  We discuss the invariant classification of vacuum Kundt waves using the
Cartan-Karlhede algorithm, and the upper bound on the number of iterations of
the Karlhede algorithm to classify the vacuum Kundt waves. By choosing a
particular coordinate system we partially construct the canonical coframe used
in the classification to study the functional dependence of the invariants
arising at each iteration of the algorithm. We provide a new upper bound $q
\leq 4$ and show this bound is sharp by analyzing the subclass of Kundt waves
with invariant count beginning with (0,1,...) to show that the class with
invariant count $(0,1,3,4,4)$ exists. This class of vacuum Kundt waves is shown
to be unique as the only set of metrics requiring the fourth covariant
derivatives of the curvature. We conclude with an invariant classification of
the vacuum Kundt waves using a suite of invariants.
