We consider the scattering of acoustic waves on a set of equidistant resistive sheets. We show theoretically and experimentally that at the Bragg frequency, the transmission presents an anomalously high peak. Moreover, the peak can be made arbitrarily narrow by increasing the number of sheets. Using the transfer matrix formalism, it is shown that this effect occurs when the two eigenvalues of the transfer matrix coalesce, i.e. at an exceptional point. Exploiting this algebraic condition, it is possible to obtain similar anomalous transmission peaks in more general periodic media. In particular, the system can be tuned to show a peak while having a subwavelength size.