We considered cases of forced oscillations of extreme waves in resonators. We now turn to the study of parametrically excited waves. The possibility of the appearance of such waves on thin layers of various media was experimentally shown by Faraday.

Faraday conducted experiments with various liquids, such as mercury, water, ink, milk, egg white, and used cuvettes of various shapes: round, square, rectangular. As a result of his experiments, Faraday came to the conclusion that the ripples almost always form a square lattice, which can deform slightly at the edges (due to the interaction of the liquid with the edges of the cuvette), and the spatial structure does not depend on either the initial conditions or the type of liquid. “Mercury on tin plate being vibrated in sunshine gave very beautiful effects of reflection” Faraday wrote in his diary.

Much later, Rayleigh linked the appearance of these waves with parametric resonance. In recent years, this problem has aroused considerable interest both among theorists and experimenters, in particular, because in the case of strong vertical excitation on the surfaces of liquid and granular media, extraordinary highly nonlinear waves and wave patterns occur.

The experimental research of parametric-excited strongly nonlinear waves is connected to the considerable difficulties, and the received results sometimes are unexpected. Therefore, in this chapter, the author at first reports about these results so that to enter the reader into a considered circle of problems. Then at the same time, experimental and theoretical aspects of the problem are studied. At the end of the chapter, ways of the further researches are discussed, and some analogies are presented.

The focus here will be on experiment and calculations of strongly nonlinear waves. We mainly used Eq. (3.1). In the case of very thin layers, Eq. (2.60) may be used. We emphasize that these equations are identical if we disregard the bottom friction. However, Eq. (3.1) describes waves u (longitudinal waves), whereas Eq. (2.60) describes waves 1 (vertical waves). Therefore, the applicability of these equations to the analysis of the Faraday waves is different. However, we do not focus on this here, because the solutions are the same, and only their interpretation and areas of application are different. Remind that the coefficients of these equations can differ very much, and therefore, these equations can describe various physical effects.

The results presented in the chapter are published mainly in our articles for the years 1998–2000 [29–32]. This determines the literature that is quoted here.