### ASA 126th Meeting Denver 1993 October 4-8

## 4aMU1. Two complex effective lengths for air columns.

**R. Dean Ayers
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*Dept. of Phys. and Astron., California State Univ., Long Beach, CA 90840
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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.