  We provide an explicit formula for the Fefferman-Graham-ambient metric of an
$n$-dimensional conformal $pp$-wave in those cases where it exists. In even
dimensions we calculate the obstruction explicitly. Furthermore, we describe
all 4-dimensional $pp$-waves that are Bach-flat, and give a large class of
Bach-flat examples which are conformally Cotton-flat, but not conformally
Einstein. Finally, as an application, we use the obtained ambient metric to
show that even-dimensional $pp$-waves have vanishing critical $Q$-curvature.
