During a lecture we were solving the Helmholtz equation for particular boundary conditions, corresponding to different shapes of an oscillating drum, as in the famous Mark Kac's problem http://en.wikipedia.org/wiki/Hearing_the_shape_of_a_drum.
For Cartesian boundary conditions we used the Fourier eigenfunctions (sines and cosines), as for the Cylindrical b.c. we used the Bessel functions (http://en.wikipedia.org/wiki/Bessel_functions). We then compared the spectra in both of those bases, interpreting the different result as a difference in the pitch and timbre, that a particularly shaped drum would have.
My question is: what do we really "hear" in a sound? I thought, that we only "hear the Fourier eigenvalues". So is there easy way of calculating Fourier eigenvalues, knowing the eigenvalues in the other basis? Or am I missing something obvious?
PS. I know that technically I'm referring to a orthonormal set of functions not a basis, but I hope that the physicists' forum won't mind.