Scattering Theory for Traveling Waves

Traveling waves in continuous media are discussed in Appendix H. 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, above, is called the wave impedance. When the wave impedance changes, from to , 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 where is the string tension and 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 pri