The problem of the transmission of plane waves along a uniform parallel-walled waveguide or channel through a periodically-spaced array of thin screens with gaps placed across the channel is investigated. Of particular interest is the existence of frequencies at which energy it totally transmitted -- shown to occur when there are two or more screens -- and how they depend upon the size of the gaps in the screen. Attention focuses on small gaps where the phenonemon of total transmission persists and where an approximate analysis of a system of full coupled integrals equations is reduced to simple linear difference equations which can be solved in closed form. Results show comparisons between exact computations, the small-gap computations and corresponding wide-spacing approximations. A discussion of total reflection is also included when the gaps are no longer assumed to be small.