A speed that hardly knows what the material is

At its defaults it draws a speed that hardly knows what the material is. The Rayleigh speed against Poisson's ratio, in units of the shear speed, found by bisection on the free-surface residual at each point rather than from a printed formula. Across the whole range a solid can have, the answer moves from 0.8837 to 0.9541 — a span of 0.0704 — while the pressure speed over the same range goes from 1.45 to 7.1 shear speeds and is unbounded at an incompressible limit. So a surface wave's speed is a shear speed to within a few per cent, whatever the rock, and the dashed curve is the customary rational approximation, which is within 0.0019 of the computed root everywhere. It is always slower than the shear wave, and it must be: a surface disturbance travelling faster than the bulk shear wave would radiate into the bulk and stop being confined.

surface-wave is one function in lib/figures/waves.js — travelling, standing, adding and shifting. Everything below came out of it during this build, at parameters taken from the essays rather than invented for this page. A figure here is the figure a reader meets in an essay, and if the generator changes, this page changes with it.

The Rayleigh speed against Poisson's ratio, in units of the shear speed, found by bisection on the free-surface residual at each point rather than from a printed formula. Across the whole range a solid can have, the answer moves from 0.8837 to 0.9541 — a span of 0.0704 — while the pressure speed over the same range goes from 1.45 to 7.1 shear speeds and is unbounded at an incompressible limit. So a surface wave's speed is a shear speed to within a few per cent, whatever the rock, and the dashed curve is the customary rational approximation, which is within 0.0019 of the computed root everywhere. It is always slower than the shear wave, and it must be: a surface disturbance travelling faster than the bulk shear wave would radiate into the bulk and stop being confined.

The speed of sound in 4 gases at 273.15 K. The short bar is Newton's √(RT/M), which assumes the compressions stay at one temperature; the long one is the same multiplied by √γ, which is what they come to if no heat crosses between a compression and the rarefaction beside it; the upright mark is the measured value. Newton's is 15.5% low for air, 22.5% low for helium, 22.5% low for argon, 12.0% low for carbon dioxide, and the corrected one is right to 0.05% for every gas here. Turning it round: γ read off each pair of bars is 1.400 for air, 1.665 for helium, 1.666 for argon, 1.290 for carbon dioxide, which is 1 + 2/f with f = 5.0, 3.0, 3.0, 6.9 ways of holding energy — three for the monatomic gases, five for the diatomic ones, and nearly seven for carbon dioxide, whose bending modes have begun to take a share at this temperature and its stretch has not. A speed measured with a stopwatch counts the ways a molecule can move.

The same 50 hertz note driven onto 2 strings at the same tension of 80 newtons but different thicknesses. The frequency is identical — it is the source's — while the wavelengths are 5.66 metres and 2.83 metres, because each string carries the wave at its own speed.

How far heat diffuses in one period, divided by the wavelength, against frequency, for air at 1013 mbar and 1 mbar. Below one, no heat crosses between a compression and the rarefaction next to it during a cycle and the wave is adiabatic, which is the assumption Laplace made and Newton did not. The ratio rises only as the square root of the frequency, so it takes 5.78·10⁹ Hz at 1013 mbar and 5.7·10⁶ Hz at 1 mbar to reach it. Beside each is the frequency at which the sound wavelength falls to the mean free path, 4.87·10⁹ and 4.81·10⁶ Hz, past which there is no continuum left to carry a wave. The two land in the same decade in every case, and they have to: the diffusivity is about a third of the mean free path times the molecular speed, and the speed of sound is about the molecular speed, so both frequencies are the collision rate to within a small factor. Newton's isothermal sound is not merely wrong for audible frequencies. There is nowhere in a gas it is right.

The speed of sound in 3 gases at 293.15 K. The short bar is Newton's √(RT/M), which assumes the compressions stay at one temperature; the long one is the same multiplied by √γ, which is what they come to if no heat crosses between a compression and the rarefaction beside it; the upright mark is the measured value. Newton's is 11.7% low for nitrogen, 12.4% low for air, 8.8% low for carbon dioxide, and the corrected one is right to 4.50% for every gas here. Turning it round: γ read off each pair of bars is 1.282 for nitrogen, 1.304 for air, 1.202 for carbon dioxide, which is 1 + 2/f with f = 7.1, 6.6, 9.9 ways of holding energy — three for the monatomic gases, five for the diatomic ones, and nearly seven for carbon dioxide, whose bending modes have begun to take a share at this temperature and its stretch has not. A speed measured with a stopwatch counts the ways a molecule can move.

The speed of a wave on a string, against the tension pulling it, for linear densities of 1.0 grams per metre and 4.0 grams per metre. It is a square root: at 180 newtons the lighter string carries a wave at 424 metres per second and the heavier one at 212.

physicscheck asserts something about surface-wave that could fail — it draws it and measures the result against a value reached some other way.

Across the library: 100 interrogated, 2 exercised only, 1 untouched, of 103. Read out of the gate's source by the gate's own two patterns — and the gate's last claim fails the build if that read disagrees with what it was handed while running.