Why are there no ellipsoidal drums? It occurred to me today that all drums I could think of have circular heads. It then occurred to me that perhaps an elliptical drumhead would produce different overtones. Do the overtones produced by an elliptical drumhead not sound pleasant? And as a more general question, how does a general drumhead reverberate? What are the equations for a drumhead described by a certain polar curve's, its standing waves?
 A: I suspect the actual answer is something boring like ease of manufacture and tuning.
However, one can work out the solutions for the wave equation in elliptic coordinates, perform a separation of variables and end up with a system of differential equations admitting Mathieu functions as solutions. The boundary value problem can be solved relatively straightforwardly [1-4]. 
(Following [3]) In elliptic coordinates $\xi,\eta$, $x=f \cosh(\xi)\cos(\eta), y=f\sinh(\xi)\sin(\eta)$ where $f$ is the distance from the origin to the foci $(\pm f,0)$. $0\leq \xi < \infty$ is the "radial" coordinate constant on ellipses and $0\leq \eta < 2\pi$ is the "polar" coordinate constant on hyperbolas. The solutions of the wave equation $$\psi(\xi,\eta,t)=T(t)R(\xi)\Theta(\eta)$$ split into the time part $$T''(t)+k^2\nu^2T=0,$$ and the two spatial parts $$R''(\xi)-(\alpha-2q\cosh(2\xi))R(\xi)=0$$ (modified Mathieu equation) and $$\Theta''(\eta)+(\alpha-2q\cos(2\eta))\Theta(\eta)=0$$ (ordinary Mathieu equation) where $-k^2, \alpha$ are the constants of separation and $q=k^2f^2/4$. 
One interesting difference from circular membranes is that for each mode there is an even and odd mode, and they oscillate with different frequencies [3]. So if you stimulate one of the modes it is likely to produce a mix of two frequencies that likely do not have a nice rational ratio, and hence does not sound very harmonious. By changing the eccentricity they can likely be made to fit [4], but I suspect this will not make the whole spectrum harmonious.
[1] E. Mathieu, M´e moire sur le mouvement vibratoire d’une membrane de forme elliptique, J. Math. Pures Appl., vol. 13, pp. 137-203, 1868. http://sites.mathdoc.fr/JMPA/PDF/JMPA_1868_2_13_A8_0.pdf
[2] http://booksite.elsevier.com/9780123846549/Chap_Mathieu.pdf
[3] http://optica.mty.itesm.mx/pmog/Papers/P001.pdf
[4] http://www.altenberg.com/peter/pdfs/mathJournalSamp.pdf
