Weak-stiffness string frequency shift (${\omega }_n=c_s{k}_n[1+\alpha k_n^2/(2c_s^2)]+O({\alpha }^2)$)

For the prescribed sine-mode model $\omega =k\sqrt{c_s^2+\alpha k^2}$, positive small stiffness shifts higher harmonics more strongly. Writing $\eta =\alpha k_1^2/c_s^2$ and $k_2=2k_1$ gives ${\omega }_2/{\omega }_1=2+3\eta +O({\eta }^2)$. At the original fundamental quarter-period, the fundamental displacement changes to first order, but the second-mode cosine is at an extremum and changes only to second order. If the quarter-period is redefined with the shifted fundamental frequency, that fundamental displacement is exactly zero. Comparing profiles requires specifying which clock convention is held fixed.