Digital Waveguide Model

Above we derived

$$f_m(t) - f_1(t) + f_2(t) = 0$$

We now perform the traveling-wave decompositions

$$\begin{eqnarray*} f_1&=&f^{{+}}_1+f^{{-}}_1\\ f_2&=&f^{{+}}_2+f^{{-}}_2 \end{eqnarray*}$$

and apply the Ohm's law relations

$$\begin{eqnarray*} f^{{+}}_i&=&Rv^{+}_i\\ f^{{-}}_i&=&-Rv^{-}_i, \quad i=1,2 \end{eqnarray*}$$

to obtain a digital waveguide model

• By symmetry, the mass must look identical from either string segment 1 or 2
• Therefore, let $f^{{-}}_2\equiv 0$ and excite from string 1

In the Laplace domain, we have

$$\begin{eqnarray*} 0 &=& F_m - F_1 + F_2 \\ &=& F_m - (F^{+}_1 + F^{-}_1) + F^{+}_2\qquad\mbox{(since $F^{-}_2=0$)} \end{eqnarray*}$$