Reflection against the thickness of the boundary

At its defaults it draws reflection against the thickness of the boundary. How much of the wave comes back, against the width of the transition between media of wavenumber 1 and 2.2, on a logarithmic scale. At zero width the curve reaches 1.406e-1, which is the abrupt-interface value computed from the two end values and nothing else — so the smooth calculation contains the sharp one as a limit rather than contradicting it. Past a width of about a wavelength the fall is a straight line on this scale, of measured slope -12.551 per unit width against the -12.566 the asymptotic form gives, which is 4π times the smaller of the two wavenumbers. Exponential, not merely small: three wavelengths of transition costs eight decades of reflection. The circles are the same quantity computed by slicing the profile into uniform layers, agreeing with the curve to a factor of 1.0115.

gradient-reflection 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.

How much of the wave comes back, against the width of the transition between media of wavenumber 1 and 2.2, on a logarithmic scale. At zero width the curve reaches 1.406e-1, which is the abrupt-interface value computed from the two end values and nothing else — so the smooth calculation contains the sharp one as a limit rather than contradicting it. Past a width of about a wavelength the fall is a straight line on this scale, of measured slope -12.551 per unit width against the -12.566 the asymptotic form gives, which is 4π times the smaller of the two wavenumbers. Exponential, not merely small: three wavelengths of transition costs eight decades of reflection. The circles are the same quantity computed by slicing the profile into uniform layers, agreeing with the curve to a factor of 1.0115.

A stack of 9 slabs of random wavenumber and random thickness, with no symmetry anywhere in it. Send a wave in from the left and the amplitude that emerges on the right is 0.6694240937; turn the stack round and send it in from the right and the amplitude is 0.6694240937. The two agree to every digit the arithmetic has. This is not a property of the stack — it survives any arrangement, any number of layers, any amount of internal reflection — but of the equation, which is unchanged when the sign of time is reversed, and the phase is protected as well as the modulus, which is the stronger statement and the one an interferometer would notice. The *reflections* are a different matter. Their moduli are equal too, at 0.742880, but only because nothing here absorbs; their phases differ by 1.621 radians, because the wave meets a different first surface from each side. What the theorem protects is the pair of ends, and not what the wave does on its way between them.

The wavenumber against position for a medium changing from 1 to 2.2, over transition widths of 0.02, 0.2, 0.6 in the same units — that is, 0.003, 0.032, 0.095 wavelengths of the longer wave. The three profiles have identical ends and differ only in how quickly they get from one to the other. The reflectances are 1.39e-1, 5.09e-2, 5.20e-4, a range of 2.7e+2 to one, computed by slicing each profile into thin uniform layers and multiplying their transfer matrices. Nothing about the two media has changed between the curves. The only thing that has changed is a length, and the length is not in the Fresnel expressions at all.

Transit-time difference between an upstream and a downstream pulse, against flow speed, for a 10 cm path in water. Reciprocity is a consequence of the wave equation being unchanged when the sign of time is reversed, so no arrangement of materials can break it — and a medium that is itself moving does break it, because reversing time reverses the flow as well. The downstream pulse arrives sooner and the upstream one later, by 2vL/c² to a part in 1e+5 across this range: 18 ns at 0.2 m/s, 46 ns at 0.5 m/s, 91 ns at 1 m/s, 183 ns at 2 m/s, 365 ns at 4 m/s. That difference is the whole of a transit-time flowmeter, and it is why the instrument needs a clock good to a nanosecond rather than a sensor in the pipe. The other thing that breaks reciprocity is a magnetic field, through the same loss of time-reversal symmetry — and neither breaks it by attenuating anything, which is why an optical isolator is a rotator *and* a polariser: the magnet supplies the non-reciprocity and the polariser turns it into a one-way door.

The amplitude of the wave through the transition, for the sharpest and the smoothest of the profiles. On the incoming side each curve is the incident and the reflected wave together, so it ripples, and the depth of the ripple is a direct measurement of how much came back: a visibility of 0.3559 for L = 0.02, against a reflection amplitude of 0.3728; and a visibility of 0.0228 for L = 0.6, against a reflection amplitude of 0.0228. The smooth transition's incoming side is almost flat — the wave passes as though nothing were there — while the sharp one's ripples visibly. Both fields are integrated from the far side, where the solution is a pure outgoing wave by construction, so the reflection is an output of the integration rather than an input to it.

How much of the wave comes back, against the width of the transition between media of wavenumber 1 and 2.2, on a logarithmic scale. At zero width the curve reaches 1.406e-1, which is the abrupt-interface value computed from the two end values and nothing else — so the smooth calculation contains the sharp one as a limit rather than contradicting it. Past a width of about a wavelength the fall is a straight line on this scale, of measured slope -12.551 per unit width against the -12.566 the asymptotic form gives, which is 4π times the smaller of the two wavenumbers. Exponential, not merely small: three wavelengths of transition costs eight decades of reflection. The circles are the same quantity computed by slicing the profile into uniform layers, agreeing with the curve to a factor of 1.0115.

physicscheck asserts something about gradient-reflection 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.