Illustrated Physics
physics, drawn
Concept

Tolerance — where it appears

The range over which a quantity can vary while an outcome still succeeds, such as the launch angles that still score or the wavelengths a device still switches. It measures robustness rather than accuracy, and a design is often chosen to widen it rather than to hit a target exactly.

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

The peak that is lost to a fraction of a wave. The height of the central peak, relative to a perfect pupil of the same size, against the root-mean-square error of the wavefront in waves, for four kinds of error — each computed from the transform and each normalised to the same rms. The dashed curve is the usual approximation, the exponential of minus the square of two pi times the error. What the figure shows is that to a good approximation it does not matter WHAT the error is, only how large it is in the mean square: four quite different shapes of wavefront give nearly the same peak. A fourteenth of a wave leaves 80 per cent of the peak, which is the conventional definition of diffraction-limited, and it corresponds to a quarter of a wave peak-to-valley for a simple defocus — which is where Rayleigh's quarter-wave rule comes from and why it is a convention laid over a computed number rather than a threshold in the physics.

How accurate a mirror has to be

The pupil's amplitude decides the rings; its phase decides the peak. A wavefront error of a fourteenth of a wave root-mean-square leaves eighty per cent of the peak intensity, which is the whole of what 'diffraction-limited' means — a convention laid over a computed number. And the number barely depends on what the error is, only on how large: four quite different aberrations of the same magnitude give nearly the same answer.

optics · Diffraction The launches that go in, and one thrower's scatter over them. Every free throw as a point: launch angle across, launch speed up. The dark curve is the launches that put the ball's centre through the centre of the hoop, lowest at the least-speed launch, 51.4° and 7.17 m/s. The shaded band is every launch that passes cleanly through, found at each angle by moving the speed until the ball touches the rim. It does not exist below 46.9°, is a hair thick near the bottom of the curve, and thickens as the launches steepen. The two ellipses are one thrower who scatters ±0.05 m/s in speed and ±1° in angle, drawn at two standard deviations and centred on two aims: the least-speed launch, and 58.5°, the aim that makes a clean pass most likely for that thrower. At the first, the ellipse lies across a band far thinner than itself; at the second, more of it lies inside, although the band there slopes more steeply.

The throw most likely to go in

A free throw can be launched at 51.4° with less speed than at any other angle, and there a small error of angle hardly moves the ball at all. It is still not the best aim. Once the question is which throw most often goes in rather than which is cheapest, the thrower's scatter has to be laid over the launches that succeed — and for a hoop the answer moves steeper, while for a board the same scatter moves it flatter.

mechanics · Projectile

Named alongside it

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

AberrationAdaptive opticsApproximationDiffractionEnvelopeError propagationFourier transformOptimisationPoint-spread functionProbability distributionProjectileScattering

All concepts