Re:What Is an Eigenvalue?
by _spduchamp on Tuesday November 08, 2022 @11:01AM
(#33520435)
I read an interview with Australian wire music composer Alan Lamb that a stringed instrument with multiple overtones vibrating on the string can be analyzed by breaking down the vibration into eigenvalues, but I've never found any reference material that explain that. I'm wondering if he was referring to FFT.
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If you discretize the string into a bunch of tiny masses, linked together by a bunch of tiny springs, you can build a mass Matrix (diagonal) M and a Stiffness matrix K (element ij = stiffness of spring that links mass I and mass j).
I can't remember the next part exactly, you can look it up in a textbook, but you multiply the matrices KMK, or similar, and the eigenvalues of this are the natural frequencies of the string. The eigenvectors represent the mode shapes, ie the displacement of each mass element.
The same technique is used in Finite Element Analysis to find the modes and modeshapes of complex structures (a car frame, a bridge, etc)
Re:What Is an Eigenvalue?
by alanbernstein on Tuesday November 08, 2022 @11:20AM
(#33520690)
Complex exponentials are the eigenfunctions of the Fourier transform. In other words, frequency component values are the eigenvalues.
https://en.m.wikipedia.org/wiki/Eigenfunction#Vibrating_stri...
Re:What Is an Eigenvalue?
by contravariant on Tuesday November 08, 2022 @12:11PM
(#33521397)
That makes no sense, the Fourier transform of a complex exponential is a delta function.
Re:What Is an Eigenvalue?
by alanbernstein on Tuesday November 08, 2022 @05:04PM
(#33524950)
Hmm, you're right, that should have been obvious. Thanks for the correction.