Digital Waveguide (DW) Scheme

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Digital Waveguide (DW) Scheme

We now derive the digital waveguide formulation by sampling the traveling-wave solution to the wave equation. It is easily checked that the lossless 1D wave equation $Ky''=\epsilon {\ddot y}$ is solved by any string shape which travels to the left or right with speed $c \isdeftext \sqrt{K/\epsilon }$ [99]. Denote right-going traveling waves in general by and left-going traveling waves by , where and are assumed twice-differentiable. Then, as is well known, the general class of solutions to the lossless, one-dimensional, second-order wave equation can be expressed as

$\displaystyle y(t,x) = y_r\left(t-\frac{x}{c}\right) + y_l\left(t+\frac{x}{c}\right). \protect$

(P.4)

Sampling these traveling-wave solutions yields

$\displaystyle y(nT,mX)$	$\displaystyle =$	$\displaystyle y_r(nT-mX/c) + y_l(nT+mX/c)$
	$\displaystyle =$	$\displaystyle y_r[(n-m)T] + y_l[(n+m)T]$
	$\displaystyle \isdef$	$\displaystyle y^{+}(n-m) + y^{-}(n+m) \protect$	(P.5)