An active mute for the trombone.

A mute is a device that is placed in the bell of a brass instrument to alter its sound. However, when a straight mute is used with a brass instrument, the frequencies of its first impedance peaks are slightly modified, and a mistuned, extra impedance peak appears. This peak affects the instrument's playability, making some lower notes difficult or impossible to produce when playing at low dynamic levels. To understand and suppress this effect, an active mute with embedded microphone and speaker has been developed. A control loop with gain and phase shifting is used to control the damping and frequency of the extra impedance peak. The stability of the controlled system is studied and then the effect of the control on the input impedance and radiated sound of the trombone is investigated. It is shown that the playability problem results from a decrease in the input impedance magnitude at the playing frequency, caused by a trough located on the low frequency side of the extra impedance peak. When the extra impedance peak is suppressed, the playability of the note is restored. Meanwhile, when the extra impedance peak is moved in frequency, the playability problem position is shifted as well.

Inverse potential scattering in duct acoustics.

The inverse problem of the noninvasive measurement of the shape of an acoustical duct in which one-dimensional wave propagation can be assumed is examined within the theoretical framework of the governing Klein-Gordon equation. Previous deterministic methods developed over the last 40 years have all required direct measurement of the reflectance or input impedance but now, by application of the methods of inverse quantum scattering to the acoustical system, it is shown that the reflectance can be algorithmically derived from the radiated wave. The potential and area functions of the duct can subsequently be reconstructed. The results are discussed with particular reference to acoustic pulse reflectometry.

The acoustical Klein-Gordon equation: the wave-mechanical step and barrier potential functions.

The transformed form of the Webster equation is investigated. Usually described as analogous to the Schrödinger equation of quantum mechanics, it is noted that the second-order time dependency defines a Klein-Gordon problem. This "acoustical Klein-Gordon equation" is analyzed with particular reference to the acoustical properties of wave-mechanical potential functions, U(x), that give rise to geometry-dependent dispersions at rapid variations in tract cross section. Such dispersions are not elucidated by other one-dimensional--cylindrical or conical--duct models. Since Sturm-Liouville analysis is not appropriate for inhomogeneous boundary conditions, the exact solution of the Klein-Gordon equation is achieved through a Green's-function methodology referring to the transfer matrix of an arbitrary string of square potential functions, including a square barrier equivalent to a radiation impedance. The general conclusion of the paper is that, in the absence of precise knowledge of initial conditions on the area function, any given potential function will map to a multiplicity of area functions of identical relative resonance characteristics. Since the potential function maps uniquely to the acoustical output, it is suggested that the one-dimensional wave physics is both most accurately and most compactly described within the Klein-Gordon framework.