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Laplace-Domain Analysis


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

is a solution for all $ s$.

Interpretation: left- and right-going exponentially enveloped complex sinusoids

General eigensolution:

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

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 for all $ A^{+}(s_i)$ and $ A^{-}(s_i)$.

Alternate derivation of D'Alembert's solution:

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``Distributed Modeling in Discrete Time'', by Julius O. Smith III and Nelson Lee,
REALSIMPLE Project — work supported by the Wallenberg Global Learning Network .
Released 2008-06-05 under the Creative Commons License (Attribution 2.5), by Julius O. Smith III and Nelson Lee
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University