Spurious start-up waves are studied for one-dimensional compressible flows in both ideal and non-ideal thermodynamic states. Numerical simulations are carried out on simple one-dimensional problems using both a Godunov scheme, the Roe scheme, and the Davis scheme. Numerical experiments show that two start-up waves are generated from an initial single shock wave for the ideal, dilute gas case. These are shown to correspond to a contact surface and an acoustic wave of the family opposed to the one pertaining to the shock. If the initial discontinuity is a contact discontinuity, no start-up waves are observed. In the non-ideal case, a similar outcome is observed for shock waves. Unlike the ideal case, contact discontinuities produce two start-up waves, one for each acoustic field, moving away from the contact discontinuity in both directions. This empirical observation is explained by the non-linearity of the jump relations across a contact discontinuity in the nonideal cases, whereby a linear relation exists linking conservative variables across a contact discontinuity in the ideal regime.