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Normal modes refer to fundamental patterns of motion of a system which oscillate at fixed, well defined frequencies. They may be used as building blocks for more complicated motions.

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Why is the energy of the harmonics in a vibrating string not infinitesimal?

First of all, one should notice that the dichotomy "eigenmode $-$ non-eigenmode" is a false one. The correct statement is that any generic motion of the string can be decomposed into superposition of …
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