Governing equations of envelope surface created by nearly bichromatic waves propagating on an elastic plate and their stability

Abstract

Plates are common structural elements of most engineering structures, including aerospace, automotive, and civil engineering structures. The study of plates from theoretical perspective as well as experimental viewpoint is fundamental to understanding of the behavior of such structures. The dynamic characteristics of plates, such as natural vibrations, transient responses for the external forces and so on, are especially of importance in actual environments. In this paper, we consider natural vibrations of an elastic plate and the propagation of a wavepacket on it. We derive the two-dimensional equations that govern the spatial and temporal evolution of the amplitude of a wavepacket and discuss its features. We especially consider wavenumber-based nearly bichromatic waves and direction-based nearly bichromatic waves on an elastic plate. The former waves are defined by the waves that almost concentrate the energy in two wavenumbers, which are very closely approached each other. The latter waves are defined by the waves that almost concentrate the energy in two propagation directions and two propagation directions are very close each other. The fact that the solution of the governing equation for wavenumber-based nearly bichromatic waves is stable is also shown.