= Two-mode string recurrence
{title2=$\omega_1t,\omega_2t\in2\pi\mathbb Z$}
For two distinct <sine> <normal modes> with nonzero <amplitudes> and initial <cosine> <phases> zero, <linear independence> of their spatial profiles means the initial shape returns exactly only when both time factors equal one. A rational <frequency> ratio $\omega_2/\omega_1=p/q$ in lowest terms gives first positive recurrence $2\pi q/\omega_1$. An irrational ratio gives no exact positive recurrence, although arbitrarily close two-mode returns exist. These statements refer to the complete spatial profile, not a displacement at one observation point.
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