Illustrated Physics
physics, drawn
Concept

Rayleigh quotient — where it appears

An estimate of a vibrating system's frequency from an assumed shape of its motion, as the ratio of the stored elastic energy to the kinetic energy for that shape. Errors in the shape give much smaller errors in the frequency, and the true mode gives the lowest value.

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

The frequency of a mass on a spring that weighs something. The angular frequency of a mass M on a spring of stiffness k and mass mₛ, in units of √(k/M), against the ratio mₛ/M on a logarithmic axis: exact, from β tan β = mₛ/M (solid); Rayleigh's estimate √(k/(M + mₛ/3)) (dashed); and the textbook answer that ignores the spring's mass, 1 (dotted). For a spring a tenth of the mass the exact value is 0.9836, Rayleigh's 0.9837 and the textbook's 1 — 1.7 per cent too fast. With spring and mass equal the exact frequency is 0.8603, Rayleigh's 0.8660: the third is good to 0.66 per cent where the textbook answer is 16 per cent out.

The third of itself a spring carries

Every textbook gives the frequency of a mass on a spring as the square root of stiffness over mass, and every real spring has mass of its own, which the formula leaves out. The repair is famous and oddly specific: add a third of the spring's mass to the load. The third is not a fudge. It is what the spring's kinetic energy comes to if each coil moves in proportion to its distance from the fixed end, and it is good to two-thirds of a per cent even when the spring weighs as much as the mass. Where it fails, it fails because the spring has stopped being a spring and become a column of waves, with notes of its own above the bounce — the notes that make the valve springs of an engine lose control.

mechanics · Harmonic approximation A glass hardly changes its note until it is half full. The frequency at which a wine glass rings when struck, relative to its empty frequency, against how high the water stands, for a glass 7 cm across with a 2 mm wall, by Rayleigh's method for three shapes of the wall's motion. The water adds mass only where the wall moves, and the wall moves mostly near the rim, so the first half of the glass lowers the note by 1.40 semitones and filling it lowers it to 0.51 of the empty frequency. The power of the fill law is set by how the wall bends: French measured close to a fourth power, which is a wall moving as the height to the power 1.5.

The glass that sings lower as it fills

Tap a wine glass and it rings; pour in water and the note falls. It falls hardly at all while the glass fills to halfway, and then it drops fast, so that the last quarter of the water does most of the tuning. The reason is where the water sits relative to where the glass moves: the wall swings most at the rim and hardly at all near the stem, and water only lowers the note where it has to move with the wall. The shape of the fill curve is a measurement of how the wall bends.

waves · Resonance

Named alongside it

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

Normal modesAdded massEffective massGlass harpKinetic energyResonanceSimple harmonic motionStanding waveStiffnessVariational principle

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