Wave Scattering

• When a traveling-wave component experiences a change in wave impedance, it will be partly reflected from and partly transmitted through the impedance discontinuity in such a way that energy is conserved. This is referred to as wave scattering.

  Figure 8: A string density discontinuity and associated traveling-wave scattering.

  

• Figure 8 depicts a string density discontinuity and associated traveling force wave components on each side of the junction.

• At the junction, we must have continuity of the medium, or there must be a common transverse string velocity at that point. Similarly, the forces on each side of the junction must be balanced.

• These constraints can be written
  $$\begin{eqnarray*} f_{0} + f_{1} &=& 0 \hspace{0.2in} \mbox{(forces on a massless... ...\ v_{0} &=& v_{1} \hspace{0.2in} \mbox{(string does not break)} \end{eqnarray*}$$

• Let $v$ denote the common transverse velocity of the string at the junction. Then the traveling-wave components of velocity can be written $v_{i} = v^{+}_{i} + v^{-}_{i}$ or $v^{-}_{i} = v - v^{+}_{i}$.

• We can then write
  $$\begin{eqnarray*} 0 &=& f_{0} + f_{1} = \left(f^{+}_{0} + f^{-}_{0}\right) + \le... ...eft(2 v^{+}_{0} - v\right) + R_{1} \left(2 v^{+}_{1} - v\right), \end{eqnarray*}$$
  which can be solved for $v$ as
  $$v = 2 \frac{R_{0} v^{+}_{0} + R_{1} v^{+}_{1}}{R_{0} + R_{1}}.$$

• Solving these expressions in terms of the outgoing traveling-wave components,
  $$\begin{eqnarray*} v^{-}_{0} &=& v - v^{+}_{0} = -\frac{R_{1} - R_{0}}{R_{1} + R_... ..._{0}} v^{+}_{0} + \frac{R_{1} - R_{0}}{R_{1} + R_{0}} v^{+}_{1}. \end{eqnarray*}$$

• It is standard to define the reflection coefficient
  $$\rho \ensuremath{\stackrel{\Delta}{=}}\frac{R_{1} - R_{0}}{R_{1} + R_{0}},$$
  which then allows us to write
  $$\begin{eqnarray*} v^{-}_{0} &=& -\rho v^{+}_{0} + (1 + \rho) v^{+}_{1} \\ v^{-}_{1} &=& (1 - \rho) v^{+}_{0} + \rho v^{+}_{1}. \end{eqnarray*}$$

• Figure 9 provides a block diagram implementation of the equations above. This is equivalent to a Kelly-Lochbaum junction representation (Kelly and Lochbaum, 1962).

  Figure 9: A traveling-wave scattering junction block diagram.

  

• The scattering equations can also be written
  $$\begin{eqnarray*} v^{-}_{0} &=& v^{+}_{1} + \rho (v^{+}_{1} - v^{+}_{0}) \\ v^{-}_{1} &=& v^{+}_{0} + \rho (v^{+}_{1} - v^{+}_{0}), \end{eqnarray*}$$
  which requires only one multiplication and three additions (as diagrammed in Fig. 10).

  Figure 10: A one-multiply traveling-wave scattering junction block diagram.