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We consider the transmission of plane waves through a one-dimensional medium whose material parameters differ from those of free space on segments which form a Cantor set. By employing self-similarity, it is shown that the transmission coefficient is obtained as a solution of an infinite recursive relation. The latter is solved numerically by taking the small fractal length limit as an 4nitialn condition. The results show that the transmission coefficient is not monotonic with increasing slab thickness. For certain values of the normalized (over the wavelength) thickness the fractal medium is practically transparent and for others it totally reflects the incident wave. A physical interpretation of these effects is suggested.