In this work we consider an indirect integral equation approach for the direct scattering problem for an obstacle satisfying Dirichlet conditions in a simple two-dimensional wavequide problem. The original problem is reformulated as an integral equation with the aid of a double-layer potential. A Nyström - type method is used to solve numerically the integral equation. To accelerate the typically slow convergence of the series representation of the Green's function we use Laplace's method based on suitable integral representation of the Hankel functions involved and on an appropriate choice of the quadrature points. For the inverse problem of shape reconstruction of an obstacle in this waveguide we will use the linear factorization method based on measurements from point source excitations. The inversion scheme is based on a characterization of solutions of an integral equation of first kind having as known term a point source. Numerical results are presented.