  In this paper, we continue the study local minimizers of a degenerate version
of the Alt-Caffarelli functional. Specifically, we consider local minimizers of
the functional $J_{Q}(u, \Omega):= \int_{\Omega} |\nabla u|^2 +
Q(x)^2\chi_{\{u>0\}}dx$ where $Q(x) = dist(x, \Gamma)^{\gamma}$ for $\gamma>0$
and $\Gamma$ a submanifold of dimension $0 \le k \le n-1$. Previously, it was
shown that on $\Gamma$, the free boundary $\partial \{u>0\}$ may be decomposed
into a rectifiable set $\mathcal{S}$, which satisfies effective estimates, and
a cusp set $\Sigma$. In this note, we prove that under mild assumptions, in the
case $n = 2$ and $\Gamma$ a line, the cusp set $\Sigma$ does not exist.
Building upon the work of Arama and Leoni \cite{AramaLeoni12}, our results
apply to the physical case of a variational formulation of the Stokes' wave and
provide a complete characterization of the singular portion of the free
boundary in complete generality in this context.
