  In this paper, we consider the numerical approximation of the Steklov
eigenvalue problem that arises in inverse acoustic scattering. The underlying
scattering problem is for an inhomogeneous isotropic medium. These eigenvalues
have been proposed to be used as a target signature since they can be recovered
from the scattering data. A Galerkin method is studied where the basis
functions are the Neumann eigenfunctions of the Laplacian. Error estimates for
the eigenvalues and eigenfunctions are proven by appealing to Weyl's Law. We
will test this method against separation of variables in order to validate the
theoretical convergence. We also consider the inverse spectral problem of
estimating/recovering the refractive index from the knowledge of the Steklov
eigenvalues. Since the eigenvalues are monotone with respect to a real-valued
refractive index implies that they can be used for non-destructive testing.
Some numerical examples are provided for the inverse spectral problem.
