Power Waves

Basic courses in physics teach us that *power* is *work per
unit time*, and *work* is a measure of *energy* which is
typically defined as *force times distance*. Therefore, power is
in physical units of force times distance per unit time, or force
times velocity. It therefore should come as no surprise that
traveling *power waves* are defined for strings as follows:

From the Ohm's-law relations and , we also have

Thus, both the
left- and right-going components are *nonnegative.* The sum of the
traveling powers at a point gives the total power at that point
in the waveguide:

(C.49) |

If we had left out the minus sign in the definition of left-going power waves, the sum would instead be a

Power waves are important because they correspond to the actual
ability of the wave to do *work* on the outside world, such as
on a violin bridge at the end of a string. Because *energy* is
conserved in closed systems, power waves sometimes give a simpler,
more fundamental view of wave phenomena, such as in conical acoustic
tubes. Also, implementing nonlinear operations such as
*rounding* and *saturation* in such a way that signal
power is not increased, gives suppression of
*limit cycles* and *overflow oscillations* [436], as discussed in the section on signal scattering.

For example, consider a waveguide having a wave impedance which
increases smoothly to the right. A converging cone provides a
practical example in the acoustic tube realm. Then since the energy
in a traveling wave must be in the wave unless it has been transduced
elsewhere, we expect
to propagate unchanged along the
waveguide. However, since the wave impedance is increasing, it must
be true that force is increasing and velocity is decreasing according
to
. Looking only at force or velocity
might give us the mistaken impression that the wave is getting
stronger (looking at force) or weaker (looking at velocity), when
really it was simply sailing along as a fixed amount of energy. This
is an example of a *transformer* action in which force is
converted into velocity or vice versa. It is well known that a
conical tube acts as if it's open on both ends even though we can
plainly see that it is closed on one end. A tempting explanation is
that the cone acts as a transformer which exchanges pressure and
velocity between the endpoints of the tube, so that a closed end on
one side is equivalent to an open end on the other. However, this
view is oversimplified because, while spherical pressure waves travel
nondispersively in cones, velocity propagation is dispersive
[22,50].

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