## Alternative Wave Variables

We have thus far considered discrete-time simulation of transverse
*displacement* in the ideal string. It is equally valid to
choose *velocity*
, *acceleration*
, *slope* , or perhaps some other derivative
or integral of displacement with respect to time or position.
Conversion between various time derivatives can be carried out by
means *integrators* and *differentiators*, as depicted in
Fig.C.10. Since integration and
differentiation are *linear* operators, and since the traveling
wave arguments are in units of time, the conversion formulas relating
, , and hold also for the traveling wave *components*
.

Differentiation and integration have a simple form in the
frequency domain. Denoting the *Laplace Transform* of by

(C.36) |

where ``'' in the time argument means ``for all time,'' we have, according to the

*differentiation theorem*for Laplace transforms [284],

(C.37) |

Similarly, , and so on. Thus, in the frequency domain, the conversions between displacement, velocity, and acceleration appear as shown in Fig.C.11.

In discrete time, integration and differentiation can be accomplished using digital filters [362]. Commonly used first-order approximations are shown in Fig.C.12.

If discrete-time acceleration is defined as the sampled version of
continuous-time acceleration, *i.e.*,
, (for some fixed continuous position which we
suppress for simplicity of notation), then the
frequency-domain form is given by the * transform*
[485]:

(C.38) |

In the frequency domain for discrete-time systems, the first-order approximate conversions appear as shown in Fig.C.13.

The transform plays the role of the Laplace transform for discrete-time systems. Setting , it can be seen as a sampled Laplace transform (divided by ), where the sampling is carried out by halting the limit of the rectangle width at in the definition of a Reimann integral for the Laplace transform. An important difference between the two is that the frequency axis in the Laplace transform is the imaginary axis (the `` axis''), while the frequency axis in the plane is on the unit circle . As one would expect, the frequency axis for discrete-time systems has unique information only between frequencies and while the continuous-time frequency axis extends to plus and minus infinity.

These first-order approximations are accurate (though scaled by )
at low frequencies relative to half the sampling rate, but they are
not ``best'' approximations in any sense other than being most like
the definitions of integration and differentiation in continuous time.
Much better approximations can be obtained by approaching the problem
from a *digital filter design* viewpoint, as discussed in §8.6.

### Spatial Derivatives

In addition to time derivatives, we may apply any number of *spatial
derivatives* to obtain yet more wave variables to choose from. The first
spatial derivative of string displacement yields *slope waves*

or, in discrete time,

From this we may conclude that and . That is, traveling slope waves can be computed from traveling velocity waves by dividing by and negating in the right-going case. Physical string slope can thus be computed from a velocity-wave simulation in a digital waveguide by

*subtracting*the upper rail from the lower rail and dividing by .

By the wave equation, *curvature waves,*
, are
simply a scaling of acceleration waves, in the case of ideal strings.

In the field of acoustics, the state of a vibrating string at any
instant of time is often specified by the displacement
and velocity
for all [317]. Since
velocity is the *sum* of the traveling velocity waves and
displacement is determined by the *difference* of the
traveling velocity waves, *viz.*, from Eq.(C.39),

In summary, all traveling-wave variables can be computed from any one, as
long as both the left- and right-going component waves are available.
Alternatively, any *two* linearly independent *physical*
variables, such as displacement and velocity, can be used to compute all
other wave variables. Wave variable conversions requiring differentiation
or integration are relatively expensive since a large-order digital filter
is necessary to do it right (§8.6.1).
Slope and velocity waves can be computed
from each other by simple scaling, and curvature waves are identical
to acceleration waves to within a scale factor.

In the absence of factors dictating a specific choice, *velocity waves*
are a good overall choice because (1) it is numerically easier to perform
digital integration to get displacement than it is to differentiate
displacement to get velocity, (2) slope waves are immediately computable
from velocity waves. Slope waves are important because they are a simple
scaling of force waves.

### Force Waves

Referring to Fig.C.14, at an arbitrary point along
the string, the vertical force applied at time *to* the portion of
string to the left of position *by* the portion of string to the
right of position is given by

(C.41) |

assuming , as is assumed in the derivation of the wave equation. Similarly, the force applied

*by*the portion to the left of position

*to*the portion to the right is given by

(C.42) |

These forces must cancel since a nonzero net force on a massless point would produce infinite acceleration.

*I.e.*, we must have at all times and positions .

Vertical force waves propagate along the string like any other
transverse wave variable (since they are just slope waves multiplied
by tension ). We may choose either or as the string
force wave variable, one being the negative of the other. It turns
out that to make the description for vibrating strings look the same
as that for air columns, we have to pick , the one that
*acts to the right.* This makes sense intuitively when one
considers longitudinal pressure waves in an acoustic tube: a
compression wave traveling to the right in the tube pushes the air in
front of it and thus acts to the right. We therefore define the
*force wave variable* to be

(C.43) |

Note that a negative slope pulls up on the segment to the right. At this point, we have not yet considered a traveling-wave decomposition.

### Wave Impedance

Using the above identities, we have that the force distribution along the string is given in terms of velocity waves by

where . This is a fundamental quantity known as the

*wave impedance*of the string (also called the

*characteristic impedance*), denoted as

(C.45) |

The wave impedance can be seen as the geometric mean of the two resistances to displacement: tension (spring force) and mass (inertial force).

The digitized traveling force-wave components become

which gives us that the right-going force wave equals the wave impedance times the right-going velocity wave, and the left-going force wave equals

*minus*the wave impedance times the left-going velocity wave.

^{C.4}Thus, in a traveling wave, force is always

*in phase*with velocity (considering the minus sign in the left-going case to be associated with the direction of travel rather than a degrees phase shift between force and velocity). Note also that if the left-going force wave were defined as the string force acting to the left, the minus sign would disappear. The fundamental relation is sometimes referred to as the mechanical counterpart of

*Ohm's law for traveling waves*, and in c.g.s. units can be called

*acoustical ohms*[261].

In the case of the *acoustic tube* [317,297], we have the
analogous relations

(C.47) |

where is the right-going traveling

*longitudinal pressure wave*component, is the left-going pressure wave, and are the left and right-going

*volume velocity waves*. In the acoustic tube context, the wave impedance is given by

(Acoustic Tubes) | (C.48) |

where is the mass per unit volume of air, is sound speed in air, and is the cross-sectional area of the tube. Note that if we had chosen

*particle velocity*rather than volume velocity, the wave impedance would be instead, the wave impedance in open air. Particle velocity is appropriate in open air, while volume velocity is the conserved quantity in acoustic tubes or ``ducts'' of varying cross-sectional area.

### State Conversions

In §C.3.6, an arbitrary string state was converted to
traveling displacement-wave components to show that the traveling-wave
representation is complete, *i.e.*, that any physical string state can be
expressed as a pair of traveling-wave components. In this section, we
revisit this topic using *force* and *velocity* waves.

By definition of the traveling-wave decomposition, we have

Using Eq.(C.46), we can eliminate and , giving, in matrix form,

*invert*the appropriate two-by-two matrix above. That is, the matrix must be

*nonsingular*. Requiring both determinants to be nonzero yields the condition

Carrying out the inversion to obtain force waves from yields

### 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

*net*power flow.

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* [432], 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].

### Energy Density Waves

The vibrational energy per unit length along the string, or *wave
energy density* [317] is given by the sum of potential and
kinetic energy densities:

(C.50) |

Sampling across time and space, and substituting traveling wave components, one can show in a few lines of algebra that the

*sampled*wave energy density is given by

(C.51) |

where

Thus, traveling power waves (energy per unit time)
can be converted to energy density waves (energy per unit length) by
simply dividing by , the speed of propagation. Quite naturally, the
*total wave energy* in the string
is given by the integral along the string of the energy density:

(C.52) |

In practice, of course, the string length is finite, and the limits of integration are from the coordinate of the left endpoint to that of the right endpoint,

*e.g.*, 0 to .

### Root-Power Waves

It is sometimes helpful to *normalize* the wave variables so that
signal power is uniformly distributed numerically. This can be especially
helpful in fixed-point implementations.

From (C.49), it is clear that power normalization is given by

where we have dropped the common time argument `' for simplicity. As a result, we obtain

and

The normalized wave variables and behave physically like force and velocity waves, respectively, but they are scaled such that either can be squared to obtain instantaneous signal power. Waveguide networks built using normalized waves have many desirable properties [174,172,432]. One is the obvious numerical advantage of uniformly distributing signal power across available dynamic range in fixed-point implementations. Another is that only in the normalized case can the wave impedances be made

*time varying*without modulating signal power. In other words, use of normalized waves eliminates ``parametric amplification'' effects; signal power is decoupled from parameter changes.

### Total Energy in a Rigidly Terminated String

The total energy in a length , rigidly terminated, freely
vibrating string can be computed as

(C.54) | |||

(C.55) |

for any . Since the energy never decays, and are also arbitrary. Thus, because free vibrations of a doubly terminated string must be

*periodic*in time, the total energy equals the integral of power over any period at any point along the string.

**Next Section:**

Scattering at Impedance Changes

**Previous Section:**

The Dispersive 1D Wave Equation