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.
$\begin{figure}\begin{center} \begin{picture}(4,0.8) \put(0,0){\epsfig{file=fig... ...,0.55){$R_{0}$} \put (3,0.55){$R_{1}$} \end{picture} \end{center} \end{figure}$

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

	$\displaystyle f_{0} + f_{1}$	$\textstyle =$	$\displaystyle 0 \hspace{0.2in} \mbox{(forces on a massless point must balance)}$
	$\displaystyle v_{0}$	$\textstyle =$	$\displaystyle v_{1} \hspace{0.2in} \mbox{(string does not break)}$

Let

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

	$\displaystyle 0$	$\textstyle =$	$\displaystyle f_{0} + f_{1} = \left(f^{+}_{0} + f^{-}_{0}\right) + \left(f^{+}_{1} + f^{-}_{1}\right)$
		$\textstyle =$	$\displaystyle R_{0} \left(v^{+}_{0} - v^{-}_{0}\right) + R_{1} \left(v^{+}_{1} ... ...ght) = R_{0} \left(2 v^{+}_{0} - v\right) + R_{1} \left(2 v^{+}_{1} - v\right),$

which can be solved for $v$

$\displaystyle v = 2 \frac{R_{0} v^{+}_{0} + R_{1} v^{+}_{1}}{R_{0} + R_{1}}.$

Solving these expressions in terms of the outgoing traveling-wave components,

	$\displaystyle v^{-}_{0}$	$\textstyle =$	$\displaystyle v - v^{+}_{0} = -\frac{R_{1} - R_{0}}{R_{1} + R_{0}} v^{+}_{0} + \frac{2 R_{1}}{R_{1} + R_{0}} v^{+}_{1}$
	$\displaystyle v^{-}_{1}$	$\textstyle =$	$\displaystyle v - v^{+}_{1} = \frac{2 R_{0}}{R_{1} + R_{0}} v^{+}_{0} + \frac{R_{1} - R_{0}}{R_{1} + R_{0}} v^{+}_{1}.$

It is standard to define the reflection coefficient

$\displaystyle \rho \ensuremath{\stackrel{\Delta}{=}}\frac{R_{1} - R_{0}}{R_{1} + R_{0}},$

which then allows us to write

	$\displaystyle v^{-}_{0}$	$\textstyle =$	$\displaystyle -\rho v^{+}_{0} + (1 + \rho) v^{+}_{1}$
	$\displaystyle v^{-}_{1}$	$\textstyle =$	$\displaystyle (1 - \rho) v^{+}_{0} + \rho v^{+}_{1}.$

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.
$\begin{figure}\begin{center} \begin{picture}(2.9,1.7) \put(0.2,0){\epsfig{file... ...0.82){$\rho$} \put (1.4,1.2){$1-\rho$} \end{picture} \end{center} \end{figure}$

The scattering equations can also be written

	$\displaystyle v^{-}_{0}$	$\textstyle =$	$\displaystyle v^{+}_{1} + \rho (v^{+}_{1} - v^{+}_{0})$
	$\displaystyle v^{-}_{1}$	$\textstyle =$	$\displaystyle v^{+}_{0} + \rho (v^{+}_{1} - v^{+}_{0}),$

which requires only one multiplication and three additions (as diagrammed in Fig. 10).

**Figure 10:** A one-multiply traveling-wave scattering junction block diagram.
$\begin{figure}\begin{center} \begin{picture}(2.9,1.9) \put(0.2,0){\epsfig{file... ... (0.5,1.5){$-$} \put (1.5,1.6){$\rho$} \end{picture} \end{center} \end{figure}$