= Solution
Write the oscillation <period> as $P=2\pi/\omega$, so locally $c_n=c_o+(a/2\pi)P$. The <first overtone> is the lowest curve above the fundamental, not the <fundamental mode> itself. Near $P=50\,\mathrm s$, Figure 1 gives approximately $c_1=5.13\,\mathrm{km\,s^{-1}}$. Representative nearby readings are about $4.96\,\mathrm{km\,s^{-1}}$ at $40\,\mathrm s$ and $5.30\,\mathrm{km\,s^{-1}}$ at $60\,\mathrm s$. Thus
$$
\left.\frac{dc_1}{dP}\right|_{50\,\mathrm s}\simeq0.017\,\mathrm{km\,s^{-2}},\qquad
\boxed{a\simeq2\pi(0.017)\simeq0.11\,\mathrm{km\,s^{-2}}
\simeq1.1\times10^2\,\mathrm{m\,s^{-2}}.}
$$
The local group-velocity estimate is
$$
\boxed{u_1\simeq\frac{5.13}{1+50(0.017)/5.13}
\simeq4.4\,\mathrm{km\,s^{-1}}.}
$$
These are graphical estimates, roughly $a=0.11\pm0.02\,\mathrm{km\,s^{-2}}$ and $u_1=4.4\pm0.1\,\mathrm{km\,s^{-1}}$, rather than high-precision fitted constants. The nearby straight-line approximation need not describe that mode over the whole plotted <period> range. No source curve is reproduced here.
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