When a fluid-filled cube rotating rapidly about an axis passing through two opposite vertices is subjected to harmonic modulations of its rotation rate (librations) at a modulation frequency that is times the mean rotation frequency, all walls of the cube have critical reflection slopes. As such, all inertial wave beams emitted from edges and vertices of the cube in response to the librations are trapped in thin oscillatory boundary layers for forcing amplitudes (Rossby numbers) below a critical value which depends on the Ekman number (ratio of rotation to viscous time scales). How the resulting oscillatory boundary layer flow, referred to as a boundary-confined wave, depends on Ekman and Rossby numbers is examined in detail over several decades. Of particular interest is how the mean flow grows with increasing forcing amplitude, leading to instability resulting from nonlinear interactions between the mean flow and waves in the oscillatory boundary layers, injecting intense small-scale structures throughout the cube.