  In this article we study the homogenization rates of eigenvalues of a Steklov
problem with rapidly oscillating periodic weight functions. The results are
obtained via a careful study of oscillating functions on the boundary and a
precise estimate of the $L^\infty$ bound of eigenfunctions. As an application
we provide some estimates on the first nontrivial curve of the
Dancer-{F}u{\v{c}}{\'{\i}}k spectrum.
