I find the problem of hearing the shape of a drum fascinating. Specifically, given two connected subsets of $\mathbb R^2$ with piecewise-smooth boundaries (or a suitable generalization to a riemannian manifold) it is in general impossible to use the spectrum of the laplacian to uniquely identify the shape of the domain.

As a simple example, reported in the Wikipedia link above, these two shapes

have identical laplacian spectra, as explained more at length by Gordon and Webb in *American Scientist* **84** no. 1, 46 (1996) (jstor). This can be shown by taking any eigenfunction $\varphi$ on $D_1$, cutting it up into the marked triangle regions, and taking the specified linear combinations on $D_2$ to make a laplacian eigenfunction with the same boundary conditions, correct matching at the internal boundaries, and the same eigenvalue.

What bugs me about this problem is that **the spectrum is not really what you hear on a real-world drum.** More specifically, the spectrum gives you the frequencies of the fundamentals and overtones that the drum is *allowed* to produce, but it does not tell you in what proportions it will do so. If you hit a circular drum in the centre, you will excite a large amplitude of the fundamental, resulting in a deeper sound, but if you drum it near the edge you will produce a rapping sound with a stronger proportion of higher overtones.

The set of sounds that a drum can produce is therefore better modelled by the pair of

- its Laplace-Beltrami spectrum, together with
the set $T$ of timbres it can produce.

By a

*timbre*I mean the sequence of eigenfunction weights when the drum is hit at $x$, i.e. $$\tau(x)=\left( \sqrt{\sum_k|\varphi_{n,k}(x)|^2} \right)_{n=1}^\infty,$$ where the $\varphi_{n,k}$ are the (possibly multiple) orthonormalized eigenfunctions corresponding to the eigenvalue $\lambda_n$, and two timbres are thought of as equivalent if they only differ by a global constant (which trivially corresponds to how hard you drum).(This quantity is meant to model the amount of energy in each eigenspace after a point excitation to the wave equation (via e.g. $\Delta \varphi-\partial_t^2\varphi=0$, $\varphi(y,0)=0$, $\partial_t\varphi(y,0)=\delta(x,y)$), and perhaps there are cleaner or equivalent definitions for $\tau(x)$ based on that PDE. However, the definition above is plenty to satisfy the intuition of how the musical timbre depends on the drumpoint from a physicist's perspective.)

The set of all timbres is then $T=\{\tau(x):x\in D\}$: that is, you're provided all the timbres, but not which place in the geometry they come from. (As pointed out in the comments, initial knowledge of the geometry renders the whole thing moot.)

In this paradigm, two isospectral surfaces $D_1$ and $D_2$ would still be 'acoustically distinguishable' if, say, $D_1$ contains a point $x^*$ which produces a timbre $\tau^{(1)}(x^*)$ that is not matched by the timbre $\tau^{(2)}(y)$ of any point $y\in D_2$. In other words, you can't see the drum, and you can't see where the drummer is hitting it, but if drummer 1 can produce a timbre that drummer 2 cannot, then the drums must be different.

(Alternatively, one might have access to a function $u\mapsto \tau(f(u))$, where $u$ lives in some standard domain $D_0$ like the unit disk and $f:D_0\to D$ is an unknown (homeo/diffeo/etc)-morphism. That is, you get a control that specifies where the drum should be hit, but you don't know exactly what it does. You can then say two drums $D_1$ and $D_2$ are indistinguishable if there's a morphism $g:D_1\to D_2$ such that $\tau^{(2)}(g(x))=\tau^{(1)}(x)$ for all $x\in D_1$, i.e. a mapping from each point $x$ in $D_1$ to an identically-sounding drum point in $D_2$.)

So, to come to my question: has this paradigm been explored in the literature? For the known isospectral-but-not-isomorphic regions (or more complex manifolds), are any pairs known to be acoustically indistinguishable as defined above? Or must acoustically indistinguishable surfaces be isometric? Does this concept have an established name in the literature? If it has been explored, are there good introductory surveys (say, pitched at a level where a physicist might understand them)? If it hasn't, are there fundamental reasons why this is a much harder problem overall?

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