The nonlinear propagation of finite amplitude waves in the non-ideal compressible fluid dynamic (NICFD) region of high-molecular weight fluids, and in particular in cases where non-classical gas dynamic phenomena like the formation of rarefaction shock waves may be expected, is of great scientific interest. Almost all the theoretical developments so far are based on the assumption that the waves propagate in a fluid kept in homogeneous conditions. Experimental activities performed with the Flexible Asymmetric Shock Tube (FAST) operated in the laboratories of the Propulsion and Power group at the Delft University of Technology have shown that obtaining such conditions is particularly challenging, especially keeping the tube at constant temperature. Stemming from this observation, this study is a part of a theoretical and numerical investigations aimed at assessing the influence of temperature gradients at the boundary on wave propagation in so-called Bethe-Zel’dovich Thompson (BZT) fluids. The full-wave Westervelt Equation is solved numerically using the Finite Difference Time Domain (FDTD) method. The steepening of the wave front is used as an indicator of shock formation. The effect of varying temperature, and the corresponding variation of the fundamental derivative of gas dynamics on the distortion of rarefaction waves is analyzed by observing the simulated behaviour of the wave as it propagates from a region of negative values of the fundamental derivative of gas dynamics to a region of positive values.