On the use of waveform images to describe the initial response of finite‐length waveguides

Abstract

The d’Alembert solution of the wave equation can be adapted to describe reflection from planar boundaries. One technique for doing so images the incident wave on the opposite side of the boundary. This concept has been introduced in a few texts, most extensively by Morse and Ingard [Theoretical Acoustics, McGraw‐Hill, New York (1964), pp. 106–115], but only for nondissipative ends (infinite or zero impedance.) This paper formalizes the procedure for the case where the boundary has a resistive impedance that is independent of frequency, and then extends it to treat waveguides of finite length. It is shown that the field that results from arbitrary initial conditions can be represented by an infinite number of images. This leads to a representation of the acoustic field as oppositely propagating wave in an unbounded waveguide, with only a limited number of images overlapping at any instant. Both mathematical and graphical descriptions of these waves are derived. In addition to assisting the student to understand the evolution of the field, mathematical analysis of the image construction leads to a number of physical and mathematical insights to fundamental acoustic phenomena. These include the fact that the field in the dissipationless case can be represented as a modal series with associated natural frequencies, and a quantitative understanding of the manner in which the field decays when either end is dissipative. A corollary of the latter analysis is an expression for reverberation time that is remarkably similar to the Norris–Eyring formula. From an instructional viewpoint, the fact that all results are derived without recourse to solving differential equations makes the image waveform concept especially useful as a way of introducing new students to fundamental acoustic phenomena.