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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

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

$\displaystyle R\isdef \sqrt{K\epsilon } = \frac{K}{c} = \epsilon c.$ (H.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} (H.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.H.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 [268].

In the case of the acoustic tube [325,301], we have the analogous relations

\begin{displaymath}\begin{array}{rcrl} p^+(n) &=& &R_{\hbox{\normalsize T}}\, u^... ...\\ p^-(n) &=& -&R_{\hbox{\normalsize T}}\, u^{-}(n) \end{array}\end{displaymath} (H.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