The paper discusses experimental characterization of a surface dielectric barrier discharge plasma sustained by repetitive, high-voltage, nanosecond duration pulses. The discharge pulse energy is controlled primarily by the pulse peak voltage and scales approximately linearly with the length of the electrodes. Images of the plasma generated during the discharge pulse, taken by a nanosecond gate ICCD camera, show that the plasma remains fairly uniform in the initial phase of the discharge and becomes filamentary at a later stage. The temperature rise in the discharge, operated in both continuous mode and in burst mode, is inferred from UV/visible emission spectra. Phase-locked schlieren images are used to measure the speed of the compression waves generated by the nanosecond pulse discharge and the density gradient in the wave. The density gradient is inferred from the schlieren images using absolute calibration by a pair of wedged windows. The results demonstrate that discharge filaments generate compression waves with higher amplitude and higher speed compared with waves produced in a diffuse discharge. The density gradient in the compression waves is compared with numerical modeling of propagating compression waves produced by short-pulse localized heating, and shows satisfactory agreement between the model and the experimental results.