The formation of the square superlattice pattern in discharge in the mixture of argon and air is investigated by using a dielectric barrier discharge device with water electrode. It is found that the bifurcation process of the patterns follows goes through the stages of square pattern，quasisuperlattice pattern，square superlattice pattern and the stripe or hexagonal pattern. Their spatial Fourier spectra are analyzed and the variation of the wave vector as a function of the applied voltage is determined. In addition，two types of square superlattice patterns with different spatial scales are observed with the applied voltage increasing. They all satisfy the standing wave condition. The role of wall charge accumulated in the dielectric layers in the formation of the square superlattice pattern is analyzed. Both the phase diagram on the plot of pattern type against the gas gap width and applied voltage and the phase diagram on the plot of the square superlattice pattern against the gas gap width and the gas pressure are obtained. In addition，the changes of the breakdown voltage and the voltage of the square superlattice pattern as a function of the air concentration are determined. It is found that they increase with the increasing of the air concentration.