Coupled Strings

• Figure 9 depicts the case of two strings that terminate at a common, non-rigid bridge of impedance $Z_{b}(\omega)$.

  Figure 9: Two strings terminated at a common bridge impedance.

  

• The constraints at the junction are:
  $$\begin{eqnarray*} f_{b} &=& f_{1} + f_{2} \hspace{0.2in} \mbox{(string forces ba... ...v_{1} = v_{2} \hspace{0.2in} \mbox{(common velocity at bridge)}. \end{eqnarray*}$$

• The bridge impedance relates the force and velocity at the bridge as $F_b(\omega) = Z_{b}(\omega) V_b(\omega)$.

• Expanding the traveling wave components in the frequency domain,
  $$\begin{eqnarray*} Z_{b}(\omega) V_{b}(\omega) &=& F_{b}(\omega) = F_{1}(\omega) ... ...\omega) - \left[V_{b}(\omega) - V_{2}^{+}(\omega)\right]\right\} \end{eqnarray*}$$
  or
  $$V_{b}(\omega) = H_{b}(\omega) \left[ R_{1} V_{1}^{+}(\omega) + R_{2} V_{2}^{+}(\omega) \right]$$
  where $R_{i}$ is the impedance of string $i$ and
  $$H_{b}(\omega) \ensuremath{\stackrel{\Delta}{=}}\frac{2}{Z_{b}(\omega) + R_{1} + R_{2}}.$$

• In the time-domain, we scale each incoming string velocity, sum them together, and filter according to the transfer function $H_{b}(\omega) = 2 / \left[Z_{b}(\omega) + R_{1} + R_{2}\right]$ to obtain the velocity of the bridge.

• The subsequent outgoing velocities are then found by
  $$\begin{eqnarray*} v^{-}_{1}(t) &=& v_{b}(t) - v^{+}_{1}(t) \\ v^{-}_{2}(t) &=& v_{b}(t) - v^{+}_{2}(t). \end{eqnarray*}$$

• This analysis can be generalized to the case of $N$ coupled strings without much difficulty.