Sampled Traveling Waves

• The lossless one-dimensional wave equation was previously derived for a stretched string as:
  $$\frac{\partial^{2} y}{\partial t^{2}} = c^{2} \frac{\partial^{2} y}{\partial x^{2}},$$
  where $c = \sqrt{T/\epsilon}$ is the speed of wave motion on the string, $T$ is the string tension, and $\epsilon$ is the mass density of the string.

• The traveling-wave solution of the wave equation was published by d'Alembert in 1747. It has the general form
  $$y(t, x) = y^{+}(t - x/c) + y^{-}(t + x/c),$$
  for arbitrary functions $y^{+}(\cdot)$ and $y^{-}(\cdot)$. A function of $(t - x/c)$ can be interpreted as a fixed waveshape traveling to the right (positive $x$ direction) and a function $(t + x/c)$ can be interpreted as a fixed waveshape traveling to the left (negative $x$ direction), both with speed $c$.

• To develop a discrete-time model or simulation of traveling wave motion, it is necessary to sample the traveling-wave amplitudes in both time and space.

• The temporal sampling interval is $T_s$ seconds, which corresponds to a sample rate $f_{s} \ensuremath{\stackrel{\Delta}{=}}1 / T_s$ samples per second.

• The spatial sampling interval is given most naturally by $X \ensuremath{\stackrel{\Delta}{=}}cT_s$ meters, or the distance traveled by sound in one temporal sampling interval. In this way, each traveling-wave component moves left or right one spatial sample for each time sample.

• Using the change of variables
  $$\begin{eqnarray*} x &\rightarrow& x_{m} = mX \\ t &\rightarrow& t_{n} = nT_s, \end{eqnarray*}$$
  the traveling-wave solution becomes
  $$\begin{eqnarray*} y\left(t_{n},x_{m}\right) &=& y^{+}(t_{n} - x_{m}/c) + y^{-}(t... ...+ mX/c) \nonumber \\ &=& y^{+}[(n - m)T_s] + y^{-}[(n + m)T_s]. \end{eqnarray*}$$

• This representation can be further simplified by suppressing $T_s$, with a resulting expression for physical displacement at time $n$ and location $m$ given as the sum of the two traveling-wave components
    $$y\left(t_{n},x_{m}\right) = y^{+}(n - m) + y^{-}(n + m).$$ (7)