... III

D. Rocchesso is with the Dipartimento di Informatica, Università degli Studi di Verona, strada Le Grazie, 37134 Verona - ITALY, Phone: ++39.045.8027979, FAX: ++39.045.8027982, E-mail: rocchesso@sci.univr.it. J. O. Smith is with the Center for Computer Research in Music and Acoustics (CCRMA), Music Department, Stanford University, Stanford, CA 94305, E-mail: jos@ccrma.stanford.edu.

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... ``tension

``Tension'' is defined here for gases as the reciprocal of the adiabatic compressibility of the gas [28, p. 230]. This definition helps to unify the scattering formalism for acoustic tubes with that of mechanical systems such as vibrating strings.

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... tube

There appear to be no tube shapes supporting exact traveling waves other than cylindrical and conical (or conical wedge, which is a hybrid) [36]. However, the ``Salmon horn family'' (see, e.g., [27,37]) characterizes a larger class of approximate one-parameter traveling waves. In the cone, the wave equation is solved for pressure $p(x,t)$ using a change of variables $p^\prime = p x$, where $x$ is the distance from the apex of the cone, causing the wave equation for the cone pressure to reduce to the cylindrical case [38]. Note that while pressure waves behave simply as non-dispersive traveling waves in cones, the corresponding velocity waves are dispersive [38].

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... power

Note that $\left\vert z\right\vert=1$ corresponds to the average physical power at frequency $\omega$, where $z=\exp(j\omega T)$, and the wave variable magnitudes on the unit circle may be interpreted as RMS levels. For $\vert z\vert > 1$, we may interpret the power ${\mbox{\boldmath$u$}}^*(1/z^*) {\mbox{\boldmath$p$}}(z)$ as the steady state power obtained when exponential damping is introduced into the waveguide giving decay time-constant $\tau$, where $z = \exp(-T/\tau)\exp(j\omega T)$ (for the continuous-time case, see [40, p. 48]).

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... real

A complex-valued function of a complex variable $f(z)$ is said to be positive real if

1)

$z\, \hbox{real}\; \Rightarrow\; f(z)\, \hbox{real}$

2)

$\vert z\vert \geq 1 \; \Rightarrow \; \hbox{Re}\{f(z)\} \geq 0$

Positive real functions characterize passive impedances in classical network theory.

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... ratio

Indeed, in the general case ${\mbox{\boldmath$\Theta$}}$ is a diagonal matrix.

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... velocity

The symbols for the variables velocity and force have been chosen to maintain consistency with the analogous acoustical quantities.

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