The paper presents the results of an investigation of an asymmetric dielectric barrier discharge (DBD) in the air at atmospheric pressure. The discharge system consisted of a plane electrode and semispherical short-radius one, and the both were coated with dielectric. The discharge was excited by a train of almost sinusoidal damped oscillations which was applied to electrodes with a frequency of 1 kHz. It has been found that the preferable to biological applications homogeneous avalanche form of the DBD is always realized in the investigated voltage range 7 – 15 kV, whereas the streamer form of the discharge is excited under identical conditions in the case of a conducting semisphere. It was also established that the positive and negative (according to the voltage sign of the first pulse of the train onto the small electrode) discharge differ greatly in their characteristics. An explanation for this difference has been proposed.