Can one hear the gauge of a string?

In 1966, Mark Kac published a paper in the American Mathematical Monthly called “Can One Hear the Shape of a Drum?”. A drum head has a set of resonant frequencies, which are the eigenvalues of the Laplacian on whatever shape the drum happens to be (just trust me). If I hand you the frequencies, can you guess the shape? Kac credited the question to Bochner and the phrasing to Lipman Bers, so I have to credit Bers as much as Kac for this blog post, since the title is what sent me down this path.

You can actually hear the shape. The asymptotics of the eigenvalues work out to the area (I learned that this is Weyl’s law from 1911). The next term then gives us the perimeter, and the term after that the number of holes. So you can actually “hear” how big the drum is as well as its edge and its holes!

The rest took until 1992, when Gordon, Webb, and Wolpert exhibited two differently shaped drums with identical spectra.

Anyway, the answer is no and I don’t own a drum.

I do play the guitar, though, badly, and I realized that, while strings are simpler, they also have a property we might reconstruct: gauges. My strings go anywhere from .009 to .052 (yes, I play a few very different styles), and maybe switching gauges changes the feel more than the sound, but there’s also something different about an E4 on my high E string and an E4 on B string, even if I wind the B string such that they’re tuned the same.

So, naturally, a string also has a spectrum. Which begs the question: can one hear the gauge of a string?

A disclaimer before we start: none of the physics here is new, I just couldn’t find anyone who actually answered my question the way I posed it. I just stick formulas in other formulas until they give me an answer.

The trivial version

Suppose we have an ideal string. It’s perfectly flexible, uniform along its length, and fixed at both ends. Then its frequencies are

f_n = (n / 2L) · √(T/μ)

Fig. 1: The frequencies of an ideal string of length L, tension T, and linear density μ.

That’s it. The fundamental and integer multiples of it.

The spectrum is an infinite list of numbers and it contains exactly one number here. Every overtone is determined by the first one. The harmonic series, the most satisfying property about vibrating strings, is trivial here and probably sounds extremely boring, just a repetition ad infinitum.

Let’s suppose we know the tension, the length, and the material. For a perfectly round wire of density ρ and diameter d we have μ = ρπd²/4, so

d = (1 / L·f₁) · √(T / πρ)

Fig. 2: Gauge from a single note, given tension, length, and material.

Hearing the spectrum is optional. You just need to hear one note and measure three things.

But this is annoying, and it also doesn’t model the real world: A .010 at a given tension and a .020 at four times that tension work out to the same fundamental and therefore are indistinguishable in this setup. An ideal string doesn’t let you hear its size. The answer is, therefore, “no”.

We can make this more interesting.

Enter Kac

The analogue of Kac’s question isn’t about a uniform string anyway. His drum could be any shape. So let the gauge vary along the length, d(x), and ask whether the spectrum determines the profile.

Before we despair at the math, the answer has already been worked out by people much more persistent than me. It turns out spectrum doesn’t determine the profile.

A single spectrum doesn’t pin down the mass density of an inhomogeneous string (yeah, try parsing that one). Bottom line: there are different strings that sound identical, and you need richer data than one list of eigenvalues to recover the density¹.

This is a one-dimensional case, Kac works in two dimensions, and even at his time the one-dimensional case had been solved already.

Enter stiffness

Ideal strings aren’t real, real strings aren’t ideal. A steel or nylon wire resists being bent (I suppose gut does as well, but luckily I don’t play baroque music). That stiffness is going to be bolted onto the wave equation as a fourth derivative, I am told:

μ ∂²y/∂t² = T ∂²y/∂x² − EI ∂⁴y/∂x⁴

Fig. 3: The stiff string, with Young’s modulus E and second moment of area I.

For pinned ends² this gives

ω_n² = (T/μ)·(nπ/L)² + (EI/μ)·(nπ/L)⁴

Fig. 4: The spectrum of a stiff string.

and now we have a second number, finally. The n² term is our friend T/μ from earlier. n⁴ term is a new guy. It grows faster and carries EI/μ, which means the overtones are no longer redundant. We’re getting somewhere!

The overtones now drift progressively sharp of the harmonic series. We get more distinguishing information about the properties of the string!

Musicians and piano technicians call this inharmonicity, and it’s usually a nuisance, unless you’re a 20th century New Music weirdo. Harvey Fletcher worked out the standard form for piano strings in 1964 with a single coefficient B:

f_n ≈ n·f₀·√(1 + B·n²),   with   B = π³Ed⁴ / (64·T·L²)

Fig. 5: Fletcher’s inharmonicity coefficient.

I’m just going to take this at face value and move on. Thanks, Harvey.

There’s a d⁴ in there, which is promising. It turns out that it does work: for a solid round wire, I = πd⁴/64 and μ = ρπd²/4. This means that the stiffness term reduces to

EI/μ = (E/ρ)·(d²/16)

Fig. 6: The stiffness term for a solid round wire.

Tension disappears (not in me, though, I’m still very tense at this point)!

Now fit the two coefficients to a recorded spectrum and know what the string is made of. The gauge is determined by how badly the overtones misbehave (or how much they rock, depending on what kind of musician you are). Estimating B from recordings is also, I am told, a solved problem in music information retrieval, and the dependence on diameter has been measured on actual plucked guitar strings. We don’t have to speculate.

Except the fit gives us A and C in ω_n² = A·n² + C·n⁴, and A still has the tension in it. So everything hangs on C, which for our round wire is

C = (E/16ρ) · (π⁴/L⁴) · d²

Fig. 7: The quartic coefficient, spelled out.

That’s d/L², not d. We get the gauge relative to the length, so someone has to supply a length. Practically speaking, we could say we “hear the gauge” if we know the instrument and its fretboard length. If we’re only given frequencies, we’re out of luck.

Still: we get a different answer!

The idealized string from textbooks hides its gauge. A real one with a “flaw” in it lets us reconstruct it. Imperfections to the rescue.

More reality checks

I play guitar, which means I know strings aren’t always that simple. We assumed a solid cylinder here, such that μ and I are both functions of the same d. This makes gauge recoverable. We have two numbers, one is unknown, job done.

A wound string breaks our formulation. For many string instruments, this replaces at least some of the strings, and they look like this: you take a thin core and wrap wire around it. I never asked myself why, because I don’t really ordinarily do physics. It’s somewhat understandable though: the mass goes up, the bending stiffness stays roughly that of the core. You get low strings that can still be played and bent and not steel cables.

But this completely screws us! The string still gives you T/μ and EI/μ, but the physics stops letting you collapse them into the diameter, because I is no longer πd⁴/64. You hear something about the core and something about the total mass, but the buck stops here for me. I don’t even know how to start this problem.

So the answer flips again, and we’re back to “no”.

Final tally

Collecting all our findings roughly in order:

Ideal uniform string, unknown tension: Nope.
Ideal uniform string, known tension, length and material: Yup, without knowing the spectrum.
Ideal string with a varying profile d(x): Nope, solved decades before the drum.
Real stiff wire, known material and length: Yup. Inharmonicity gives it away.
Real stiff wire, spectrum and nothing else: Nope. Gauge and length are degenerate together.
Wound string: Maybe? A nope for me, dog.

Fin

Fletcher published the stiff string in 1964, two years before Kac’s paper that tipped me off appeared on the scene, and the inverse problem for the inhomogeneous string is even older than that.

What I couldn’t find was anyone phrasing it as a question, and going through the ladder of findings. And it’s quite cool: the ideal problem is unsolvable, the real world makes it solvable, and then it quickly becomes too complicated again.

I don’t have a grander conclusion than that. I’ve been letting guitar strings bite my hand for almost exactly 20 years and the question had never occurred to me before.

If you do work out the wound string, do tell me. I’m curious if dubious that I’ll understand it.

Footnotes

1. The entry point is G. M. L. Gladwell, “Inverse Problems in Vibration”, if you want the engineering version rather than the analysis. H. P. W. Gottlieb’s “Isospectral strings” comes with examples. I am not going to pretend to have read the analysis literature properly or understood any of it, and I don’t need to. We press on.

2. Pinned ends are convenient, but of course not quite what a real string is. Clamped ends add corrections. There is literature on that, too, and I don’t understand it either. For the argument here it doesn’t matter, since the n⁴ term stays.