Periodic potentials can be described in two contexts, classical and relativistic. In each category there is a wide assortment of actual potentials, but in both cases three stand out for their simplicity and generality.
In the Kronig-Penney model , the potential starts out as a step function, alternating between one value in part of a unit cell, but another for the rest of the cell. The advantage lies in having a plane wave in each interval, requiring only some trigonometry to match the boundaries and derive energy levels and band structures.
However, these authors noticed that the algebra could be simplified still further when the potential was very large over a limited interval, allowing it to be approximated by a Dirac delta function. In physical terms, the particle receives a pair of jolts as it passes a certain point, one accelerating it and the other slowing it down again. Or the reverse, if the sign of the potential is changed.
Finally, the third choice avoids a potential with discontinuities; a simple periodic function such as the cosine can be selected. Classically, this leads to Mathieu's equation, which has been extensively studied in various places for a variety of reasons; for example, after separation of variables for wave equations in elliptical coordinates [1,2,3].