A finite two dimensional photonic crystal with a triangular lattice of air columns in a dielectric background is designed with a waveguide and a resonant cavity. By a time-domain solution of Maxwell's equations, we observe the existence of impurity modes inside the photonic band gap. An electromagnetic wave launched through the waveguide, with an appropriate frequency, can enhance the field inside the cavity and yield a peak in the transmission. Considering a manufactured structure, in which the columns are not perfectly equal, we repeat our calculations and analyze how the impurity modes are modified. For both cases, periodic and real, we measure the quality factor of the cavity.