Directional beams have extensive applications in communication and sound reproduction. This paper investigates the theoretical maximum directivity of infinitely flanged open-ended waveguides and the radiation pattern synthesis. We derive a rigorous solution for the maximum directivity factor of a flanged aperture with arbitrary shape by projecting its surface velocity on the waveguide modes, enabling the creation of a directional beam in any desired direction. We present case studies for a three-dimensional circular waveguide and a bidimensional waveguide. The theoretical beam that is obtained in a subspace spanned by all the propagating modes can then be synthesized by a group of incident modes or a point-source array within the waveguide. The optimality of the beam is demonstrated by comparing it with Gaussian shaded modes radiated from the waveguide. If the evanescent modes are taken into account, the maximum directivity factor increases with considerable loss to the radiation efficiency. Nevertheless, the optimum aperture velocity dominated by its evanescent components is capable of precise beam steering in extreme directions and could be useful for designing material-filled horns. Our work provides benchmark directivity factors and patterns for the practical design of horn antennas. In addition, we present a generalized form of Bouwkamp's impedance theorem.