- its Laplace-Beltrami spectrum, together with
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). The set of all timbres is then $T=\{\tau(x):x\in D\}$, i.e. 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.)
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.)
