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Physical Audio Signal Processing
    Digital Waveguide Theory
       Alternative Wave Variables
          Wave Impedance

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Wave Impedance

Using the above identities, we have that the force distribution along the string is given in terms of velocity waves by

$\displaystyle f(t,x) = \frac{K}{c} \left[{\dot y}_r(t-x/c) - {\dot y}_l(t+x/c) \right], \protect$ (C.44)

where $ K/c \isdef K/\sqrt{K/\epsilon } = \sqrt{K\epsilon }$. This is a fundamental quantity known as the wave impedance of the string (also called the characteristic impedance), denoted as

$\displaystyle R\isdefs \sqrt{K\epsilon } \eqsp \frac{K}{c} \eqsp \epsilon c.$ (C.45)

The wave impedance can be seen as the geometric mean of the two resistances to displacement: tension (spring force) and mass (inertial force).

The digitized traveling force-wave components become

\begin{displaymath}\begin{array}{rcrl} f^{{+}}(n)&=&&R\,v^{+}(n) \\ f^{{-}}(n)&=&-&R\,v^{-}(n) \end{array} \protect\end{displaymath} (C.46)

which gives us that the right-going force wave equals the wave impedance times the right-going velocity wave, and the left-going force wave equals minus the wave impedance times the left-going velocity wave.C.4Thus, in a traveling wave, force is always in phase with velocity (considering the minus sign in the left-going case to be associated with the direction of travel rather than a $ 180$ degrees phase shift between force and velocity). Note also that if the left-going force wave were defined as the string force acting to the left, the minus sign would disappear. The fundamental relation $ f^{{+}}= Rv^{+}$ is sometimes referred to as the mechanical counterpart of Ohm's law for traveling waves, and $ R$ in c.g.s. units can be called acoustical ohms [261].

In the case of the acoustic tube [317,297], we have the analogous relations

\begin{displaymath}\begin{array}{rcrl} p^+(n) &=& &R_{\hbox{\tiny T}}\, u^{+}(n)... ...p^-(n) &=& -&R_{\hbox{\tiny T}}\, u^{-}(n) \end{array} \protect\end{displaymath} (C.47)

where $ p^+(n)$ is the right-going traveling longitudinal pressure wave component, $ p^-(n)$ is the left-going pressure wave, and $ u^\pm (n)$ are the left and right-going volume velocity waves. In the acoustic tube context, the wave impedance is given by

$\displaystyle R_{\hbox{\tiny T}}= \frac{\rho c}{A}$   (Acoustic Tubes) (C.48)

where $ \rho$ is the mass per unit volume of air, $ c$ is sound speed in air, and $ A$ is the cross-sectional area of the tube. Note that if we had chosen particle velocity rather than volume velocity, the wave impedance would be $ R_0=\rho c$ instead, the wave impedance in open air. Particle velocity is appropriate in open air, while volume velocity is the conserved quantity in acoustic tubes or ``ducts'' of varying cross-sectional area.

Previous: Force Waves
Next: State Conversions

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


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