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The first overtone is the lowest oscillatory normal mode above the fundamental mode. It is an ordering by frequency, not a claim that its frequency is an exact integer multiple of the fundamental. If the fundamental is labeled $n=0$, the first overtone is $n=1$; another common convention labels them one and two.

For a string fixed at both ends, the frequencies are $f_n=nc/(2L)$ with $n\ge 1$, so the first overtone is the second harmonic at twice the fundamental. For an ideal pipe closed at one end, the next allowed mode is at three times the fundamental. Geometry and boundary conditions determine the relation.

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.