Governing Equations of Envelope Surface Created by Directional, Nearly Monochromatic Waves

Abstract

This paper focuses on the envelope surface of directional, nearly monochromatic waves and the governing equation of the envelope surface is reduced. The spectrum of nearly monochromatic waves is almost concentrated on a single frequency. Such waves create the envelope in a one-dimensional system. In a two-dimensional system, the envelope surface instead of the envelope in a one-dimensional system can be considered. Two-dimensional nearly monochromatic waves in the meaning of the frequency create the envelope surface. This is simply expanded to a two-dimensional system from a one-dimensional system. The author expands the frequency-based nearly monochromatic waves to the propagation-direction-based nearly monochromatic waves. We have the new concept of directional, nearly monochromatic waves, whose energy is almost concentrated in a single propagation direction. It is shown that directional, nearly monochromatic waves also create the envelope surface, which is featured by time-invariant in a linear system. However the fact that a non-linear system makes the envelope surface time-variant is presented.