Scattering Theory for Traveling Waves

Traveling waves in continuous media are discussed in Appendix G. However, we will summarize the main facts here. While this section is concerned with applying scattering theory to lumped modeling, it is clearest to derive the basic scattering relations in the traveling wave case.

In a traveling wave, force is in phase with velocity. For left-going waves on a string, the minus sign takes care of the fact that a given force (which is proportional to string slope) acts to the left and right with opposite signs. For waves in an acoustic tube, the minus sign properly accounts for longitudinal velocity waves in each direction.

The ratio of force to velocity in a traveling wave, $R_0$ above, is called the wave impedance. When the wave impedance changes, from $R_0$ to $R_1$, say, scattering occurs at a junction connecting the two impedances, i.e., the traveling wave splits into reflected and transmitted components. This follows immediately from the basic traveling-wave relations above and from physical continuity.

In vibrating strings, the wave impedance is given by $\sqrt{K\epsilon }$ where $K$ is the string tension and $\epsilon$ is mass density. Thus, one way to change the wave impedance along a stretched string is to change the string density by adjoining two strings of different material or thickness. It is more difficult to change the string tension since a ``frictionless vertical guide rod'' is necessary, in principle. At a junction between two wave impedances on a string, the physical continuity constraints are that velocity is unchanged across the junction (to avoid breaking the string) and the net vertical force at the junction, obtained by summing the force applied by each string endpoint at the junction, must be zero (to avoid accelerating a zero mass at the junction). Therefore, the junction is formally a series connection of the two ports representing the string endpoints which are joined.

In acoustic tubes, the wave impedance is given by $\rho c/A$ where $A$ is the cross-sectional area of the tube (and the velocity variable is volume velocity). Thus, the easy way to introduce a scattering junction in an acoustic tube is to change the cross-sectional area discontinuously. This is why the vocal tract is modeled as a piecewise cylindrical acoustic tube in speech modeling [275,82]. At an area discontinuity in an acoustic tube, the physical continuity constraints are that the pressure must be continuous across the junction (to avoid accelerating a massless plane of air at the junction), and the volume velocities (which are taken as positive when flowing into the junction) must sum to zero at the junction (so that air particles are not created or destroyed). Thus, in acoustic tubes, parallel junctions naturally arise between sections of two different wave impedance.

It is quick to derive the scattering relations for either the ideal string or acoustic tube. Let $f^{{+}}_0,f^{{-}}_0$ denote the traveling force wave components immediately to the left of a junction in a string, and let $f^{{+}}_1,f^{{-}}_1$ denote the components on the right, as shown in Fig. J.12. Similarly define the velocity wave components. The physical continuity constraints can be written

$$\begin{eqnarray*} f_0+f_1&=&0\qquad\mbox{(forces on a massless point must balance)}\\ v_0&=&v_1\qquad\mbox{(string does not break)} \end{eqnarray*}$$

Figure J.12: Two ideal strings of differing wave impedance joined at a point by a frictionless guide rod.

Let $v$ denote the common velocity of the string endpoints meeting at a junction. (All velocities are taken as positive in the upward direction.) Then since $v_i=v^{+}_i+v^{-}_i$, continuity implies $v^{-}_i = v-v^{+}_i$. Therefore, we have

$$= f_0+f_1 = (f^{{+}}_0 + f^{{-}}_0) + (f^{{+}}_1+f^{{-}}_1)$$ 0 (J.7)

$$= R_0(v^{+}_0-v^{-}_0) + R_1(v^{+}_1-v^{-}_1) = R_0(2v^{+}_0-v) + R_1(2v^{+}_1-v)$$ (J.8)

which implies

$$v = 2\frac{R_0v^{+}_0 + R_1v^{+}_1 }{R_0+R_1}.$$

This expresses the junction velocity in terms of the incoming velocity-wave components and the two wave impedances meeting at the junction, and so it can be used to implement a scattering junction in practice. All that's left is to compute the outgoing traveling waves, again using continuity, as

$$v^{-}_0 = v - v^{+}_0 = -\frac{R_1-R_0}{R_1+R_0}v^{+}_0 + \frac{2R_1}{R_1+R_0} v^{+}_1,$$ (J.9)

$$v^{-}_1 = v - v^{+}_1 = \frac{2R_0}{R_1+R_0} v^{+}_0 + \frac{R_1-R_0}{R_1+R_0}v^{+}_1.$$ (J.10)

It is customary to define the reflection coefficient by

$$\rho \isdef \frac{R_1-R_0}{R_1+R_0}.$$

In terms of $\rho$, (J.10) becomes

$$v^{-}_0 = -\rho v^{+}_0 + (1+\rho) v^{+}_1$$ (J.11)

$$v^{-}_1 = (1-\rho)v^{+}_0 + \rho v^{+}_1.$$ (J.12)

A scattering diagram is shown in Fig. J.13.

Figure J.13: The series velocity scattering junction in Kelly-Lochbaum form.

This is the so-called Kelly-Lochbaum form [275]. However, it is important to notice that 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 is diagrammed in Fig. J.14.

Figure J.14: The series velocity scattering junction in one-multiply form.

Thus, a scattering junction fundamentally requires only one multiplication and three additions.