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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

$\displaystyle y'(t,x)$ $\displaystyle \isdef$ $\displaystyle \frac{\partial}{\partial x}y(t,x)$  
  $\displaystyle =$ $\displaystyle y'_r(t-x/c) + y'_l(t+x/c)$  
  $\displaystyle =$ $\displaystyle -\frac{1}{c} {\dot y}_r(t-x/c) + \frac{1}{c}{\dot y}_l(t+x/c),
\protect$ (C.39)

or, in discrete time,
$\displaystyle y'(t_n,x_m)$ $\displaystyle \isdef$ $\displaystyle y'(nT,mX)$  
  $\displaystyle =$ $\displaystyle y'_r\left[(n-m)T\right]+ y'_l\left[(n+m)T\right]$  
  $\displaystyle \isdef$ $\displaystyle y'^{+}(n-m) + y'^{-}(n+m)$  
  $\displaystyle =$ $\displaystyle -\frac{1}{c} \dot y^{+}(n-m) + \frac{1}{c}\dot y^{-}(n+m)$  
  $\displaystyle \isdef$ $\displaystyle -\frac{1}{c} v^{+}(n-m) + \frac{1}{c}v^{-}(n+m)$  
  $\displaystyle =$ $\displaystyle \frac{1}{c} \left[v^{-}(n+m) - v^{+}(n-m) \right].
\protect$ (C.40)

From this we may conclude that $ v^{-}= cy'^{-}$ and $ v^{+}= -cy'^{+}$ . That is, traveling slope waves can be computed from traveling velocity waves by dividing by $ c$ 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 $ c$ .

By the wave equation, curvature waves, $ y''= {\ddot y}/c^2$ , 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 $ t_0$ is often specified by the displacement $ y(t_0,x)$ and velocity $ {\dot y}(t_0,x)$ for all $ x$ [320]. 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,

one state description can be converted to the other.

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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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2015-05-22 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University