Lossless digital waveguide with observation points at $x=0$ and $x=3X=3cT$

• Recall:
  $$\begin{eqnarray*} y(t,x) &=& y^{+}\left(\frac{t-x/c}{T}\right) + y^{-}\left(\fra... ...ight) \\ &\downarrow& \\ y(nT,mX) &=& y^{+}(n-m) + y^{-}(n+m) \end{eqnarray*}$$
  

• Position $x_m = mX= m cT$ is eliminated from the simulation
• Position $x_m$ remains laid out from left to right
• Left- and right-going traveling waves must be summed to produce a physical output
  $$y(t_n,x_m) = y^{+}(n-m) + y^{-}(n+m)$$

• Similar to ladder and lattice digital filters

Important point: Discrete time simulation is exact at the sampling instants, to within the numerical precision of the samples themselves.

To avoid aliasing associated with sampling:

• Require all initial waveshapes be bandlimited to $(-f_s/2,f_s/2)$
• Require all external driving signals be similarly bandlimited
• Avoid nonlinearities or keep them ``weak''
• Avoid time variation or keep it slow
• Use plenty of lowpass filtering with rapid high-frequency roll-off in severely nonlinear and/or time-varying cases
• Prefer ``feed-forward'' over ``feed-back'' around nonlinearities and/or modulations when possible

Interactive Java simulation of a vibrating string: http://www.colorado.edu/physics/phet/simulations/stringwave/stringWave.swf