We present results from time-dependent hydrodynamic calculations of the interaction between internal gravity waves and the mean radial differential rotation in the solar tachocline. Such waves are thought to be generated by turbulent fluid motions at the base of the convection zone. Our simplified model treats the effects of wave forcing, produced by radiative damping of downward propagating disturbances, on the rotational shear flow in the region immediately below the convection zone. We have used the model to investigate the dependence of the computed flow properties on the values assumed for the wave frequency, the horizontal component of the wavevector, the initial wave velocity amplitude, and the viscosity of the background medium. Our results indicate that if the first three of these quantities are held fixed, stationary shear flow solutions are obtained for viscosities larger than a parameter-dependent critical value. If the viscosity is continuously decreased from this value, the flow undergoes a succession of dramatic transformations, first becoming periodic, then quasi-periodic, and ultimately chaotic when the viscosity is made sufficiently small. We discuss the implications of these results for the recently reported time variability of the angular velocity of rotation within the solar tachocline.