  Light-cone-like propagation of information is a universal phenomenon of
nonequilibrium dynamics of integrable spin systems. In this paper, we
investigate propagation of a local impact in the one-dimensional $XY$ model
with the anisotropy $\gamma$ in a magnetic field $h$ by calculating the
magnetization profile. Applying a local and instantaneous unitary operation to
the ground state, which we refer to as the local-impact protocol, we
numerically observe various types of light-cone-like propagation in the
parameter region $0\leq\gamma\leq1$ and $0\leq h \leq2$ of the model. By
combining numerical integration with an asymptotic analysis, we find the
following: (i) for $|h|\geq|1-\gamma^{2}|$ except for the case on the line
$h=1$ with $0<\gamma<\sqrt{3}/2$, a wave front propagates with the maximum
group velocity of quasiparticles, except for the case $\gamma=1$ and $0<h<1$,
in which there is no clear wave front; (ii) for $|h|<|1-\gamma^{2}|$ as well as
on the line $h=1$ with $0<\gamma<\sqrt{3}/2$, a second wave front appears owing
to multiple local extrema of the group velocity; (iii) for
$|h|=|1-\gamma^{2}|$, edges of the second wave front collapses at the origin,
and as a result, the magnetization profile exhibits a ridge at the impacted
site. Furthermore, we find by an asymptotic analysis that the height of the
wave front decays in a power law in time $t$ with various exponents depending
on the model parameters: the wave fronts exhibit a power-law decay $t^{-2/3}$
except for the line $h=1$, on which the decay can be given by either $\sim
t^{-3/5}$ or $\sim t^{-1}$; the ridge at the impacted site for
$|h|=|1-\gamma^{2}|$ shows the decay $t^{-1/2}$ as opposed to the decay
$t^{-1}$ in other cases.
