Green’s Function (GF) for the Two Dimensional (2D) Time Dependent Inhomogeneous Wave Equation

Abstract

Interference effect that occurs when two or more waves overlap or intersect is a common phenomenon in physical wave mechanics. A carrier wave as applied in this study describes the resultant of the interference of a parasitic wave with a host wave. A carrier wave in this wise, is a corrupt wave function which certainly describes the activity and performance of most physical systems. In this work, presented in this paper, we used the Green’s function technique to evaluate the behaviour of a 2D carrier wave as it propagates away from the origin in a pipe of a given radius. In this work, we showed quantitatively the method of determining the intrinsic characteristics of the constituents of a carrier wave which were initially not known. Evidently from this study the frequency and the band spectrum of the Green’s function are greater than those of the general solution of the wave equation.  It is revealed in this study that the retarded behaviour of the carrier wave described by the Green’s function at some point away from the origin is much greater than the general wave solution of the carrier wave at the origin.  The Green’ function is spherically symmetric about the source, and falls off smoothly with increasing distance from the source. The anomalous behaviour exhibited by the carrier wave at some point during the damping, is due to the resistance pose by the carrier wave in an attempt to annul the destructive tendency of the interfering wave. Evidently it is shown in this work that when a carrier wave is undergoing attenuation, it does not consistently come to rest; rather it shows some resistance at some point in time during the damping process, before it finally comes to rest.