### Signal Scattering

The digital waveguide was introduced in §2.4. A basic fact from acoustics is that traveling waves only happen in a*uniform medium*. For a medium to be uniform, its

*wave impedance*

^{3.17}must be

*constant*. When a traveling wave encounters a

*change*in the wave impedance, it will

*reflect*, at least partially. If the reflection is not total, it will also partially

*transmit*into the new impedance. This is called

*scattering*of the traveling wave.

Let denote the constant impedance in some waveguide, such as a stretched steel string or acoustic bore. Then signal scattering is caused by a change in wave impedance from to . We can depict the partial reflection and transmission as shown in Fig.2.33. The computation of reflection and transmission in both directions, as shown in Fig.2.33 is called a

*scattering junction*. As derived in Appendix C, for force or pressure waves, the

*reflection coefficient*is given by

That is, the coefficient of reflection for a traveling pressure wave leaving impedance and entering impedance is given by the

*impedance step over the impedance sum*. The

*reflection coefficient*fully characterizes the scattering junction. For

*velocity*traveling waves, the reflection coefficient is just the negative of that for force/pressure waves, or (see Appendix C). Signal scattering is

*lossless*,

*i.e.*, wave energy is neither created nor destroyed. An implication of this is that the

*transmission coefficient*for a traveling pressure wave leaving impedance and entering impedance is given by

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