Elastography is a non-invasive imaging technique that can assess in vivo tissue stiffness. Shear wave elastography imaging uses the acoustic radiation force (ARF) to produce shear waves. Numerical models rely on different formulations of the ARF. We present three existing formulations for the ARF: its full expression in the second-order approximation and two simplified formulations using a quasi-plane wave and an attenuated plane wave approximation. Analytical expressions for the ARF are derived for the special cases of a spherical concave source and for a quasi-Gaussian beam. They show discrepancies between the different formulations. For strongly divergent or highly focused beams the result from the second-order approximation differs from both simplified formulations. However the second-order and quasi-plane wave approximations create identical shear displacements. The k-Wave simulation package is used to compute the ARF and the ensuing transient displacement produced by an ultrasound probe. The second-order approximation and the quasi-plane wave approximation give different forces but identical displacements. The results using the plane wave approximation significantly differ. It is concluded that for highly focused transducers only the second-order approximation accurately estimates the ARF. For estimating shear displacements the second-order or quasi-plane wave approximation are equivalent and preferable to the plane wave approximation.