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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

{\dot y}& \,\mathrel{\mathop=}\,& sy & \quad & y'& \,\mathrel{\mathop=}\,& vy \nonumber \\
{\ddot y}& \,\mathrel{\mathop=}\,& s^2y & \quad & y''& \,\mathrel{\mathop=}\,& v^2y \nonumber

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)
$\displaystyle \,\,\Rightarrow\,\,\frac{K}{\epsilon }$ $\displaystyle =$ $\displaystyle \frac{s^2}{v^2} \isdef c^2$  
$\displaystyle \,\,\Rightarrow\,\,v$ $\displaystyle =$ $\displaystyle \pm \frac{s}{c}.$  


$\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.

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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4
Copyright © 2022-07-01 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University