In this paper, we present a simplified model for the string-bridge interaction in guitars or other string instruments simulated by digital waveguides. The bridge model is devised for the displacement wave representation in order to be integrated with other models for string interactions with the player and with other parts of the instrument, whose simulation and implementation is easier in this representation. The model is based on a multiplierless scattering matrix representing the string-bridge interaction. Although not completely physically inspired, we show that this junction is sufficiently general to accommodate a variety of transfer functions under the sole requirement of passivity and avoids integration constants mismatch when the bridge is in turn modeled by a digital waveguide. The model is completed with simple methods to introduce horizontal and vertical polarizations of the string displacement and sympathetic vibrations of other strings. The aim of this paper is not to provide the most general methods for sound synthesis of guitar but, rather, to point at low computational cost and scalable solutions suitable for real-time implementations where the synthesizer is running together with several other audio applications.