Perfectly matched layers (PMLs) are widely used for the numerical simulation of wave-like problems defined on large or infinite spatial domains. However, for both time-dependent and time-harmonic cases, their performance critically depends on the so-called absorption function. This paper deals with the choice of this function when classical numerical methods are used (based on finite differences, finite volumes, continuous finite elements and discontinuous finite elements). After reviewing the properties of the PMLs at the continuous level, we analyze how they are altered by the different spatial discretizations. In the light of these results, different shapes of absorption function are optimized and compared by means of both one-dimensional and two-dimensional representative time-dependent cases. This study highlights the advantages of the so-called shifted hyperbolic function, which is efficient in all cases and does not require the tuning of a free parameter, by contrast with the widely used polynomial functions.