The method of waveform images for finite waveguides with resistive terminations subject to arbitrary initial conditions

Abstract

Reflection at normal incidence of a plane wave can be described by imaging the incident wave profile on the opposite side of the boundary. This concept has been introduced in a few texts, but only for nondissipative conditions. In this paper the procedure to describe a purely resistive boundary is generalized, and then the concept is extended to describe the transient response of a one-dimensional, finite length waveguide. The field generated by arbitrary initial conditions is characterized by an infinite number of images, which leads to a representation of the acoustic field as oppositely propagating waves in an unbounded waveguide. Both graphical and mathematical descriptions of these waves are derived, with the former shown to provide significant insights. Mathematical analysis of the image construction leads to identification of several fundamental acoustic phenomena, including acoustic modes and reverberation time. From an instructional viewpoint the ability to explore fundamental acoustic phenomena without recourse to solving differential equations makes the waveform image concept especially useful as an introductory tool.