The dent that raises the note

A resonator has a set of frequencies, and those frequencies are decided by its shape. Change the shape and they change. The question this rung asks is narrower and more useful: change the shape slightly, at one place, and which way does a given frequency move?

The answer is not “up” and it is not “down”. It is both, and which one it is depends on where the change is made — with a set of places where the change does nothing whatever, and those places are the nodes.

Why a node is free #

The zeros in that figure are the whole rule in its simplest form, and they need no calculation at all.

A node is a place the mode does not move. Put a mass there and it is never accelerated; it takes no part in the motion, contributes nothing to the kinetic energy, and cannot change the rate at which the energy sloshes back and forth. So the frequency is unchanged — not approximately, but exactly, and the figure verifies this against the exact solution rather than against the formula that predicts it.

Between the nodes the mass does move, and it moves in proportion to the mode’s amplitude there. The kinetic energy it adds goes as the square of that amplitude, and the fractional change in frequency turns out to be exactly that square, scaled by the mass ratio: $\delta\omega/\omega = -(m/M)\sin^2(k a)$ for a string of mass $M$ carrying a mass $m$ at position $a$.

There is a reason a first-order answer needs only the unperturbed shape, and it is worth having rather than accepting. A mode’s frequency is the stationary value of a ratio — the potential energy of a trial shape divided by its kinetic energy — over all shapes satisfying the boundary conditions. Stationary means the ratio has zero derivative there, so a trial shape wrong by a small amount $\epsilon$ gives a frequency wrong by order $\epsilon^2$. Perturb the system and the true shape shifts by something of order the perturbation; using the old shape instead therefore costs only the square of it. The frequency is easy to get right precisely because the shape is hard, and that trade is why first-order perturbation theory is useful at all rather than merely available.

This is the same statement as only some notes fit read at one point rather than end to end. There, the whole string’s mass and tension set the whole spectrum; here, a pinch of extra mass at one place is felt in proportion to how much of the mode’s motion is happening there.

The exact problem is solvable, and that matters for what comes later. A point mass on a string gives a transcendental frequency equation — $\sin kL = (m/\rho)\, k \sin ka \, \sin k(L-a)$ — whose roots can be found numerically to any precision. Every curve here that claims to be exact is a set of those roots, and the first-order formula is what they are compared with. A figure that plots an approximation next to itself proves nothing about the approximation.

The other sign #

Mass is not the only thing that can be added at a point. A spring can be too, and it does the opposite.

Put the two curves together and the rule stops being about masses and springs at all. A mode stores kinetic energy where it moves and potential energy where it bends. Anything added to the kinetic side slows it, anything added to the potential side speeds it up, and both are felt in proportion to how much of that kind of energy the mode was already keeping at that place.

That is a statement about energies rather than about materials, which is what lets it move to a different subject without changing. In the mechanical case the two sides are inertia and stiffness. In the electromagnetic case they are the magnetic field and the electric field, and the rule reads identically.

A cavity, and the map on its wall #

A microwave cavity is a box with conducting walls, ringing at a frequency set by its dimensions. Denting a wall inwards changes those dimensions slightly — and the same rule says the answer depends on where.

The electromagnetic version is Slater’s, and it says that a small inward dent shifts the frequency in proportion to the magnetic energy density minus the electric energy density at that point on the wall. The magnetic field is the kinetic side of an electromagnetic oscillation and the electric field the potential side, exactly as in the term that made light, where the two fields take turns generating one another. So a dent raises the frequency where the magnetic field is strong and lowers it where the electric field is.

The contour is the part worth carrying away. It is the locus where the two energy densities are equal, and a tuner placed on it has no authority at all: a screw driven in there changes the frequency by nothing to first order, however far it is driven. Cavity tuners are therefore placed deliberately — near an end wall to raise, near the middle to lower — and a tuner that has drifted a few millimetres onto the contour is a tuner that has stopped working while remaining perfectly intact.

The numbers are small and entirely workable. A superconducting accelerating cavity at 1.3 GHz tunes at roughly three hundred kilohertz per millimetre of length change, and its loaded bandwidth — the width the accelerating field actually has, set by how hard it is coupled to its power source — is a few hundred hertz. One micrometre of length is therefore about one bandwidth. A cavity is a metre-long object being held to a micrometre by a mechanical tuner, against the slow creep of its own stresses and the pressure of the helium bath around it, and the whole business of doing so is a control problem rather than a design one. The map above is what decides where the tuner acts; the fact that a shift can be a hundred bandwidths while being a part in ten thousand of the frequency is what makes both the tuning and the measurement possible.

A whole-cavity squeeze is a sum over this map and says nothing about any point on it. Shrink every dimension by one part in a thousand and every frequency rises by one part in a thousand, because a resonant wavelength is a length. That true statement is often turned into the false general rule that making a cavity smaller raises its pitch. The map is positive over part of its area and negative over the rest; the uniform squeeze happens to sum positive.

Reading the field out of the shifts #

Once the rule is trusted, it can be run backwards. Instead of asking what a known perturbation does to an unknown mode, put a small known object in and measure the shift, and the shift reports the field where the object is.

This is the bead-pull, and it is the standard field measurement on an accelerating cavity: a small bead on a nylon thread, drawn along the axis while a network analyser tracks the resonance, and a field profile that comes out of the arithmetic without any probe inside the beam path. It measures the quantity a designer actually wants, which is the field the particles will see rather than the field a simulation predicts.

Two limitations are structural rather than technical. The shift depends on the square of the field, so the sign of the field is not recoverable — a mode and minus that mode produce identical readings, and the phase has to come from a separate measurement or from knowing which mode is being looked at. And what the bead reports depends on the bead: a metallic sphere has both an electric and a magnetic polarisability, in a ratio fixed by the geometry of a sphere rather than by the cavity, so its shift mixes the two fields. Choosing the bead is choosing the quantity measured, and reporting one as the other is the commonest way the method is misused.

The same rule at a size where it fails #

A first-order formula is a slope. It knows how the answer starts to change and nothing about where it is going, and a resonator makes that unusually easy to see, because the destination can be worked out independently.

The destination is the interesting part, and it is one rung lower than a first guess suggests. Clamping the string at a point raises every frequency, because a clamp is a constraint. Yet loading it with mass lowers them. Both are true, and they are reconciled by a mode leaving through the bottom of the spectrum: the bead bouncing on the string as a mass on a pair of springs, at a frequency that goes to zero as the bead gets heavier. So the third harmonic descends onto the second mode of the clamped pair, the fourth onto the third, and the count works out.

That accounting is a small thing to check and a good habit to have. When a perturbation is taken to a limit where it is no longer a perturbation, the modes have to be tracked and counted rather than followed one at a time, and a spectrum that has gained or lost a member is the usual sign that something has been mislabelled.

Two modes that share a frequency #

Everything so far assumed the mode being perturbed was alone at its frequency. When two are not, the rule changes character entirely.

A square drum has pairs of modes with the same frequency — the shape with two half-waves across and one down, and the shape with one across and two down. Because they share a frequency, every combination of them is also a mode, and the drum has no preference among them. Nothing in the unperturbed problem picks a pair to work with.

A point load has rank one, and that is a strong statement. It means one combination absorbs the entire shift and its orthogonal partner is left exactly where it was — and the partner is the combination with a node at the load, which it must be, since a mode with a node there does not know the mass is present.

The freedom being removed is worth naming, because it is not the freedom to have any shape at all. The pair is two-dimensional: any combination of the two drawn shapes is a mode of the unloaded drum, and there are infinitely many of them, all with the same frequency. What the load does is pick out the one orthogonal direction in that two-dimensional space which has a node at the load, and hand the entire shift to everything else. The drum after loading has one frequency slightly lowered and one exactly unchanged, and no combination of the two is a mode any more — the freedom has gone, and it has gone at first order in a load that changed the frequencies by a fraction of a per cent.

A degeneracy is fragile in a way a single frequency is not. A perturbation far too small to matter to any frequency still decides, permanently, which two shapes the drum will ring in, because it removes the freedom to choose. This is why a real bell or a real drum has two closely spaced frequencies where the ideal one has a doubled frequency, and why the split is audible as a beat: no physical object is symmetric enough to keep a degeneracy. The related quantum statement — that a perturbation coupling two states pushes their energies apart rather than letting them cross — is the crossing that never happens, and it is the same first-order matrix, diagonalised.

The same arithmetic in another subject #

The formula for the shift in a mechanical mode is the mode shape squared, integrated against the perturbation. The formula for the first-order shift in a quantum energy level is the wavefunction squared, integrated against the perturbing potential. They are not analogous; they are the same calculation, because both problems are eigenvalue problems for a self-adjoint operator, and first-order perturbation theory is a statement about eigenvalues rather than about physics.

That is why the results transfer without translation. A quantum level is unshifted by a potential placed at a node of its wavefunction. A degenerate pair of levels is split by a perturbation into one combination that feels it and one that does not. A perturbation series stops being useful when the perturbation stops being small compared with the spacing to the next eigenvalue — which, in the string, is the spacing that closes as the bead approaches the mass of the string itself.

A wine glass is the everyday demonstration. Its rim rings in a pair of modes with four nodes each, identical but for a rotation of forty-five degrees, and a perfectly circular glass would have them at exactly the same frequency. No glass is perfectly circular, so the pair is split by a few tenths of a hertz to a few hertz, and a struck glass excites both. What is heard is not two notes but one note whose loudness rises and falls at the difference — a wobble of a second or two, audible on any glass worth striking, and a direct reading of how far from circular that particular glass is. The same measurement on a bell is how a founder decides where to grind metal away, and the map that says where grinding will help is the one drawn above with the drum’s mode shape in place of the cavity’s field.

The one place the correspondence needs care is the meaning of “small”. In the quantum problem the comparison is with a level spacing. In the resonator it is with the same thing under a different name: how far the perturbed frequency has moved relative to the distance to its neighbours. A dense spectrum, like a large room’s acoustics or how many ways there are to vibrate at high frequency, has no room for a perturbation to be small in that sense, and first-order results there are worth very little.

Where the model stops #

The perturbation is assumed not to move the mode shape. First order changes the frequency using the unperturbed shape, and the shape’s own change contributes only at second order. That is why the mass at a node gives exactly zero — the exact statement — while the shift elsewhere is exact only to first order.

The cavity map treats the wall as a boundary and the dent as small compared with a wavelength. A dimple deep enough to reach where the fields are structured is not a boundary perturbation any more, and a tuner driven far into a cavity eventually becomes a new geometry with its own modes.

Loss is ignored throughout. A real cavity has a resonance of finite width, and a shift smaller than that width is not measurable however exactly it is computed. The whole bead-pull technique depends on the shift being resolvable against the linewidth, which is why it is done on high-quality superconducting cavities and is hopeless on lossy ones. The width that is a lifetime is where that limit comes from.

And the degenerate case assumes exactly two. Three or more modes at one frequency give a larger matrix whose rank is still one for a point load, so one combination moves and the rest do not — but which of the survivors the physical system settles into is then decided by whatever comes next, and the ordering of small effects starts to matter.

What the pictures cannot show #

The cavity map is drawn as a flat rectangle and a cavity wall is a surface of a box. What the map leaves out is the other five walls, each with its own map, and the fact that a real tuner enters through a hole whose own effect on the mode is not small. The map is the right guide to where a tuner should go and not a prediction of what a particular tuner will do.

The degenerate panels show two mode shapes and cannot show what actually happens to a struck drum, which is that both are excited and the pair beats against itself at the difference frequency. The splitting drawn as two static pictures is heard as a wobble.

Where the ladder goes next #

The standing-wave ladder began with only some notes fit, where a boundary picks a discrete set out of a continuum. It went on to counting those modes in how many ways there are to vibrate and to the shapes a boundary chooses in the drum that has no harmonics, and to the node that is not standing still, where a node turns out to be an interference rather than a place. This rung asks what a mode does when its boundary is changed slightly and finds a rule with two signs and a set of places where nothing happens at all.

The rung after it is what happens when the change is not slight and not slow: a boundary moved rapidly compared with the period, which does not shift a mode but couples modes to one another and can pump energy into them from nothing. The habit that carries forward is the one this rung is built on — before asking how much a frequency moves, ask where the mode was keeping its energy.