Illustrated Physics
physics, drawn
Concept

Mathieu equation — where it appears

The equation of an oscillator whose stiffness is modulated periodically in time. Its stability chart, a set of tongues in the plane of modulation depth and frequency, says which modulations make an oscillation grow, as in a parametrically driven pendulum, an ion trap or an internal tide.

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

Also named here as parametric resonance — the same set of essays touches all of them, so they are one junction rather than several.

The potential a shaken pivot creates. The effective potential of a pendulum whose pivot is shaken vertically, against the angle from hanging, for shaking rates of 8, 14, 20, 30 times the pendulum's own frequency at an amplitude of 0.12 of its length. The shaking averages to no force at all — it is up as often as down — and yet it adds a term to the potential, because the pendulum's position is correlated with the phase of the shake rather than independent of it. The added term is proportional to sin²θ, so it is largest sideways and zero at both the hanging and the inverted positions, and it turns the maximum at 180° into a minimum once 14× and 20× and 30× the natural frequency is reached. The criterion is (aΩ)² > 2gL: the shake speed must beat the speed a fall through the pendulum's own length would give. Upside down then becomes a stable equilibrium, with a restoring force and a period of its own.

Held up by a force that averages to nothing

Shake a pendulum's pivot up and down fast enough and the pendulum will stand upside down, balanced, and push back if it is nudged. The shaking supplies no average force at all — it is up as often as it is down — and the reason it nevertheless holds is that the pendulum's position and the phase of the shake are not independent.

mechanics · Pendulum The latitude past which the tide cannot shed its energy by halves. Frequency in cycles per day against latitude. The curve is the inertial frequency, 2Ω sin(latitude), below which no internal wave can oscillate; the shaded region above it is where internal waves exist. The horizontal lines are the semidiurnal and diurnal tides and the frequencies half of each, where a parametric instability would put the waves the tide decays into. Each line ends where it meets the curve, which is its critical latitude: M2, semidiurnal at 1.932 per day, 74.5°; M2 ÷ 2 at 0.966 per day, 28.8°; K1, diurnal at 1.003 per day, 30.0°; K1 ÷ 2 at 0.501 per day, 14.5°. Equatorward of 28.8° the semidiurnal tide can feed waves at half its frequency; poleward of it those waves cannot exist and that route is closed. The diurnal tide's subharmonic is confined within 14.5° of the equator, and the diurnal tide itself cannot propagate as a free internal wave poleward of 30°.

The latitude past which a tide cannot split

The ocean's internal tide carries about a terawatt, and somewhere it has to be broken into waves small enough to mix the water. One of the ways it breaks is by pumping waves at half its own frequency, the way a child on a swing pumps at twice the swing's. Those half-frequency waves cannot exist where the planet's rotation forbids oscillations that slow — poleward of 28.8° for the semidiurnal tide — so the route has an edge on the map, fixed by the Moon's period and the Earth's spin.

fluids · Stratification

Named alongside it

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

Parametric resonanceAveragingCoriolis forceCritical latitudeEffective potentialEquilibriumGroup velocityInertial oscillationInternal wavesPendulumSeparation of timescalesStability

All concepts