 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 isAt $L/c$ 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.