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I was wondering when eigenmodes (modes of vibration) were first mentioned in mechanics. Any reference would be appreciated.

I expect them to date back to the second half of the 19th century because of the breakthrough in linear algebras, but I'm not sure.

Additional notes This question was initially asked on physics.SE. I was suggested to post it here, but I'm keeping track of the comments below:

  • [DanielSank] Fourier's big publication was 1822, and although the big example there was heat diffusion it definitely gets at the idea of normal modes. Interestingly, Wikipedia points out that decomposition into basis functions was kind of sort of known in antiquity, albeit without any true understanding of what was going on. But I don't think there was anything about vector space structure at that time some we can't speak "modal analysis", which is more general: any motion of a linear n-dimensional system (even non-periodic) can be decomposed as a linear combination of n modes.
  • [CuriousOne] You may want to look at the references in Gram-Schmidt wiki. I think one should link an understanding of eigenmodes to an interest in orthogonalization of bases in linear spaces, which would put it formally closer to the end of the 19th, early 20th century. As it has been pointed out, these fairly trivial ideas have been invented multiple times, so it's not clear who was "first", since you have to pick based on the context.
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    $\begingroup$ Was the version on Physics deleted? $\endgroup$ – HDE 226868 Dec 28 '15 at 17:33
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    $\begingroup$ @HDE226868 Yes, I deleted it, as crossposted is not recommended. $\endgroup$ – anderstood Dec 28 '15 at 17:39
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As every question which involves "who was the first" or "when was the first", this one is unlikely to have a definite answer. (Read the first chapter of Th. Mann, "Joseph and his brothers" where this is explained in great detail).

But it is clear that the modes where well known long before the 19th century. (I mean the phenomenon itself, I am not discussing the term "mode"). An exact solution of the 1-dimensional wave equation in the form of superposition of eigenfunctions was obtained by Daniel Bernoulli in 18th century.

However, the fact itself (as an experimental fact) was certainly known to Marine Mersenne, (1637) and to Kepler (1619), and in some form to the Pythagoreans.

In the ancient times this fact was considered as belonging to music theory, but music theory was a part of mathematics, probably since Pythagoras himself.

So the fact is indeed known since antiquity, but for a rigorous mathematical formulation probably Bernoulli can be credited.

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