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Physical Audio Signal Processing
    Digital Waveguide Theory
       Alternative Wave Variables
          State Conversions


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State Conversions

In §C.3.6, an arbitrary string state was converted to traveling displacement-wave components to show that the traveling-wave representation is complete, i.e., that any physical string state can be expressed as a pair of traveling-wave components. In this section, we revisit this topic using force and velocity waves.

By definition of the traveling-wave decomposition, we have

\begin{eqnarray*} f&=&f^{{+}}+f^{{-}}\\ v&=&v^{+}+v^{-}. \end{eqnarray*}

Using Eq.$ \,$(C.46), we can eliminate $ v^{+}=f^{{+}}/R$ and $ v^{-}=-f^{{+}}/R$, giving, in matrix form,

$\displaystyle \left[\begin{array}{c} f \\ [2pt] v \end{array}\right] = \left[\b... ...ay}\right] \left[\begin{array}{c} f^{{+}} \\ [2pt] f^{{-}} \end{array}\right]. $

Thus, the string state (in terms of force and velocity) is expressed as a linear transformation of the traveling force-wave components. Using the Ohm's law relations to eliminate instead $ f^{{+}}= Rv^{+}$ and $ f^{{-}}=-Rv^{-}$, we obtain

$\displaystyle \left[\begin{array}{c} f \\ [2pt] v \end{array}\right] = \left[\b... ...d{array}\right]\left[\begin{array}{c} v^{+} \\ [2pt] v^{-} \end{array}\right]. $

To convert an arbitrary initial string state $ (f,v)$ to either a traveling force-wave or velocity-wave simulation, we simply must be able to invert the appropriate two-by-two matrix above. That is, the matrix must be nonsingular. Requiring both determinants to be nonzero yields the condition

$\displaystyle 0 < R < \infty. $

That is, the wave impedance must be a positive, finite number. This restriction makes good physical sense because one cannot propagate a finite-energy wave in either a zero or infinite wave impedance.

Carrying out the inversion to obtain force waves $ (f^{{+}},f^{{-}})$ from $ (f,v)$ yields

$\displaystyle \left[\begin{array}{c} f^{{+}} \\ [2pt] f^{{-}} \end{array}\right... ...ft[\begin{array}{c} \frac{f+Rv}{2} \\ [2pt] \frac{f-Rv}{2} \end{array}\right]. $

Similarly, velocity waves $ (v^{+},v^{-})$ are prepared from $ (f,v)$ according to

$\displaystyle \left[\begin{array}{c} v^{+} \\ [2pt] v^{-} \end{array}\right] = ... ...[\begin{array}{c} \frac{v+f/R}{2} \\ [2pt] \frac{v-f/R}{2} \end{array}\right]. $

Previous: Wave Impedance
Next: Power Waves

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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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