Green's function of the three-dimensional convective wave equation for an infinite straight pipe

Abstract

Green's function of the three-dimensional convective wave equation for an infinite straight pipe of arbitrary (but constant along its length) cross-sectional shape and area, having either acoustically rigid or acoustically soft walls or the walls of a mixed type, is obtained. This function is represented by a series of the pipe acoustic modes. Each term of the series is a superposition of the direct and reverse waves propagating in the corresponding mode downstream and upstream of the acoustic source, respectively. In the Green's function, the effects of a uniform mean flow in the pipe are directly reflected. The effects become more significant as the flow Mach number increases, causing, in particular, the appearance and the further growth of the function asymmetry about the pipe crosssection in which the noted source is located. Vice versa, a decrease of the Mach number results in a decrease of the effects and, in particular, a decrease of the indicated function asymmetry. In the absence of a flow, the Green's function is symmetric about the noted cross-section. A transformation is suggested that allows one to reduce the one-dimensional convective Klein-Gordon equation to its classical one-dimensional counterpart and, by proceeding from the known solution of the later equation, to obtain a solution to the former one.