The count that cannot be cheated

A uniform string is a fortunate special case, and almost everything memorable about standing waves is learned from it.

Its frequencies are in the ratios $1:2:3:4$, its modes are sines, its nodes are evenly spaced, and the whole spectrum is one number multiplied by the integers. None of that survives making the string heavier in one place than another, and a string with a lump in it is a much better representative of the systems physics actually contains.

What does survive is a count. The fourth mode of the lumpy string has three interior zeros, exactly as the fourth mode of the uniform one does, and the theorem behind that is stronger and stranger than the uniform case suggests.

What the lump destroys #

The essential change is that the wave speed is no longer the same everywhere. On a string of tension $T$ and mass per unit length $\rho(x)$ the local speed is $\sqrt{T/\rho}$, so the wave travels slowly through the heavy region and quickly through the light one.

Three consequences follow immediately from the figure. The shapes are asymmetric, because there is nothing left to be symmetric about. The oscillations are compressed over the heavy region, because a slower speed at the same frequency means a shorter local wavelength. And the frequencies come out at $1.70$, $4.01$, $6.10$ and $8.12$ in the units of the calculation, which are in the ratios $1$, $2.36$, $3.59$, $4.78$ — not integers, not even close, and not any obvious sequence.

A string like this would not sound like a violin. Whole-number ratios are what make a note a note, and their absence is why a bell, whose thickness varies over its surface, has a clangorous quality that no amount of tuning removes.

The local wavelength is worth dwelling on, because it is the one part of the uniform picture that generalises honestly. Where the medium is slow the wave fits more oscillations into a given length, so the crossings crowd together; where it is fast they spread out. That is the short-wavelength approximation, and it says that the number of zeros in a stretch is the integral of the local wavenumber over that stretch, divided by π. Applied to the whole string it predicts the total count, and applied to a piece of it predicts where the crossings sit — which is how the medium’s own speed shows up in a shape rather than in a number.

The approximation gets the count right for high modes and can be off by one for the lowest, which is exactly where Sturm’s theorem is exact and the approximation is not. The two are complementary: the theorem knows the count and nothing about the positions, the approximation knows the positions and can miscount.

So the lump has taken away the ratios, the symmetry and the spacing. It is reasonable to expect the ordering of the modes to be a matter of computing them and sorting, with no structure in it.

What survives, and it is exact #

The theorem is Sturm’s, from 1836, and in one dimension it is an equality rather than a bound. For a system of the form

$$\frac{\mathrm{d}}{\mathrm{d}x}\left(p(x)\frac{\mathrm{d}y}{\mathrm{d}x}\right) + \lambda \rho(x) y = 0$$

with $y$ vanishing at both ends and $p$ and $\rho$ positive, the eigenvalues are simple and increasing, and the eigenfunction belonging to the nth of them has exactly $n-1$ zeros strictly inside the interval.

Nothing in that statement mentions what $p$ and $\rho$ are. They can vary by any amount, in any manner, smoothly or otherwise, and the count is unaffected. A string that is a hundred times heavier in the middle than at the ends has a fourth mode with three interior zeros; so does a string whose density is drawn at random.

The modes here are not asserted to have those counts. They are found by shooting — integrate from one end with unit slope, and search for the frequencies at which the far end comes back to zero — and the zeros are then counted on the computed shape. That is what makes the theorem a measurement in this essay rather than a quotation.

How the modes were found, which matters here #

A spectrum computed by a method that assumes the answer would prove nothing, so it is worth saying what was done.

Each mode here comes from shooting. Pick a trial frequency, start at the left-hand end with zero displacement and unit slope, and integrate the equation across the string with a fourth-order Runge–Kutta step. The far end will generally not come back to zero; the value it does reach is a smooth function of the trial frequency, and the eigenvalues are its zeros, found by scanning for a sign change and then bisecting.

Nothing in that procedure knows about node counts, orthogonality or ordering. It knows the density profile and the boundary conditions. The zeros are then counted afterwards by walking along the computed shape and looking for sign changes, and the theorem is what the two independent operations are found to agree about.

The same code run with a constant density returns $\pi, 2\pi, 3\pi, 4\pi$ to six figures, which is the check that the shooting is right rather than merely self-consistent.

Why the count cannot slip #

The mechanism is a comparison, and it is worth having because it explains why no medium can defeat it.

Take two solutions belonging to different eigenvalues, $\lambda_1 < \lambda_2$. Between any two consecutive zeros of the lower one there must be at least one zero of the higher. The argument is a page of calculus in full and one sentence in outline: form the Wronskian of the two solutions, differentiate it, and find that its derivative has a definite sign wherever both solutions do — so the Wronskian cannot return to zero at the second zero of the lower solution unless the higher solution has crossed the axis in between.

Run that up the ladder and the count is forced. The first mode has no interior zero, because a solution starting and ending at zero with none in between is the only thing a positive $\lambda$ smaller than every other can produce. Each subsequent mode must interleave its zeros with the previous one’s, which adds exactly one, and there is no room for two because the eigenvalues are simple.

The interleaving is the part that makes the result robust. It is a statement about pairs of solutions and it never refers to the medium, so a lumpy string, a tapered string, a string in a gravitational field and a quantum particle in an arbitrary potential all obey it identically.

Orthogonality, and the weight nobody mentions #

The modes of a uniform string are orthogonal in the sense everyone remembers: multiply two of them together, integrate along the string, and the answer is zero. The lumpy string’s are not.

What is true is that they are orthogonal against the density as a weight: $\int \rho(x)\, y_m(x)\, y_n(x)\,\mathrm{d}x = 0$ for $m \ne n$. Physically the weight is counting each element of the string in proportion to the mass it carries, which is what the kinetic energy does, and the reason the relation holds is that the operator is self-adjoint with respect to that inner product and not with respect to the plain one.

Everything a decomposition into modes is used for rests on this. A plucked string’s shape is expanded in modes; its energy is a sum over modes with no cross terms; the amplitude of each is extracted by projecting with the weight. Get the weight wrong and the cross terms do not vanish, and an energy computed mode by mode does not add up to the energy that is there.

The uniform case hides all of it, because a constant weight divides out and the two inner products differ by a factor nobody notices.

The same theorem, wearing quantum clothes #

The time-independent Schrödinger equation in one dimension is a Sturm–Liouville problem: $-\psi'' + V(x)\psi = E\psi$, with $p = 1$ and $\rho = 1$ and the potential absorbed into the eigenvalue. Every statement above therefore applies to it unchanged.

The same theorem wears quantum clothes without changing. A particle in a box has a ground state with no node, a first excited state with one, and a sequence that continues — which on a string is an observation about pictures and here is a statement about energies. The two are the same theorem because the two equations are the same equation, and nothing about quantum mechanics was needed to prove it.

The consequence physicists use daily is that the ground state of a one-dimensional system has no node, ever, for any potential. That is not a fact about square wells; it is Sturm’s theorem with $n = 1$.

It also explains why, in a way the picture alone does not. A node is a place where the wavefunction changes sign, and changing sign requires curvature, and curvature is kinetic energy. A trial state with a node therefore pays an energy penalty that a nodeless state does not, so the lowest state cannot afford one. That argument is the variational principle and the oscillation theorem meeting from opposite directions, and they agree.

It is also how a numerical solver knows which level it has found. Counting nodes in a computed wavefunction identifies the state, and the energy does not — because the solver does not know the energy in advance and the node count is available by inspection. That is a practical use of a theorem that looks like a curiosity, and it is why the count is worth having as a theorem rather than as an observation.

That last point is practical rather than decorative. Solving for the tenth eigenvalue of an awkward potential is a search, and a search needs to know when it has landed on the right thing. Counting nodes answers it: the state with nine interior zeros is the tenth, whatever its energy turns out to be. Every shooting method and every Sturm-sequence bisection uses this, which means the theorem is load-bearing in software rather than only in textbooks. It is also how a chemist reads a computed molecular orbital: the number of times it changes sign along a bond says whether it is bonding or antibonding, and that ordering is the same counting argument applied to a chain of wells.

Hydrogen’s radial equation is a Sturm–Liouville problem too, so at fixed angular momentum the states are labelled by their radial node count. The familiar quantum numbers are, in that sense, node counts wearing different names — and the ordering of the levels follows from the counting rather than from anything about the Coulomb potential in particular.

The hydrogen atom’s quantum numbers carry the count openly. At fixed angular momentum the radial function belonging to principal quantum number $n$ has $n - \ell - 1$ radial nodes, and that formula is not a piece of spectroscopy but Sturm’s theorem applied to the radial equation.

Where the count is used on things much larger than strings #

The Earth rings after a large earthquake, and it rings in normal modes. The relevant equation is the elastic one in a sphere whose density and stiffness vary enormously with depth — a lumpy string in the most literal sense, with a liquid core in the middle of it — and the modes are labelled by their angular pattern and by a radial overtone number that is a node count.

Seismologists measure frequencies and want the profile, which is the inverse problem, and the node count is what makes the bookkeeping possible: a measured line at a certain frequency has to be assigned to a mode before it can constrain anything, and the assignment is made by matching computed node counts to the depth sensitivity the data show. The same applies to the Sun, where helioseismology reads a spectrum of millions of modes and each is labelled by exactly this pair of integers.

The physics in those problems is nothing like a string. The theorem is the same theorem, because the radial equation is of the same form, and that is the sense in which a nineteenth-century result about ordinary differential equations is doing work in a modern observatory.

Two dimensions, where the equality fails #

A drumhead is not a string, and the first thing that goes is the notion of a node as a point.

In two dimensions the equality fails, and it fails because the zero set changes character. A membrane’s nodes are curves rather than points, and curves divide the disc into regions — so the object to count is no longer a number of crossings but a number of domains. That is a different count, and there is no reason for it to keep the exact equality the one-dimensional theorem enjoys.

The zero set of a two-dimensional mode is a collection of curves, and what those curves do is cut the shape into regions. Counting regions is the natural generalisation of counting points, and the natural guess is that the nth mode makes $n$ of them.

The guess is wrong in a specific and interesting way. Courant proved in 1923 that the count is at most $n$, and the inequality is usually strict: the counts here run $1, 2, 4, 2, 6, 4, 8$ against ranks of $1, 2, 4, 6, 7, 9, 11$. The second radial mode, two concentric rings, sits at rank six and has two domains where it is allowed six.

The counts are found by flood-filling the sign of the computed mode on a grid, which knows nothing about where the nodal lines ought to be, and the degeneracy is counted properly — every mode with angular structure is two modes at one frequency, so the rank advances by two.

Pleijel showed in 1956 that the gap is not an accident of the disc: for any shape, all but finitely many modes fall strictly below the bound, and the ratio of domains to mode number tends to a constant less than $0.7$.

The constant is worth a sentence because of where it comes from. Pleijel’s argument bounds the area a nodal domain can have from below, using the fact that the lowest mode of any region of a given area cannot be arbitrarily small — an isoperimetric statement, Faber and Krahn’s — and then divides the shape’s area by that minimum. Counting modes below a frequency by Weyl’s law supplies the other half, and the ratio of the two is a number involving the first zero of a Bessel function. So the limit on how finely a mode may cut a shape is set by a fact about the lowest mode of a small region, which is an unexpectedly circular-sounding route to a very concrete bound.

What has failed is the interleaving argument. In one dimension the zeros of consecutive eigenfunctions must alternate, because there is nowhere else for them to go on a line. On a surface a nodal curve can close on itself, meet another curve, or wander, and there is no ordering along which anything has to alternate. The theorem loses its equality precisely where the geometry stops being one-dimensional, and that is the whole of the difference.

Where it stops #

Simplicity of the spectrum is doing more work than it looks. In one dimension no two modes share a frequency, which is what makes “the nth mode” well defined. In two dimensions modes are routinely degenerate — the disc’s angular modes come in pairs — and a degenerate pair can be recombined into any linear combination, each with a different nodal pattern. Asking how many domains “the sixth mode” has is then a question about which combination was chosen.

The staircase makes the difference plain. In one dimension the number of modes below a given frequency rises by one at each step, so the step number is the node count and the two quantities are the same list. In two dimensions the staircase rises irregularly, degeneracies arrive together, and the correspondence between position in the list and number of nodal domains breaks — which is where an exact theorem becomes an inequality.

The boundary condition has to be the same at both ends. Sturm’s theorem is stated for a fixed–fixed string, and a free end changes the count by half a wavelength — the mode shapes acquire an antinode rather than a node there, and the sequence of interior zeros shifts. The theorem still holds in the appropriate form; the naive count does not.

A nodal line is not a stationary line either. On a Chladni plate the sand collects along the nodal curves, which makes them visible and makes them look like the important part of the pattern; what is actually happening is that the sand is thrown off everywhere else and lands where the motion is least. The curves are where nothing moves and the energy is everywhere else, and a mode that lives in a defect is the case where that intuition fails hardest, since the pattern collapses onto a region rather than spreading over one.

And a node is not a place where nothing happens. The string is stationary at a node only in the sense that its displacement is always zero there; the tension is not zero, the slope is not zero, and the energy flux through the node is not zero — a standing wave carries no net power and its two travelling halves each carry a great deal. A node in a quantum wavefunction is likewise not a barrier: a particle in the second state of a box is found on both sides of the node with equal probability and never at it, which is a fact about a probability density rather than about a journey.

The ladder from here #

Later rungs on this anchor: Sturm–Liouville theory in full, with the completeness of the eigenfunctions and what it takes to prove that an arbitrary shape really can be expanded; the inverse problem, which asks what a spectrum determines about the medium and whose answer is famously “not the shape”; nodal domain counts for quantum billiards, where the statistics of the count distinguish integrable systems from chaotic ones; and the discrete analogue, where a matrix’s eigenvector sign changes obey a version of the same theorem.

The neighbouring ladders are only some notes fit, which is the uniform case; the drum that has no harmonics, where the two-dimensional spectrum first stops being a ladder; and how many ways there are to vibrate, which counts modes below a frequency rather than nodes within one.