Sign in

username:

password:



Not a member?

Search Online Books



Search tips

Free Online Books

Chapters

Physical Audio Signal Processing
    Digital Waveguide Theory
       Alternative Wave Variables
          Spatial Derivatives

Chapter Contents:

Search Physical Audio Signal Processing

  

Book Index | Global Index

Would you like to be notified by email when Julius Orion Smith III publishes a new entry into his blog?

  

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:

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

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

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 normally specified by the displacement $ y(t_0,x)$ and velocity $ {\dot y}(t_0,x)$ for all $ x$ [325]. Since displacement is the sum of the traveling displacement waves and velocity is proportional to the difference of the traveling displacement waves, one state description can be readily obtained from 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