### 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.

**Next Section:**

Force Waves

**Previous Section:**

Higher Order Terms