= Solution
The natural interpretation is the fundamental transverse <normal mode>, $n=1$. Indeed
$$
\omega_{c,1}=\frac{\pi v}{b}=62.832\,\mathrm{rad\,s^{-1}},\qquad
\omega_{c,2}=125.66\,\mathrm{rad\,s^{-1}},
$$
so only $n=1$ propagates over the main part of the given <Gaussian distribution> of <angular frequencies>. Its centre travels at
$$
v_{g,0}=2\sqrt{1-(62.832/100)^2}=1.5559\,\mathrm{m\,s^{-1}},\qquad
t_0=\frac{10\,\mathrm m}{v_{g,0}}=6.4271\,\mathrm s.
$$
Differentiate the <group velocity> with respect to <angular frequency>:
$$
\frac{dv_g}{d\omega}
=\frac{v\omega_c^2}{\omega^3\sqrt{1-(\omega_c/\omega)^2}}.
$$
At the centre this is $0.010149\,\mathrm m$. The one-standard-deviation <group velocity> spread is consequently
$$
\sigma_v\simeq\left|\frac{dv_g}{d\omega}\right|_{\omega_0}\sigma_\omega
=0.05075\,\mathrm{m\,s^{-1}}.
$$
Over the central travel time the associated spatial shear is
$$
\boxed{\sigma_{x,\rm disp}\simeq t_0\sigma_v=0.326\,\mathrm m.}
$$
Equivalently, use representative components at $\omega_0\pm\sigma_\omega$, for which $v_g(95)=1.5001\,\mathrm{m\,s^{-1}}$ and $v_g(105)=1.6024\,\mathrm{m\,s^{-1}}$. Their separation after $t_0$ is
$$
\boxed{\Delta x_{[-\sigma,+\sigma]}\simeq[v_g(105)-v_g(95)]t_0=0.658\,\mathrm m\simeq0.65\,\mathrm m.}
$$
At a fixed detector $10\,\mathrm m$ away, the corresponding arrival-time difference is $10/v_g(95)-10/v_g(105)=0.426\,\mathrm s$.
The paper asks for an estimate of spreading, without specifying a spatial width convention or the initial spectral <phase>. A <Gaussian distribution> has no finite full range. Thus $0.326\,\mathrm m$ is a one-standard-deviation dispersive contribution, whereas about $0.65\,\mathrm m$ describes the separation of the $\pm\sigma_\omega$ representatives. Neither is automatically the increase in the packet's original width. For an initially unchirped <Gaussian wave packet> in the narrow-band approximation, independent initial and dispersive contributions combine as $\sigma_{x,f}^2\simeq\sigma_{x,0}^2+\sigma_{x,\rm disp}^2$; a packet with an initial <chirp> can behave differently. The modal assumption and width convention must accompany the numerical answer.
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