The research is concerned with the study of dispersive shock waves (DSW), also termed undular bores, on the surface of fluids, DSWs arise due to the dispersive resolution of step, or near-step, initial conditions and are a common waveform in nature. They consist of a modulated dispersive wavetrain linking distinct levels ahead and behind it. In this talk, we will present some preliminary results related to the accuracy of DSW solutions of the Hamiltonian Bidirectional Whitham compared with fully nonlinear results. All research on DSWs to date has been based on weakly nonlinear approximations of full systems of equations, for instance, the water wave equations. The research of this project is the first attempt for the study of fully nonlinear DSWs. We will use the Whitham-Boussinesq (W-B) model that I introduced in my doctoral thesis as the bridge between the water wave equations (the free-surface Euler equations) and in which the "Shock fitting method” can be applied. This "Shock fitting method", derived by G. El and his collaborators, is a general method for determining the leading and trailing edges of DSWs, based on the dispersion relation for the governing nonlinear, dispersive wave equation.