We report on simulations of the near field for arrays of Ag nanowires and nanocolumns excited by plane waves of light at normal incidence. The results show a systematic variation of the local electric field with spatial period and incident polarization, which is confirmed experimentally in fluorescence enhancement experiments. The distribution of the local field and the dependence of local field intensity versus spatial period and polarization indicate that the excitations in the Ag nanostructres are surface plasmon standing waves.