Do eigenvalues in a cylindrical symmetric problem tell us anything about the Fourier spectrum?

Question

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.

Answer 1

what do we really "hear" in a sound? I thought, that we only "hear the Fourier eigenvalues".

This is involves biology, so I'll just quote Wikipedia:

The basilar membrane of the inner ear spreads out different frequencies: high frequencies produce a large vibration at the end near the middle ear (the "base"), and low frequencies a large vibration at the distant end (the "apex"). Thus the ear performs a sort of frequency analysis, roughly similar to a Fourier transform.

Let's suppose the analogy is perfect, in that case you detect each frequency $\omega_n$, which would correspond to the trigonometric Fourier descomposition (using sines and cosines) of the detected wave.

In the case of an ideal string, the sound would be a sum of sines, where each possible frequency is a multiple integer of a fundamental one. In the case of a circular membrane, the solutions are sines and cosines, where the frequencies are related to the zeros of the Bessel functions.

Anyway, we detect the frequencies of the Fourier decomposition of the wave, not the eigenvalues of the differential equation, unless they both coincide.

Then, you ask:

So is there easy way of calculating Fourier eigenvalues, knowing the eigenvalues in the other basis?

The eigenvalues are:

$$Lf_n=\lambda_n f_n$$

where $L$ is the Sturm-Liouville operator. But a change of basis won't modify the eigenvalues.