When a vessel containing liquid is made to vibrate vertically with constant frequency and amplitude, a pattern of standing waves on the gas–liquid surface can appear. For some combinations of frequency and amplitude, waves appear; for other combinations the free surface remains flat. These waves were first studied in the experiments of Faraday (1831), who noticed that the frequency of the liquid vibrations was only half that of the vessel. Nowadays, this would be described as a symmetry-breaking vibration of a type that characterized the motion of a simple pendulum subjected to a vertical oscillation of its purpose.

The first mathematical study of Faraday waves are due to Rayleigh (1883a, 1883b) but the first definitive study is due to Benjamin and Ursell (1954; hereafter BU) who remark that “The present work has been made possible by the development of the theory of Mathieu functions.”

Faraday's problem is a rich source of problems in pattern formation, bifurcation, chaos, and other topics within the framework of fluid mechanics applications in the modern theory of dynamical system. Under the excitation of different parameters governing the Faraday system, different patterns, stripes, squares, hexagons, and time-dependent states can be observed. These features have spawned a large recent literature on Faraday waves. The experiments of Ciliberto and Gollub (1985) and Simonelli and Gollub (1989) on chaos, symmetry, and mode interactions are often cited.