Illustrated Physics
physics, drawn
Concept

Sturm liouville — where it appears

A class of second-order equations, (p y′)′ + q y = −λ ρ y on an interval, with conditions at its ends. It covers strings, membranes, heat flow and the Schrödinger equation; its eigenvalues form an increasing list and its nth eigenfunction crosses zero exactly n − 1 times.

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

Four modes of a string that is heavier in one place. The first four modes of a string whose mass per unit length rises to 5 times the light end's over a smooth bump centred 62 per cent of the way along, drawn beneath them. Nothing is symmetric any more: the shapes bunch up over the heavy region, where the local wavelength is shorter, and the amplitudes there are smaller. The frequencies are 1.70, 4.01, 6.10, 8.12, which are in the ratios 1.000, 2.361, 3.590, 4.779 rather than 1, 2, 3, 4 — this string has no harmonics and would sound like a bell rather than a violin. What has not changed is the one thing an ordering needs: the interior zeros, marked, run 0, 1, 2, 3 exactly as they do on a uniform string. That is Sturm's theorem, and the nodes here are counted on the computed shapes rather than assumed.

The count that cannot be cheated

A string with a lump in it has no harmonics, no symmetry and no obvious order to its modes. It has one thing left: the nth mode crosses the axis exactly n−1 times, whatever the string is made of. Ordering by frequency and ordering by node count turn out to be the same operation, and in two dimensions the equality quietly becomes an inequality.

waves · Standing waves How a hanging chain can swing. The shapes of the first four ways a uniform hanging chain can swing, side by side, against height from its free end: J₀(j₀,ₙ √(y/L)), fixed at the top and largest at the free bottom end. For a 2-metre chain they sound at 0.424, 0.973, 1.525, 2.078 hertz. The first swings as a whole, like a pendulum, with its bottom moving most. The higher modes have nodes, and the nodes are not evenly spaced: they crowd towards the bottom, where the tension is least and waves travel slowest, so a stretch near the free end holds more of each wave than a stretch of the same length near the top.

The chain that swings in Bessel's notes

Hang a chain from a hook and set it swinging. It can swing as a whole, like a pendulum, or in shapes with still points along it, as a guitar string does — but its higher notes are not two, three and four times its lowest. They are 2.295, 3.598 and 4.903 times it, the zeros of a function that Daniel Bernoulli met for the first time in exactly this problem in 1732, a century before Bessel's name was attached to it. The reason is that a hanging chain is a string whose tension grows from nothing at its free end to its whole weight at the top, and a wave crossing it changes speed all the way.

waves · Standing waves

Named alongside it

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

EigenvalueNormal modesBessel functionBoundary conditionDegeneracyGround stateInhomogeneous mediumNodal domainNodeOrthogonalityOscillation theoremPendulum

All concepts