R. Dean Ayers
Dept. of Phys. and Astron., California State Univ., Long Beach, CA 90840
In the context of a plane-wave approximation for acoustic waves in an air column, the pressure reflection coefficient R is related to the relative input impedance Z/Z[sub c] by a simple bilinear transformation. At any location along the air column, the complex effective length to an open, downstream end of an equivalent cylinder is defined as L[sub o](identically equal to)(j/2k[sub 0])ln(-R), where k[sub 0] is the undamped propagation number, and that to a closed end is defined as L[sub c](identically equal to)(j/2k[sub 0])ln(R). The real parts of these quantities are just the conventional effective lengths to the corresponding ends, which have been used in the study of musical wind instruments. The imaginary parts incorporate the effects of damping within the air column and at its downstream termination. The common theme here is a simple extrapolation of interfering, damped plane waves, with k[sub 0] serving as an artificially large attenuation constant. L[sub c] turns out to be more useful than L[sub 0] for analyzing the brass instruments [R. D. Ayers, J. Acoust. Soc. Am. Suppl. 1 88, S163 (1990)], and an earlier treatment of undamped horns made implicit use of a real L[sub c] [R. W. Pyle, Jr., J. Acoust. Soc. Am. 57, 1309--1317 (1975)]. This new treatment is more realistic, computationally simpler, and conceptually more straightforward.