  We eliminate the existence of cusps in a class of \textit{degenerate}
free-boundary problems for the Alt-Caffarelli functional $J_{Q}(v, \Omega):=
\int_{\Omega}|\nabla v|^2 + Q^2(x)\chi_{\{v>0\}}dx,$ so-called because $Q(x) =
\text{dist}(x, \Gamma)^{\gamma}$ for $\Gamma$ an affine $k$-plane and $0<
\gamma$. This problem is inspired by a generalization of the variational
formulation of the Stokes Wave by Arama and Leoni. The elimination of cusps
implies that the results of [Mccurdy20] in fact describe the entire
free-boundary as it intersects $\Gamma$.
