Illustrated Physics
physics, drawn
Concept

Harmonic series — where it appears

The set of frequencies at whole-number multiples of a lowest one: one, two, three, four times the fundamental. A clamped uniform string and an open pipe sound it exactly, and any periodic signal is built from it, which is why a repeating waveform has a definite pitch.

Named by 3 essays across 3 fields — each of them below, with the objects they name alongside it.

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

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

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

waves · Standing waves One turn of a circling charge, as it arrives. The radiation field reaching a distant observer in the plane of a charge's circular orbit, over one full turn, for speeds of 0.2c, 0.6c, 0.9c, each scaled to its own peak and plotted against arrival time as a fraction of the orbital period. At 0.2c the flash arrives 20.9 per cent of a turn wide at half height, having been emitted over 25.4 per cent; at 0.6c the flash arrives 6.0 per cent of a turn wide at half height, having been emitted over 14.2 per cent; at 0.9c the flash arrives 0.7 per cent of a turn wide at half height, having been emitted over 6.3 per cent. The slowest charge sends something close to a sine wave, with a trough 0.44 of its peak; the fastest sends a spike whose peak, divided by its speed, is 100 times a slow charge's, and a trough of 0.0028 of that peak. The field integrates to zero over every turn, as a radiation field must.

The flash a circling charge sends once a turn

A slow electron going round a magnet broadcasts one radio frequency. The same electron at nearly the speed of light, on a circle of the same kind, sends X-rays. Nothing about the mechanism changes between the two. The charge chases its own light, and the chase compresses one power of γ worth of emission into three.

astrophysics · Radiating charge Eight blocks leaning past the edge of a table. 8 identical blocks of length 1 stacked on a table edge (the vertical line), each set as far out as balance allows: the dot on each interface marks the centre of mass of all the blocks above it, which sits exactly on the edge of the block beneath. The top block overhangs the second by 1/2, the second the third by 1/4, the third the fourth by 1/6, and the k-th by 1/(2k); together the stack reaches 1.359 block lengths beyond the table — the top block lies wholly beyond it, since 1.359 is more than one. Every interface is balanced on its edge, so the stack stands, but with no margin at all.

The stack that leans past its own table

Stack identical books on the edge of a table, each pushed out as far as it will go without falling, and four of them are enough for the top book to lie entirely beyond the table's edge. Keep going and the overhang grows without limit — the stack can reach as far out as anyone likes — but only as half the harmonic series, so that two book-lengths need 31 books, three need 227 and five need more than twelve thousand. The arithmetic is the same that settles whether a ladder leaning on a wall slips or a wardrobe topples, and the slow growth is the price of having nothing but the stack's own weight to hold it back.

mechanics · Free-body

Named alongside it

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

Boundary conditionsCentre of massCorbelCyclotronFourier transformLarmor formulaThe Lorentz factorNormal modesOverhangQuantisationRelativistic beamingRetardation

All concepts