Any initial shape repeats periodically only if all modes are harmonic

Evidence

Equation 8.15 carries another surprise: any shape we choose for the string initially will be repeated periodically at the frequency f1 = c/2L. This is simply because all the cosines in equation 8.15 are periodic with the same period τ = 1/f1. This can be verified in Loaded String using a large number of beads, and being careful not to excite the highest frequency modes. The reason for this exclusion is that these modes are increasingly inharmonic, and only if the modes have frequencies that are integer multiples of a given fundamental frequency will the shape be perfectly periodic. If, instead, 20 or 30 beads are taken and plucked rather sharply, the shape is only approximately regained after one period, and the subsequent attempts degrade after that. — Eric J. Heller, Why You Hear What You Hear- An…, 853

Categorization

• DEFINE: Explains why inharmonic higher modes break the exact periodicity of a plucked string.