Nothing propagates below 8.98 MHz

At its defaults it draws nothing propagates below 8.98 mhz. Frequency against wavenumber for a wave in a plasma of 1.0e+12 electrons per cubic metre, in units of the plasma frequency. The curve is ω² = ωₚ² + c²k², so it starts at ωₚ with zero slope and becomes the light line far above it; the whole band below ωₚ has no real k at all, which is the shaded region. At k = 3.19e-1 m⁻¹ the phase velocity read off the curve is 1.1607c and the group velocity 0.8615c, whose product is c² to a part in 10¹⁰ — the crests outrun light and the signal does not, which is the same arrangement as any other medium with a cutoff.

plasma-response is one function in lib/figures/extremes.js — radiation, cross-sections and self-gravity. 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.

Frequency against wavenumber for a wave in a plasma of 1.0e+12 electrons per cubic metre, in units of the plasma frequency. The curve is ω² = ωₚ² + c²k², so it starts at ωₚ with zero slope and becomes the light line far above it; the whole band below ωₚ has no real k at all, which is the shaded region. At k = 3.19e-1 m⁻¹ the phase velocity read off the curve is 1.1607c and the group velocity 0.8615c, whose product is c² to a part in 10¹⁰ — the crests outrun light and the signal does not, which is the same arrangement as any other medium with a cutoff.

The dc self-bias against the phase between the fundamental and its second harmonic, each of amplitude A, for three chambers: symmetry parameter ε = 0.5, 1, 2, where ε = 1 is a chamber with equal sheaths on both sides. In the symmetric chamber the bias runs from −0.438 A at 0° through zero at 45° to 0.438 at 90°, and back — −43.8 V to 43.8 V at A = 100 V — staying within 0.011 A of a straight line between 0° and 90°. An asymmetric chamber adds a fixed offset of the kind unequal electrodes produce and leaves the phase control in place, so the geometry and the waveform add rather than compete.

A voltage made of a fundamental and its second harmonic, each of amplitude A, over two cycles of the fundamental, for a phase between them of 0° and 90°. At 0° the waveform reaches +2.000 A and −1.125 A, so between two equal electrodes it settles at a bias of −0.4375 A; at 90° the waveform reaches +1.125 A and −2.000 A, so between two equal electrodes it settles at a bias of 0.4375 A. With A = 100 V that is −43.8 V and 43.8 V. Neither waveform has any direct current in it and both have the same power spectrum; what differs is only which way up the tall peak points, and the chamber is identical on both sides.

The voltage across the sheath at the driven electrode and at the grounded one, over one cycle, for the two-frequency waveform at a phase of 0° in a chamber whose two electrodes are the same size. Each sheath collapses to zero once a cycle — the moment it lets electrons through to cancel the ions it has collected — and at other times holds the voltage. The driven sheath peaks at 1.562 A and averages 0.800 A; the grounded one peaks at 1.562 A and averages 0.362 A. With A = 100 V the ions arriving at the driven electrode fall through about 80 V on average and those at the grounded one through 36 V. The two sheaths always add to the applied voltage plus the bias of −0.4375 A.

The mean voltage across each sheath — close to the energy per charge of the ions that cross it without colliding — against the phase between the two harmonics, at A = 100 V in a chamber with equal electrodes. At 0° the driven electrode's ions arrive with about 80 eV and the grounded electrode's with 36 eV, a ratio of 2.21 to one; at 90° the two have swapped, 36 and 80 eV. The applied amplitudes never change along the axis, so the power put into the plasma, and with it the ion flux, stays roughly fixed while the energy each ion carries is moved by a phase. With a single frequency the two are tied together, because the only way to raise the bias is to raise the voltage.

Waveforms built from the first N harmonics of one frequency with amplitudes falling as (N − k + 1)/N, each scaled to the same peak-to-peak voltage, over one cycle, for N = 1, 2, 4, 8. As harmonics are added the waveform narrows into a tall positive spike over a long shallow negative stretch, and the bias between equal electrodes grows as −(N − 1)/(N + 1) of half the peak-to-peak voltage: 0.000 with one frequency, −0.333 with 2 harmonics, −0.500 with 3 harmonics, −0.600 with 4 harmonics, −0.714 with 6 harmonics, −0.778 with 8 harmonics. With eight harmonics the driven sheath's mean voltage is 11.44 times the grounded one's. The asymmetry that the areas of the electrodes used to supply can be written into the waveform, and the more harmonics, the more of it.

physicscheck asserts something about plasma-response 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.