A sine on its way to a vertical face

At its defaults it draws a sine on its way to a vertical face. One period of a sine, followed by the equation whose only nonlinearity is that the local speed depends on the local height. The profiles are at σ = t/t_b of 0, 0.3, 0.6, 0.9, 0.995, each obtained by solving the implicit relation u = u₀(x − (c₀+βu)t) for u at every point by bisection — and checked against the partial differential equation itself, which it satisfies to 2.6e-7. The crest travels faster than the trough, so the descending front leans forward and the ascending one leans back; the wave stays exactly as tall as it started and exactly as long, and only its shape changes. The steepest gradient grows as one over (1 − σ) — measured here as 9.83 times its initial value at σ = 0.9, against ten — so it is infinite at σ = 1 and the curve has a vertical tangent. The picture cannot be drawn past that point, which is not a failure of the drawing: the solution genuinely becomes three-valued, and what actually happens is a jump whose width is set by the dissipation this equation does not contain.

nonlinear-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.

One period of a sine, followed by the equation whose only nonlinearity is that the local speed depends on the local height. The profiles are at σ = t/t_b of 0, 0.3, 0.6, 0.9, 0.995, each obtained by solving the implicit relation u = u₀(x − (c₀+βu)t) for u at every point by bisection — and checked against the partial differential equation itself, which it satisfies to 2.6e-7. The crest travels faster than the trough, so the descending front leans forward and the ascending one leans back; the wave stays exactly as tall as it started and exactly as long, and only its shape changes. The steepest gradient grows as one over (1 − σ) — measured here as 9.83 times its initial value at σ = 0.9, against ten — so it is infinite at σ = 1 and the curve has a vertical tangent. The picture cannot be drawn past that point, which is not a failure of the drawing: the solution genuinely becomes three-valued, and what actually happens is a jump whose width is set by the dissipation this equation does not contain.

The signal arriving at a fixed distance from a point source that emits a single short pulse, in one, two and three dimensions, each normalised to its own peak. In three dimensions the arriving signal is the emitted pulse, unchanged in shape: it arrives, and then there is silence. In two dimensions the same kick arrives at the same moment and then keeps arriving — a tail falling as one over the time, which is still at a tenth of its peak 12.1 pulse-lengths after the front has passed. In one dimension it never comes back down at all; the medium is left displaced. The wave equation is the same equation in all three, and the source is the same source; what differs is only how many dimensions the disturbance has to spread into. Sharp arrival is the exception rather than the rule — it happens in three dimensions, and in five, and in seven, and in no even number of them — and every argument that treats a wavefront as the whole of the signal is an argument that has quietly used the fact that we live in three.

Why a two-dimensional wave has a tail, drawn as a three-dimensional picture. A point source in a plane — a splash, a ripple, a wave on a membrane — is the same thing as an infinite line of sources in space, all firing at once. The listener stands at one distance from the nearest point of that line and further from every other point on it, so the sound of the nearest source arrives first and the sound of the rest keeps arriving afterwards, for ever. The circles mark which part of the line is being heard at three successive moments: at each time the listener hears the two points at slant distance equal to the elapsed time, and they run off to infinity as the time goes on. Adding up the whole line numerically — every source a sharp three-dimensional arrival, and nothing about two dimensions assumed — reproduces the exact two-dimensional tail to 0.5 per cent at the three times checked. The tail is not a property of the medium and not an approximation. It is the far end of a line of sources, arriving late.

The signal arriving at a fixed distance from a point source that emits a single short pulse, in one, two and three dimensions, each normalised to its own peak. In three dimensions the arriving signal is the emitted pulse, unchanged in shape: it arrives, and then there is silence. In two dimensions the same kick arrives at the same moment and then keeps arriving — a tail falling as one over the time, which is still at a tenth of its peak 17.0 pulse-lengths after the front has passed. In one dimension it never comes back down at all; the medium is left displaced. The wave equation is the same equation in all three, and the source is the same source; what differs is only how many dimensions the disturbance has to spread into. Sharp arrival is the exception rather than the rule — it happens in three dimensions, and in five, and in seven, and in no even number of them — and every argument that treats a wavefront as the whole of the signal is an argument that has quietly used the fact that we live in three.

The two-dimensional signal after the front has passed, on logarithmic axes, at 3 distances. Every curve straightens to a slope of -1.00, -1.00, -1.00, measured over the last part of each — that is a tail falling as one over the time, which is slow. A signal that decays as an exponential has a time after which it is gone; one that decays as a power has only a time after which it is small, and the difference decides whether a second pulse can be distinguished from the first one's remains. Ten times the delay buys a tenth of the amplitude, and no more. The curves are parallel and offset, because the amplitude of the tail at a given multiple of the arrival time falls as one over the distance: further away is not quieter in the tail than at the front, only later. This is why ripples on a pond continue to arrive long after the first ring has passed, and why the vibration of a struck plate outlives the strike by far more than its damping would suggest.

3 short pulses are emitted in quick succession and received 6 pulse-lengths away, in three dimensions and in two, each normalised to its own peak. In three dimensions what arrives is what was sent: the gaps between the pulses come back to 0.000 and 0.000 of the peak, which is silence. In two dimensions each pulse leaves a tail that the next one arrives on top of, so the gaps fill in to 0.26 and 0.41 and the sequence arrives as a single swell with bumps on it. Nothing has been absorbed, nothing has been scattered and there is no echo: the medium is uniform, lossless and unbounded, and the smearing is a property of the number of dimensions alone. It is worth noticing what this means for a two-dimensional world — a membrane, a shallow layer, the surface of a pond. Sound in it could not carry speech, because a message is a sequence of events and each event would be heard on top of the ones before it. The sharpness we take for granted, and the whole idea of an echo as a separate arrival, are consequences of a three that could have been a two.

physicscheck asserts something about nonlinear-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.