The problem of two dimensional internal wave scattering by a vertical barrier in the form of a submerged plate, or a thin wall with a gap in an exponentially stratified fluid of uniform finite depth bounded by a rigid plane at the top, is considered in this paper. Assuming linear theory and the Boussinesq approximation, the problem is formulated in terms of the stream function. In the regions of the two sides of the vertical barrier, the scattered stream function is represented by appropriate eigen function expansions. By the use of appropriate conditions on the barrier and the gap, a dual series relation involving the unknown elements of the scattering matrix is produced. By defining a function with these unknown elements as its Fourier sine expansion series, it is found that this function satisfies a Carleman type integral equation on the barrier whose solution is immediate. The elements of the scattering matrix are then obtained analytically as well as numerically corresponding to any mode of the incident internal wave train for each barrier configuration. A comparison with earlier results available in the literature shows good agreement. To visualize the effect of the barrier on the fluid motion, the stream lines for an incident internal wave train at the lowest mode are plotted.