= Solution
To recover the original signed displacement profile, orthogonality requires both cosine time factors to equal one. Since $\omega_3=3\omega_1$, the smallest positive time is
$$
\boxed{t_{\rm return}=\frac{2\pi}{\omega_1}=\frac{2L}{c}.}
$$
At $L/c$ both factors are $-1$, giving the negative of the initial profile, rather than the same profile. More generally, simultaneous recurrence requires commensurate excited-mode frequencies and undiminished amplitudes. The ideal nondispersive, fixed-end string has integer-multiple frequencies, whereas damping or a dispersive correction can destroy exact recurrence.
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