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




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),
![$\displaystyle y(t,x) \eqsp \int_0^{x} y'(t,\xi)d\xi
\eqsp -\frac{1}{c}\int_0^{x} \left[v_r(t-\xi/c) - v_l(t+\xi/c)\right]d\xi,
$](http://www.dsprelated.com/josimages_new/pasp/img3474.png)
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.
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Force Waves
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Higher Order Terms