Illustrated Physics
physics, drawn
Concept

Bessel function — where it appears

A family of functions that solve wave and diffusion problems in round geometries, as sines and exponentials do in straight ones. The modified function K₀ gives the shape of a meniscus round a small floating object, and the field round a line source screened over a finite distance.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

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 Like menisci attract, unlike menisci repel. Cross-sections of a water surface with two small floating bodies, drawn with menisci of the shape K₀(r/ℓc) that a small floater makes, exaggerated vertically. Left: two light bodies, each pulling the surface up round itself as buoyancy pushes it up against the surface; each sits on the other's slope and is drawn towards it, as a bubble rises — the two climb together. Right: a light body that raises the surface beside a heavy one that depresses it; each sits on the other's slope the wrong way and they drift apart. The deformations reach a few capillary lengths, 2.71 mm for water, and the rule is the whole of the Cheerios effect: cereal clumps, and cereal gathers at the wall of a bowl the milk wets.

The cereal that gathers at the edge of the bowl

The last few pieces of cereal in a bowl of milk clump together and drift to the side of the bowl. Nothing is pulling on them but the milk's surface. Each floating piece bends the surface round itself, and a second piece sitting on that bend has somewhere to slide: like menisci attract, unlike menisci repel, and the reach of the force is a few capillary lengths, 2.7 millimetres for water. The arithmetic is the arithmetic of charged lines in two dimensions, screened by gravity, and it says why leaves gather on a pond, why bubbles on coffee drift to the cup's rim, and why particles smaller than about ten micrometres do not clump this way at all.

fluids · Capillarity

Named alongside it

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

AttenuationCapillarityCapillary lengthCutoffGuided wavesMeniscusMode conversionNormal modesScreeningSelf assemblySkin depthSurface tension

All concepts