= Weak-stiffness string frequency shift
{title2=$\omega_n=c_sk_n[1+\alpha k_n^2/(2c_s^2)]+O(\alpha^2)$}
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.
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