Numerical simulations of acoustic radiation from a jet engine inlet are performed using advanced computational aeroacoustics algorithms and high-quality numerical boundary treatments. As a model of modern commercial jet engine inlets, the inlet geometry of the NASA Source Diagnostic Test is used. Fan noise consists of tones and broadband sound. This investigation considers the radiation of tones associated with upstream-propagating duct modes. The primary objective is to identify the dominant physical processes that determine the directivity of the radiated sound. Two such processes have been identified. They are acoustic diffraction and refraction. Diffraction is the natural tendency for an acoustic duct mode to follow a curved solid surface as it propagates. Refraction is the turning of the direction of propagation of a duct mode by mean flow gradients. Parametric studies on the changes in the directivity of radiated sound due to variations in forward flight Mach number, duct mode frequency, azimuthal mode number and radial mode number are carried out. It is found there is a significant difference in directivity for the radiation of the same duct mode from an engine inlet when operating in static condition versus one in forward flight. It will be shown that the large change in directivity is the result of the combined effects of diffraction and refraction.