Beam–plasma instability of inhomogeneous media is an important problem associated with practical applications in space and laboratory plasmas. Previous investigations of the spatially inhomogeneous beam–plasma instability problem assumed one-dimensional velocity and wave number space. This paper extends the available theory to two-dimensional velocity and wave number space. A fully self-consistent set of quasilinear particle and wave kinetic equations is solved for two-dimensional velocity and wave number space in both time and one-dimensional spatial inhomogeneity scale length. The analytical equations include induced and spontaneous emission processes, as well as convective and dispersive effects.