The beamwidth is a primary directivity metric for the design of a constant directivity horn. To date, investigations on this property have predominantly been restricted to the half-space radiation or idealized geometries. This paper examines the beamwidth behavior of an axisymmetric horn mounted in a finite cylindrical enclosure by proposing an elegant multimodal solution to the far-field directivity pattern. The variation of beamwidth is examined for the frequency, dimensions of the enclosure, and shape of the horn. At low frequencies, a fitted model is proposed to precisely depict the intrinsic beam narrowing governed by the enclosure diffraction. The asymptotic behavior of the beamwidth is explored as the flange width increases. In the high-frequency range, the horn profile is a determinant of the directivity characteristics. We report the possibility of extending the bandwidth of a constant directivity horn by leveraging the enclosure diffraction effects. The proposed analytical method is highly accurate and much faster than the finite element method for wideband analysis. It allows for an arbitrary velocity distribution at the mouth of the horn and incorporates idealized flange configurations such as an infinite baffle, a zero-thickness closed baffle, and an infinitely long enclosure as limit cases.