 Two-mode string recurrence

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.