In this paper we consider scattering theory on manifolds with special cusp-like metric singularities of warped product type g=dx2+x−2ah, where a>0. These metrics form a natural subset in the class of metrics with warped product singularities and they can be thought of as interpolating between hyperbolic and cylindrical metrics. We prove that the resolvent of the Laplace operator acting on p-forms on such a manifold extends to a meromorphic function defined on the logarithmic cover of the complex plane with values in the bounded operators between weighted L2-spaces. This allows for a construction of generalized eigenforms for the Laplace operator as well as for a meromorphic continuation of the scattering matrix. We give a precise description of the asymptotic expansion of generalized eigenforms on the cusp and find that the scattering matrix satisfies a functional equation.