The two-dimensional time-harmonic acoustic scattering by a semi-infinite waveguide composed of two parallel rigid plates is considered. An original mode matching method is developed to avoid the use of the Wiener–Hopf technique, generalizing usual mode matching methods to the case of unbounded media. A brief description of the method is given by means of Fourier decomposition leading to a well-posed variational problem with unknown being the trace of the solution on a well-chosen interface. From the numerical point of view, a local approximation is first considered by using Lagrange P1 finite elements on a segment of the interface. This leads to the computation of oscillatory integrals involving Fourier transform and complex square root functions. As a matter of accuracy, a special function is added to the finite element space, in order to take into account the asymptotic behavior of the solution. Finally, this method is extended to deal with local perturbations of the media by coupling the previous method to a classical integral one.