Sampled Waves and Lumped Filters

We have that the wave equation $Yd''=\epsilon \ddot{d}$ is obeyed by any pair of traveling waves

$$d(t,x) \;=\;d_r\left(t-\frac{x}{c}\right) \;+\; d_l\left(t+\frac{x}{c}\right)$$

• $d_l(\cdot)$ and $d_r(\cdot)$ are arbitrary twice-differentiable displacement functions

• $c=\sqrt{K/\epsilon }$ for transverse waves, and $c=\sqrt{Y/\rho}$ for longitudinal waves, where $Y$ is Young's modulus = ``spring constant'' for solids (stress/strain $\mathrel{\stackrel{\scriptscriptstyle\mathrm{\Delta}}{\scriptstyle=}}$ force-per-unit-area / relative displacement),
  $\rho$ is mass per unit volume (rods), and
  $\epsilon$ is mass per unit length (ideal strings)

• We can sample these traveling-wave components to obtain the super-efficient digital waveguide modeling approach for strings and acoustic tubes (and more)

• Any acoustic ``ray'' or propagating wave can be implemented digitally using a simple delay line followed by linear filtering to implement loss and/or dispersion: