Three geometries, three exponents

At its defaults it draws three geometries, three exponents. Amplitude against distance for a wave spreading in one, two and three dimensions, on logarithmic axes, over 3 decades. The same power crosses every surface round the source, so the intensity falls as one over the area of that surface and the amplitude as one over its square root. The fitted slopes are 0.000 along a line, -0.500 over a cylinder, -1.000 over a sphere, each fitted by least squares to the drawn curve rather than written on it. A ripple on water is the middle case and a sound in a room is the last, which is why a ripple stays visible so much further than a shout carries.

spreading-loss 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.

Amplitude against distance for a wave spreading in one, two and three dimensions, on logarithmic axes, over 3 decades. The same power crosses every surface round the source, so the intensity falls as one over the area of that surface and the amplitude as one over its square root. The fitted slopes are 0.000 along a line, -0.500 over a cylinder, -1.000 over a sphere, each fitted by least squares to the drawn curve rather than written on it. A ripple on water is the middle case and a sound in a room is the last, which is why a ripple stays visible so much further than a shout carries.

Amplitude against distance for a wave spreading in one, two and three dimensions, on logarithmic axes, over 3 decades. The same power crosses every surface round the source, so the intensity falls as one over the area of that surface and the amplitude as one over its square root. The fitted slopes are 0.000 along a line, -0.500 over a cylinder, -1.000 over a sphere, each fitted by least squares to the drawn curve rather than written on it. A ripple on water is the middle case and a sound in a room is the last, which is why a ripple stays visible so much further than a shout carries.

Two losses on the same axes: the geometric one, which is 6 dB per doubling of distance and therefore a straight line against the logarithm; and absorption at 200 dB per kilometre, which is a straight line against the distance itself and so climbs without limit on this plot. They are equal at 239 metres, found by bisection on the two drawn curves. Below that distance a source is quiet because it is far away; above it, because the medium has eaten the sound. At 10 m the two are 20.0 and 1.8 dB; at 100 m the two are 40.0 and 19.8 dB; at 1000 m the two are 60.0 and 199.8 dB.

Two losses on the same axes: the geometric one, which is 6 dB per doubling of distance and therefore a straight line against the logarithm; and absorption at 4 dB per kilometre, which is a straight line against the distance itself and so climbs without limit on this plot. They are equal at 21.68 km, found by bisection on the two drawn curves. Below that distance a source is quiet because it is far away; above it, because the medium has eaten the sound. At 10 m the two are 20.0 and 0.0 dB; at 100 m the two are 40.0 and 0.4 dB; at 1000 m the two are 60.0 and 4.0 dB.

Amplitude against distance for a wave spreading in one, two and three dimensions, on logarithmic axes, over 4 decades. The same power crosses every surface round the source, so the intensity falls as one over the area of that surface and the amplitude as one over its square root. The fitted slopes are 0.000 along a line, -0.500 over a cylinder, -1.000 over a sphere, each fitted by least squares to the drawn curve rather than written on it. A ripple on water is the middle case and a sound in a room is the last, which is why a ripple stays visible so much further than a shout carries.

Two absorption curves for air against frequency, both logarithmic, in decibels per kilometre. The lower one is the classical Stokes–Kirchhoff result computed from air's viscosity and thermal conductivity alone, and it goes as f^2.000 — exactly two, because the loss per cycle is fixed and the number of cycles per metre is proportional to the frequency. The upper one is the measured atmospheric absorption at 20 °C and 50 per cent humidity, which fits f^1.42 and is 30 times larger at 1 kHz and 211 times at 125 Hz. The excess is not a correction to viscosity: it is nitrogen and oxygen storing energy in vibration and giving it back late, at a rate the water vapour sets, and it is the mechanism that actually removes the treble from a distant sound.

physicscheck asserts something about spreading-loss 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.