  The Kolmogorov $N$-width $d_N(\mathcal{M})$ describes the rate of the
worst-case error (w.r.t.\ a subset $\mathcal{M}\subset H$ of a normed space
$H$) arising from a projection onto the best-possible linear subspace of $H$ of
dimension $N\in\mathbb{N}$. Thus, $d_N(\mathcal{M})$ sets a limit to any
projection-based approximation such as determined by the reduced basis method.
While it is known that $d_N(\mathcal{M})$ decays exponentially fast for many
linear coercive parametrized partial differential equations, i.e.,
$d_N(\mathcal{M})=\mathcal{O}(e^{-\beta N})$, we show in this note, that only
$d_N(\mathcal{M}) =\mathcal{O}(N^{-1/2})$ for initial-boundary-value problems
of the hyperbolic wave equation with discontinuous initial conditions. This is
aligned with the known slow decay of $d_N(\mathcal{M})$ for the linear
transport problem.
