 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 $c_1=5.13\mathrm{km}{\mathrm{s}}^{-1}$. Representative nearby readings are about $4.96\mathrm{km}{\mathrm{s}}^{-1}$ at $40\mathrm{s}$ and $5.30\mathrm{km}{\mathrm{s}}^{-1}$ at $60\mathrm{s}$. ThusThe local group-velocity estimate isThese are graphical estimates, roughly $a=0.11\pm 0.02\mathrm{km}{\mathrm{s}}^{-2}$ and $u_1=4.4\pm 0.1\mathrm{km}{\mathrm{s}}^{-1}$, 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.