Partial discharges can occur in different types of electric equipment and cause progressive insulation deterioration, so there is interest in monitoring partial discharges for assessing the state of the isolation of an electric system and to predict failures. Techniques to detect partial discharges, such as detecting ultrasonic emission, have been proposed in the literature, but as various effects can co-occur during a discharge, identifying characteristics in sound and pressure waveforms and correlating then with discharge characteristics is difficult. Simulations can assist in these correlations by allowing the isolation of the different phenomena. In this work the drift diffusion model, including photoionization, is coupled with the linearized compressible Navier-Stokes equations to simulate ultrasonic waves produced by partial discharges. Previous works have used the incompressible Navier-Stokes equations, so they can simulate ionic wind produced by the Trichel pulses, but no sound. In the literature, simulations have focused either on streamers or Trichel pulses. In this work both discharges and produced sound waves are successfully simulated for the needle-plane geometry in air. The electric current and charge per pulse are compared with experimental results reported in the literature for the same discharge conditions. The simulations have demonstrated that the sound waves depart from the electrode tip for the Trichel pulses, and for the streamers two sound waves are produced, one from the electrode tip and the other from the whole discharge length. Differences in the wave front with respect to the relative position to the electrode tip were analyzed, showing that near the discharge spot the sound wave is not a spherical wavefront. The sound wave for one of the discharges in the Trichel pulse regime was compared with experimental results in the literature. Results are in good agreement with the experimental data found in the literature. Both current and sound waves were successfully predicted and correlated with the discharge, results that can be used to help in the detection of partial discharges.