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Physical Audio Signal Processing
    Digital Waveguide Theory
       Traveling-Wave Solution
          Wave Velocity

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Wave Velocity

Because $ e^{st}$ is an eigenfunction under differentiation (i.e., the exponential function is its own derivative), it is often profitable to replace it with a generalized exponential function, with maximum degrees of freedom in its parametrization, to see if parameters can be found to fulfill the constraints imposed by differential equations.

In the case of the one-dimensional ideal wave equation (Eq.$ \,$(C.1)), with no boundary conditions, an appropriate choice of eigensolution is

$\displaystyle y(t,x) = e^{st+vx}$ (C.12)

Substituting into the wave equation yields

\begin{displaymath} \begin{array}{rclcrcl} {\dot y}& \,\mathrel{\mathop=}\,& sy... ...\quad & y''& \,\mathrel{\mathop=}\,& v^2y \nonumber \end{array}\end{displaymath}

Defining the wave velocity (or phase velocityC.2) as $ c \isdeftext {s/v}$, the wave equation becomes
$\displaystyle Kv^2y$ $\displaystyle =$ $\displaystyle \epsilon s^2y$ (C.13)
$\displaystyle \,\,\Rightarrow\,\,\frac{K}{\epsilon }$ $\displaystyle =$ $\displaystyle \frac{s^2}{v^2} \isdef c^2$  
$\displaystyle \,\,\Rightarrow\,\,v$ $\displaystyle =$ $\displaystyle \pm \frac{s}{c}.$  

Thus

$\displaystyle y(t,x) = e^{s(t\pm x/c)} $

is a solution for all $ s$. By superposition,

$\displaystyle y(t,x) = \sum\limits_i^{} A^{+}(s_i) e^{s_i(t-x/c)}+ A^{-}(s_i) e^{s_i(t+x/c)} $

is also a solution, where $ A^{+}(s_i)$ and $ A^{-}(s_i)$ are arbitrary complex-valued functions of arbitrary points $ s_i$ in the complex plane.

Previous: String Slope from Velocity Waves
Next: D'Alembert Derived

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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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