Auralization of traffic noise can be of great value for city planners and in the communication to citizens. In inner city environments, auralization of traffic noise based on predictions might require a wave-based method for the low frequencies. In this paper, a method to auralize a car pass-by using binaural impulse responses (BIRs) computed with the wave-based pseudospectral time-domain method (PSTD) in two dimensions is presented. A dry synthesized car signal is convolved with the BIRs for different locations in the street, and cross-fade windows are used in order to create a smooth transition between the source positions. Auralizations were performed for the simplified scenarios where buildings are absent, and for an environment where a long flat wall is located behind the car, opposite from the receiver. A same-different listening test was carried out in order to investigate whether increasing the angular spacing between the discrete source positions affects the perception of the car pass-by auralizations. Signal detection theory (SDT) was used for the design and analysis of the listening test. Results showed that differences exist, although they are difficult to notice. On average, 52.3% of the subjects found it almost impossible to spot any difference between auralizations with larger angular spacing (up to 10°) compared to the reference auralization (2° angular spacing).