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

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

For velocity waves, the transmission coefficient is , which is perhaps more intuitive.

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University