged · the correspondence reading path

“Thermal light is a stream of photons arriving independently at random, like raindrops.”

Only at frequencies far above kT/h, where a mode is nearly always empty. One mode holding five photons on average has a variance of 30, not 5, and is empty 17 per cent of the time against 0.7 per cent for independent arrivals; the extra spread is the wave term, n̄².

Tested in The flicker that is half wave and half particle · the photon reading path

“When the travelling twin turns round, the stay-at-home twin's clock jumps forward by years from the traveller's point of view.”

Only on the recipe that borrows each momentary rest frame's simultaneity, which jumps the home clock from 3.2 to 6.8 years at a turnaround at 0.6c. Timed by radar, which the traveller can actually do, the home clock advances continuously: it reads 4, 5 and 6 years at the traveller's radar times 3.5, 4 and 4.5.

Tested in The now a travelling twin can measure · the simultaneity reading path

“Radiation reaction is too small to matter for anything a laboratory can do to an electron.”

For an electron at rest in a laser field it is under a millionth of the applied force. For a 511 MeV electron meeting a ten-cycle pulse of 0.8 µm light head-on it takes away 24 per cent of the electron's energy at a₀ = 30 and 78 per cent at a₀ = 100, because in the electron's frame the field is larger by γ(1 + β) ≈ 2000.

Tested in The light that takes back an electron's energy · the relativistic dynamics reading path

“When a hanging spring is released, the whole spring starts to fall at once, the bottom more slowly than the top.”

The bottom does not move at all until the collapsing top reaches it, √(M/3k) after release — 0.261 s for a spring that hangs a metre. Everything below the front of collapsing coils hangs exactly as it did, because the tension there has not changed.

Tested in The spring whose bottom does not know it has been dropped · the momentum reading path

“Newton's laws with Coulomb friction always determine how a rigid body in contact with a surface will move.”

For a uniform rod sliding with its lower end moving out from under it, the contact's acceleration falls as the normal force rises once μ exceeds (tan²θ + 4)/3 tan θ — never below 4/3, first at tan θ = 2 — and then the rigid equations have no solution when the rod is pressing down and two when it is lifting.

Tested in The rod that friction cannot decide about · the friction reading path

“How strongly a vapour absorbs at a spectral line measures how many atoms are in the light's path.”

Only while the line is weak. Calibrated where it is, the absorption of sodium's D₂ line in a 10 cm column reports 0.038 of the true density at 10¹² atoms per cubic centimetre and 0.0011 at 10¹⁴, because the line's centre has gone black and only its wings can grow. The hooks return the true density to within a per cent over the same range.

Tested in The hooks that weigh an absorption line · the dispersion reading path

“Total internal reflection is a perfect reflection, so a prism that uses it cannot degrade an image.”

It loses no light but shifts the phases of s and p differently — by up to 46.4° in crown glass. In a roof met at 60°, as in an Amici prism, the two halves of the beam leave in different polarisations whose overlap is 0.392, and a point's image falls to 0.70 of the peak an undivided pupil gives.

Tested in The roof that splits a binocular's image · the total internal reflection reading path

“Quicksand can suck a person under until they drown.”

Liquefied quicksand at a sand fraction of 0.41 is 1.68 times as dense as water, so a person with a mean density 1.03 times water's floats with 0.61 of their volume submerged — sinking to above the waist and no further. Sand that has packed down to a fraction of 0.6 would hold more than half the body out.

Tested in The sand that lets a body in and will not let it out · the rheology reading path

“A granular material has one friction coefficient, its angle of repose, and flows whenever a slope exceeds it.”

Moving grains have a friction that rises with the inertial number I = γ̇d/√(P/ρ): from tan 20.9° when barely moving to tan 32.8° when flowing fast, for glass beads. Steady uniform flow exists only between those angles; below 20.9° nothing flows, above 32.8° the layer accelerates.

Tested in The sand whose friction depends on how fast it flows · the granular matter reading path

“Measuring the ratio of a gas's heat capacities needs calorimetry.”

A steel ball oscillating in a tube on a flask of the gas does it with a stopwatch: the period is 2π√(mV/γPA²), so on a 10 litre flask a 16 mm ball swings in 0.983 s on helium, 1.072 s on air and 1.117 s on carbon dioxide, and γ = 4π²mV/A²PT² follows from one timing.

Tested in The ball that weighs how a gas stores heat · the equipartition reading path

“Petrol engines run at compression ratios near ten because higher ratios would gain little efficiency.”

The ideal efficiency on air keeps rising — 56 per cent at a ratio of 8, 63 at 12, 70 at 20. The limit is set by knock: in the engine modelled here, 95-octane fuel's end gas ignites by itself while more than a tenth of the charge is unburned once the ratio passes about 11.4.

Tested in The squeeze a petrol engine is not allowed · the heat engines reading path

“A Doppler radar measures speed by measuring the frequency shift of its echo.”

A pulsed Doppler radar measures how far the echo's phase turns from one pulse to the next, 4πv/λ·PRF radians, and a phase is known only modulo 2π. A 10 cm radar pulsing 1000 times a second records identical samples for targets closing at 10 and at 60 m/s, and reports both at 10.

Tested in The speeds a pulsed radar folds into one · the doppler reading path

“A radar's range resolution is set by the length of its pulse.”

For an unmodulated pulse it is, c·T/2. A chirp lasting T but sweeping a bandwidth B, received through a matched filter, compresses to a spike 0.878/B wide at its half-power points — fifty times narrower than the pulse for TB = 50 — so two targets 3/B apart, whose long echoes overlap almost entirely, come out as two separate spikes.

Tested in The long pulse that arrives as a spike · the wave packets reading path

“Two electron beams that are each stable on their own make a stable system when they pass through each other.”

Each cold beam alone has only real frequencies, ω = kv₀ ± its own plasma frequency. Two equal beams at ±v₀ have a root that grows for every wavenumber below ωₚ/v₀, fastest at kv₀ = √(3/8) ωₚ, where it grows at ωₚ/(2√2) = 0.354 ωₚ — by e in less than half a plasma period.

Tested in The two beams that unravel each other · the plasma oscillation reading path

“A generator needs a permanent magnet or a separately powered electromagnet to make its field.”

A disc whose current runs through a coil wound round it makes its own field once its speed exceeds R/M, the circuit's resistance over the mutual inductance between coil and disc. Above that speed any stray field grows exponentially; a disc driven to 1.5 and 2.5 times the critical speed settles at exactly the critical speed with a current of 0.71 and 1.22 in units of √(G/M), from a seed of a millionth.

Tested in The dynamo that reverses with nothing pushing it · the flux freezing reading path

“Joining a charged capacitor to an empty one with a superconducting wire would transfer the charge with no loss.”

Charge conservation fixes the final state at half the voltage on each, which holds half the energy whatever joins them. With no resistance the loop's inductance makes the charge slosh instead of settle, and it settles only as the loop radiates; the lost half then leaves as radio waves rather than heat.

Tested in The half a wire always takes · the field energy reading path

“Air breaks down wherever the field exceeds about 30 kV/cm, so a conductor glows as soon as its surface field reaches that.”

An avalanche needs room to grow, and round a thin conductor the field falls away within a distance comparable to its radius. Peek's onset field is 32.9 kV/cm on a 10 cm conductor, 39.0 on 1 cm, 59 on a 1 mm wire and 316 on a 10 µm filament.

Tested in The glow a thin wire wears before it sparks · the dielectrics reading path

“The periodic table must end at element 137, because Dirac's formula for the innermost electron fails there.”

The formula √(1 − (Zα)²) assumes a point nucleus. Solved for a nucleus of realistic size, the innermost level continues past 137, crossing zero near Z = 148 and reaching −mc² only near Z = 170.

Tested in The nucleus that would make positrons from nothing · the atomic spectra reading path

“Light whose photons each carry less than an atom's ionisation energy cannot ionise it, however intense.”

It can, by absorbing several photons at once or by tunnelling. Hydrogen needs 13.6 eV and an 800 nm photon carries 1.55 eV, so nine are needed with no field — and as the intensity rises the threshold itself rises by the quiver energy, 5.98 eV at 10¹⁴ W/cm², so that by 2 × 10¹⁴ seventeen are needed.

Tested in The atom a light wave opens like a wall · the tunnelling reading path

“A stack of blocks on a table edge cannot lean out further than one block length, since the blocks must rest on something.”

Each block must rest on the one below, not on the table. With the k-th block from the top overhanging the next by 1/(2k), four blocks reach 1.042 block lengths beyond the edge and eight reach 1.359; the top block lies wholly in the air.

Tested in The stack that leans past its own table · the Free-body reading path

“Water flowing out of a bottle must slow down as the bottle empties, because there is less water pushing it out.”

Only if the top is open. A sealed bottle 10 cm across with a tube through its stopper reaching 10 cm above a 1.5 mm outlet delivers 5.81 mL/s, unchanged, for 4.5 minutes while its level falls from 30 cm to 10 cm; an open bottle from the same fill starts at 10.46 mL/s and slows continuously.

Tested in The bottle that pours at one rate however full it is · the hydrostatics reading path

“Any push given to a pendulum alters its period, so a driven pendulum cannot keep perfect time.”

A small impulse given at the bottom of the swing changes the amplitude and not the phase: integrated through the kick, a pendulum's next beat shifts by less than one per cent of the shift from the same kick half-way out, and to first order by nothing at all (Airy, 1826).

Tested in The kick that does not change the beat · the pendulum reading path

“In a symmetric collider, with equal beams meeting head on, every collision happens at rest in the laboratory.”

Only the protons' total is at rest. The hard collision is between two constituents carrying fractions x₁ and x₂ of the protons' momenta, and it moves along the beam with rapidity ½ ln(x₁/x₂) — different in every event, which is why distributions are drawn in a variable a boost only shifts.

Tested in The angle a collider counts in logarithms · the Velocity-addition reading path

“Relativity makes a satellite clock run at a fixed offset from a ground clock, which is set once before launch.”

Only on a circular orbit. On an eccentric one the clock's height and speed both vary, and it gains and loses periodically by −2√(GMa) e sin E/c²: ±23 ns for a GPS satellite at e = 0.01, ±367 ns for the stranded Galileo satellites at e = 0.156, every 12.9 hours.

Tested in The satellites launched into the wrong orbit · the time dilation reading path

“An ice sheet that is losing ice at its edge retreats in proportion to how much it loses, and stops when the loss stops.”

Not where its bed deepens inland. On the MISMIP benchmark bed, softening the ice from A = 1×10⁻²⁵ to 3×10⁻²⁵ Pa⁻³ s⁻¹ moves the grounding line from 1,376 km not to a nearby position but across the whole inland-deepening stretch to 722 km, because no balanced position exists on that stretch.

Tested in The grounding line with nowhere to stop · the buoyancy reading path

“A twisted liquid-crystal display turns light's polarisation by guiding it round, layer by layer, with the molecules.”

Only a thick cell does. Displays use the first Gooch–Tarry maximum, u = 2dΔn/λ = √3, where the light inside lags behind the molecules, becomes elliptical by up to 30° and arrives linear along the exit molecules by an interference coincidence; at u = 12 the light does follow within a few degrees.

Tested in The twist that carries light round a corner · the polarisation reading path

“Light scattered by a transparent crystal comes back at the colour it went in.”

Almost all of it does, but a fraction comes back shifted by the crystal's vibrational frequency, redder or bluer. For silicon's 520 cm⁻¹ vibration the bluer (anti-Stokes) line is 7 × 10⁻⁴ of the redder (Stokes) line at 100 K, 0.103 at 300 K and 0.36 at 600 K.

Tested in The light that comes back bluer when it is warm · the scattering reading path

“A system governed by deterministic, reversible rules cannot show an irreversible approach to equilibrium.”

Kac's ring is deterministic and exactly reversible, and its excess of white over black decays as Boltzmann's (1 − 2μ)^t, its coarse-grained entropy rising to within a hair of ln 2 within a few dozen ticks — for 400 ticks, before it recurs.

Tested in The ring that forgets and then remembers · the entropy reading path

“A solid formed where a diffusing reactant meets another must be spread smoothly behind the advancing front.”

Only if it can form wherever it is made. If the product must reach a nucleation threshold above its solubility before any solid starts, each band drains the product around it and the next can only start further on: the computed tube shows 23 sharp bands, the last two 26 units apart against 0.8 between the first two.

Tested in The rings a reaction lays down in a gel · the diffusion reading path

“Salt clumps in damp air because a little water sticks to the outside of its grains.”

Above a sharp humidity it does not stop at sticking: the crystal dissolves completely in water drawn from the air. For sodium chloride the threshold is the water activity of its saturated solution, computed as 0.752 at 25 °C — 75.2 per cent humidity — and above it the drop holds 2.79 g of water per gram of salt, 4.24 g at 85 per cent and 11.5 g at 95.

Tested in The salt that dissolves in the air's own water · the chemical potential reading path

“The bands seen when two meshes overlap are a defect in one of them.”

They belong to neither: they are the difference term of the two gratings' product, with period p₁p₂/|p₁ − p₂|. Gratings of 10 and 11 units make bands every 110 units, eleven times coarser than either, though neither contains that period.

Tested in The pattern two gratings make that neither contains · the superposition reading path

“A liquid shaken at a frequency responds at that frequency, like any driven oscillator.”

Shaken vertically it is driven parametrically, through a modulated gravity, and the first waves to grow oscillate at half the drive frequency: a single mode at four times threshold, followed numerically, makes 30 cycles in 60 drive periods.

Tested in The ripples that answer at half the shaking · the standing waves reading path

“When the signal from a sample of nuclear spins fades, the spins have lost their magnetisation.”

Usually they have only fanned out. Spins whose precession rates are spread by 20 Hz lose their combined signal within about 10 ms while each still has almost all of its own; a π pulse at 30 ms brings 0.472 of the signal back at 60 ms, exactly the e^(−2τ/T₂) the true 80 ms relaxation allows.

Tested in The echo that gives back everything but the wandering · the spin reading path

“The scatter in reading out N atoms, √N/2 atoms either way, is a fundamental limit of quantum mechanics on how well N atoms can measure a phase.”

It is the limit for atoms that answer independently. Twisted for a moment before they are read, 100 atoms scatter by 1.25 atoms instead of 5.0, and a 0.1 rad phase error that independent atoms misjudge 31% of the time is misjudged 2.3% of the time.

Tested in The atoms that measure better together · the measurement reading path

“A magneto-optical trap compresses atoms more tightly the more of them it is given, as a deeper trap or a colder gas would.”

Only until rescattered light takes over. With a 200 Hz trap at 100 μK the peak density follows the trap's Gaussian up to about 10⁴ atoms and then stops at 2.5 × 10¹⁰ per cm³; 10⁸ atoms fill a sphere 0.99 mm in radius at that density, where the Gaussian would have put them at 541 times it.

Tested in The light that makes a cold cloud repel itself · the radiation pressure reading path

“A neutral insulator like liquid helium cannot bind a free electron, since it has no charge to attract it.”

It binds it by polarising. An electron above helium (ε = 1.0572) is attracted by an image of 0.0278 of its charge, a potential −Λe²/4πε₀z with Λ = 0.00695, which holds it in levels at −0.657, −0.164 and −0.073 meV, hovering 11.4 nm above the surface in the ground state.

Tested in The atom whose nucleus is a reflection · the potential reading path

“A geyser erupts when the water at the bottom of its tube, being the hottest, reaches boiling.”

The water at the bottom is the hottest and the furthest below its own boiling point. In a 22 m conduit at 2,240 m elevation, water warming by 0.8 °C a metre under a 4 m cooler layer reaches its boiling curve first at 4 m, at 103.6 °C, while the water at 22 m is still 13.7 °C below its 131.8 °C.

Tested in The column that boils from the middle · the phase change reading path

“A narrow outlet makes a hopper empty more slowly, by letting fewer grains through each second.”

While it flows it passes grains at the full Beverloo rate for its size; it stops more often. At 3.5 grain diameters the outlet would pass 421 grains a second while flowing and delivers 84 on average if each jam is cleared in half a second, 5 if clearing takes ten.

Tested in The outlet that jams by chance · the granular matter reading path

“Old window panes are thicker at the bottom because glass is a very slow liquid that has flowed over the centuries.”

For a pane 3 mm thick and 0.5 m tall to thicken at the bottom by 0.5 mm in 800 years under its own weight, its viscosity could be no more than about 10^15.3 Pa·s, which soda-lime glass reaches only at about 490 °C. The pane would have had to stand in an annealing oven for its whole life.

Tested in The drop that takes a decade · the viscosity reading path

“A tall cylinder draining through a hole in its floor makes a water clock with evenly spaced hour marks.”

Its level falls as a parabola in time, fast at first: a cylinder holding 25 litres to 35 cm drains through a 0.8 mm hole in 16.0 hours, is half empty in depth after 32% of the run, and its first hour lowers the level 6.0 times as far as its second-last. An evenly marked scale is 202 minutes fast at worst.

Tested in The vessel whose level falls evenly · the hydrostatics reading path

“A falling chimney breaks at its base, where the bending is greatest, as a leaning column does.”

A column held still at a tilt has its greatest bending at the base, mgL·sinθ/2. Toppling freely about its base, it carries none there; the moment along it is mgL·sinθ·s(L − s)²/4L³, largest at a third of the height and 13.5 times smaller than the static value at the base.

Tested in The chimney that breaks a third of the way up · the Free-body reading path

“A whip cracks because the arm moves it fast enough.”

The arm moves the handle at about 11 m/s. The tip passes 343 m/s because the energy the arm gave, 15 J here, stays in a moving part whose mass shrinks from 250 g to the 0.2 g of the cracker as the loop runs out, so its speed √(2E/M) rises to 387 m/s.

Tested in The crack at the end of a whip · the momentum reading path

“A swinging spring set going with a pure bounce swings back and forth in a fixed plane, since nothing turns it.”

Only if it is perfectly planar. With an angular momentum about the vertical of 0.6 × 10⁻⁵ — a trace of sideways motion — each swing returns turned by 43.6°, nine identical steps in 1,500 units of time, though no torque about the vertical ever acts.

Tested in The swing that turns a step at a time · the chaos reading path

“A better microscope — sharper, with more resolving power — would show a transparent cell clearly in ordinary light.”

A perfect image of a cell that delays its light by 0.3 rad and absorbs none is a perfect image of uniform brightness: through an objective of numerical aperture 0.25 the cell varies between 0.998 and 1.001 of the background. The information is in the light's phase, which brightness does not record.

Tested in The specimen that is invisible until its light is delayed · the imaging reading path

“A beam whose core keeps a constant width over a long distance has escaped diffraction.”

Its core does not spread because it is rebuilt at every distance from a fresh ring of the aperture. Behind a 0.5° axicon the core keeps a half-width of 13–14 μm over 57 cm, 209 Rayleigh ranges of a matched Gaussian, but holds only 1.16% of the power; the other 99% is in about 69 rings.

Tested in The beam that does not spread and the price it pays · the diffraction reading path

“A distant mountain disappears because too little of its light reaches the eye.”

A black ridge sends no light at all and is still seen, by being darker than the sky behind it. It vanishes when sunlight scattered into the line of sight by the air in front of it fills it in: its contrast against the sky falls as e^(−βx), and reaches the eye's 2% threshold at 3.912/β — 337 km in clean air at 550 nm.

Tested in How far the air lets a mountain be seen · the scattering reading path

“Gas bathed in brighter light gets hotter.”

Gas heated and cooled only by scattering settles where the gains and losses balance, 4kT = the light's energy-weighted mean photon energy, which depends on the light's spectrum and not its brightness: 1.7 × 10⁷ K for equal power per decade up to 100 keV, however bright. Brighter light only gets it there sooner.

Tested in The temperature a gas takes from the light around it · the relativistic thermodynamics reading path

“A particle's mass can only be measured by reconstructing the total energy and momentum of what it decays into.”

When one decay product escapes unseen, a quantity blind to the unmeasurable motion along the beam still carries the mass: the transverse mass √(2 × the electron’s transverse momentum × the neutrino’s × (1 − cos Δφ)) never exceeds the W's mass in any decay and piles up to an edge at 80.37 GeV.

Tested in The mass weighed without one of its pieces · the Mass-energy reading path

“Boltzmann's constant is a conversion factor between units, so it could only ever be agreed, never measured.”

Before 2019 it was measured, from a gas whose energy per atom is fixed by counting: argon's speed of sound at 273.16 K is √((5/3)kT/m), 307.8 m/s, and a 62 mm sphere of it rings at 3,551 Hz. Measurements of that pitch took its uncertainty from 8.5 ppm in 1986 to 0.37 ppm in 2017, before the value was fixed.

Tested in The note that fixed the kelvin · the equipartition reading path

“Only a heavier wall can block sound better than the mass law allows for its weight.”

At one frequency a light one can. A 2 kg/m² panel carrying 1 kg/m² of resonators tuned to 300 Hz blocks 35 dB there, the loss the mass law gives a wall of 26 kg/m², where the same 3 kg/m² rigidly joined gives 16.8 dB.

Tested in The wall made heavy only at one note · the impedance reading path

“An oscillation that has not started cannot be read from a machine's quiet running; below onset the mode is silent and there is nothing to measure.”

The mode is silent on average but never still: turbulence kicks it continually, and its variance is 2D/|σ|, its correlation time 1/|σ| and its noise peak |σ|/π hertz wide. All three measure the distance to onset. In the simulations drawn, the variance rises from 0.051 at a decay rate of 40 per second to 0.72 at 1.5, and the memory from 25 ms to 374 ms.

Tested in The noise that warns a tube is about to sing · the resonance reading path

“A sound that cannot be heard at fifty kilometres cannot be heard at two hundred.”

In the standard atmosphere with a 50 m/s stratospheric wind along the path, rays launched shallowly from a ground explosion are bent back down by the warm, windy air near 45 km and land 193–232 km away, beyond a zone of silence in which nothing arrives.

Tested in The thunder that passes over the listener's head · the huygens reading path

“Radiation pressure is far too feeble to hold anything as large as a mirror in place.”

In a metre-long cavity of finesse 10,000 with a kilowatt circulating, the radiation force changes by 1.63 × 10⁵ newtons for each metre the mirror moves: four-fifths of the stiffness of a steel wire a metre long and a square millimetre thick. A 1-gram mirror on that spring alone would oscillate at about 2,000 hertz.

Tested in The mirror held on a spring of light · the radiation pressure reading path

“A source passing exactly behind a point mass is magnified without limit.”

Only in geometric optics. The exact wave result for the factor on the power is πw/(1 − exp(−πw)), with w = 8πGMf/c³: 3.28 at w = 1 and 31.4 at w = 10, finite at every frequency, because a wave cannot be brought to a point.

Tested in The lens a long wave walks round · the light deflection reading path

“Protons get their mass from the Higgs field, like every other particle.”

The rest masses of the two up quarks and the down quark in a proton, which do come from the Higgs field, add to 9.0 MeV against the proton's 938.3 — about one per cent. The rest is the energy of the strong interaction's field and of the quarks' motion, set by the scale where the strong coupling runs off.

Tested in The nineteen decades a slow coupling buys · the planck scale reading path

“A metal reflects any electromagnetic field, static ones included.”

A metal of finite conductivity reflects a static electric field perfectly and a static magnetic field not at all: induced currents decay and the field soaks through, as through a copper sheet. That is the Drude description's zero-frequency TE reflection of 0; the plasma description's −0.993 at a 3 μm gap is a superconductor's behaviour.

Tested in The static field a metal may or may not reflect · the retardation reading path

“A charge released in an electric field moves along the field line through its starting point.”

It sets off along the line and then leaves it. Released at rest 0.32 charge-separations from the positive end of a dipole, at 20° to 70° from the axis, a light positive charge swings outside its line at every bend and escapes, although every one of those field lines ends on the negative charge.

Tested in The path a charge takes is not its field line · the the field concept reading path

“A type I superconductor expels any magnetic field weaker than its critical field.”

Only a long rod held along the field does. A sphere concentrates the field to 3/2 of its applied value at its equator and turns partly normal from two-thirds of the critical field; a flat plate across the field can expel none of it, and from the first trace of field a fraction H/Hc of the plate is normal.

Tested in The stripes a superconductor makes of a field · the superconductivity reading path

“A charge thrown across a uniform electric field keeps its sideways speed, since the field pushes only along itself.”

It keeps its sideways momentum. Its sideways speed is that momentum divided by its energy over c², and the field keeps adding energy: launched at 0.6c, the charge crosses at 0.231c by t = 3 mc/qE and ever more slowly after, with no sideways force at any moment.

Tested in The throw a uniform field bends into a chain · the relativistic dynamics reading path

“An electric field that exceeds cB just makes the drift faster than light, which shows the drift formula is wrong.”

Above E = cB there is no drift at all: a frame moving at c²B/E sees no magnetic field and an electric field E√(1 − c²B²/E²), and in it the charge runs away in a straight line. At E/cB = 1.05 its Lorentz factor reaches 152 by t = 400 m/qB, where at 0.95 it never exceeds 19.5.

Tested in The orbit that opens when E reaches cB · the field transformation reading path

“An entangled state always violates some Bell inequality.”

Werner's mixture of a singlet with weight p and noise with weight 1 − p is entangled for every p above 1/3, yet for p up to 1/2 an explicit local model — a random direction carried by each pair, one analyser answering −sign(a·λ) and the other +1 with probability (1 + b·λ)/2 — reproduces its correlation −p cos θ for every pair of projective measurements, and later models extend that to about 0.66.

Tested in The entangled pair a list of answers can imitate · the entanglement reading path

“Any interaction with the environment destroys a particle's interference.”

Only an interaction that records which path it took. A photon emitted in a random direction leaves the fringe visibility multiplied by sin(kd)/kd: 0.935 for a photon ten times longer than the separation of the paths, zero for one twice the separation. Molecules at 1000 K emit 0.111 photons each in a 2 ms passage, nearly all too long to matter, and their fringes survive.

Tested in The molecule that interferes until it glows · the matter waves reading path

“Once a pendulum clock is regulated, its rate depends only on the quality of its escapement.”

Its rate depends on its length, and its length on the temperature. A steel rod loses 0.50 s a day for every kelvin it warms and a brass rod 0.82, so a room that swings 10 °C between seasons drives an uncompensated steel pendulum through 5 s a day.

Tested in The pendulum that must not grow when it warms · the pendulum reading path

“A raw egg spins badly because its liquid sloshes and absorbs the push, so it turns more slowly than a cooked one.”

Given the same angular impulse it first turns faster: its shell carries only 16 per cent of the moment of inertia and starts at 19.2 revolutions a second where the cooked egg starts at 3. It then drags its contents up within about a second and falls to their common speed, losing 84 per cent of the flick's energy on the way.

Tested in The egg that keeps turning after it is stopped · the rotation reading path

“A liquid surface at rest is flat, apart from the size of its molecules.”

It is rippled by heat at every wavelength, each ripple holding ½kT. For water at 20 °C the ripples add up to 0.16 nm of rms roughness over a 10 nm patch, 0.26 over a micrometre and 0.390 nm beyond a few millimetres — the thickness grows with the size of the patch measured, because every decade of wavelength adds 0.0204 nm² to the mean-square height.

Tested in The surface whose thickness depends on the view · the surface tension reading path

“The wind is what makes the doors of a tall building hard to open in winter.”

With no wind at all, a 100 m building at 21 °C in air at −20 °C has its ground floor 95 Pa below the outside pressure, because its warm air column weighs less than the cold one beside it: a 2 m² pair of lobby doors is pressed shut by 191 N, and air rushes in when they open.

Tested in The building that breathes through its doors · the hydrostatics reading path

“A brighter-looking or thicker-looking cell in a phase image holds more material.”

The phase at any one point depends on the cell's thickness and concentration there. A model cell of 300 pg delays light by 3.41 rad at its centre when rounded up and by 0.55 when flattened, yet the delay integrated over its image is 536 rad·μm² in every shape, and gives back 300.0 pg.

Tested in The cell weighed by the light it delays · the imaging reading path

“Nanostructured flat lenses can be made achromatic at any size by designing their structures carefully enough.”

Any structure L thick whose building blocks span a group index Δn can delay light by at most LΔn/c, so R(1 − √(1 − NA²))/NA ≤ LΔn: a micrometre-thick lens with Δn = 1 can be achromatic only up to 40 μm across at NA 0.1 and 7.5 μm at NA 0.5.

Tested in The delay a flat lens has no room for · the refraction reading path

“The hot gas an engine exhausts carries energy that no design of the cycle could recover.”

Part of it is work left unextracted. An Otto engine at a ratio of 10 opens its exhaust valve on gas still at 3.8 bar and 1213 K; expanded until it reaches the intake pressure, the same gas has done 136 kJ/kg more work and leaves at 859 K with no rush through the valve.

Tested in The engine that expands further than it compresses · the heat engines reading path

“The length of a geyser's eruption tells something about when the previous one happened.”

It tells about the next. The interval after an eruption follows interval = 33.8 + 10.5 × duration, a correlation of 0.94, because a long eruption empties more of the conduit; plotted against the duration of the eruption that follows, the same intervals show a correlation of 0.04.

Tested in The geyser that remembers one eruption · the phase change reading path

“Sound travels along a metal sheet at the speed of sound in that metal.”

There is no single such speed. Aluminium carries compression at 6.32 km/s and shear at 3.13 km/s in bulk, and a thin sheet carries neither: its stretching wave travels at the plate velocity — twice the shear speed times √(1 − (shear speed/compression speed)²) — 5.44 km/s, and its bending wave at a speed that depends on frequency and thickness — 0.96 km/s at 100 kHz in a 1 mm sheet.

Tested in The plate that carries two kinds of sound · the guided waves reading path

“Any solid surface can carry a surface wave of any polarisation.”

A uniform half-space carries the Rayleigh wave, which moves in the vertical plane, and nothing that shakes purely sideways: a horizontal shear wave at a free surface reflects completely as itself, with no conversion and no decaying solution to bind it. Love waves need a slow layer over faster rock, which holds them by total reflection whenever they meet its base more obliquely than arcsin(3.5/4.5) = 51.1° for a typical continental crust.

Tested in The shaking a uniform Earth could not carry · the wave motion reading path

“An insulator spinning in a magnetic field cannot produce a voltage, because no current can flow through it.”

No current has to flow. Each piece of the material moves through the field and is polarised by v × B; a cylinder 2 to 4 cm in radius of an insulator with ε = 6 and μ = 3, turning 50 times a second in 0.5 T, holds 89 mV between its surfaces, 94 per cent of what a metal cylinder would give.

Tested in The voltage a moving magnet adds to an insulator · the maxwell equations reading path

“An expanding universe with a Hubble law needs general relativity; special relativity, with its fixed flat spacetime, cannot describe one.”

A cloud of particles thrown from one event into empty flat spacetime, sliced by the particles' own clocks, is uniform, expands in proportion to the clock reading, and obeys recession rate = distance ÷ τ exactly — a Hubble law. Its space has constant negative curvature: a circle of radius cτ has 1.18 times the Euclidean circumference.

Tested in The universe one explosion makes in empty space · the spacetime diagram reading path

“A row of clocks synchronised in their own frame still shows one time when it moves past a laboratory, only a dilated one.”

On one laboratory instant the clocks of a row moving at 0.6c read less the further ahead they are, falling by one tick every 1.33 light-periods of a clock: the moving row's instants are tilted across the laboratory's, and the readings make a wave.

Tested in The wave a row of moving clocks leaves on an instant · the simultaneity reading path

“A nucleus with too many protons decays by emitting a positron.”

Only if its atom outweighs the atom one proton down by more than two electron masses, 1,022 keV. Beryllium-7 is heavier than lithium-7 by 862 keV and can only capture an electron; potassium-40, 1,504 keV above argon-40, emits a positron in about one decay in a hundred thousand.

Tested in The parabola that decides which way a nucleus decays · the Mass-energy reading path

“A gravitational wave carries so little energy that a detector can only ever catch a tiny amount because there is little to catch.”

The waves carry a great deal: a burst that left one kT of room-temperature energy in Weber's 1.4-tonne bar would have carried about 2 × 10⁷ joules per square metre. The bar catches little because its cross-section, (8/π)(GM/c)(v/c)², is suppressed by the square of the speed of sound over the speed of light — 2.3 × 10⁻²⁵ m² Hz for Weber's bar.

Tested in The bar a wave of gravity barely touches · the gravitational waves reading path

“A speck of dust in a plasma picks up one or two charges, like an ion.”

It floats at a potential set by the balance of electron and ion currents, 2.4 kTₑ/e below the plasma in argon — 7.2 V for 3 eV electrons — and its charge is that potential times its capacitance: about 500 electrons at 0.1 μm, 5,000 at 1 μm and 25,000 at 5 μm.

Tested in The dust a plasma freezes into a crystal · the plasma oscillation reading path

“Any slight net charge on ordinary matter would be negligible against gravity for small objects and dominate only for very large ones.”

If charge follows mass, Gauss's law makes the electric repulsion of a body's parts a fixed fraction of their gravity, (δ/δc)² with δc = 8.9 × 10⁻¹⁹ of an electron's charge per nucleon, at every size and inside and out: a quarter of gravity at every radius of a layered planet when δ = δc/2.

Tested in The imbalance of charge that would outweigh gravity · the Gauss's law reading path

“Magnetising or demagnetising a piece of iron changes nothing about its rotation.”

The magnetic moment of the electrons comes with angular momentum, μ/γ′. Reversing the saturated magnetisation of an iron rod 2 mm thick and 10 cm long changes the electrons' angular momentum by 6.1 × 10⁻¹² kg m²/s, and the rod, hung freely, turns the other way at 0.005 rad/s.

Tested in The rod that turns when its magnetism flips · the spin reading path

“For two observables whose commutator has zero expectation in some state, the uncertainty principle allows both to be sharp in that state.”

Robertson's bound for a qubit's x and z is |⟨Y⟩|, zero for every state on the great circle through x and z. The entropic bound of Maassen and Uffink still holds there: H(Z) + H(X) is never below one bit, and equals it exactly where one answer is certain and the other a fair coin.

Tested in The sharpness a variance cannot see · the uncertainty reading path

“A quantum system whose classical counterpart is chaotic behaves classically once its quantum numbers are large.”

A rotor kicked with strength K = 5 heats classically by about 6.3 units per kick without limit; the quantum rotor with effective Planck constant ƛ = 1 follows that growth for a few tens of kicks and then stops near an energy of 58, with its momentum distribution frozen into an exponential of localisation length about 15 quanta.

Tested in The heating a quantum rotor refuses · the correspondence reading path

“A hot plume keeps rising until it has cooled to the temperature of the air around it, and then stops.”

It reaches the level where its density matches its surroundings — 4.37 F₀^(1/4)N^(−3/4) for an entrainment coefficient of 0.1 — but its momentum carries it on to 5.75 in the same units, heavier than its surroundings, before it stops and falls back to spread at the neutral level.

Tested in The plume that stops below the surface · the stratification reading path

“A hot-air balloon flies by making its air as hot as possible; the outside temperature hardly matters.”

Lift per cubic metre is the outside density times one minus the ratio of absolute temperatures, and the envelope is limited to about 120 °C. At that limit a cubic metre lifts 0.444 kg on a −10 °C morning and 0.248 kg at 35 °C, so a 2,800 m³ balloon's useful load falls from 980 kg at 0 °C to 569 kg at 35 °C.

Tested in The balloon a cold morning lifts · the buoyancy reading path

“Any equation of motion can be derived from a principle of stationary action if the right Lagrangian is found.”

The damped oscillator's motion is not stationary for the ordinary action ∫(T − V)dt — the slope at the true path is −0.062 for γ = 0.2 over six time units — and it is stationary for ∫e^(γt)(T − V)dt only because the weight makes the Lagrangian depend explicitly on time, which changes what momentum and energy mean.

Tested in The friction an action can hold only by changing its clock · the least action reading path

“A bigger circular loop is gentler on its riders.”

For a circular loop entered at any speed, the seat force at the bottom exceeds the force at the top by exactly 6 g, whatever the radius: the climb takes 4gR from v², the curvature 1/R turns that into 4 g, and gravity adds 1 g at the bottom and takes 1 g at the top. A loop with 0 g at the top has 6 g at the bottom at 5 m radius or at 40.

Tested in The loop that is not a circle · the circular motion reading path

“A perfectly built laser, with perfectly rigid mirrors, would emit a single exact frequency.”

Spontaneous emission into the lasing mode adds photons of random phase, and the gain holds the amplitude but nothing holds the phase, so the phase random-walks and the line has a Lorentzian width proportional to the square of the cavity's linewidth and inversely to the output power. For a milliwatt helium–neon laser with a 1.6 MHz cavity that is 2.5 mHz: tiny, and not zero.

Tested in The width a perfect laser still has · the coherence reading path

“A grid of parallel wires passes light polarised along the wires, the way a fence passes a rope shaken along its slats.”

The field along the wires drives currents in them, and the currents re-radiate a wave that cancels the incident one beyond the grid: that polarisation is reflected. The field across the wires can move charge only through a wire's width and passes. Hertz's grid of 1888 reflected about 96% of the power along its wires and transmitted all but a negligible fraction across them.

Tested in The grid that stops the light along its wires · the polarisation reading path

“Particles that clump together must attract each other through some force between them.”

In the Asakura–Oosawa mixture no pair of particles attracts: the small ones pass through each other and merely cannot enter the large ones. A direct simulation of two large discs among 270 such small ones, in which every move that does not put a small particle inside a large one is accepted, finds the discs touching about 2.6 times as often as apart — the gain in the small particles' room, counted.

Tested in The attraction that is the small ones' room · the entropy reading path

“The third law requires every mixture to separate completely into its pure components as absolute zero is approached.”

It requires the entropy to vanish, and a degenerate Fermi gas has a unique ground state whatever its density. Helium-3 dissolved in helium-4 is such a gas, with an entropy proportional to temperature, so the dilute phase can keep 6.6% helium-3 at absolute zero without any entropy left over.

Tested in The mixture that stays mixed at absolute zero · the third law reading path

“The chirp from the stair of El Castillo must have been built in deliberately, by tuning the steps.”

Any long regular staircase does it. A clap returns from each riser, the reflections arrive at intervals of twice the extra path to the next step divided by the speed of sound, and that extra path grows from 22 cm to 36 cm as the line of sight climbs a stair of 26 cm steps — a pitch falling from about 770 Hz to 480 Hz for a listener at 10 m, set by geometry alone.

Tested in The chirp a staircase sings back · the periodic media reading path

“A source moving round a circle at constant speed produces no Doppler shift, because on average it is neither approaching nor receding.”

The shift is instantaneous, not averaged: a horn mouth 15 cm from its axis turning 6.7 times a second moves at 6.3 m/s, and a listener level with it hears every note swing ±32 cents as the mouth approaches and recedes. Only directly above the axis, where the mouth never approaches, does the vibrato vanish.

Tested in The loudspeaker that goes round · the doppler reading path

“The Sun's gravitational redshift is too small to measure, which is why it took until the 1960s to confirm.”

Six hundred metres a second is easily resolved; solar spectrographs measured shifts of tens of metres a second early in the twentieth century. The difficulty was that the granulation blueshifts lines by a comparable amount — 231 m/s for a line formed low in the photosphere in the two-stream model, 31 m/s for one formed 300 km higher — so different lines gave different answers.

Tested in The redshift the Sun's boiling hides · the gravitational redshift reading path

“Only the Sun bends light enough to matter; the planets' deflection is too small to measure.”

Jupiter bends a grazing ray by 16.3 mas and a ray 14 of its radii away by 1.14 mas, which radio interferometers measured in 2002 to a few microarcseconds; the Earth's own limb deflection is 0.57 mas. At a single-star precision of 20 µas, Jupiter's deflection must be corrected for out to 3.5° from the planet.

Tested in The bend a moving planet gives · the light deflection reading path

“A conductor's sharpest point multiplies a field by an amount set by how sharp the point is.”

The tip of a grounded spheroid in a uniform field carries 49 times the field at a length-to-width ratio of 10 and 2,326 times at 100 — nearly the square of the slenderness. The tip's radius of curvature is b²/a, so the enhancement depends on the whole length as well as on the point.

Tested in The charge that follows no local rule · the conductors reading path

“A transformer hums at the frequency of the mains, because the alternating current shakes it at that frequency.”

The magnetostrictive strain is even in the magnetisation: the core stretches equally when magnetised either way. Driven at 50 Hz it stretches at every positive and every negative flux peak, four times in two cycles, so the hum's fundamental is 100 Hz and the strain contains no odd multiple of 50 Hz at all.

Tested in The hum at twice the mains · the magnetisation reading path

“An induction hob heats a pan in proportion to how well the pan conducts electricity, so a copper or aluminium pan should heat best.”

For a thick base the heating for a given coil current goes as the surface resistance √(πfμρ), which rises with resistivity. At 25 kHz carbon steel offers 9.9 × 10⁻⁴ Ω and aluminium 5.1 × 10⁻⁵ Ω, so steel heats 19 times as hard, and aluminium takes only half the power while the coil takes the other half.

Tested in The pan a better conductor will not heat · the induction reading path

“A gas cooled towards absolute zero shrinks towards a point at the bottom of its trap, whatever its atoms are.”

Only a classical gas does. Ideal bosons condense below 0.52 of the Fermi temperature into the trap's ground state; ideal fermions level off at an rms size of 1/√8 of the Fermi radius, 0.354 Fermi radii, and at a quarter of the Fermi temperature they are already 1.17 times the classical size and barely changing.

Tested in The cloud that stops shrinking · the exclusion reading path

“A steady electric field accelerates an electron in a crystal steadily, slowed only by collisions with the lattice.”

In a perfect lattice a steady force makes the crystal momentum grow steadily, but the velocity is the slope of the band, which reverses at the zone edge. The electron swings over 4J/F — 16 lattice spacings for a force of a quarter of the hopping energy per site — and returns after the Bloch period h/Fa, with no collisions at all.

Tested in The push that makes an electron swing · the bands reading path

“In a circular accelerator, a particle with more momentum always completes a turn sooner, because it is moving faster.”

It is moving faster by δ/γ² for a fractional momentum excess δ, but its orbit is longer by α δ. For a 0.1% excess in a 628 m ring with transition at γ = 6.1, the higher speed saves 605 ps a turn at γ = 2 and only 5.2 ps at γ = 20, while the longer orbit costs 56 ps at every energy: below γ = 6.1 the faster proton arrives first, above it last.

Tested in The energy above which faster arrives later · the relativistic dynamics reading path

“The limiting measurement in testing E = mc² is the weighing, because a mass difference is a small difference of large numbers.”

Penning traps measure single mass ratios to parts in 10¹¹, so a mass difference that is 3 × 10⁻⁴ of the atom is known to a part in ten million. The limit is the gamma-ray energy: a 8.47 MeV gamma ray is diffracted through 381 µrad, and a precision of 4 × 10⁻⁷ needs the angle to 152 picoradians.

Tested in The test of mass and energy that needs a ruler · the Mass-energy reading path

“A drop resting on a pool of the same liquid is held up by the air beneath it, as a raft is by water.”

The air holds nothing up; it is merely slow to leave. Under Reynolds's lubrication law the film thins as dh/dt ∝ −h³, so a 0.38 mm silicone-oil drop pressed at twice its weight takes 0.65 s to drain the film from 10 µm to the 100 nm at which van der Waals forces merge the surfaces — 156 times longer than a 4.1 ms bounce.

Tested in The drop that will not merge · the surface tension reading path

“A soft plant stays upright because its tissue is stiff, and wilts because the tissue weakens when it dries.”

The tissue's stiffness comes mostly from the pressure in its cells. A membrane tube carries a bending moment of πPr³/2 before its compressed side goes slack, in proportion to the pressure and independent of the membrane's own stiffness; at a turgor of 0.74 MPa a 1 mm tube carries 1.17 N·mm, and with no pressure nothing.

Tested in The stem that stands by pressure · the osmosis reading path

“A coin toss is random because its motion is chaotic, so that the smallest difference in the throw is magnified until the outcome is unpredictable.”

Nothing is magnified exponentially. Two coins launched with spins a thousandth apart, 0.24 rad/s, drift apart in orientation in proportion to time and differ by 0.12 radians when caught after 0.49 s — still the same outcome. The randomness is the hand's own scatter set against bands of heads and tails only 2.7% of the spin wide.

Tested in The toss that is random without chaos · the chaos reading path

“In a station spun to give one g at its floor, a thrown ball behaves exactly as it would on the Earth.”

Thrown at 5 m/s and 45° in a 10 m station, a ball lands 1.52 m away spinward and 3.06 m away antispinward, against 2.55 m in every direction on the Earth; thrown straight up a metre, it lands 96 cm from the hand. The floor's push is like gravity; the ball's free flight is not, because nothing pulls it.

Tested in The throw a spinning station bends · the projectile reading path

“A rain sensor sees the raindrops on a windscreen, like a small camera.”

It sees nothing outside the glass. Infrared light trapped in the glass at 45° by total internal reflection — beyond the 41.2° critical angle for glass and air — leaks out wherever water touches the outer face, because glass and water have a critical angle of 61.3°. The sensor measures how much light the outer surface still traps.

Tested in The windscreen that feels the rain · the total internal reflection reading path

“The eye's lens does its focusing, and the cornea is a transparent cover.”

In a simplified model eye the front of the cornea contributes 48.2 dioptres, the back −6.2 and the relaxed lens 19, of about 61 in all. The cornea's front surface owes its power to the step in index from air to tissue, 0.376; under water the step is 0.043 and 42.7 dioptres vanish, while the lens, bathed in the eye's own fluids, is unaffected.

Tested in The eye that loses its lens in water · the imaging reading path

“When two metals diffuse into each other, equal numbers of atoms cross in each direction, since each atom that leaves a site is replaced by one from the other side.”

That is what a direct exchange of neighbours would require, and it is not what happens. With one species diffusing three times as fast as the other, the fast species' flux exceeds the slow one's wherever the gradients are equal, and the difference is carried by a counter-flow of vacancies; inert markers at the original interface drift into the fast species' side, by 1.63 units in the model at the last time drawn.

Tested in The markers that move because atoms are unequal · the diffusion reading path

“Newcomen's engine was wasteful because it was crudely made, and better workmanship would have fixed it.”

The waste was in the cycle. In the model 53 cm cylinder, cooled to 45 °C by the injection, 2.10 kg of steam condensed on the walls each stroke against 0.33 kg needed to fill it — 6.4 times as much — and no precision of boring could change that while the same vessel had to be both hot and cold.

Tested in The cylinder that must not be cooled · the heat engines reading path

“A heat pipe works by conduction through especially conductive metal.”

A solid copper rod 300 mm long and 6 mm across needs 796 K of temperature difference to carry 30 W, and even diamond 159 K. A water heat pipe of the same size carries it with 0.8 K, almost all of it across the wet wicks at its ends, because the heat travels as latent heat in vapour that needs almost no temperature difference to flow.

Tested in The pipe that carries heat as vapour · the phase change reading path

“Noise recorded at two places far apart carries no information about the ground between them, since it comes from sources that have nothing to do with either.”

Averaged over many directions, the cross-correlation of the two records is the wave that would travel between the stations: for stations 60 km apart in a medium carrying surface waves at 3 km/s, pulses at ±20 s. The sources that matter are those nearly in line with the stations — 230 of 720 round a ring contribute coherently and the rest cancel.

Tested in The echo hidden in random noise · the guided waves reading path

“A quartz watch is accurate because quartz vibrates at a fixed natural frequency that does not change.”

Its frequency changes with temperature along a parabola of −0.034 ppm per °C² about a turnover near 25 °C: at 31 °C on a wrist the watch loses 0.11 s a day, at 10 °C in a drawer 0.66 s, at 0 °C 1.8 s and at 60 °C 3.6 s. The accuracy comes from the curve being flat at a temperature chosen to match a wrist.

Tested in The fork inside a watch · the resonance reading path

“Fusion needs a temperature, and a hot enough plasma will burn.”

Temperature alone fixes only a floor near 4.3 keV, below which radiation beats heating at any density. Above it a plasma still goes out unless its density times its confinement time exceeds about 2 × 10²⁰ s/m³ at 14 keV; a plasma at 100 million kelvin held for a millisecond at ordinary tokamak density produces next to nothing.

Tested in The product a fusion plasma has to beat · the plasma oscillation reading path

“A magnetic field holds a plasma because charged particles spiral along its lines and cannot cross them.”

In a torus with no twist they do cross them. The field's curvature and gradient drive ions up and electrons down at about 600 m/s at 10 keV, the separated charge sets up an electric field, and the crossed fields throw the whole plasma outward: at ITER's size and 14 keV it reaches the wall in about 3 µs.

Tested in The twist a bottle of plasma has to have · the flux freezing reading path

“A laser fusion capsule is crushed by the pressure of the light falling on it.”

At a cavity temperature of 300 eV the radiation pressure is about 0.37 million atmospheres. The pressure driven by the capsule's surface boiling off is about 86 million, over 200 times larger, because hot plasma carries far more momentum per joule than light does.

Tested in The fuel held together only by its inertia · the radiation pressure reading path

“Inertial fusion heats its fuel to ignition temperature.”

Heating one milligram to 5 keV would cost 580 kJ, while compressing it cold to a thousand times solid density costs about 13 kJ. So almost all of the fuel is compressed and kept cold, and only a central spot of a few per cent of the mass is heated, by the fuel's own convergence.

Tested in The fuel held together only by its inertia · the radiation pressure reading path

“A point dipole is fully described by its moment, so any two dipoles of equal moment are physically the same object.”

The mean field inside an enclosing sphere is −1 in units of (1/4πε₀)p/R³ for a charge pair and +2 in the matching magnetic units for a loop, at every radius that encloses them. That difference survives any amount of shrinking and appears in the theory as a term concentrated at the origin, which an s electron feels directly.

Tested in Two dipoles that agree everywhere but inside · the the field concept reading path

“A vacuum has no resistance, so the current in a vacuum tube is limited only by how many electrons the cathode can give off.”

At low voltage a tungsten filament at 2200 K and one at 2800 K carry the same current, on the same three-halves-power line, though the hotter one offers about 260 times more electrons. The charge in the gap reduces the field at the emitter to zero, and the extra electrons are sent back.

Tested in The current a vacuum will not carry · the Gauss's law reading path

“Spontaneous emission is a fixed property of an excited atom, set by its own wave functions.”

A dipole a tenth of a wavelength from a perfect mirror emits at 1.85 times its free rate if it points across the mirror and much less if it points along it, and between two mirrors less than half a wavelength apart a dipole along them does not emit at all. Karl Drexhage measured the oscillation around 1970, and caesium atoms between mirrors were seen to live over twenty times longer in 1985.

Tested in The lifetime that depends on where the mirror is · the decay reading path

“Stimulated emission is a rare effect that only occurs in lasers, and lasers were its discovery.”

It was deduced in 1916–17 from the shape of the thermal spectrum, because absorption and spontaneous emission alone give Wien's law and not Planck's. In the microwave background at hydrogen's 21 cm line it outruns spontaneous emission about 39 times, with no inversion at all.

Tested in The third process equilibrium demands · the blackbody reading path

“A fast particle's decay products have energies peaked near the typical boosted value.”

For two photons from a neutral pion, the energy is linear in the cosine of the decay angle, and an isotropic decay is uniform in that cosine. 200,000 samples at 400 MeV fill the range from 11.7 to 388.3 MeV evenly to within statistics, with no peak anywhere.

Tested in The flat spectrum a fast pion must paint · the relativistic dynamics reading path

“A photon with more than 1.02 MeV can turn into an electron and a positron.”

Not by itself, at any energy. A single photon has energy equal to its momentum times c; a pair has more energy than that for any momentum, on a curve that approaches the photon's line and never touches it. Another particle must take up the difference.

Tested in The partner that sets the price of a pair · the Mass-energy reading path

“Wetter sand is stronger sand, up to the point where it turns to soup.”

Once the first trace of water has formed bridges at the contacts, tensile strength is nearly flat with further water: Rumpf's estimate gives 864 Pa for 0.3 mm sand, and a bridge with a thousand times the volume pulls only 7 per cent less. Added water only begins to matter when the bridges run together into pools.

Tested in The sandcastle a spoonful of water holds · the granular matter reading path

“Lake Nyos had filled with carbon dioxide until it was saturated, and then overflowed.”

Water in a representative survey holds about 0.35 mol/L at 200 m, where it could hold 0.75, so it is under half saturated at the depth where the gas is stored. It can stay that way indefinitely. It reaches saturation only if lifted, near 88 m in this profile.

Tested in The lake that holds its gas until it does not · the stratification reading path

“A bicycle stays up because its wheels are gyroscopes.”

Removing the front wheel's spin from the benchmark machine does take its stable band away, up to 14 m/s, but that is one design. In 2011 Kooijman and colleagues built a working bicycle with counter-rotating wheels, so no net spin, and it was self-stable. Gyroscopic terms help some machines and are not required.

Tested in The bicycle that steers itself upright · the harmonic approximation reading path

“Trail is what makes a bicycle stable, since a castering front wheel follows its own direction.”

Trail affects stability strongly, and the benchmark's band shrinks to 2.9–3.1 m/s without it. It does not explain the effect. The same 2011 machine had negative trail, which was believed to be fatal, and it recovered from falls on its own at the right speed, because its mass distribution provided the coupling instead.

Tested in The bicycle that steers itself upright · the harmonic approximation reading path

“A yo-yo de-spin device needs a cord matched to the spin rate, so a stage must be spun at a known speed.”

The cord length for zero spin is √(1 + I/mR²) times the radius, with the starting spin nowhere in the expression. A direct integration of the equations of motion, using neither conservation law, reproduces the closed form to a part in a thousand. Tension scales with the square of the spin, so it matters for the cord's strength, not for its length.

Tested in The weights that stop a spinning satellite · the energy reading path

“The chain rises because its momentum carries it up, as a thrown object goes up.”

Every link is at the chain's speed at the top and bottom alike, and the leg in the air is in steady state. What must be explained is the force. A steady arch needs the tension at the pick-up to be below the momentum flux λv², by exactly the weight of the chain in the arch. That requires an upward push from the pile with a share α = gh₁/v².

Tested in The chain that climbs out of the jar · the momentum reading path

“The glory is a rainbow, a little different in form, made by rays bending in drops.”

A rainbow is a true ray phenomenon at 138°. A rim ray in water turns through only 165.6° after one internal reflection, falling 14.4° short of straight back, and only with an index above √2 could a rim ray reach 180°. Near the antisolar point only a narrow bundle near the axis returns, and it cannot explain brightness about 78 times the surface reflection.

Tested in The rings no ray can reach · the scattering reading path

“Insects see poorly because their eyes have too few facets, and evolution could simply give them more.”

More facets on the same head means narrower facets, and past D = √(λR) each one's diffraction blurs more than the finer spacing gains. On a 1 mm eye the best achievable blur is about 1.8°, set by the size of the head, and no facet width can get below it.

Tested in The eye a fly cannot afford · the diffraction reading path

“A ring-down cavity gains its sensitivity only through its long path.”

The long path buys a factor of ten thousand at 99.99 per cent mirrors. In the simulation, however, the scatter falls by a factor of about a million against a direct measurement over a metre, since fitting the slope of a logarithm discards the source's 5 per cent brightness noise entirely.

Tested in The absorption measured with a stopwatch · the dispersion reading path

“Kelvin made a mistake in his calculation of the Earth's age.”

t = T²/πκg² from a gradient of 36 K/km, a start of 3,870 °C and a diffusivity of 1.2 × 10⁻⁶ m²/s gives 97 million years, and an independent numerical integration of a uniform body gives the same. The error was in assuming the interior conducts heat like a solid.

Tested in The age a cooling Earth gives · the diffusion reading path

“A microwave oven works by matching a resonance of the water molecule.”

Water's rotational relaxation is broad and peaks near 20 GHz at room temperature. At 2.45 GHz it is on the shoulder of a broad relaxation, not at any sharp resonance, and this frequency was chosen because of its licensed band and its depth of about 1.9 cm in water. A resonant frequency would be absorbed at the surface and heat nothing inside.

Tested in The oven that cooks in spots · the standing waves reading path

“Waves come in straight because the wind on a coast blows onshore.”

A 10 s swell arriving at 50° offshore reaches the beach at about 5° with no wind at all, from Snell's law in a speed that falls with depth. At 80° offshore it is still only 16° in 2 m of water. The offshore direction is nearly erased, whichever way the wind is blowing.

Tested in The waves that always arrive square to the beach · the wave motion reading path