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This paper develops the theory of EM radiation from metallic enclosures with apertures, excited by an internal source at frequencies below the fundamental resonance of the enclosure. The enclosure with an aperture is analyzed from two different points of view: as a cavity with a small aperture in a wall, and as a waveguide section short-circuited at one end and open at the other. Rectangular geometries are used throughout, since these are by far the most commonly encountered in practical enclosures and cabinets. Using the corresponding dyadic Green's functions, the fields generated inside the enclosure by some simple sources are determined. The fields radiated through small apertures in a cavity are determined using Bethe's theory of diffraction by small holes. The radiation from an open waveguide is calculated with the help of field-equivalence theorems, with assumptions applicable to the case of evanescent waves. The final step is to derive expressions for the " insertion loss" of the shield, defined as the ratio of the field strength at a point external to the shield, before and after the insertion of the enclosure. To do so, the effect of the shield upon the input impedance of the antenna is analyzed, and expressions obtained for the applicable cases. The resulting insertion-loss expressions are numerically evaluated for some representative cases, and graphically compared with a series of measurements performed to obtain experimental confirmation. Very good agreement is obtained in all cases, establishing the validity of the analysis.