Illustrated Physics
physics, drawn
Concept

Complex wavenumber — where it appears

A wavevector with an imaginary part, describing a wave whose amplitude changes as it propagates. Its real part says where the phase goes and its imaginary part where the amplitude does, and in an absorbing medium those are different directions — so there is no single refracted angle to quote.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

The year, arriving underground at four different times. Temperature against depth in soil of diffusivity 0.5 mm²/s, at four times of year, measured from the annual mean. The surface swings by ±12 K; the swing underground is smaller and later, and both are governed by one length, δ = √(2D/ω) = 2.24 m. The amplitude falls as e^(−z/δ) — the dashed envelope — and the phase lags by z/δ radians, so the curves lean over as they go down and cross the axis at different depths. At 7.0 m the lag is half a year: the ground there is at its coldest in August and its warmest in February, in antiphase with the sky, with a swing of 0.52 K left. That is a cellar, and it is also why a water main below about a metre and a half does not know that it froze last week.

The summer that reaches the cellar in December

Drive the diffusion equation at its boundary instead of releasing something into it and the solution is a decaying, lagging oscillation with a single length in it. That length governs both the shrinking and the delay, which is why the depth at which the ground is coldest in August is fixed by the same number as the depth at which the seasons stop being felt at all.

thermodynamics · Diffusion A wave with two directions in it. Light at 60° entering silver, 550 nm, whose index is 0.055 + 3.32i. Phase matching along the boundary fixes the transmitted wave's tangential wavenumber and leaves the normal one to the medium, which supplies a complex answer. The real part decides where the phase goes and the imaginary part where the amplitude does, and they are not the same direction: the surfaces of constant amplitude are parallel to the interface, because the decay is entirely into the metal, while the surfaces of constant phase are tilted by 86.5° from the normal. There is therefore no single refracted angle to quote. The familiar picture, in which one set of parallel planes carries both, requires the absorption to be exactly zero.

The angle that is two angles

Snell's law survives a complex index by giving a complex answer, and a complex angle is not an angle. What the phase-matching argument actually fixes is the tangential wavenumber, and when the medium absorbs, the surfaces of constant phase and the surfaces of constant amplitude stop being parallel. In silver at 550 nanometres the phase fronts run within four degrees of the surface while the amplitude decays straight into it.

optics · Refraction

Named alongside it

The objects these essays reach for when they reach for this one.

Boundary conditionSkin depthAbsorptionAttenuationDiffusion equationDispersionEvanescent waveHeat conductionInhomogeneous waveMetalsPenetration depthPeriodic forcing

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