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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.

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    $\begingroup$ If you are solving this problem: en.wikipedia.org/wiki/Vibrations_of_a_circular_membrane. You don't get only Bessel functions, but trigonometric functions too. You can see that the frequencies are related to the zeros of Bessel functions, so they don't have a nice proportional form. So in both cases there a sine waves (since Bessel functions only take into account the radial part). $\endgroup$ – jinawee Jun 2 '14 at 19:34
  • $\begingroup$ OK, assuming we can hear only the time-dependent part of a particular solution and perform a Fourier analysis of such, this answers my question. But what about a more general case - a object is making sounds by a process, for which the time-dependent part of the solution is not easily solved in terms of trigonometric functions? Is it something we can hear, and do we have to calculate the Fourier decomposition of such a solution? $\endgroup$ – Lurco Jun 3 '14 at 8:56
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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.

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  • $\begingroup$ I know the eigenvalues don't change, but when you solve the equation in different variables (suitable for the symmetry of the problem) you get different set of eigenfunctions because technically the form of the operator has changed. Anyway, we hear "a sort of frequency analysis" which I understand as frequencies with their respective amplitudes. The frequencies number the solutions (eigenfunctions), whereas the amplitudes correspond to the eigenvalues. Theoretical distinction between 2 sounds is given by distinction in frequencies or the amplitudes (if the base frequency is the same)... $\endgroup$ – Lurco Jun 3 '14 at 8:37
  • $\begingroup$ ... So my question concerns the second case. But we only know eigenfunctions (frequencies) and eigenvalues in different variables for 2 different drums - for one in the Cartesian coordinates (trigonometric functions) and for the second in Cylindrical coordinates (Bessel functions). So my question is: how do we compare them, to determine which (on a theoretical level) sounds differently? $\endgroup$ – Lurco Jun 3 '14 at 8:42
  • $\begingroup$ @Lurco If the time dependent part is an arbitrary function, you must do the trigonometric series of that function, and you will detect the corresponding frequencies. $\endgroup$ – jinawee Jun 3 '14 at 10:43

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