Mathieu functions, a visual approach

The behavior of the Mathieu functions is illustrated by using a variety of plots with representative examples taken from mechanics. We show how Mathieu functions can be applied to describe standing, traveling, and rotating waves in physical systems. Some background is provided on notation and analogies with other mathematical functions. Our goal is to increase the familiarity with Mathieu functions in the scientific and academic community using visualization. For this purpose we adopt a strategy based on visual recognition rather than only looking at equations and formulas.