= Solution
The <phase velocity> in Figure 1 is measured along the <Earth>'s surface. Spherical geometry makes the local horizontal phase <speed> at radius $r$ equal to $cr/R_E$. In the ray approximation, squared radial shear <wavenumber> is
$$
q_r^2=\omega^2\left[\frac1{\beta(r)^2}-\frac{R_E^2}{c^2r^2}\right].
$$
The <spherical surface-wave evanescence threshold> immediately below the discontinuity is therefore
$$
\boxed{c<5.08\frac{6371}{6371-410}
\simeq5.43\,\mathrm{km\,s^{-1}}.}
$$
Equality is the grazing limit. Using $5.08\,\mathrm{km\,s^{-1}}$ directly would omit the requested spherical correction. Immediately above the discontinuity the corresponding boundary is $4.87(6371/5961)\simeq5.20\,\mathrm{km\,s^{-1}}$. Hence the interval $5.20<c<5.43\,\mathrm{km\,s^{-1}}$ specifically describes propagation just above the jump and evanescence just below it. Slower modes can already be evanescent above $410\,\mathrm{km}$.
Intersecting the plotted branches with $5.43\,\mathrm{km\,s^{-1}}$ gives the following approximate upper <periods>. Below each upper <period>, that branch is evanescent immediately beneath the discontinuity:
* <fundamental mode>: throughout the displayed $10$–$100\,\mathrm s$ range;
* first and second <overtones>: approximately $67$ and $38\,\mathrm s$;
* successive higher curves: approximately $27$, $21$, $18$, $15$, $13$ and $11\,\mathrm s$.
For these <overtones> the displayed ranges thus start near $10\,\mathrm s$ and end at their respective crossings. Any still higher curve already above the threshold at the lower plotted <period> has no evanescent interval within the displayed range. Readings of the closely spaced higher curves are only accurate to about a second, and the plot does not establish their continuation to unshown <periods>. Nor does the local inequality prove that a mode remains evanescent at every greater depth in an unspecified <Earth> model.
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